West Side Geometry

2011-11-13T21:05:00.001-05:00

This Will Revolutionize Education

Professor Hefferanan at WPI developed a program a few years ago, called AssistMents. Under the race to the top initiative, he donated the program to schools in Maine, Massachusetts, and Vermont. I was checking it out recently. (I recognize certain areas that I will need training in...)

This is a very powerful tool, for giving assessments, including homework, and tests or quizzes. Its power rests in the way it can be used to teach students while they work, and not having to wait until they are done working.

There have been many other programs written like this, such as Carnegie and... whatever.

Here, you can use some existing problems. Students at WPI put in more than 30k problems. Teachers (mostly from districts already using this, such as near Worcester), have input over 70k problems. (They are not restricted to math problems)... from 5th grade through 12th, (mostly through 9th or 10th grade...)

You can input your own problems.

You can layer the problems, and layer the number and kinds of hints you provide along the way. So if a kid gets one question incorrect, you could link him to a hint. Or (I think this works this way, but I need some help figuring it all out), you can lead him to the same type of problem on an easier level.

If a student were to do an assignment, rather than have to wait until the next day to find out what to do next, they could work through a series of hints... Or rather than wait to find out they were incorrect, they could get instant feedback (and hence, try again...).

You can link tutorials to the problems, so if more help is needed, it can be available.

The whole process of doing homework would be useful. That is why it is revolutionary.

It would be free. Revolutionary.

It would be geared for the specific course or classroom. Revolutionary.

The help. Revolutionary.

It's revolutionary.

It's not without price. Oh, it's free alright. But preparing things for the classroom takes time. It also takes skill. If I'm having a little struggle with it, and I'm among the fairly well versed ones at the school, as it pertains to technology, think of the person who is a technophobe. Oh, they probably wouldn't use technology even if you made it easy and set it on a platter and ... of all things... trained them!

There is also an issue of math type. When kids input their answers, they need to be able to write math. As it is now, this would be hard. Sigh.

2011-09-27T22:30:00.002-04:00

Does it improve student learning?

I was talking to a senior, former student who I had as a sophomore. He says that the geometry we had then now seems easy. The math course he is currently in is much harder. AND he was talking to his girlfriend, now in college, who says the jump from senior year to college is about 10 times harder still! When I had him, he said what I was asking was hard...

I'm feeling somewhat overwhelmed, still unexplainable. A stab at why:

Per the administration, all websites are to be through our school's google site; I'm moving lodgings of my personal website... (This blog stays put).
My schedule runs 5 or 6 periods straight. Exhausting. Followed by after school meetings.
I'm grading a lot more this year. Giving feedback. Research says it may mostly be ignored. So, I'm trying to write meaningful stuff, that hopefully won't be ignored. The weekly open responses are a killer to grade. Hours. I'm sure the kids are feeling the pinch with the work load.
I think the reason for pressure may be a seminar I took this summer. I'm making manipulatives, working with groups, creating link sheets and warm-ups and etc. all of which take time to conceive, create and develop. I sure hope it pays off.

My mantra this year is:

Does it IMPROVE STUDENT LEARNING?

Everything I do with the students has this goal in mind.

I wrote that this blog would discuss student work. It would be a shame to not do so here.

So, we were discussing midpoint. On a quiz, a student solved for midpoint and came up with:

(4/2 , 4/2).

When he simplified it, he got:

(1/2, 1/2).

He took that further still to get:

=1

My wife says to smile. The kids at this age are hormonal.

2011-09-11T23:18:00.003-04:00

Pythagorean Theorem

Here is a graphic organizer I got from some summer training I went to:

When dealing with formulas, such as the Pythagorean Theorem, which we've been using all week, following this format really works.

Here is a problem from our first quiz. I gave a picture, but purposefully did not say that the side length 61 was the hypotenuse.

This first student did not follow the organizer. No squared symbols...

Next is a person who set up the problem correctly.
This was a no calculator quiz. Aside from the math errors, which happen, is the issue of reasonableness.

Here the person set it up correctly.
The issue is not knowing a radical from a multiplication.

The person can multiply, but substituted incorrectly...
If they'd written the formula, this might have been different.

I see the person's work on the side for 61 x 61.
Too bad he didn't also do it for 60 x 60.
And again does he ask, "Is my answer reasonable?"

A little problem with math facts and sqrt (121).

Good work, mostly. The worker has good habits, mostly.
Anything you do to the left of the equation, you must do to the right.
Here, he left out the radical sign when taking the root of both sides.

This is really pretty good. I would like to see things lined up.
Neatness counts inasmuch as staying organized gets you to make fewer mistakes.

I think this is the best of the lot.
I ask that students write the units. This is so that at the end of the year, when we're dealing with surface area or volume, I don't get the usual comment, "Is that inches squared or inches cubed." Having dealt with it all along, you know.

Final Exam 2010-2011

2011-06-24T01:24:00.000-04:00

This year's geometry final exam had 40 multiple choice questions and 5 short response. It covered 49 of about 60 key concepts covered throughout the year. Unfortunately, it was the first test we gave where each multiple choice question had selection (e): None of the Above. Further, way too many of the questions originally had this as an answer. I altered the test I gave so that only 4 of the questions had (e) as an answer. I meant there to be only 3, max...

A good teacher reflects on his practice. I analyzed the student results. This was a good test to analyze in that each of the concepts was separated (essentially one topic per question), and labelled (as to what the concept was -- a big hint to one taking the test). It was also good because I taught these kids this. I want to see where MY teaching needs to change. The real question I had was:

Where was my teaching ineffective?
What need I teach differently next year?

I came up with 7 topics that the students either forgot about, or were confused about... Things that need to be done differently. I'll post them next time. It is too late to do so now. Tomorrow.

Best Math Class All Year

2011-06-19T21:00:00.000-04:00

"This was the best math class all year." ... so said a student last Friday. We did a little math. We spoke a little about logic. Mostly, we processed what took place the previous period.

This was our last class together before the final exam. We've been reviewing all week. It was hot. The kids were tired and hungry. In walks one usually very talkative and boisterous girl. Somewhat somberly, she began to rehearse her previous period. I sat at my desk. Everyone listened.

In Health, the kids have to give a presentation. One girl chose to talk about drunk driving. For her presentation, she brought the mother and sister of her good friend, who was killed last year in a drunk driving accident.

I really can't do justice to the story in writing or in a few words. Let me preface it by saying in the end all the girls, and all the boys were in tears.

The girl was set to go to college. It was the end of the school year. There was a bon-fire. The girl told her mother where she was going. But when it came to the bon-fire, that was somewhat impromptu. She didn't call. At the fire, the girl fell asleep in the back seat of a vehicle. Later, some other kids got in. At least 2 of them were drunk, one being the driver. They took off. As they were getting on to the freeway (I-95), they fish-tailed into the adjacent lane. An oncoming semi ran into them. She was killed instantly.

The next day, the driver and the girl came to the house of the mother who's daughter was killed, to apologize. "That doesn't bring my daughter back."

As a teacher, I opted to use the class time to process this with the kids, rather than to try to do math, when their minds and hearts were wrapped up in this story.

After the story, I asked two questions. First was, "How do you feel now about drinking and driving?" (Several of the kids planned a trip to the Cape for the weekend to... ) My second question was, "How do you feel about texting while driving?" I know that texting is as serious as drinking when it comes to inattentiveness to the road and hazards or dangers you create. It is a year after tragedy struck this woman's family, and she is still grieving as if it were yesterday. I hope she gets the help she needs to carry on. The younger sister could be missing on the relationship if the parent remains prisoner to the feelings generated from all this.

One youth said, "It must be so hard to lose a child this way." I pointed out that it is hard to lose a loved one at any age, under any circumstances, even a parent or grandparent when they are 85 years old... To quantify who suffers more is not a productive or worthwhile endeavor.

Our discussion led to talk of faith in the afterlife, and near death experiences had by two of the kids.

A young man said, "I don't believe in God." Now, I'm deeply religious. I have a witness and faith in God. I am always very cautious to teach religion or anything like it, for I am in a position of authority. As a public servant, I am not paid to teach religion. Supporting one belief system may step on the toes of another. Here, this young man opened the door for what I said next.

Blais Paschal, a noted scientist, mathematician and philosopher, once used logic to denote what has become known as Paschal's Wager. He said, even if you don't believe in God, it is a better wager to behave as if you did. In the end, if you die and come to find out there is no god, what have you lost? A few hours of service. A few hours of worship. A few hours of doing good. On the other hand, if you behave as if there is not god, and in the end you find out there IS a God, ... you've got real problems.

The same boy asked if I had had any experience with the devil.
"I would really be interested in hearing about experiences with the devil," he said.
I sat quietly a moment, then said, "I do not satisfy your curiosity in this manner."
"Would you tell your children?"
V- "Yes, but I'm not your father.... But know this. Evil (the devil) is real. It is not something to mess around with."

One student declared, "I wish we had a class like this to take in school. I would love to study different religions and philosophy." I was amazed to hear there is no philosophy course. I took philosophy in school. My kids attended private school where they had comparative religions and philosophy. Heck, one of coordinate planar geometry's heroes was a philosopher, Rene Descartes. He invented (discovered... brought to use) the Cartesian coordinate plane. He also said, "Je pense donc Je suis."

Story has it, he walked into a bar. The bartender asked if he would like a whisky. He replied, "I think not," and disappeared.

One student asked me what my philosophy was, given that I had so many years of experience and so much knowledge. I told her flattery would not get her any gains. She relented. So, I said, "If a person lives his or her life and in the end cannot answer three questions, her life will have been pretty much wasted."

"What are those three questions?"
V- "What do you think they should be?"
"Are you happy? Have you found happiness?..."
V - "Those are good questions. We all pursue happiness. The questions I had in mind are... Where did I come from?..."
"I came from West Springfield..."
V - "Why am I here?"
"I'm here because the law requires me to be in school."
V - "Where am I going?"
"I'm going...."
V - "These are life long questions that require a lot of consideration. They are not questions you can answer flippantly..."

Some of the kids have to give a mock interview for one of their classes.
"What do you wear to an interview?"
V - "You wear Sunday clothes?"
"You mean what I've got on?"
V - "Do you go to church?"
"No." (About 70% of the class does not attend church.)
V - "What does it mean to you if I said, 'Wear church clothes.'?"
"When I go I wear what I've got on." (He was wearing a tee-shirt and shorts.)
V - "Have you heard the expression, 'Sunday best.'?... Wear a shirt and tie. If you have a sports coat, wear it. Some interviews require you to wear a suit. If you're a woman, wear a skirt or a dress."
"I don't own a dress."

I think she was lying. Teenage social drama. She doesn't own a dress she likes or that she likes to wear. I could be wrong. I don't get the feeling I'm wrong...

"I'm religious. I don't go to church. But I believe in God. Do you think I'm going to hell?"
I wouldn't touch that one. I sat quietly a minute, then said, "Life is full of discoveries. If you want answers badly enough, you'll probably find them..." The story of Socrates came to mind. A pupil curious about becoming wise asked what it took to be wise. Socrates took him to the sea, stuck the pupil's head under the water until he started to drown. Flailing and near breathless, he broke free and exclaimed, "Why did you do that?!" "When you want wisdom as badly as you wanted air, you'll find it."

How to Motivate Teachers

2011-06-11T00:10:00.000-04:00

What good are teacher reviews by administrators? What are they used for?

The obvious answer is they provide evidence with which to fire a teacher. They also provide evidence to withhold a raise. The likely answer (from the administrator) is it is a tool to "help" the teacher improve.

If I were an administrator (heaven forbid!), some of the things I would like to see from my teachers are:

Reflect on their teaching.
Reflect on student interaction.
Keep accurate attendance.
Don't violate the law, particularly with regard to privacy issues.
Engage students consistently.
Teach with rigor.
Teach so that students can understand and comprehend.
Differentiate.
Volunteer to help out.
Volunteer time to make the school a community.

Assuming the teacher is fairly competent and not needing to be fired, the way to get the good, desired behavior from teachers is to focus on 1 or 2 things that are needed most. If there is to be a negative comment, make it on that one or those two things. Once is enough. Otherwise, lavish with praise. Notice everything you possible can about what the person is doing right. Forget about all the rest.

I know some administrators who've never heard of and who do not practice the oreo principle. For every negative comment you give, you must couch it between two positive things.

Here are some comments that have come down:

You need to reflect more on your teaching practice.
That's oddly invalidating when said to a person who has blogged regularly for the past 4 years. If it were more specific, such as "reflect on...." some specific aspect that needs more consideration, well that might be acceptable. Pshaw.

You don't keep accurate attendance records.
Some schools use Power School, an online database for tracking attendance, tardies, passes etc... At my school, we were introduced to this under the guise of it replacing other related correspondence such as cut slips. There is no written policy for how to deal with cut slips. Is "attendance" the same thing as "cut slips"? Why not say, "cut slips." A comment about not keeping attendance infers contractual negligence, whereas a comment about cut slips would be a bad reflection on administrative oversight. Ahah.

Most of the time you give students engaging problems and explorations.
What is being done the rest of the time? Implications... A back-handed "compliment."

He really knows his materials. He challenges students. Some students may not be able to follow.
Really? Are you kidding? Differentiation, by definition, means that you cater to the low fliers as well as the stars. I recall an observation where I gave easy, medium, and hard questions in succession, knowing full well not everyone would get it all. If I hadn't done so, the comment would come back, "Does not differentiate..." You can't have it both ways. It is talking out the sides of your mouth.

You challenge students well. But you are flunking too many students. We'll watch their progress.
At my school, teachers are encouraged to do all they can to pass a student. We are told not to give grades lower than a 50. We are told to make accommodations to "help" the students. And they know it. For all of middle school and 9th grade, they are used to waiting until the last minute to ... turn in homework or projects ... do tests make-ups etc... and they will be passed along. Nobody wants a 16 year old hanging out around middle school kids (because he "flunked.") Oh yeah. If a kid flunks, he can take the full year course in 4 weeks of 2 hour per day sessions, and get an easy passing grade. (Still not knowing the skills.) You can't have it both ways. You can't get rigor without demanding excellence and paying for it.

Students can't understand your accent. ... or worse yet ... Students can't understand you...
It would be nice if administrators knew more about the curriculum. When teaching quadratic equations, to get a remark on a review that it was a great lesson on the distributive property is more a reflection on administration.

You didn't inform administration early enough of a struggling student.
What about the reports sent to the psychiatrist. What about the grades, posted on Power School. What about cut slips or disciplinary referrals that were sent. Where is it spelled out what "soon enough" means.

You need to stay after more often.
My first three years, I stayed until 6:00, or often until 8:00. I know other new teachers who do this. A comment such as this demonstrates administrative ignorance to what is going on in the school, and ... the feeling it imparts in the teacher is one of not being appreciated, and not wanting to do any more than the contract calls for.

You need to participate in an after school club.
I've been the head of the choir, a Bible study group, the school musical, MCAS prep class, and... contractual obligations such as curriculum planning, an after school function (I did the honor society indoctrination this year). I know one teacher who was pregnant and called me on the day she was delivering (she was in labor) concerned about the plans for her students. A comment such as this does not endear anyone to work more, EVEN IF the good work that was done was acknowledged.

And this is the worst of all...

I teach using research based practices. I'm probably the most well read of anyone in my department on the math journals, what works, why etc. I am a 10 year member of the National Council of Teachers of Mathematics, and I subscribe to the journals and... I actually read them and apply what they come out with. (Specifically, I used Math's Mate this year as a part of mixed practice, I have students write in sentences in their open responses, I give problems that make students think -- I've been accused of giving word problems, but this is a misnomer. The problems have words that kids must read, but these are not word problems like the ones you're used to from school days. These are problems without pictures: challenging tasks.) Bottom line, I get results.

Rather than adopt my practices, in the interest of standardizing what we do for curriculum, I've been asked to follow a text book, weak in algebraic content, weak on challenge. I've been asked to return to practices that don't require students to think. I predict the results will be as has been shown historically: students learn a concept, package it up, set it on the shelf and forget about it (or worse yet, toss it in the trash). They don't learn the materials.

I can't tell you how many students (about 20 or so) who have come to me.... "I made an 'A' in ___'s class, but I didn't learn anything. I don't remember any of what was taught. But I remembered everything you did with us. As I was taking the SAT, I had your voice in my head. ... Thank you."

Helas.

How Students Think: Pentagon Angles and Apothem

2011-06-10T23:25:00.000-04:00

Problem: Given a regular Pentagon, draw lines from the center to each of the vertices. Find the angle measure of one of the triangles at the vertex. Find the apothem.

Teachers approach this problem that the central angles sum to 360 degrees, and there are 5 of them, so each is 72 degrees. The sum of the angles of a triangle is 180 degrees, this is an isosceles triangle, so x + x + 72 = 180 etc...

What a student did in my class today was notice that since this is an isosceles triangle, if you draw an altitude from the center f the pentagon to the side, it will intersect at the midpoint, so it will be perpendicular. This smaller triangle has an angle that is half as gib as at first, so x + 90 + 36 = 180, or x = 54. Yeah. Quicker. Elegant.

The problem comes in what is next. Watch the clip to see how the discussion went.

How Students Think 1
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Math Department Motto

2011-05-25T21:53:00.002-04:00

For some odd reason, we've been asked to come up with a motto for the math department at our school - something you might wear on a tee-shirt. Here are some of the suggestions that have been submitted so far:

Westside Math:

Math: from chaos to clarity.

Go figure!

It all adds up.

Opportunities Squared

Exponential Possibilities

Exponential Potential

Figure it out

Figure of speech

Figuratively speaking

Can you figure that?

How do you figure?

R U A Figure?

Got math?

It ain't rocket science.

Math: gotta do it.

f(u) = au^2+bu+c ... Get it?

Holy shift. Get a load of the assymptote on that mother function.

and a very, very dark one:
There is no safety in numbers, or anything else.

I suppose my favorites are the first three.

I have a question

2011-05-25T21:45:00.001-04:00

Here is a recent quiz I gave students on apothem. Sorry for the handwritten document. I was pressed for time.

The kids I teach are smart. They just forget to show it.
I find that they often disengage themselves from the learning process. They want to be told... everything!

Teacher Teacher! I have a question:

V- "I'll pass out the quiz, then I'll pass out calculators."
Student - "Can I have a calculator?"
V - )(*&#$%@ "Grrrr."

Student - "Are you going to give us the length of p?"
Uh. I'm asking for p. It is not a length. It is in degrees.
V- "Please use your powers of observation. The answer to your question is on the page."

Student - "Is there a question to number 2?"
Let's see... This quiz is all about ... area ... finding area using apothem...
V - "Please use your powers of observation. The clues are on the page."

Student - "What does 'n' stand for?" (speaking of the formula Area = 1/2 x number of sides x apothem x side length)
V - " ... " (silence).
Student - "Aren't you going to answer me?"
V - "No. That is not a question I can answer in this assessment."

Student - "Is this an equilateral triangle?"
(Apothem, which is what this quiz is on, only works with 'regular' polygons: all angles are congruent, all sides are congruent.)
V - " ... Please use your powers of observation. The clues are on the page."

Student - "What is the formula for apothem?"
V - "I can't answer that."

Student - "Do we have to use the formula for apothem to find the area?"
V - " ... " (silence).

Student - "Can you tell me how to..."
V - "No. A quiz is for you to tell me how to..."

Oh, brother.

Order of Operations

2011-05-15T21:44:00.003-04:00

PROBLEM: Simplify the expression
4 ( 2^3 - 7 ) - 5^2

This is an order of operations problem:

How many ways can we mess it up?

I'll highlight the source of error in red.

Let's start with the simple errors, such as transcription:

Or forgetting a negative sign:

Or forgetting the ten's column, dropping the 2:

Those simple math errors are understandable. We all make them.

Next is a misunderstanding of exponents. An exponential expression is made up of the base and the exponent. Do you recognize the difference? Do you keep track of which is which? This person didn't:

I'll point out the difference in correctly identifying the base and not. It is a simple math sign, which leads to big errors down the road.

Now we come to a very serious error, one in which the number sense is missing.

I think I understand what is going on in the student's head. They know 2 squared is 4, and since I used up part of the expression, I'm left with another 2.

Instead of multiplying the 4 by the left over 2 to get 8,

they multiply the 4 two times (by itself), incorrectly resulting in 16.

Same mistake, compounded by slopiness. Notice how the "negative" sign morphs into a "times" sign.

Ahhnowtheinfamousrunonmathsentencecombinedwiththeexponenterror.

In this error, the student either made a simple error, or doesn't have number sense to know 2^3 = 8.

I saved the most disturbing for last. In this example of student work, the student distributes and multiplies 4 times 2 (the base of the exponent) before expanding the exponent. That is a huge number sense error.

Order of operations, which is what this problem is about, is among the early topics discussed in algebra 1a. If a student is still messing up with this, what can I conclude?

I believe the single most important thing a young person can do that will improve their math abilities later in life is to learn the multiplication tables. (preferably by 8 years old). That forms the basis for fractions, proportion, percent, decimals, and all that comes afterward.

I can ask one question of a student on the first day of school as a good indicator of what the student's success will be. Amazingly:

"What is 7 times 8?"

Grades

2011-05-11T00:53:00.000-04:00

I just finished writing a final exam for senior geometry students. 26 questions, 4 points each, max score 100 (so, one bonus question), all short response. All but 2 are single concept (skill level of understanding). It is certainly among the easier tests I've written. I ran in by the other two geometry teachers to verify we're on the same page. We are. Short response is good because it lets me know more about what the student knows than multiple choice questions.

I've been inputting grades for students. I know many are very capable. Much more than their grades show them to be. One girl has a 50. She doesn't turn in the work. She never has. Her previous teacher warned me of such. i've spoken with her about it at least twice one-on-one. Oh well. Since grades are calculated by the computer, there is no fudging. It is what it is.

I find that as a teacher, I feel like taking the grade personally. Intellectually, I realize that I can no more take blame for failure than I can take credit for success. The assessments I've given are fair.

In one class, one loud-mouthed girl said, "Why do you give us work? Nobody's gonna do it anyway." She didn't realize how much more a reflection on her personally this was than on the class. Several kids are trying. "You can't flunk everybody. Is anybody passing your class?" Well, a lot of folks are not passing that class. Way more than I would like to see. But, they have yet to demonstrate competence. So. Oh well. It is what it is.

I'm teaching to a standard. I'll meet students more than half-way. But I will not give grades away. I do not want incompetence polluting next year's geometry classes. Students must demonstrate that they have the basic skills before being allowed to take the next level that this course is a pre-requisite for.

I am bothered by the grades.

21st Century Skills

2011-05-06T07:52:00.003-04:00

The other day in a teacher meeting, we met in groups. Previously, the groups generated terms to describe a Westside graduate in the categories: academic, social, and civic.

[I'm not sure I grasp the significance of separating social and civic.... And I think academic could be better broken into literacy and problem solving...] So, among the words on my group's list were... problem solving, which we selected. I suggested we break out "literacy," which we did.

One person in my group suggested we choose, "motivated learner," as an encompassing phrase. As for "motivated learner,"... that works fine for some kids. I have a problem with my taking responsibility for the student's agency. There are some kids I cannot motivate, no matter what kind of song and dance I do... no matter how engaging I try to make the material. This is not a phrase we should adopt to describe our graduates, in my opinion. It implies control that is not there.

In the meeting, we walked around and looked at others' work. I was rather disappointed to see the number of catch phrases that were being used. Like, "life long learner," "critical thinker," "higher order thinking," "engaged," ...

[I was involved in the interview process for vice principal at the school. Those candidates who used catch phrases were ones we weeded out pretty quickly. The man who was selected, when asked how he would do his job, replied, "Boots on the ground." I know this is an army term. But it gives a visual picture. It is not a catch phrase. (well, maybe it IS. But not one generally used in education circles.)]

At the end of the meeting, we were asked what 5 things stood out most for academics. Among the 5 were literacy, problem solving, critical thinking. Someone asked, "Can we just lump problem solving and critical thinking into 'twenty-first century' skills?"

That pushed a button with me.

Fifty years ago, we could envision some things. Dick Tracy wore a wristwatch communication device, not unlike our cell phones of today. But most things of the future eluded the main population. We couldn't see cell phones, microwaves. Heck... the hand-held calculator was out of reach. I remember in 1971, my father spending about $375 (a lot of money then) for a desktop calculator that could add, subtract, multiply, and divide. And do percents! Wow. Oh, if you bumped it, it went cookey.

My point is, who knows what changes 10 years from now will bring? A prophet. That is to say, not most of us.

My son-in-law's profession (bioinformatics)... when he selected it as a major when a sophomore, he'd never heard of it before. Lots of people still haven't heard of it today.

I think "21st century skills" is a buzzword for "use technology."

I probably use technology as much or more than anyone at the school. Yet, my modeling the use of it doesn't means it transfers to my learners. Seeing me use technology doesn't mean they will be able to, or be good at it, or sophisticated in their application.

So you can use the computer? You can google-search? ...but all you did was cut and paste from the first two from a million hits, and then you call it your own writing. It's plagiarism. It is also not very sophisticated. Why not narrow the search by using more sophisticated means, such as putting quotes in your search? Get 25 good sources.

My son-in-law, again, suggested that 21st Century skills would be to change the oil in the car, or change the brake pads. Something useful that everyone should know. He said, "All high school students should be required to take an auto-mechanics course..."

I think what probably is meant (but doesn't always come across that way) by "21st century skills" is the ability to think... to synthesize ... to do more than repetitive machine-labor.

In the past, manufacturing jobs required rote skills. One could leave high school and not do math, yet have a job that could provide for a family. Most of those type of jobs have gone overseas or been replaced with machines. Twenty-first century skills does not mean that you can text or type an email. There is more to technology than plugging it in. We want students who think.

But ...

Wasn't that a 20th century skill, too?

... and 19th ... and ...

More on "When Will I Ever Use This?"

2011-05-06T07:23:00.001-04:00

The question arises, "When will I ever use this?"

It is revealing about the asker.

Usually, the person who asks the question is one who wants to stop the lesson. Why? Because he doesn't want to learn. Why? Because he is unmotivated (lazy).

That is the quick answer true most of the time.

But a more thorough explanation would include:

Either the asker presupposes the person he is asking is God... In order for me to know when you will use the math I teach, I would have to know what your life has in store for you. I'd have to be God, or at least a prophet, to tell the future like that. Not happening.

Or, there is the possibility the person who is asking thinks he is God. In order to presuppose that the stuff you are learning will never be used implies you will have to know your whole life and set of life choices before they happen. That is not human.

Or, the most likely reason a person asks this is that the asker is lazy. Being indolent or unambitious, a person may find that the only thing worthwhile, and therefore worth learning, are things he knows TODAY WILL BE used by him in the FUTURE.

There is another reason the person is lazy. He may think himself clever to distract the planned lesson so that it gets cut short, and he is no longer responsible for what was to have been taught, because it wasn't. He wants to do less work. Lazy. So, complain about something to defer from what should be going on to entertainment (watching the teacher squirm). Not happening.

There are alternatives.

A much better question than "When will I ever use this" is,

"What Can You Do With That?"

"If I didn't learn this, what can I not do?"

"How can this be useful to me?"

Or...

Making the problems more challenging

2011-05-04T01:00:00.000-04:00

My father-in-law, a former educator, told of a math teacher who used to say that geometry is not math. It is logic with pictures.

I suppose that is one way of looking at it. As I was preparing a geometer's sketchpad file of "exercise machines," it occurred to me that if you can add and multiply, you can do geometry. It seems everything either adds to 180 or to 90. (That is not true... but several of the problems, in their basic form, rely on mere addition...)

What can we do to make the problems harder? ... to get students ready for algebra 2?

I've been teaching the concepts of circles. I start with simple numbers. To make things more complex, I add a variable, or make the problem into a linear system, or into a quadratic that students must use factoring to solve.

Today, one such problem, was a quadratic that was not factorable. What to do?

I reminded students of an ALGEBRA 1 task that I am currently teaching... the quadratic formula! We could find the roots using it. And so we did.

My son recently told me that when he took algebra 1, he learned all this "a priori" stuff that made no sense. He asked himself, "When will I ever use this." Later, in algebra 2, he had a computer lab where everything came together. He could see how circles, elipses, parabolas, conic sections fit into the bigger picture. Now, the formula made sense.

I'm doubtful that the utility of algebra 1 makes sense while taking algebra 1, unless you have other things in mind, like moving to algebra 2, or ... graduating...

I want my students to be able to do 8 things with quadratic functions:

1. From standard form, identify a, b, and c.
2. Tell whether the graph opens up or down (a>0, or a<0).
3. Write the equation of the axis of symmetry (x = (-b)/(2a) )
4. Find the vertex. (Plug in the x value of the axis of symmetry, solve for y. The vertex is the (x,y) coordinate.)
5. Find the discriminant. (b^2 - 4ac. If positive, 2 real roots. If zero, 1 real root. If negative, no real roots).
6. Find the roots (using the quadratic formula).
7. Set up a table of values, including at least 2 points on either side of the vertex.
8. Graph the equation.

I recently watched an 8th grade middle-school teacher explain all of this on you-tube. Middle school. It is not rocket science. I didn't ask students to derive the quadratic formula... just to use it. Plugging in numbers into an equation is an EARLY ALGEBRA 1A skill. (substitution, evaluate).

On a recent re-test, I gave two quadratic functions and asked students to do these 8 tasks. Both of the functions had "nice" numbers (integers). Neither were exceptionally challenging. One was among the many examples worked in class. Another was the same question from the first test. Both were problems taken from the homework (lifted directly). The test results were very disappointing. Quite frankly, I'm not sure what more I could have done for the students, except hold their hand, or perhaps lifted their pencil for them, or write the solution.

I don't know what will motivate these kids to use discipline.

sigh.

When Will I Ever Use This

2011-05-04T00:43:00.000-04:00

You may have heard it said that there is no such thing as a stupid question. I disagree. "When will I ever use this?" is pretty silly. (I'm not sure "stupid" is the right word, unless your vocabulary is limited and "stupid" fits the bill to get the meaning across.) It is not to say that a person who makes such a comment is themselves "stupid." It speaks volumes to the person's ambitions.

Who knows the answer?

Even if you have decided on your career, say, build swimming pools, for instance. You never know when you may need or use the math that is part of the 10th grade curriculum. You may find yourself teaching someday. Who knows?!

Your career is chosen by yourself. In making such a choice, doesn't it make sense to want to have the most tools in your toolbox to make that choice?

On the first day of class, lecture 101, I tell students a few things - specifically so I can refer back and say, "On the first day of class, I told you..." For instance, I tell them that electronics are to be turned off and put away before coming to class. The "off and away" policy is in force, and they should consider themselves fairly warned. I'll not be giving another "warning" before confiscating any devices openly used.

One thing I mention is that this question, "When will I ever use this?" comes up. The fact is, they may never use what I teach in geometry, or for that matter, in algebra. You could lead a successful career (like a former Italian teacher here at the school, or ... like the parents of many of the students I teach at what is more and more an inner-city school), and never use the distributive property.

What you will use, and this, nearly daily, is logical thought processes. You must make decisions based on logic and reasoning. Math is the shortest path to learning this. It is unforgiving, black and white, spot on. You can learn it from other disciplines. It's just that math is the shortest path.

What you will use is discipline and following rules. Try showing up to work in a factory without your hardhat or without your steel-toed shoes. Try showing up 5 minutes late. You'll find yourself saying, "Paper or plastic," or "Want fries with that?" pretty quickly. Math requires that you follow the rules. If you lie (or break the rules), you get a wrong answer. If you don't have the discipline to follow the rules, well... we'll see you again next year trying again.

I welcome your feedback. How do you respond to this stupid question?

Sixth Grade Math

2011-04-30T08:29:00.000-04:00

I was speaking with a 6th grader yesterday. She proudly quizzed her 8th grade brother, "Solve this!" The equation she showed was one she had been studying in school. ... Regarding ... FRACTIONS!

n / (6-1/2) = (1-4/6)

That is: n divided by six and a half equals one and four sixths.

I think this would be a nice question to put on the pre-course test. To solve it, you should be able to simplify fractions, to do the inverse operation to isolate the variable, and to multiply fractions. Maybe make it even easier by changing 6-1/2 to 7-1/2.

The valedictorian at Sarah's graduation quoted John Lennon, "Living is easy with eyes closed."

Nooooooo

2011-04-27T20:03:00.000-04:00

A student (from a rowdy crop) saw my pink pen yesterday and pestered, "Can I use it?" "Nooooo." "Pleeeeeease?" "Noooooo." "C'mon. Just let me use it." "Noooooo." "Why not?" "Noooooo."

So, I told her this story:

An aardvark walks into a bar. The bartender asks, "Would you like a beer?"
"Nooooooo," replied the aardvark.

"How about a whiskey?"
"Nooooooooo."

"Can I interest you in a scotch?"
"Noooooooooo."

The bartender thought for a moment and asked, "Say... What's with the long no's?"

....

And now the rest of the story

...

The student asked, "What's an aardvark?"

More on Sweller, Clark and Kirschner, "Teaching Geneal Problem-Solving Skills is Not a Subsitute for, or a Viable Addition to, Teaching Mathematics."

2011-04-20T21:22:00.000-04:00

Since my last post, I've read numerous articles on the controversy presented in that article regarding constructivist vs. guided teaching.

The long and short: research shows that short term memory is affected by cognitive load. Trying to learn too much novel information hampers getting the same stored into long term memory (what we call "learning.") To reduce the cognitive load, guided "worked-examples," are proven to be helpful.

I will certainly take all of this into account in my instruction, particularly of the kids who don't always have the firm foundation of the fundamental prerequisites. The door is open for more or added "worked examples."

However, I still believe that the scaffolding and threading of inquiry-based instruction is generally, and certainly can be modified to be specifically, such that the cognitive load is not overburdened by the selection of problems or by what is to be learned. IBL is flexible enough to allow for guided instruction, depending upon the crowd.

There is solid research into cognitive load theory to suggest that short term memory should not be overburdened. This has to be taken into account when teaching. The research is NOT clear when it comes to constructivist theories, including inquiry-based instruction, and I'm not so sure that it ever will be. Does it work? How do you design a controlled experiment to show constructivism improves problem solving abilities. Any controlled study that ignores the other factors in teaching... inquiry at the expense of on the fly modifications or ignores making changes to meet the needs of those being taught (which may included guided instruction)... might further research, but not result in the teaching we want.

So, there was this fellow, de Groot, in the Netherlands, who in 1946 published an article about chess players. He showed the setup of a chess board of a known play maneuver for 5 seconds. Experts could reproduce this about 70% of the time. Weekend players, only 30%. The study was reproduced in 1973 with similar results. It was extended to show that when the board was setup randomly, the experts performed same as the weekend players, with about 30% of the time the correct setup. What this says is that those experts, instead of having better skill, had more stored in their long term memory.

Do constructivist pedagogies result in better students?

I don't know.

I do know that it is more interesting for the students. Listening to worked examples can be boring.

Kirschner, "Why Minimal Guidance During Instruction Does Not Work..."

2011-04-19T22:58:00.005-04:00

I read an interesting article recently by Paul A. Kirschner, Utrech University, the Netherlands, et al, entitled, "Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-based, Experiential, an Inquiry-based Teaching."

The article slams constructivism. The research it quotes comes from brain study and how the brain works. In short, there is long term memory and working memory. Once information is in the long term memory, it can be retrieved generously. The working memory, however, is limited. When overloaded with novel or new information, the learning, or getting that information into long term space, is hindered.

Working memory has limits when dealing with unfamiliar information. Those limits disappear when working with familiar information. "The onus should surely be on those who support inquiry-based instruction to explain how such a procedure circumvents the well-known limits of working memory when dealing with novel information."

He discusses the faulty rationale that makes no distinction between "the behaviors and methods of a researcher who is an expert practicing profession and those students who are new to the discipline and who are, thus, essentially novices." The way one works in his domain (epistemology) is not the same as how one learns in that area (pedagogy). Educators sometimes seem to confuse the teaching of a discipline AS inquiry with the teaching of the discipline BY inquiry.

Kirschner points out that this applies to the novice to intermediate level performer in the field. I read it to mean that he is saying constructivist approaches may work better or well with those persons having sufficient background information and expertise.

If I am to put this into perspective of what I am trying to accomplish in my classroom, I must not ignore a separate body of research that supports the use of inquiry-based approaches to problem solving. What I do in my classroom is not entirely an inquiry-based approach. I give direct instruction at times. I also attain explicitly each concept after first giving students a chance to try to figure it out. The problems are selectively chosen to be within the zone of proximal development. That is to say, I'm trying to give problems that they should have sufficient background information to be able to solve, all the while pressing the new information to its limits. On a particular concept, I may give a basic problem, and move to progressively more challenging problems, requiring students to draw on information they've seen...

For instance, we start with Pythagorean triples. Find the hypotenuse given two legs (must apply the theorem). Next, find the leg, given the other leg and the hypotenuse (must identify the hypotenuse as the longest side). Move to finding the hypotenuse when it is irrational (must simplify radicals). Next, substitute variables: [one leg is 8, the other is 5 more than a number. The hypotenuse is 2 more than the longer leg.] (must be able to multiply two binomials -- FOIL or box or ...). Next, substitute variables, but increase the complexity (must factor and solve a quadratic): [one leg is 3 more than 3 times the other side. They hypotenuse is one more than the longer leg.] Next, use radicals for side lengths (must know how to multiply radicals).

As you can see, I am not teaching a "novel" concepts. At least... none of this SHOULD be novel. Each of these tasks are from algebra 1 or before. However, here we have to put it to use at the same time. Novel information is the 30-60-90, 45-45-90 right triangles. Novel information is the tangent ratios and how they work. "But we only studied that one day, and now you're testing us on it?!!!" (Actually, it was 3 days, but who's counting?)

I think that is where a lot of what I'm trying to do is train the brain to retrieve stuff from the longer term memory.

The problem I've seen in the classroom is consistent with Kirschner. Novice learners who have insufficient background skill are not successful at the problem solving aspect of this approach. Those who don't have the information in the long term memory to begin with... who "learned" (but didn't "learn" ... not getting it into the long term memory... not having enough practice or the right kind of practice) ... who "learned" (short term) then threw the compartmentalized package away, struggle, having to re-learn (e.g. multiplying radicals), and then some.

The article quoted a citation (1992, cited in Bernstein, Penner, Clarke-Stewart, Roy & Wickens, 2003) which said, "... large amounts of guidance may produce good performance during practice, but too much guidance may impair later performance. Coaching students about correct responses in math, for example, may impair their ability to later retrieve correct responses from memory on their own."

Students come to me with ingrained fixation of what math is. They think of it as a series of worksheets about concepts they are to master, test on, and promptly forget. They show from the questions they ask in class (which are answer focused rather than process oriented) and from how they ask them (usually abrupt and accusatory) that they have been Molly-coddled, coached, told what to do, and how to do it. They have not been required to think for themselves, to read, to draw, or to figure out on their own how to use the knowledge they've been given.

Starting with day 1 and the first problem I assign, they must read, comprehend what they read, follow instruction: they must think. If this is inquiry, then I must reject many of Kirschner's assumptions.

I can appreciate what he wrote about working memory, overload with novel information, and getting information into long term memory, with the huge implication long-term memory has on "learning."

I draw on information students are to have as prerequisites for my course, using inquiry-based approaches. But, perhaps what goes on is not inquiry-based, at least not in-total. It is differentiated, graduated, higher-order...

My neighboring teacher is from Rumania. She says that as students they do soooo much more practice than we do. Same in China and other "successful" countries where math is taught. It is not so much that the students are smarter or that they are better problem solvers. It is that they've seen more because they've had more practice. They've been exposed to more. It is in their long term memory and they can draw on it.

This makes me wonder if I should adopt what Howard A. talked about in teaching the book 3 times during the year. Once to breeze through the material. Once to be given intermediate problems, and a third time to get the very challenging problems. The model comes from a Florida champion teacher who develops champion students. This is a pedagogy I feel I have to try sometime before I leave this profession. I'm not at a state of mind to want to do this just yet. I don't know what to do with low fliers - students whose skills don't measure up to the minimal prerequisites. How do I deal with them in this situation. My fear is that they would drag down the class to be always at the very low level - a place they're use to being in and a place they're used to directing the class toward.

In any case, I know that what students are getting from my classroom is different from most other math classes at my school. If they do as I ask, I believe in my heart they are well served. I recently subbed for a teacher who was out during my prep period. I observed the interaction of students. It was more of the same that they had had as students before coming to me. There was little carryover from how I taught. I feel like an island. I believe my ideas are progressive. Perhaps innovative. The school is same-oh, same-oh as it has been the past 50 years.

Pythagoras challenges us.... how?

2011-04-16T11:51:00.003-04:00

Students learn the Pythagorean Theorem in middle school. They come to me in 10th grade and can all recite, "It's a-squared plus b-squared equals c-squared." Fine. On the first day of school, we do a problem involving connecting dots and rearranging pieces... that prove the Pythagorean Theorem:

Thus begins our journey through geometry.

Some scholars believe the Pythagorean Theorem is the fundamental theorem of geometry. That's what I tell the students. It is the over-arching under-girding foundational principle of nearly everything we do in geometry. Yes, it is that important.

What's so hard about it?

Beats me.

"It's just a-squared plus b-squared equals c-squared."

Here are some things that make it hard... or ways of setting up the problems that make them more challenging:

1. Simplifying the radical. If the legs of my triangle are 4 and 8, the hypotenuse is sqrt(80). We have to know how to write radicals in simplest form. This means no perfect square factors of the radicand, no fractions or decimals in the radicand, and no radicals in the denominator.

2. Multiplying Radicals. How do you, for instance, square 2*sqrt(2)? I explain (in one ear...) that when multiplying, you can keep that which is outside the radical together, and you can keep that which is under the radical sign together. The only way to get out from under the radical is to be a perfect square. So our instance becomes 2 * 2 * sqrt(2 * 2). Easy enough, yes?

We know that a square root, squared, is the radicand.

3. Writing out the math. We know Pythagorean triples have whole numbers for sides, and we're familiar with several, such as 3, 4, 5, or 5, 12, 13. What if the sides were sqrt(5), sqrt(12) and sqrt(13)? Is that triangle right, obtuse, or acute? ... You really should do the a-squared plus b-squared equals c-squared out on paper.

How about this one, where you first must determine whether it is a right triangle:

4. Can you square a binomial? We've discussed this since the first month of school, with multiple repetitions. The box... FOIL... Ms. Dubois' multiplication... All different ways of doing this without dropping the middle terms.

5. Can you factor. If I wanted to make this even harder, you not only square the binomials, but you have to solve the quadratic... Again, stuff we've practiced. (I would not put this kind of problem on a 10th grade WSHS test on right triangles... That would just be cruel).

6. Can you draw? I gave the following on a test:

PROBLEM: Given triangle ABC, angle A measures 45 degrees, angle C measures 30 degrees, and BC = 20. Find AC.

There is no 30-60-90 or 45-45-90 triangle here. But if you draw it, perhaps you could see that you could make one of each easily enough. If you know properties of special right triangles, or if you know how to use a formula card which has that information on it, this is reduced to a really easy problem.

So, you can see, there are ways of increasing the challenges associated with the Pythagorean Theorem.

Interesting Problems Engage Students

2011-04-12T20:27:00.003-04:00

PROBLEM:

We're having a test this Thursday on right triangles. It covers the following topics:

Pythagorean Triangles (right triangles)
Is it right, obtuse or acute (other uses of PT)
Pythagorean Triples
Special Right Triangles: 30-60-90, 45-45-90
Tangent Ratios

In preparing for today's review, I wanted to select more than the problems we've been doing from the book. I looked to other sources. I found and used this handful of problems from Art of Problem Solving's Geometry Introduction by Richard Rusczyk. I highly recommend the book. I wish we used it here. None of the problems are, "just the same with different numbers." I see the challenge from these problems are what engage the students.

Throughout today's entry, I share some of what we covered during one period of class.

This was like the first, but different. When a student said, "I have no clue," I could ask where they got stuck. I note here that it was the students who didn't write things down that got stuck. Learning a work ethic... how to work... is a classroom culture I've tried to nurture since the first day.

Students come to me with poor skills. Primarily this is because they are used to being given exercises where they can look at it and know the answer long before their pencil hits the paper. Why should they write things down. I explain what is needful. "You'd be a fool not to write this down in your attempt to solve it."

So, where did she get stuck? It was in multiplying the radicals. I revealed the solution piece by piece... and explained how we deal with radicals. What is under the radical, you deal with under the radical. What is not, you deal with separately.

The next problem was a lot like this, yet different.

"How do I do this?" one asked. "How do you?" I responded. Wait.... wait... Don't jump in to give her the answer... wait a little longer... for what seems like an eternity, even though it is only about 5 seconds... "Oh, I get it. You just..." "That's right."

The next few problems were to see how students used their knowledge of Pythagorean Triples, right triangles whose sides are whole numbers. Would you believe that there are only 50 Pythagorean triples with a hypotenuse of less than 100. There are only 16 relative prime such triangles. (I got this by reading Waclaw Sierpinski - of Serpinski triangle fame - ' s book on Pythagorean Theorem.)

Solutions:

You could just multiply things out using the Pythagorean Theorem, like some students did:

but that doesn't use what you know about Pythagorean Triples. Instead, note how we found the scalar multiple, then the third side of the triple, then the third side of our triangle. The 3,4,5 is by far the most common triple encountered by high school students.

The second most common, and a good one to remember is the 5, 12, 13:

I like (and introduce a problem the second week of school that uses) the 8, 15, 17.

Continuing with our review, I ask students to apply what they know about special right triangles to solve this:

We have some problems where the students answer whether the triangle is right, obtuse or acute. I got from Beth Atwood, a school teacher who moved back here after teaching in Texas for some time, this idea of putting your hands together, tips of right hand to palm of left to form a right triangle. These represent the legs. The hypotenuse (or "would be" hypotenuse, if it is a right triangle) is the invisible connection.

Focus on the square of the hypotenuse. Is it longer or shorter than the sum of the squares of the "legs"?

Our next treat was a reading problem. I'm saving the reading and writing problem for tomorrow. Oh, this one needs a drawing, too, but the one(s) for tomorrow are pretty much impossible for young high schoolers to visualize without drawing them out.

We have access to formula cards, I remind students. Several kids "get it." To those who don't, I remind them to, "...think half an equilateral... the hypotenuse is double the shorter leg... knowing this will keep you out of trouble." (from a little ditty I wrote).

If I can only get them to write this equation, solving it is less a challenge.

And the area is...

So, I said, "Would you like another one that is more the same, or would you like an interesting problem?" "When you say, 'interesting,' you mean harder. I want an easier problem." "Well, OK, then. Here is a problem that is either very very easy, or else it it very hard... depending on how you approach it."

Some kids set up the proportion I wrote here, then checked in, "Am I doing this right?"

I encouraged them to stick with it. Perseverance is one of those math habits of the mind I want to foster. Sometimes, bull work... just doing all the grunt stuff... pays off. There were some math errors along the way, which threw kids off... But, then, the last 3 minutes of class, I moved a few things around, and the kids were wow'd.

All along, I explain to students how mathematicians think. We don't start by knowing the answer. We do this by looking at it... Staring at it, sometimes, trying to think of where to start. We don't just write down the answer... "If all I gave you were easy problems that you knew how to do, what kind of a challenge would that be? How would that stretch you? I want to lead you to learn how to figure things out by giving you problems that require you to think, to write, to do the grunt work in writing it out..."

During most of my periods, the majority of students were engaged and trying. Nobody slept. They worked hard.

And now, as a final note, I had a couple of nay-sayers. One student, during one of my classes accused me of not following the curriculum. I squinted my eyes, shook my head, and pointed to the Massachusetts Frameworks posted in the back of my room. "Oh, but I do. I teach from the frameworks." She went on to say, "But you don't follow the book. You wrote your own book. It's not the same as everyone else. Didn't you have to get it approved?" I wonder where she got this from. It sounds like she's repeating something that a parent told her. I can hear the conversation now. "You're flunking because he's not teaching you what everyone else is teaching..." She went on to say something along the lines of how she wished the administrator who observes me teach could see the way things are normally: "You'd be fired." Again, where is this coming from? This doesn't sound like the voice of a student.

I teach the curriculum. I teach the same topics in the same scope and sequence as the other geometry teachers. (In fact, we are always within 1-2 sections of each other!) I do not teach the same way as other teachers. (We are all different. I lean more on trying to get the kids to take ownership of their learning, to think, to work, not just me give them the answers.) While it is true that I try to be selective about the problems I have students solve, I do so out of their best interest. I know they are not going to go home and do 100 exercises, let alone 10, or even 3. So, I must accomplish more with less. All the problems I select are within the zone of proximal development. They will lead the kids to better understanding, if they work them out. (which includes writing things down -- something this student did not do all during class -- ... trying to solve them using the algebra they've been taught.)

If this student were speaking for herself (and I believe she was merely regurgitating what some adult said to her, not thinking for herself), how dare her accuse me in this manner. The thinking is provincial. It is not progressive. It will not take her to the learning that should take place in a 21st Century classroom.

You don't get smart by doing what you already know how to do.

Synapses

2011-04-12T19:34:00.000-04:00

PROBLEM:

We're studying right triangles in conjunction with the Pythagorean Theorem. In fact, we started with the Pythagorean Theorem the first day of school. I explain it is the fundamental theorem of geometry. The more I teach it, the more I see it throughout high school geometry. It is everywhere.

I have this one period (not an honors geometry class), where a handful of the kids consistently perform at levels above the rest of the class. Today, in an effort to differentiate, while I reviewed with most of the class (which I will discuss later), I sent two boys and two girls out in the hall for a little friendly competition. I pitted the girls against the boys with this problem, which I took from the Art of Problem Solving by Richard Rusczyk.

The girls came back after only 1/3 of the period, with this correct response.

"Is the answer 72?" they asked.

Normally, the process is much more important to me than the final answer. In this problem, I knew you wouldn't arrive at 72 unless your procedure were correct. When I said, "Yes," you could feel their excitement! "Doesn't it feel good?!" I asked "It feels great!" the one girl beamed. "Yeah. Not only to be correct, but to beat the boys..." I muttered with a grin.

The boys did not finish. They realized a mistake they'd made and had to start all over. They ran out of time. They would have completed the problem if they'd had time.

I am so proud of these kids. Their perseverance paid off in a big way. They not only did the problem correctly (and differently than I would have done it...), but they also got to feel the rush from personal accomplishment.

I've worked all year to bring someone to this moment.

Kudos to them.

How do I find x?

2011-04-09T20:56:00.000-04:00

PROBLEM: A right triangle has one acute angle measuring 25 degrees. The opposite side is 10. The adjacent side is x. Find x.

After some discussion of tangent ratios (SOH-CAH-TOA etc... tangent is opposite over adjacent...) I gave this problem for bell work. Students during one of my periods balked. "How do you do this? When are you going to show us what to do?"

I get the feeling that up to this point in their young math lives, they've been hand held and Molly-coddled. They've been told, "Write this..." They've not had to think.

I insisted they could put this together with what they knew. "What is the tangent of 25?" One person read off a chart: "0.4663." "What is the tangent ratio?" Another said, "10 over x."

So the equation looked like this:

0.4663 = 10/x

"How do you solve that?" "You never showed us that." "When are you going to teach us?" The conversation was a mass of confusion. I insisted that the students stop complaining. I expressed my frustration when all that was coming from their mouths was negative commentary about the classroom dynamic. I wanted to see them attempt to solve this.

(Surprisingly, all my other class had done so, and they did so with much success! It is surprising what you can accomplish when you.... write things down for starters.)

One particular mouthy student wouldn't take no for an answer - she insisted I walk her through every step.

My reply was to put a much simpler problem on the board. "Can you solve this?"

3 = 10 / x

Wouldn't you know it. She couldn't. I was flummoxed. Algebra 1 is a pre-requisite for this course.

I know one question that will be on my geometry readiness test next year.

Financial Literacy in the US

2011-04-09T20:41:00.003-04:00

In May, 2010, the Economist published an article on the financial literacy of the nation. They tied many of the defaulting subprime borrowers to lack of financial literacy. How good are you? Here is a shortened version of the quiz:

In a sale, a shop is selling all items at half price. Before the sale, a sofa costs $300. How much will it cost in the sale?: $200; $150; $450; $120

If the chance of getting a disease is 10 per cent, how many people out of 1,000 would be expected to get the disease?: 100; 10,000; 1,000; 200

A second hand car dealer is selling a car for $6,000. This is two-thirds of what it cost new. How much did the car cost new?: $9,000; $4,000; $8,000; $9,999

If 5 people all have the winning numbers in the lottery and the prize is $2 million, how much will each of them get?: $100,000; $500,000; $400,000; $200,000

Say you have $200 in a savings account. The account earns ten per cent interest per year. How much will you have in the account at the end of two years?: $224; $240; $242; $220 .

"... I expect to hear back from you ASAP..."

2011-04-06T23:25:00.000-04:00

As we near the end of the term (tomorrow), kids are nervous about their grades. "What can I do to improve my score... by tomorrow...."

Parents are retentive. I received an email this evening in which a parent wrote, "... I expect to hear back from you ASAP..."

The principal at the school has a motto, "It's on us." I am not alone in wondering how this translates. It feels like, "It's on YOU... to pass these kids..." That's probably not the way it is meant, but it IS the way it feels.

"... what are you doing to help the kids succeed?"

What am I doing? Are you kidding me?

...far more than is expected, or than is practical, or than is reasonable. Here's a partial list:

Tests and Quizzes (50% of the grade)

Post links to practice tests (that look and feel like the department tests for each unit).
Give extra credit, up to 50 TEST POINTS bonus, on the term project (flagpole), along with much additional assistance at taking measurements, working calculations, setting up the project as per the requirements.
Allow make-up well beyond the one week deadline.

Open Response (25% of the grade)

Post links to question specific help.
Allow students to submit work online, receive comments, and modify their work.
Allow late work to be submitted.
Allow, nay encourage, students to redo their work for a better grade.

Math's Mate (15% of the grade)

Allow late work to be submitted.

Other

Post class notes daily online.
Post numerous helps that I either wrote, developed or found.
Post homework assignments.
Hold extended office hours 3 days per week instead of the mandatory 1 day.
Provide access to infinite exercises on a program I wrote.
Provide tutorial, practice, and extra credit for successful proficiencies on specified topics at the Khan Academy.
Input grades into the PowerSchool grade portal for parents and students to have immediate feedback.

Aside from the additional 2 hours per week I spend inputting scores into the online grade portal (which was supposed to relieve us of time communicating grade specific information to parents and students, who both have access), I spend on average an additional 2 hours per week answering parent requests regarding their child's grade. (One wrote that as a working single parent, she didn't have time to learn how to go online, instead I am to provide her scores to her individually. -- and her children will be in school for the next few years... And the administration informs us it is our duty to respond to all such communication... Thus it is -- how my time is valued.)

I'm doing my part.

What are you (students) doing?

Special Right Triangles

2011-04-05T23:53:00.000-04:00

A month or so ago, I taught about special right triangles: 45-45-90 and 30-60-90. I taught ho to simplify the radical and (as part of that) how to rationalize the denominator.

Since we had discussed it before, today, I taught it again. Sort of a refresher. To some, you'd have thought it was the first time they'd seen it.

I use geometer's sketchpad, and a little program I wrote, that generates unlimited exercises. Once students seemed to get it, I gave them a problem (such as the 45-45-90 adjacent triangles where the hypotenuse of the one is the leg of the next...)

The 30-60-90 triangles were sequenced like this:

First were the problems with the shortest leg given.
Next I gave them the hypotenuse.
Finally, I gave the longer leg as a whole number. Students had to set it equal to y*sqrt(3) and solve for y...

Once they got comfortable with this, I gave them a problem. (Up to this point, they were only exercises).

Finally, I gave a problem involving both 30-60-90 and 45-45-90 triangles mixed up. It was clever. It felt good to see the students working and trying hard. I think many made connections. Many started to see this as the easy thing it is.

Memorizing - Important to Learning

2011-04-05T23:46:00.000-04:00

Problem: Use a calculator to simplify each of these expressions.

I teach mostly 10th graders. I do not provide graphing calculators. If students don't have a calculator of their own, they may use a little "chicklette." It has basic functions, including the square root. It does not let you type in what you see, as most graphing calculators do.

I grew up with a Hewlett - Packard, state of the art at the time (it sent the shuttle to the moon - so they advertised). It uses RPN - reverse Polish notation. It is actually quite logical, once you get used to it... But if you're not used to it, it is a mass of confusion. "Where is the equal key?" If a student asks to use my calculator, and if I'm particularly frustrated that day, or perhaps feeling sarcastic, I'll loan it to them, but say something along the lines, "It's a smart person calculator." They punch a few buttons and give up.

Back to my story... The kids were unable to solve these problems, using a calculator. They did not know what buttons to push, or what order. I wrote the sequence on the board. My solution for the students was to have them do the radical multiplication, first, write it down... then punch the buttons for the rest of the calculation. "2 * 3 sqrt =". Quite a few kids pushed the 3 and sqrt sign, but did not push the "=" button, even though I stated and wrote that these are the buttons you push, in this order. So, their solution kept coming up the same number... sqrt(3). Oh well. So much for learning the calculator.

Some forgot that the "plus or minus" sign meant two different operations, even though we'd discussed it earlier, again yesterday, and today already... Sigh.

So, to accompany what I have to say about all this, let me tell a short story.

This weekend, my wife and I watched our surrogate grandchildren. Our grandchildren live in Houston. My wife feels particularly matronly. So, we have these friends from Botswana. He's a doctor at Bay State Medical. His wife is pregnant. Good friends. We watched their 7, 5, and 2 year old, to give the parents a break.

This weekend, we had the kids for a sleep-over. They were remarkably well behaved. However, they were not very nice to their host. I got out the "original memory game." The box said "Ages 3-6." You may recognize this game. It is 50 little cards with pictures on them. We used to call it the matching game, or "Concentration." I remarked to my wife, "Should I put down about half the cards." "Put them all down," she said without hesitation. She teaches k-4 and knows this age well.

These kids had played this game at home (with Mickey Mouse cards). ... so it wasn't fair. In the end, I collected 6 cards, my wife 4 (she started late in the game), the little 5 year old had 16 cards, and the 7 year old had 24! They were not very nice to the host. Of course, I'd beat my own kids at Shutes and Ladders... I was not holding back anything. They wiped me out! Their memories are phenomenal.

I reject the notion that it is because their brains are not so full. We only use the nth of our capacities... I think they have done the drill. They are well versed, and practiced. Their brains have the synapses - they've made the connections.

After the kids left, I tried this on my own. I found the only way I could remember the cards from one to the next, was to put a storyline with it...

Well. In math, kids need to know their multiplication tables. If they don't know them, they cannot find prime factors, they cannot find least common denominators and do fractions, they have no number sense! Parents and 3rd grade teachers (for 3rd grade is when students learn their multiplication tables) do their kids a huge disservice if they fail to drill their kids and assist them in learning - MEMORIZING - their multiplication tables.

I can ask a student on the first day as she walks in the room, "What is 7 x 9?" or "What is 8 x 7?" If they can answer this directly, it is an incredibly accurate indicator of the success she will experience (or not).

I asked a student, "What is 180 minus 79?" The student replied, "Do you have a calculator?" When we give students the calculator at an early age, they use it as a crutch. They don't learn math facts. They are handicapped.

I teach an after school MCAS Prep class. Last week, I observed kids struggling with multiplication. I had them make a multiplication table and start to fill it out. One student could do the 1's and the 2's. Another could also do the 3's. We're talking high school here! This was very disheartening. Especially when you consider those same kids to sing every word to some 1,000 songs in their ipods!

When I was in high school, I swam on the swim team and played tennis. We did these drills, called, "suicide drills," because they nearly killed you. Run from one end of the court to the other, pick up a ball etc... It was murder. When done, we were exhausted. Yet we did the drills. Why? Because doing so built strength! We were able to outperform our opponents.

In math, we need to do skill drills to keep the synapses going. To freshen the brain, and keep things sharp. We need to do exercises. We also need to practice problem solving to become good at it.

Just like the little kids could do the memory game, because they practiced it, so, too, my students need to do lots of practice exercises. Hundreds. If they do so, it becomes second nature.

Yesterday, a student asked once again, "When will I ever use this?" I sat down on a stool placed in front of her. I returned to lecture "101." from the first day of class. "You may never use this... But you will always use logic thought processes. That is what I'm teaching you." Memorizing is a little piece of that.

"Finding the Glory in the Struggle" by Suzanne Sutton

2011-04-01T00:00:00.007-04:00

Suzanne Sutton published and article in “Bulletin” (Feb. 1997) – a periodical for the National Association of Secondary School Principals entitled, "Finding the Glory in the Struggle." I learned of it from Audrey Weeks (who is known for her work with geometer's sketchpad, among other things).

Let me try to give a synopsis:

Learning mathematics is a struggle at some level for all of us. We can see this as something we want to avoid. OR We can recognize that the OPPORTUNITY BEGINS WITH THE STRUGGLE. We can use the math to learn how to work through struggles.

Students have a perception of their own capabilities.

(For instance, my students will make comments like, "I never was any good in math," or "I never liked math." I've also seen those who are proud of their capabilities. They become complacent because for years they could look at a “problem” and know the answer without writing anything down. They develop bad habits, instead of seeking out more challenging problems.)

Regardless of how they see their aptitude, they can learn they have what it takes to meet the challenge.

Struggling is to math what sweating is to basketball; it is part of the process; it is a sign of being in the game.

With math, you have to think in ways you may not be used to. To be rigorous and to be clear, means you have to read, write, and speak in ways the may stretch you. In any case, the onus is on you. You have to be honest in your pursuit; there are no shortcuts.

How you approach math struggles can strengthen how you approach life's problems. Again, Opportunity begins with the struggle.

As for parents...

"Without knowing a factor from a function, the parent, more than anyone, is in a position to help the student engage in the struggle of mathematics. Parents don’t need to fear this struggle, nor do they need to take it on themselves; it is an essential and important part of learning mathematics. If the parent accepts the struggle, the youngster can. And even more important, if the parent values the struggle, and sees math as more than just a series of right answers, the young person can approach mathematical learning in a way that will not only make success in mathematics more likely, but carry over to pursuits far beyond the mathematics classroom."

My final comment is to say that if I can give students problems that require them, out of necessity, to know a concept (as opposed to telling them the concept and then giving them an exercise), they’ll hold on longer to the concepts.

Similarity and Proportion

2011-03-30T21:57:00.001-04:00

PROBLEM: How tall is the flagpole?
Answer: 53' 6". How do I know? Mr. Bernard measured it. He had the janitorial staff lay it down on its side, and he measured it. (Technically, he measured its length, not its height... but...)

We just finished another year of the Similar Triangles Project. Students are required to use at least 3 of 7 methods for taking measurements at ground level and calculating the height of the flagpole. It is a practical application of the math we're doing in the classroom. The shock of going from abstract work with numbers, to concrete, with tape measurer, baffles many.

The seven methods include:

MIRROR - look at the reflection of the flagpole in the mirror. Measure the distance of the mirror from the flagpole, the distance of the mirror to you, and your eye-height (actually, we used a stick of known height and lined it up.)

WILE - Here you take a stick as long as your arm and hold it up to make a 45-45-90 right triangle. Measure the distance to the flagpole and your eye-height.

TWO RULER - With this, you hold a ruler a known distance from your face, align the bottom of the flagpole with a ruler, and measure where the top falls. You also need to know how far you are from the flagpole.

HYPSOMETER - With this, we made our own hypsometers and took a measurement after sighting the flagpole. We also knew the width of the hypsometer was 10 cm, and the length to the flagpole was... (along with eye-height)

TRIGONOMETRY - Here we measured an angle, standing at a known distance, and used tangent ratio to solve for the height (along with our eye height)

SHADOWS - We aligned the shadow of a stick with the tip of the shadow of the flagpole, and took measurements of how long each shadow was.

GEOMETRIC MEAN - I've never been too happy with this method. The way I was doing it was not very practical. This year, Mr. S came up with an idea that made sooooo much sense. You need your eye height, a piece of cardboard, and weighted string. You stand at a known distance, and sight the flagpole along the edge of the cardboard. Then, turn the cardboard around, and sight it again, up the other side of the cardboard, such that the angle of the string is the same as the first time. Take a measurement. This was surprisingly accurate!

I showed the kids early on how they will come across only six configurations (in high school geometry) of similar triangles:

It is comforting to know this is something you can put your arms around. If you recognize the configuration and reorient the picture to the #1 configuration, setting up the proportion is a piece of cake.

A few things to point out. #5 configuration is a lot like the hypsometer (a hard one for the kids I teach to grasp). #6 is the geometric mean. Discussion of this takes a day. The "Side-Splitter" Theorem involves #2. Not shown is the "Angle Bisector" Theorem, all of which are worthy of discussion.

I think in all my years as an engineer, the most useful of all I teach in geometry is proportion (and similarity).

We start out with problems with easy numbers, then we add "x." Most kids know how to cross multiply. I don't know why they cross multiply here, when to isolate the variable all that is needed is the inverse operation (multiply by 15)...

Most of the proportions we go over require cross multiplication. Early mistakes include failure to distribute. (Kids leave out the parenthesis around 20 - x ...)

The problems get harder a little at a time. In this one, you had to multiply two binomials. We discussed back in October 3 ways to do this: FOIL, the BOX, and DUBOIS' way. We've used this all along in the problems I've assigned. How quick we are to forget. I see my job as getting the kids ready for algebra 2... "Don't embarrass me when you get to Ms._'s class."

Finally, the most challenging problems I offer the standard class is where they must solve the quadratic equations by factoring. This is an algebra 1 skill, which few students retain by the next year (geometry). I know it is discussed in more detail in algebra 2...

... If we go over enough of these...

... students start to remember how to do them....

And another...

...And another...

Exercise Machines

2011-03-20T14:15:00.000-04:00

I've been taking the course: Teaching Algebra with Geometer's Sketchpad, in preparation for moderating the course. I've been a moderator for the Teaching Geometry counterpart course for some time, and I'm fairly good with Sketchpad. The final week of the project requires us to write a sketchpad project that we'll use in the classroom. So, with all the talk about Khan Academy, I came up with some exercises.

I think my students who choose to look these over will enjoy them and come to see how easy geometry is. What I notice as I'm going through the exercises is that if I don't add algebra back in (sort of force fit), then the exercises all come out as easy. You can nearly look at the figures and come up with the proper relationship. Almost all of them involve simple addition or simple multiplication to solve.

See the relationship => solve

Easy!

I'm hopeful that my students who look over these exercises will see the relationships (that many have missed up to this point), and "get it."

More on Khan

2011-03-13T21:35:00.003-04:00

Have you been to the Khan academy site? Check it out:

www.khanacademy.org/

The videos are clever and easy to listen to.
But videos are only the beginning!

What I really like is the practice, the hierarchy of problems, and the record of work done, correctness etc.

I see all kinds of potential for this! It revolutionizes education.

Typically, I lecture a group of 28 students, most of whom don't want to be there. They all have huge gaps in their math learning through the years. It's like building a foundation on Swiss cheese, there are so many holes. They all learn at different rates and are all at different levels. We give our all to differentiate the classroom learning experience, but still I'm going too fast for some and too slow for others.

With the Khan Academy, students can go at their own pace. They can re-listen to the lecture as many times as they need. They get the practice on the concept. If they get 10 problems in a row correct, they are then considered "proficient" on the concept at that level. So, for instance, if they are doing 5th grade addition, get 10 correct, they are proficient at 5th grade addition. By the time they've finished the practice, they are getting pretty good at the skill, at the particular level. Levels are broken into modules... so you can get good at addition, at the 5th grade, move on... to the 10th grade level... etc.

Students are supposed to come to me with the skills in hand! If students were to do the practice on the KhanAcademy, I could then proceed with the interesting math problems. The ones that are fun to discuss as a group. (I differentiate between "exercises" and "problems" as follows: Exercises are what students do when applying algorigthms or routines to come up with a solution. Problems, on the other hand, require one to give more than a mechanical, memorized response. My goal with all students is to help them become problem solvers. If they don't have the skills, this is pretty hard to do. To get the skills, they need to do exercises. They need practice with exercises... just what Khan offers! He even calls them "exercises"!!!)

As for the record keeping in Khan. Wow! The software in the Khan Academy allows me to follow a student's progress, either with the broad brush stroke, or up to the granular level. (I could tell which problems a kid got wrong, when he did them, how long he spent on them). You would pay good money on some math sites for what Khan offers for free.

I could use this to pair up an "expert" student with someone who needs help, and have her explain it to him. They both benefit. You think about the math differently when you have to explain it. So the explainer benefits. Students tend to listen to their peers with different ears than they listen to adults. He would hear her. So they both benefit. An alternative would be to have the students listen to Khan explain it on his video tutorial... Once you get it (relisten as much as you need to), then you continue with the PRACTICE.

I get to see how much work has been done. I get to see what exactly they got correct, or specific problem (granular level!) of what they missed. So I can focus my time on the students who are really struggling.

I can see this used as a tool for summer school.

Heck, I can see it used as a huge part of my Algebra 1B course. Those students who are currently failing the class could do this work, demonstrate competence, pass a comprehensive test, and pass. Why not?! The kid benefits by getting credit instead of blowing off the full year. I benefit in that the kid learns the skills (so passing him on to the next year will not be just scooting the problem along, but fixing it before it moves). The principal benefits because the student scores better on the state test.

The geometry part of the academy needs work, more problems, more of a fix. But even with what is there, I could see students using this to demonstrate to me a degree of proficiency.

And this really does very little for my true goal of getting the kids to be problem solvers. EXCEPT, that if they know the skills and are not struggling just to get by with the skills, then they will be able to better attack and learn the problem solving skills through the problem sets I give them.

The real question I have is this: How motivated will the students be to take ownership of their own learning and do this work? How much time will they devote to watching the videos and doing the practice? Will they put out? We'll see.

I will dangle this carrot.

No excuse after this...

Khan Academy

2011-03-12T11:44:00.001-05:00

Salman Khan started the Khan Academy a few years ago. Here he explains his self paced learning ideas at a Bill Gates led TED conference.

I love this idea that teacher time is no longer the lecture, but the interaction. He speaks of using technology to personalize the classroom. I think he hits the target on the head.

My mentor in the band world, H. Robert Reynolds, said in an interview that the purpose of the band rehearsal was not so that you could work on practicing the kids' part... (a kid comes to rehearsal, not knowing how to play his part, thinking the band director will go over it enough times for him to learn it in rehearsal... good for the kid, bad for everyone else)... rather, the purpose is to DO THE THINGS YOU CAN'T ACCOMPLISH ON YOUR OWN, BUT ONLY AS A GROUP. You come to rehearsal knowing your part. The rehearsal is to put it all together.

In the same way, the classroom should be where you come to do things you can only do in the group setting. Listening to me lecture is NOT my idea of fun. Having kids interact and present their work and discuss their work, and teach each other... That's fun. That's learning.

Teaching Special Right Triangles

2011-02-22T00:48:00.001-05:00

The days leading up to our discussion on the two special right triangles: 30-60-90 and 45-45-90, I have students do some problems that require fixing the denominator, such as (1)/(1/4).

This lesson started with this warm-up.

PROBLEM: Given square ABCD with [ABCD]=25, find AB. Find AC.
ANS: AB = 5. AC = 5 x sqrt(2).

This is a RIGHT ISOSCELES triangle, also called a 45-45-90 because of its angles.

All 45-45-90 right triangles have the same shape. They may be larger or smaller, but the shape is the same. We look at the GENERAL CASE (using an abstract "number" called "s", which could be any positive number):

So the general case to remember or be able to come up with the way we did here is:

Here is a key to solving special right triangle problems. If you draw this configuration in the orientation of your problem, and match up the sides so that you can solve for "s", you're done. The hypotenuse is sprt(2) * s.

After this example, I give students a few easy examples:

Easy enough. So, we try the harder problem (harder because it requires some kind of algebra to solve it). Solve for "s". Put your answer in simplest radical form:

Some students make the attempt by matching up the sides. This requires us to be able to rationalize the denominator. This is my preferred method:

Most students find this second method to be easiest. Write out the Pythagorean formula and solve it:

I can't help but also mention Danny Z, a former student, who came up with Z's theorem:
Start with a 45-45-90 right triangle, given the hypotenuse, such as = 8. He says, draw an altitude from the right angle. Since the one angle is 45, the other is 45. This forms two congruent triangles, whose legs are both 4. Follow the pattern for 45-45-90 triangles, given the leg, the hypotenuse is sqrt(2) times it, to get:

Now we turn our attention to the other special right triangle. Start by being given that it is a right triangle the hyptenuse double the measure (2s) of one of its legs (s).

Reflect this over the longer leg. By SAS, you have two congruent triangles.
So the hypotenuse of the other one is 2s. The two short legs add to give s + s = 2s.
You have an equilateral triangle.
How much are the angles? 60 degrees.
So the small triangle is 30-60-90.

What is its longer leg? Here is where it gets interesting. The kids always fall, predictably, into the same traps:

So, after we get that straight we come across our second math trap:

The answer students will give is b^2 = 4, which is incorrect.

So we solve and get the general case:

As with the other problems, if we draw this configuration in the same orientation, we can match up sides and solve for s. Then we're done.

Here's another:

And the "hard" problem (because it requires the algebra to "fix" -- rationalize the denominator):

When we solve this:

This is not a rational number, but is nothing to fear, either.

Finally, we come to this week's open response question. You are given the kitchen triangle and asked to find d, h, and the area of the triangle. If we match the triangle up with the known special right triangle configuration, we solve for "s" and plug in the other values.

d = 2s = 2 (2) = 4 ft.
h = s*sqrt(3) = 2*sqrt(3) ft.

We plug the values into the area forumla to get the area. Note that base and height are [ft] x [ft] for units of [ft*2].

What I learn from all of this is that I can put my congruent triangle knowledge to test to have students come up with the configuration for the 30-60-90 right triangle. Coaching is warranted, given the common mistakes that creep in.

The drawing of the correct special right triangle next to your problem and matching up the sides is an algorithm that seems to work.

Crows Can Learn by Observation

2011-02-08T19:31:00.001-05:00

I ask my geometry students to memorize the definition of a polygon. Memorization is a lower level thinking skill on the Bloom's taxonomy. However, it has its place.

A polygon is a closed figure with three or more coplanar segments called sides. Two sides with a common endpoint are non-collinear. Each side intersects exactly two sides, but only at the endpoints.

If you leave out any part of the definition, I can draw a counterexample: a figure the fits your definition that is NOT a polygon.

So today, a student said, "Mr. V. I did a google search last night. The definition of polygon isn't nearly as complex as what you told us."
V: "Did it leave out some of the parts?"
"Yeah."
V: "Then it was wrong."
"You're telling me that your definition is right?"
V: "Yeah. There's a lot of junk on the internet. Maybe what you saw was for grade school."
"So, you're some kind of expert."
V: "Yes. I am."

All year long students have had the task of recognizing patterns.

If it is true that crows practice facial recognition....

... there is hope my kids may come to recognize properties of parallelograms.

Ball Diameter

2011-02-07T22:41:00.001-05:00

How many rotations (or partial rotations) will the ball make at the distance they are set apart?

Why is it a mistake to set the balls one complete revolution apart?

http://www.youtube.com/watch?v=uGT-EXhnUYs

Trapezoids: Easy to Hard

2011-02-04T23:58:00.000-05:00

PROBLEM: Solve for x.

This quadrilateral program was a refresher to bring us to trapezoids.
The first questions were easy. Starting with angle problems:

In this one, a student drew the linear pair to 150 to find y. When a female student in a later class couldn't visualize that the same side interior angles were supplementary, I drew this in red to emphasize the alternate interior angles. Her response, "Oh yeah. I get it."

More trapezoid angles. The problems get a little harder as we move our discussion of area. Find the area of triangle 1 and triangle 2. Combine them to find the area of trapezoid PQRS.

After this, I had students combine the two general formulas for the areas of the triangles to come up with the formula for the area of a trapezoid. (We derived it, thus). And so, our discussion continued with area of a trapezoid, only no pictures. First one that was straight forward plug and chug:

Next was the problem where you had to solve for one of the variables, again no picture was given:

The beauty of trapezoids! If I give the bases and the diagonals, or the bases and the sides, or the bases and one side and one diagonal, I can find the area. This next part is way cool.

I found that if I warmed the kids up to it first, they stepped up.
First I had them draw the figure and mark it.

I explained we were going to find the area of the trapezoid. I further explained that it was going to take some time and perseverance, but that none of the individual steps was beyond their capability. It wasn't hard, just long.

"I'm stuck."
V: "OK. Here is your first hint. 'The Fundamental Theorem of Geometry.'"
Some kids took off.
"I'm still stuck."
V: "Can you come up with two Pythagorean equations?"

V: "Now, simplify each... What is 13-squared? ... How do you square a binomial?... What do you see?"
"I see both equations have an x-squared and an h-squared?"
V: "So?"
"So you can substitute."
V: "Very good!"

Students carried it out to solve: x = 5.

Not done yet. Plug it back in to solve for h.

Not done yet.
Plug this in to solve for the area.

You must remember, I had warmed students up to the idea that we'd be doing 4 problems of greater difficulty today, and that these would take longer to do, but they wee not hard.

"I could never have come up with all that on my own."
V: "Now that you've seen it, could you do it again, on your own?"
"Maybe."
V: "We'll do another one. But first let's try an easier hard problem."

One girl in my class who usually complains about how things are too hard, volunteered.
"Oh, I think I get this. ..."

She explained that HI = 6 because it is isosceles. She than came up, unprompted, to draw the rectangle down from GH. She pointed out that the small segments were each 1. "Why?" "Because the middle section is 8, and if you split the remainder in half, each part is 1."
V: "And you know they are the same because these two triangles are congruent. What shortcut tells you it is so?... HL... It is commendable that ___ set this up..."

She picked right up and responded well to compliments.
From here, writing the Pythagorean equation was easy. The only problem was that h turns out to be a radical! How do you square a radical?....

Anyway... you get the picture.

After this problem, I gave students the option to go to another problem like the first or to go to a considerably more difficult one. They chose the easy one first:

A = 300.

And finally, the more difficult one.

After staring at it for a while. Some kids suggested connecting the two parallel sides.

Folks, these were regular Joe high school kids. This is not honors students or kids from a private accelerated school. These were kids who are learning great math habits. They are part of what demographers call the "gimmie" or "instant gratification" generation. They want it now. They don't want to work for it or, heaven forbid, to have to show both tenacity and perseverance.

These kids showed both. I was so very proud of them. I think they are starting to see some benefits of all this work. I did assure them that they would not se anything so challenging on our upcoming test.

I don't think I could have fared so well with my 7th period class; too tired. Helas.

Show the Two Angles Are Congruent

2011-02-01T09:25:00.004-05:00

PROBLEM:

I assigned this for homework a couple of nights ago, thinking students would look for congruent triangles. A few did.

When we discussed it in class, the initial student response from all classes was to mention SLOPE.
"You find the slope."
V: "How would you use slope to show the two angles are the same?"
"They have the same slope?"
(None of these segments have the same slope. If you connect the dots to make two triangles, only two of the segments have the same slope. So it is clear at this point the student said "slope" not knowing how to use it and not having pondered this with sufficient depth to conquer it. He didn't know what he was talking about.)
V: "What are the slopes?"

Students found the slope of each of the segments.
An interesting pattern emerged. Two pairs had slopes that were opposite reciprocals.
"The segments are perpendicular."

From here we see that since angles BAC and EAD are complements of the same angle, angle CAD, the two angles must be congruent.

After the response "slope," the second most common knee-jerk response was:

"Find the length." or

"Use the distance formula," or

"Use the Pythagorean theorem"

all of which are synonymous.

V: "Did you find the length of each of the segments? What are they? What do you see?"

We find that three of the segments are congruent:

"That was a waste of time."

V: "Oh no! It didn't take us where we thought it would. But it wasn't a waste. We sometimes may need to retract and try a different path... But the act of exploring and searching for patterns or clues is not a waste."

One student threw out the idea of looking at the midpoint of segment AC. We explored that further.

We found two congruent triangles, by the SSS criteria. Since the triangles are congruent, all the corresponding parts of congruent triangles are congruent (CPCTC), so angles BAC and DAE are congruent:

I asked students to look for another way to solve this problem. Here it is:

Again the triangles are congruent by SSS, so by CPCTC, the two angles must be congruent.

"You can draw lines like that?"

V: "Segments. Yes. They are 'auxiliary' lines or segments: ones you add by extending existing lines, or drawing parallel or perpendicular lines or etc..."

Yet one more for those who don't yet consider congruent triangles. How about drawing two squares and cutting them in half using AC and AE...

As you can see, this question is another multi-layered thinking problem with multiple access points from which students can enter the discussion...

PEMDAS - Order of Operations - Student Work

2011-01-27T01:44:00.002-05:00

PROBLEM: Simplify the expression

The first concept taught in algebra 1 is "order of operations," also known as PEMDAS, or PEDMAS. This stands for "parenthesis" "exponent" "multiplication division" or "division multiplication" (whichever is first when placed side by side), "addition" "subtraction."

If one plays by the rules, one gets this:

My students seem to be a law unto themselves. That is to say, they make up and follow their own rules. I got 21 different incorrect responses (a total of 26 incorrect responses out of 87 students). My knee-jerk response is to ask what those #$%^* math teachers before me were teaching. But I know my colleagues. I know how and what they teach. They are doing their part.

So here is some actual student work (and all the examples are from actual student work):

(just click on the image to enlarge it for viewing).

Note the math error at the end. The third example was actually a PEMDAS error.

I can't tell you how many students count on their fingers! The temptation is to ask, "Do you need to take off your shoes?" (not enough digits on the fingers alone...) These are basic math facts! I think we do our students a disservice by introducing calculators early on. They become dependent instead of learning multiplication tables, or division.

I've found that playing games (an old person sport, apparently lost on much of the upcoming generation), such as scrabble or yatzee, which require you to add numbers at the end, enhances my ability to add numbers. Lots of practice. Rote stuff... but still needed, especially at the elementary grades.

These next three examples show one error, none of which are PEMDAS errors.

These next two involve PEMDAS errors. The one person got stuck:

Some students do the "inverse operation" - but there is NO EQUATION! The problem is to "Simplify," not "evaluate" (where you plug in numbers) or "solve" (where you have an equation). There is also the PEMDAS error.

Next, we have two PEMDAS errors, and instead of multiplying, the person adds. A lack of attention to detail which is all too common.

And again, there are two PEMDAS errors on this next example of student work. I point them out.

So, what is the deal. What utility is there in looking at all these problems. Actually, there is a lot to say when students view each other's work. They can be more critical than the teachers! And, more importantly, if they can identify errors, perhaps they will see what they themselves are doing wrong.

So, on these next few examples, I do not tell you the errors. I want to see if you can find them yourself, same as I might ask of students. I think all of them involve PEMDAS errors. We start with 2 errors per problem:

Can you find the errors?

How about these next 3 examples. They each have 3 errors:

Another two examples with 3 errors. The second involves a division error, too!

And finally, these last three problems each involve 4 errors, PEMDAS and careless lack of attention to detail.

What do we learn from all of this?

My purpose was not to show that West Side students can't do math... We all make errors.
What do 21 incorrect responses tell us about the student body?

I'm torn.

I have two opposite views on what is going on and how to fix it.

I'll repeat that any of my 108 students can tell you all the rules governing order of operations. If I point out their inconsistencies, they'd say, "Oh yeah." If I ask them to find the errors, they may do it... MAY do it... for one or two of the incorrect responses... then they'd blow it off. Too much work. Or... "I can't do this." Not that they "can't," it's just that they won't. It requires effort. So this means there is a discipline problem in doing work. The WORK ETHIC NEEDS FIXING.

My first of the two opposing views is that students could be bored with it all. "What, again... another one of these... When will I ever get out of having to do math... I hate math..." This kind of problem is uninteresting. It doesn't connect. Of course, it is a foundation that you have to have to be able to go anywhere with math... You can't have the cake until the wheat is harvested... Perhaps what is needed to FIX IT is more interesting problems that will captivate the attention. I'll be looking at the Math Forum problems for some of these.

My second of the opposing view is from the Chinese. They are among the top scorers on math worldwide. The president says we should model our instruction after them to improve our standing in the world. (US ranks near the bottom of industrialized countries on math scores... I'll get you my source of information if you don't believe it)... However, on the evening news we learn that the Chinese students themselves say their system has a lot to be desired. First, they track, so if you are on the college track, you are surrounded by others on the college track. Second, they have a much longer school day, lots of homework, and very long school year. The pressure is tremendous to get those coveted high paying jobs (which means you have to get the higher education, which means you better do well on the exam, which means study during high school). I think that they do so many problems, that ... it's not that they're particularly smart (don't get me wrong. China may have more geniuses than the US has students, just from the sheer number of people ... ) but they've done so many problems and they've seen so many variants that the problems look familiar.

In short, we either give fewer harder problems,

OR

we give lots more practice.

In any case, we need to develop the math habits of the mind.

What Every Parent Wants Their Kids To Know

2011-01-20T19:32:00.002-05:00

PROBLEM: Two parallel lines cross over two other parallel lines to form 4 intersections. Which angles are corresponding? (Could you identify which pair are "corresponding" if each of the angles were numbered and several pairs were listed).

This is a question from our mid-term exam that I find absurd. Who cares, really? What is far more important to me is the relationship between those two angles and what you will do with that information.

Corresponding angles (formed by a transversal crossing parallel lines) are congruent. My problem might be to give two equations for the two angles (such as 7x - 10, and 5x + 14) and ask what the angle measure is. Students would have to know the angle relationship, solve for x, and substitute their result to find the angle measure. Not ridiculous, but certainly more of a thinking problem than the original, and certainly more pertinent than the vocabulary. In my opinion, the original question should never have found place on a mid-year exam.

No better:
PROBLEM: Given the picture with several points on it and two connected to form a line, the question reads, Name the line and the plane.

Or try this:
PROBLEM: given a polygon with one exterior angle given, find x and y, two of the interior angles. But notice that there is only one correct answer among the multiple choices for which x, the adjacent angle, is correct. Why bother with the polygon; a linear pair would suffice!

Here is another.
PROBLEM: Angles AOB and BOC are adjacent angles with measures 35 and 42. Determine the measure of angle AOC. I'm sure there are folks who will rush through this and not get it correct. But it is not a thinking question. It is, by all intents and purposes, an exercise.

Just one more:
PROBLEM: Two triangles are given to be congruent. (They look the same!) On side is marked "4." The same side on the other triangle is marked, "c." Find c.

I estimate that half of the 40 multiple choice questions on this year's mid-term exam are low level thinking questions like these. The rest barely rise to an intermediate level. Only about 2 of those are challenging. All came from a test generator.

I originally titled this entry, "What I stand for." Later, I wanted to call it, "Disgusting Putrid Stinking Questions." I've settled on the current title for good reason.

I have four children. My wife and I raised them to value education and learning for its own sake. Learning is its own reward. They truly believe that. Two speak fluent Arabic, one is fluent in French. One is studying to be a psychologist. Anecdotal, true. Irrelevant, hardly.

I want for my classroom students what I want for my own children: the best education they are willing to receive.

My title is "What Every Parent Wants Their Kids to Know." You may add... about math... What do you want your kids to know? Do you really want them answering questions like the ones presented at the top of this entry? Really...?

I want to raise a generation of students who think. I want them to think on their feet, to think for themselves, to solve problems, to transfer with synergy what they learn. I want them to persevere when stuck, to call upon resources at their disposal, to figure things out. I want them to be tenacious. I want them to show attention to detail.

You don't get this from asking lower level thinking questions.

Such questions don't motivate, either. Kids just get sloppy, develop bad habits, and don't care. Oh, sometimes higher order thinking questions don't motivate, either.

The task: match the questions to the capacity for learning.

I don't see my attitude changing on this any time soon.

Altough I will administer the department exam, I will continue to feel and express my discontent.

In my classroom, I will continue to press all students to achieve their maximum potential, whatever that may be! I'll ask the hard questions (only to have students return next year to say, "Geometry was easy. What we're doing in Algebra 2 is so hard..."

More Pencasts

2011-01-16T21:41:00.001-05:00

On my website, you can find a link to helps with the open response questions that are due weekly. Here is an example.

open response - Ryans Summer Earnings - Linear Equations
brought to you by Livescribe

Livescribe Pencast: Graphing Inequalities

2011-01-13T23:42:00.003-05:00

Here is one of my early pencasts using a pulse smartpen and livescribe desktop software.

a1.6.1.Graphing Inequalities
brought to you by Livescribe

The jury is still out on how well I like this tool for explaining things. It has promise.

Linear Systems

2011-01-09T16:54:00.004-05:00

PROBLEM:
Find all points on the line y = 2x - 3 five units away from the x-axis.

Graphing the line went well.
Most found (4, 5) OK. None that I'm aware of graphed the line y = 5 (all points 5 units from the x axis in the positive direction). Likely, they said, "Well this looks like it is 5 units away.... Let me count..."
Only a handful found (-1, -5), five units away from the x axis in the negative direction. Likely they thought, "It has to be 5 units away. Count up 5." Counting down 5 never occurred to them. Or if it did, they may have said, "Not negative five.

Here is a second problem, similar, more complex.
If you did the first, you may be able to do this one.

PROBLEM:
Find all points on the line 2x + y = 8 that are equidistant from the coordinate axes.

Again, graph it, no problem.

"I'm stuck at this point. I have no idea how to solve this."
V: "Let's DECODE the problem. What don't you understand."
"Everything."
V: "OK. Let's see. 'Find all points.' Do you know what a point is?"
"Yeah, of course."
V: "Then you know more than 'nothing.' You probably know what a line is, too, and how to plot it, right?"
"Yes. I can do that."
V: "Good. We're making some progress. The problem says to find points '...that are equidistant.' Do you know what 'equidistant' means?"
"It sounds like equal distance. So I suppose it has to be the same distance."
V: "Yes. How about the 'coordinate' plane?"
"It's everything here."
V: "Right. And the axes? ... there's the x-axis and the y-axis..."
"So?"
V: "So find all points on the...."

At this point, amazingly enough, I get the same incorrect results from each of my classes. The student goes to the board and plots the point (2, 4) on the line.

V: "How far is that from the x-axis? ... from the y-axis? That's equidistant for sure..."
"Oh, it has to be a point like (3,3), or (4,4)."
V: "Right. Will you plot several points that are equidistant from the axis? Plot as many as you can... Is that all?"

The student then plots the points on the line y = x. When I ask what the equation of the line is, they stop. Writing the equation did not cross their minds. Why not? To be honest, the first time I did this problem, I don't think I thought of the problem that way.

So the student solves the linear system and thinks she's done.

BUT...

We go back to the question, and ...

...'all points'...

We plot the points on the line y = - x. Those, too, are equidistant from the axes. At the intersection of this and our given line, we find a lattice point!

These problems sustain momentum. Most curricula cover a topic, then move on. So do students, dropping what they've learned. They package it up and either neatly stack it on the shelf or trash it. But they struggle to retain and USE what they learned. I'm seeing more and more the importance of looping. We revisit previously taught topics frequently, doing lots of graphing. These problems are more likely to sustain student interest. Don't get me wrong... Not all students were interested. Not all were able to call upon those topics of old. Not all could solve these problems. But all will remember it a little more, having had the discussion in class and gone through the exercise.

These problems fit the criteria:

Short / Elegant / Tiered.
Mathematically viable.
Connected to current learning.
Make you think.

Differentiated Instruction

2011-01-03T20:59:00.000-05:00

I taught my DI lesson (from this graduate level course I'm taking) on polygon area to four periods: two regular geometry classes, and two algebra 1B classes. This lesson was stand-alone in the sense that it could be taught at any time during the curriculum since most students had prior exposure to triangle area.

I received some flak from ONE of the algebra 1b courses.

"Isn't this an algebra course?"

V: "This is algebra 1b. Which means I am to prepare you for the MCAS, including finding the area of a triangle. Manipulating formulas is an algebra 1b skill. Today we are finding the area of triangles."

Several things went to my liking. First, students were able to think for themselves with minimal prompting.

A) I did not initially give the formula for the area of a triangle. The geometry students found it not necessary. When some algebra students asked how to find the area of a triangle, I did not say anything, but just wrote the formula on the board, “A = (1/2) b*h.” This was enough; they ran with it.

B) Similarly, later in the lesson, I gave a short demonstration showing how the area of a triangle fits the formula. A question on the worksheet had students find the area of an obtuse triangle. Some individuals who struggled with this asked how it was to be done. I asked them back, “What is the base? … What is the altitude?” Invariably, all my students correctly came up with the height of the triangle.

C) When I asked students to find the area of the parallelogram, without giving them the formula for the area of a parallelogram (the only thing I supplied was the formula for the area of a triangle), they not only did so, but came up with four unique yet correct ways to accomplish this, including one way I had not thought of.

They creatively drew on prior knowledge and experience to do this. There were about 7 students in all the classes (including the inclusion class) who were unable to do anything with this.

I am well versed in the act of giving hints or less direct instruction to allow students to construct their own learning, like I mentioned in the first paragraph. So, it came as a surprise to me that what I wrote was a big success, ironically, was also a big failure. The problems that I had selected as beginner level turned out to be more intermediate (and vice versa).

By putting these tougher problems first, some students were lost, with no way to catch up, unless I just told them outright what to do. No amount of prompts could bring them where I wanted them to be. So, I ended up on two of the “beginner” problems, giving several kids direct explanation, “Do this…” or something along those lines. If instead, I had switched the basic and intermediate rankings, I think more students would have had success constructing their own learning.

Further, there was a problem in my lesson of trying to do too much in the allotted time.

A) Before entering the starting gate, I recognized right off that what I had planned was way too much. So, I cut back the part about student presentations.

B) What I didn’t notice until reflection after the fact was that the innards of the lesson were too compact. My lesson was (big picture) about finding the area of a triangle. I brought up two points. First, one could draw a rectangle around the polygon and subtract off triangles to get to the desired area. Second, the base can be any side of a triangle, not just the bottom; the “height” is really the “altitude,” not just what goes up and down. That was too much. If I downplayed the second item, this lesson would have gone smoother.

I am learning that DI means we have a mastery objective and we assess, formatively and summatively. I’m learning that I need problems for the four levels mentioned in our training (for simplicity I call them introductory, basic, intermediate, and challenging levels of mastery). If I can quickly figure out (or have students self-assess) where they fall, they can work on questions on the appropriate level of mastery where they are.

When I do this lesson next time, I will reorder the problems. I will simplify. I will not crowd the lesson with too much to do in one sitting. And I will work toward grouping students so they can work on their own mastery level without going through all levels of problems. Reordering the problems will allow students to achieve more on their own before teacher intervention. Simplifying will minimize distractions from a key solution method I’m want them to grasp. And grouping students will enable them to excel at their own rate.

More from Carmel Schettino

2010-12-30T19:31:00.000-05:00

Carmel teaches in Troy, NY at the Willard Academy, an all girls school. She has been an inspiration to me regarding my use of inquiry based practices. Here is a post I lifted from her blog in which she discusses some feedback that came to her in a round about way.

"I received an email from a colleague a few weeks ago, that was amazingly touching. She had been meeting with an advisee and asked the thoughtful question, "Can you think of a course or a moment that changed your academic experience in a significant way." One would think that most high school sophomores would either take that question lightly, or would at least need to pause and reflect on the depth of this question. My colleague said that her advisee responded without hesitation, "Geometry last year. I hated it at first because I couldn’t do what I had always done and do well. But by the beginning of the second semester, I had started to figure out what it was all about. And this year in my other classes, like English and history, I’m THINKING better, I’m analyzing differently because of that Geometry class.” This student seemed to be able to connect improvement in her critical thinking skills in other disciplines to the work she had done in her problem-based learning class. Did she have empirical evidence that this was the cause? Of course not, but something in her intuition and learning process was telling her that the struggle she had undergone to move through a course that challenged her in so many different ways, allowed her to grow intellectually like no other course had. For this student to even recognize this was very mature, and for her to attribute her success and skill in other courses to the learning that had occurred in this course was remarkable.

"Throughout the year when teaching with PBL, I struggle with the comments I receive from students. I often wonder when they ask me to go up to the board and give more direct notes, "Why don't I just go up and make them happy" instead of asking another scaffolding question? Why do I continue to push them out of their comfort zone and let them sit with the unknown for just that extra amount of time grappling with their peers, instead of relieving the tension and anxiety by giving them want they desire? And then a moment comes when they realize something on their own, and clarity, true understanding takes over. It is a moment of true joy on both our parts, because I know not only did they come to that understanding through their own agency and empowerment, but the way they came to it has allowed them to be the relievers of their own anxiety. They have transformed their vision of what is possible in coming to make meaning in mathematics. It is in that moment that we are together changing their understanding of what a mathematics classroom is, just a little more.

"And then the next week, they are back to asking me for more direct instruction, while at the same time there is just a glimmer of increased curiosity and I know that by the end of the year, there will be a changed student sitting in front of me who may have a different belief system about learning mathematics."

Open Response Questions Help

2010-12-28T10:41:00.000-05:00

Here are some students' comments regarding the weekly open response questions I have them complete:

"The open response questions helped me to put my math into writing. They help me put what I was doing into words. They also helped me on the MCAS open response questions. i feel that the open response questions helped, but on the MCAS I didn't have enough room to write as much as I wanted. I feel I wrote what was needed, but MCAS had very limited space. Writing about math helped me improve my understanding by helping me to explain. I could not only write what I was doing, but I could speak it as well."

"My feeling about open response questions is that they prepared me for the MCAS. I think I did really wel on the MCAS because of the journals. I felt that the thought process of doing the open response questions for class was lot harder than most of the open response questions on the MCAS. So I think come time for the MCAS, it was so easy! Writing about math made it a lot easier. It also helps you think of different ways to solve the problem. It helps the process and keeps your mind clearer! Talkng about math is a lot harder to do, but if you keep trying, you get really good at it."

"I think the opne response questions have a really good effect on our math experience. That is because it makes us realize ll of the steps in each problem. Aso, it makes you more organized with math problems. I say that you should still give these out next year and actually, I think put even more problems in them. The one best thing about the open response questions is that it totally helps with MCAS, because MCAS is based on open response questions (each open response is 4 points). I think writing all about math does very much improve the understanding of math by that of trying to put so much information on solving the problem into writing. Also I think the open response questions made me put writing math in more detail as possible. The way that I started to write the open response questions is to write as you are explaining the problem to a really young person (which makes you have to explain everything.)"

Class Culture

2010-12-21T21:29:00.000-05:00

I am very proud of two of my classes. They are rapidly adapting good math habits. They persevere. They attempt hard work. They get it when it comes to what is expected of them.

It has taken months to develop this culture.

If we were to go to semester courses, just as we are getting started, it would all stop. I'd get a new group and start over again. Hardly worth the effort. The big payoff comes after time spent doing.

So today, in one of the classes, I asked the question,

V: "What is negative three times one."

"Three."

V: "That is incorrect."

"Oh, It's one."

I wonder what is going on in some students' lives. Is he on drugs. Where has he been up to this point that people allowed him to pass and get into my class? Is the person ... a low flier?

V: "What is negative three times negative 8?"

"Uh... seventeen."

(This really happened.)

V: "No. That is incorrect."

"Is it negative 24?"

V: "No."

"Well I don't know."

...

I feel like I'm a warm body overseeing ... a class.

It is hard to teach those who don't want or can't learn.

Late Work

2010-12-19T08:52:00.003-05:00

Last Friday, students received their grades for the middle of the term. I wrote earlier of what I expect and how grades are marked. Now, with the Power School program, I put the values in, and it does the calculation of the grade. Those who did not submit their work, for whatever reason, had a grade that reflected it.

Should students be shocked?

Did the system fail them?

Should a note have been sent to the parent saying, "Your child is not submitting his work?"

Where does the student's responsibility begin?

What is meant by "It's on us"? (Our principal's theme)

I asked the students to reflect and write about the following:

"If a student does all their work correctly, they deserve an 'A', right?"
"What if the work is late?"
"... by 1 day?"
"... by 1 week?"
"... by 1 month?"
"What if the work is turned in the day grades are being posted by the teacher?"
"What criteria are you using to determine your answer?"

Most students said that late work should be accepted, up to a point. Some said a day, others a week.

Most students said there should be a penalty for late work. Some said minus 10 points every day it is late. Others minus 5. Some said 50% off for each day it is late. Others 25%. Others said for each week... But almost all said there should be some penalty. There is a sense of propriety when it comes to right and wrong.

************
Here are a few of the comments:
************

"Depends on the excuse..."

"If it's more than a month late, the work should not be counted."

"... people work at different speeds. One person could work faster than another but they could both have very good work. But it shouldn't be too late."

"... the person could have went through a change in life or thought that made them realise (sic) that they need to do the work in order to pass and make their parents happy. I think the option to turn in work should be open until the day before grades close and then once grades close, no work should be allowed due to the fair warning that states that 'grades will close and work will not be accepted after grades close.'"

"... a day late, an A should be given. Even a week an A should still be given but more time than that it is jut not fair. So the grade should decrease."

************
A surprising some said there should be no penalty whatsoever, even up to the end of the course:
************

"The person deserve (sic) an A"

"Yes they should get an A, because at least they did it."

"Yes, that person should get full credit for doing so. That is because they actually tried it and had put effort into the work. There could be ample reasons why their homework was done late."

"Kids have a lot of things to do after school and you give entirely too much homework."

"If it's late, it still deserves an (sic) just because its (sic) late the work is still the same if its (sic) correct then it deserves an A."

I have deadlines, usually Mon, Wed, and Fri, for specific assignments. I've allowed work to be submitted all by Friday. It is an overwhelming amount if you wait until Thursday night to begin, but a small task if done a little at a time - as it is assigned. I've even allowed late work... up to the time I'm inputting grades.

Apparently, I've been unreasonable.

Yes. Unreasonably generous. The students know what is right and what is wrong. I will have to change my grading policies to be more strict.

What is amazing to me is that in all my years of teaching, this particular group wants more credit and grade compensation for less work completed, not to mention work completed with minimal effort. (In homework, I am less concerned about correctness as I am concerned about the honest effort made. For instance, if a problem gives points with coordinates, yet the student makes little or no effort to plot the points, that is not a worthy or honest effort).

"May I do extra credit?"
V: "I'd settle for you doing the work for credit. Let alone the 'extra.'"

Flow

2010-12-16T23:35:00.001-05:00

Wow.

Flow, as you know, if you've been reading my blog, is a topic that has been on my mind a lot of late.

I just went through about 80 problems with pictures and put them in an order that I think has good flow.

See...

Exercises are "problems" that you know how to do. It is work like the example the teacher gave, but with different numbers. It is like our textbook; same ole, same ole. (There are some interesting problems, but you have to look for them. I find the workbook to be on an even lower level.) I mean, once you've been introduced to, say, the exterior angles theorem (in a triangle, an exterior angle equals the sum of its remote interior angles -- the exterior angle equals the two angles of the triangle that don't include the one adjacent to the exterior angle itself).... once you've been introduced to this, how many problems do you need to see, that look like the seminal problem, before you know what to do?

Problems, on the other hand, are "problems" that it may not be readily apparent to you how to solve. You may not "see" what to do right off. In my ranking these problems to get the flow correct, I actually give two examples of the original exterior angle theorem problem, but on separate pages. The next ten problems, however, increase in difficulty, incrementally. They are true problems. The last few are hard. So, these are differentiated, truly... by concept.

In the past, I've interspersed problems of many types, saying that they are "spiraled" throughout the curriculum. That is a nice way to say that they are mixed in with each other. Don't get me wrong. There is an order to it all... I mean, I tried to get the easier problems first, but the concepts were all mixed up. When I get "flow" right, there will be more of a grouping before interspersing the prior concepts. Also, the more challenging problems are "tiered"; which is to say, they require you to know two or more concept to solve them.

The student comments come back to me through their journalling, that they get confused by all the skipping around.

Another way of saying that is the problems didn't flow.

This is why I've been working feverishly to get it right (for next year, and possibly for more this year...) Sure, the problem sets work with an honors class. I'm trying to write the sequence of problems that work with any group. Truly differentiated. I've been working with my own classes on this.

A true test would be to get one of my peers who teaches geometry to try it out with his class. If he doesn't usually teach in this way, all the better. I'd like to see how they fare after trying it out. Perhaps a good thing would be to give a pre- and post assessment to see how much they learned by doing the problems. The students would be none the worse off for having tried. The curriculum is the same, concepts are the same, it is only the problems and their order that differ.

Further, I have two student interns working for me. They seem to be a little bored at times. (I've had them teach, make copies, circulate to help students etc...). This could be a project they could provide great insights to. If they do the problems and they like the flow, or identify problems with the flow... it will help my cause.

.

Wishful Thinking

2010-12-15T01:31:00.001-05:00

In a recent poll of teachers at my school by my principal (early October), we were asked about the climate. A small group of us were discussing the results today. One teacher said, "There were no categories where we all STRONGLY AGREE on anything."

I pointed out that was not so. There were two:

1. The teaching staff is committed to doing well for the students.

and

2. The teaching staff believe in the students.

I was one that voted this way, and I truly believe it. Although, some days, I wonder whether I'm delusional. I gave a re-test today. At least two of my peers say they don't do re-tests because the students never do any better. Well, neither did mine. In fact, many did worse.

I find myself either having to dumb down the test, or curve the grades. Curving leaves a putrid taste in my mouth. I'm teaching a standard. I've done this for years. If this group of kids doesn't get it, how much of this is me... and how much is the kids? I did curve the grades, but reluctantly so.

Look it.

There are 3 things I ask of my students weekly:

1. A basic skills worksheet.
2. An open response page, and
3. Two pages of homework, 10 problems per page (about 4 per night).

I grade based on:

50% - Tests (Quizzes count as 1 test)
25% - Open Response (huge part of grade).
15% - Homework (Effort based!), and
10% - Classwork.

I give numerous helps, including my website, the class notes, video tutorials. I stay after school. I give comments on open response and encourage revision. I gave a retest for T3, and for T7, I offered "grade recovery" for kids to complete (at home) similar questions to those on the test -- for which I offered after school help!

I find myself, as a teacher, being vested in the student's success.

I'm gearing up for the time I am called to justify why so many kids are failing my class... (more than 3...). If a student doesn't do the work, am I to give him credit? I don't think so...

My first step is to contact parents and let them know why the grade is what it is. Usually, missing work.

A peer of mine recently gave a strategy for calming the storm.

"I believe my expectations are reasonable. If you have advice, I will gladly listen to your input. But, I need real, concrete evidence that what I'm doing is not reasonable. What I ask is that you come observe my room for a week. At the end of the week, I'll give an assessment. You can see for yourself whether it is reasonable. If not, you will be equipped to give me firsthand advice..."

I was recently told by an administrator that I teach too many ways to solve a problem. I was told that I should teach one way. I asked myself why this was. The only thing that could come to mind was a parent must have complained after my instruction regarding linear systems. My response was to point to the frameworks, which state that students will know how to solve linear systems by graphing, substitution, and elimination. I was then asked (and I find this incredulous), whether other teachers at the school teach multiple ways of solving linear systems. They are if they are doing their job.

Again, a peer suggested I could have said instead that I would teach one way, as requested. I would choose elimination, of course, for it works in most cases. And if I were instructed to do this, the whole department should be, too. What is more, I would need an administrator's written request to bypass the state frameworks in this manner.

See, the state does not say you have to teach the frameworks. You can teach anything you want. But, come test time, I am responsible that students have been taught the information needed to pass the MCAS. If I taught only elimination, and a question showed the intersection of two lines on a graph and asked you to identify the linear system, this may throw kids for a loop if they had never seen it this way.

I may have been asked to teach only one way because of one student. Should I cater to the needs of one at the expense of the learning of the others? I suppose I could have assessed differently. I was assessing the ability to use specific methods. I could have asked one to solve it any way the person liked. What is really ironic, is that the student studied this material last year with a different teacher, who taught (and assessed) all three ways, yet there were no complaints last year. Oh well.

What do you see?

2010-12-12T08:53:00.002-05:00

PROBLEM:

V: "What do you see?"

"Two triangles."

"Really?"

"No. I see three."

"You do?"

"Yeah. There's the big one."

silence

V: "What do you see? ... I mean, what more is there?"

silence

"Well... They are isosceles triangles."

"I don't know how to do these problems. Why don't you give us worksheets."

V: "This is not a b**** and moan session. I can't do your thinking for you. I want you to learn how to observe. 'You see, Mr. Watson, but you don't observe.' You said they are 'isosceles triangles'?"

"Yeah. So?"

silence

"You know that those two angles would be the same."

"And the other two."

V: "Let's mark that."

"The three angles of a triangle sum to 180 degrees. So you could write an equation."

We do so...

silence.

"Well, that's 2y and 2z. But how is that going to help us? Does y have to be 45 degrees?"

V: "It can be, but it doesn't have to be. There are other things it could be."

"How is this going to help us?"

"Yeah. You have one equation, but two unknowns."

"When are you going to show us what to do?"

silence.

V: "What is it you're looking for?"

" 'x'. ... So... Are you going to help us?"

silence.

V: "What is it you're looking for?"

"Oh... I see."

This, by the way, is called "Thales Theorem," named after one of the early Greek mathematicians, circa 600 BCE, from Miletus (current day Turkey, Asia Minor). It says that the diameter of a circle and ANY third point on the circle form a right angle at that point.

Cool, eh?

After reading the NCTM article entitled "Are there any places that students use their heads?" I'm constantly on the lookout for "thinking" questions. I'm always looking at how the problems I ask flow toward what I want the students to learn. Do they always make the connections I want them to? Well... that's another story for another day.

Focusing on a Problem - Indicator of Success

2010-12-10T00:05:00.003-05:00

PROBLEM:

This is a classic problem, near guaranteed to be on the commonwealth's MCAS test, or on our midterm exam, or on our final exam, or the SAT, or all the above etc...

I was thrilled today to see students engaged in thought. Things were quiet. There were only minimal side conversations. When I perceived students getting impatient with not knowing what to do, I offered little tid-bits of hints.

V: "What do you see?"
"I see two triangles, an x and a 52 degrees."
V: "Anything else?"
"I see tick marks. So those angles are the same. But we don't know anything!"

I want to say, "You see, Watson, but you do not observe." That would be too callous.

V: "What do those tick marks have to do with anything? Why are they there? How can we use them to help us solve the problem?"

In between each of these, there were long breaks, and lots of focus on the problem. I think that was what was going on. It is possible that they were sleeping with their eyes open... but I don't think so.

"We're still stuck. Can you give us another hint."
V: "OK. This one will give away the store." I mark one of the angles in one of the pair of congruent angles, 'y'.
The kids want to say, "Your hints s***." But they were restrained, at least within my earshot.

NC says, "Oh, we'll mark the other side 'z' and x + y + z = 180 because it's a triangle."
I write all this on the board but give only minimal acknowledgement. (If I were to say, "Good job!" I fear she or others would shut down, thinking they'd done their share of the work. The problem is not over. Their work is not done. They must have confidence in what they are doing, regardless of looking at me for confirmation. Mine is a poker face.)

We stare at the board another 5 minutes... about 12 minutes has gone by.

AW says, "Well, the other angle would be 'y' and that one would be 'z'. So the large triangle is... 52 + 2y + 2z = 180."
I write it down.

Another long pause and lots of studying what is on the board.

Like popcorn, work starts. Soon, several kids scramble to write things down. Pretty soon, everyone is doing something.

There were two basic approaches. One was to substitute the expression for x + y into the second equation. The other was to set up the system for elimination...

x = 116.

It was pretty cool, watching everyone at work. Wow.

During the class period, I presented a number of problems... How many can we do in an hour sort of thing... Some were thinking problems. Most of the problems had a picture associated with it, which the kids viewed as a relief from all the word problems.

The day before a student asked, "I do better with worksheets. When will you give us worksheets?" I had 24 problems prepared to hand out. Today, I asked that crowd, "Did the problems help?" "Oh yes." "Good, here are another 20 problems..." They were glad, yet sad... They didn't exactly receive the worksheet readily, but they weren't quite reluctant. (Knowing the alternative is a return to the problem sets.)

That was geometry.

Algebra 1b... well...

In my 3rd period class, I presented new material, regarding inequalities. We started with easy problems. {x: 3}, {x: x<= 5} ...

Amazingly, the kids struggled with this next problem when they first saw it:

PROBLEM: Represent the domain on the number line: {x: 2x < - 4}.

Once I showed them how to simplify the equation, they were on a roll...

... Until we first saw the equation where they divided by negative 1...

The fact is, they were very attentive and seemed to grasp a lot.

Now, I contrast this with my sixth period class. The class is like an informal geometry class of old. The kids in this class, speaking collectively, not individually, are not disciplined, do not focus, refuse to ponder math. In general, their performance is well below standard, either the state's standards or the department's. I attribute it in part to years of bad habits. Others were unsuccessful in getting through. So am I.

It's on us, alright. Indeed!

Number Sense

2010-12-08T21:18:00.000-05:00

Here are a few scenes from some of my classes today:

PROBLEM: A man is twice as old as his son.
Write an equation expressing this.

I sat with a student for about five minutes. This person struggled with this immensely. I suggested putting a variable for the man, say, "M." The kid wrote "M"... after a time... " = "... I was patient. The person looked and stared, looked back at me (social referencing), back at the paper, and wrote, "S" (for son). So the net was:

"M = S"

I said, "This says the man's age (M) is (=) the son's age (S)."

The kid stared at it a while. Then wrote a 2 in front of the S.

PROBLEM: One number is two more than another number.
Write an equation expressing this.

A girl wrote, "x = 2y" She was astonished when I said it was incorrect. Flabbergasted. I said, "If I plug in numbers, I would say, 'Eight is two more than ___" to which she replied, "six."
V: "How did you get that?"
"I just added six to two..."
V: "Then why did you multiply here?"

PROBLEM: p + j = 3, p - 5 = 3j + 2

"Where do I begin?"
V: "You have two equations and two unknowns. This is a linear system. You can solve this graphically, or by substitution, or by elimination... like we've discussed."
"But what do I do first?"
V: "You could try to get the 'p' by itself. We call that 'isolating the variable.'"
"How? I don't know?"
V: "How to move 'j' to the other side of the equation?"
"Yeah."
V: "Do the inverse operation. Instead of adding, you..."
"Huh?"
At this point I write down below the 'j', '- j' ... The student then continues correctly.
Why did she not see this?

I think students aren't doing algebra. They see easy equations like, 2x = 8, and ask themselves (usually in an instant), "What do I multiply by 2 to get 8? 4."

This is NOT algebraic thinking. Algebra would be, "How do I isolate the variable? I do the inverse operation. I have to divide both sides of the equation by 2..."

When I give them harder problems, or problems they don't recognize, they fall apart. I mean easy problems like: A = (pi) r^2. If Area = 24, and (pi) = 22/7, solve for r. They don't know to do the inverse opertion to get r^2 by itself... This is no less difficult than 2x = 8, if you are doing it algebraically. If you solve it mathematically (instead), it suddenly becomes inaccessible.

PROBLEM: (3/4) x + x = ?
Here are some student responses:
   = (3/4) x ^2. She multiplied instead of adding.

   = x. She added 3x plus x to get 4x. 4x divided by 4 is x.

   = 1/2 x. I don't know what this person did.

   = (3/4) x.
V: "How can (3/4)x plus x equal (3/4) x? I thought (3/4)x plus zero equals (3/4) x."
V: "What about LCD?"
Several students chime in, "Least Common Denominator!"
... what to do with that information... ??? Nobody knows.

Would you believe we went on for 4 minutes like this? Amazing. The number sense and particularly understanding of fractions is waning.

One x plus three-quarters x equals...
One and three-quarters x.

Or (7/4) x,

or (3/4)x + (4/4)x (The "identity property of multiplication" to get (4/4)...)

PROBLEM: 4 ( - 3/4 x)
= (-3) x

"Why don't you multiply the 4 by the (- 3/4) and the x?"
V: "We distribute multiplication over addition. We don't distribute over multiplication."
I then gave an example that made sense using just numbers. This is not the first time this students and a few friends came up with this. Their first math teachers were not exactly taskmasters.

PROBLEM: (3/4) = [(3 - w) / (2w)]
"How do you solve that?"
"Can you leave it like it is?"
V: "I want you to solve for w."
"But how?"

...

V: "You could cross multiply."
"Huh?"
V: "Have you never done that?... Well multiply to get rid of the fractions. First we'll multiply everything by what number?..."
"Four."
V: "Very good."...

How Do We Reform Math Education?

2010-12-07T20:00:00.000-05:00

I grew up in a world of arrogant supremacy. The US is the best. At everything. So we were taught. So we believed.

Now, this clip, just in from NBC News:

Visit msnbc.com for breaking news, world news, and news about the economy

The Chinese are world dominant. They've been among the better in math for decades. We've fallen behind.

Today, I gave a standard geometry assessment, created by a peer, based on the materials we were supposedly studying. I thought there would be dramatic breakthroughs with my students. There weren't. Some kids did surprisingly well, others not so well. Sigh. Why?

This was a traditional assessment. I've been teaching using non-traditional means. Why didn't I get the results I anticipated?

Was it because ...
...the students don't respond to inquiry-based methods?
...to be successful with IBL requires massive student buy-in? ... involvement?...
...I didn't have the right flow?
...I didn't explain connections? ... or have students explain them to me?
...Was it because we hadn't studied "proof" enough (and there were 2 out of 25 proof questions)?

What went wrong?

What should be done about it?
Does this mean I should change what I've been doing? How should it be changed? What should be different?

I've considered teaching like a friend of my mentor from Florida. This fellow's students repeatedly win competitions. He teaches the book 3 times in a year. The first time, he goes through it rapidly, perhaps in a month or month and a half, only looking at the easy questions on the concepts. The second time, they look at more intermediate questions, taking about 2 months. The third time around, they spend more time, and go into greater depth on each of the problems. Perhaps this is the way to alter what I'm doing.

Maybe there is nothing wrong with what I'm doing. The students see it for the first time with me. They aren't used to it. But, what they are learning will stick with them longer. Maybe I need to do more of it, and do it better. Currently, I'm looking very closely at two things: FLOW is really important. And EXPLICITLY ATTAINING THE CONCEPT is the second thing. The students need to know ... THIS is the concept you are to be learning... and be able to tell me what it is and how it works.

Sorely disappointed.

Simply Complex

2010-12-06T23:56:00.001-05:00

Problem: Determine whether the triangle formed by the three lines below is isosceles or right.
   y = (3/4)x + 2
   y + 1 = (- 4/3)(x + 4)
   y + 9 = 7 (x - 2)

A little different twist on the problem would be:
Show that the sides of the triangle fit the Pythagorean equation.

This is not the same as:
Show that it is a right triangle.

On the second, just show two of the slopes are opposite reciprocals.

The first requires you to determine each of the side lengths, then put them into the Pythagorean equation. So, you must know how to square a radical, how to multiply radicals, and how to add...

I like this problem because it is a culmination of all that we've been doing in class for some time. One must know how to plot points that are on a line, how to solve linear systems, use the distance formula, tell something of slope, deal with fractions and radicals multiplication and addition. If one recognized the point-slope form of a line, the problem is greatly simplified. Here is where the rubber hits the road.

Next year, when I do this, I think I'll differentiate as follows: I'll offer 3 different versions of this same problem, and tell students your grade for successful completion of the one is a C, the next is a B, or finally an A. The "C" problem would have integers for y-intercepts, and lattice points for the vertices. The "B" problem would have non-integers for some of the y-intercepts, and less complicated fraction calculations. The "A" problem would still have lattice points for vertices, but, perhaps none of the y-intercepts would be integers, and there would be some fraction calculation (if you choose to solve it by elimination)... For each, I would keep the point-slope option available...

I gave this to my regular geometry classes. In the one period, a full 1/3 of the class correctly responded to completion. In another, I had 1/5, and in the third, only 1 person out of 20.

The problem is short. It is worded simply. The ideas behind it are fairly simple and straight-forward. But solving it requires that one pull a lot of knowledge about a lot of things together.

I gave this as a quiz. Beforehand, I told students it would be like problem 11.10 from the problem sets. I also gave two sample problems for them to attempt on their own. I'm disappointed (but not surprised) at how few students tried the practice problems.

I found it rather incredulous that it was the class with 1/3 of students performing at an advanced level, that the most vocal females complained about how this problem was impossible, that no-one could do it, that "no one gets it." Apparently, quite a few do.

   y - 2 = (6/7) (x - 1)
   y - 2 = (-7/6) (x - 1)
   y + 4 = (1/13) (x + 6)

   y = 2x - 7
   x + 2y = 16
   3x + y = 13

I though 'b' is 'one'

2010-12-02T22:00:00.000-05:00

Yesterday, while teaching about exponents (and this is our second, nay third week of this), I reverted to the basic rules. The first rule was the Product of Powers: if you have a product of two powers, and if they have the same base, you can add the exponents.

We did a number of simple problems using this concept.

I had the idea of throwing in a question with a zero exponent:

J responded by saying, "I thought you said that b to the zero power is 1."

V: "That's right."

J: "Why didn't you change it to 1?"

V: "I suppose you can do that. There is more than one way to approach this, so long as we follow the rules. You could have written:"

J: "Isn't it just '1'? !!! You said that b was one!!!"

Notice the lack of attention to detail. This is typical. Students at this level often do not have discipline to attend to detail. I'm not speaking in a denigratory manner of J's work. It is just that she didn't hear what I said.

What I said was, "Any number (except zero) raised to the zero power is 1." OR "'b' raised to the zero power, is '1'." I did NOT say, "'b' is one." I said, "'b' TO THE ZERO POWER is one."

A big difference. the final answer is NOT one. It is, of course, b cubed:

I wish I had identified this during class. It came to me in the middle of the night what J meant by saying that "b is one."

I expect I will see this again... and again... and again... each year.

Are there any questions that make students think?

2010-12-01T21:01:00.000-05:00

In my Algebra 1B classes, we've been discussing exponents. The textbook has 8 sections. Five (of many) rules are discussed in the first two chapters:

Product of Powers:
If two powers with the same base are multiplied, add the exponents.

Power of a Power:
If a power is raised to a power, multiply the exponents.

Power of a Product:
If you have a product raised to a power, you can raise the individual parts of the product to the power.

Zero Exponent:
Any number to the zero exponent is 1, except zero, which is undefined.

Negative Exponent:
To change a negative exponent to a positive exponent, take the reciprocal (flip it), and change the sign of the exponent.

We've done hundreds of practice problems as a class, as individuals... I gave students 50 practice problems, with solutions. They don't follow rules. They have no attention to detail. They mess up big time. I don't know were to go with this. I think I will give 10 practice problems with each concept. Drill and kill. It is sooo bad. If the kids can't (or don't) think with the easy problems, how can I give them hard ones? sooooooo bad.

Using Your Head Questions

2010-11-30T22:01:00.000-05:00

PROBLEM: Prove the Pythagorean Theorem as follows: draw and mark the following drawing. (I didn't show the kids the measure of the side length of the inner square). Determine the area of the large square. Determine the area of the four triangles. Find an expression for the length of the side of the small square. Determine the area of the small square. Add the areas of the small square and the triangles to get the area of the large square...

This problem was a struggle for many students. I really had to hold back from giving the answer. When students go to the front of the room, others in the room want to "help" by chiming in with their solution. I have to nicely quelch that...

V: "Please attempt this on your own at your seat while ___ does it up at the board. We'll comment on the work in a minute."

The kids get stuck, or quit prematurely.

"I have no clue."

V: "You'll think of something."

or, later...

V: "Here's a clue: Mark the side lengths of each of the triangles..."

Some did it carefully and correctly. Others sloppily, so they couldn't make heads or tails of what they'd done...

When we determined the length of the side of the small square, the kids claimed to be stuck, not knowing how to find the area...
V: "What is the area of a square?"
"Length times width."
V: "What is the length?... What is the width?..." Amazing.

So, in large measure, this was about multiplying two binomials.

To get the FLOW right, I followed it up with a homework problem that was to expand...

This was followed by some problems where kids were to factor...

Finally, we got to this problem:

PROBLEM: A 9x12 picture is framed by a border of uniform width. The combined area of frame and picture is 180 square units. Find the width of the border.

This turned out to be a surprisingly difficult and challenging problem. I walked the classes through it, step by step. About 3 kids followed all that we did. Oh well. Differentiation.

I love how these problems that make kids think accomplish this task. Many of the kids are starting to see the value in it, I think. There are still several who don't see that by their treating lightly the homework (and NOT thinking about the problems), their learning suffers. So, I'm using class time to teach them how to approach these things so that they learn how to think.

In my Algebra 1B class, the kids are the same age as the geometry students, but their maturity level is way low. They have virtually no discipline (perhaps a handful of students have some discipline). They won't stick with the EXERCISES long enough to do them... let alone any THINKING problems! There is no discipline or attention to detail to follow the rules. So, I find us going through problem after problem, step by step... showing what to do. Still there are numerous mistakes.

I received an email from the teacher's union regarding a survey ... on how the Commonwealth will grade its teachers. Several of the questions were about whether to or how to rank teacher performance based on student scores and student performance. It is such a misnomer! This is not a factory. This is not a business model. I'm dealing with real people here... with habits they've spent a lifetime developing (good or BAD!). How well the students I have perform depends in large measure on WHICH students I am assigned. It is just stupid... to tie my performance into theirs... You're blind if you don't see the problem with this. It sounds good in theory. But what formula will you use to equalize. If I have students who are underperformers, and they make gains better than their peers who are underperformers, how do you compare them... how indeed, to those who are high fliers who make tremendous gains. I am an engineer and a math teacher. I find it hard to come up with an algorithm that would fairly treat this! Stupid.

I wish I could ask my algebra students thinking questions.

Sigh.

Are There Any Places That Students Use Their Heads?

2010-11-26T22:52:00.002-05:00

This title, "Are There Any Places That Students Use Their Heads," is part of the title from an article in the Journal for Research in Mathematics Education (JRME), published by the National Council of Teachers of Mathematics (NCTM). I think this article is going to change dramatically what I am doing as a teacher.

At the end of the article were two cogent points that are aligned with what I believe and am doing: 1) attain explicitly the concept, and 2) give mathematically challenging tasks for kids to wrestle with.

The article addresses Japanese teaching from an atypical approach. The researchers monitor mentor comments to student teachers, before, during, and after the instructional experiences. In the US, there is a wide range of what is accepted as good instruction. It usually centers on classroom management, supplies and materials, and "engaged," meaning, are students actively involved in the activities. In Japan, there is a very narrow range of acceptable beliefs in what is good teaching, and this is generally accepted by most Japanese teachers. It centers on whether the lesson was clear, connected, and were the activities mathematically important.

Whereas US teaching focused on the skills attained, the Japanese focused on skills AND understanding. More higher order thinking questions, requiring reliance upon two or more skills, in previously unfamiliar settings, comes to play.

The Japanese spend a lot more time planning. One could look at the plan and determine whether it met criteria for a good lesson. Inherent are the kinds of questions that are to be asked, the flow (which is perhaps most important for my purposes), and anticipated responses. The textbooks may follow logical sequences and order, from an adult perspective. The Japanese lessons focus on getting the flow right from the student's perspective. Student think.

One math teacher, at the end, said that even though one may plan the perfect lesson and implement it perfectly according to plan, it still may not meet criteria for good teaching. One must always reflect on the lesson to make it better. Lots of metacognitive influence.

I have a good set of questions. Higher order. The questions often drive what is to be learned -- "Oh I need to know that to be able to solve this problem... so... I need to learn..."

But...

My order does not have the flow I want. Flow would make explaining the concept recognizable and clear. I am rethinking this now. I have got to get the flow right in order to make my problem sets work. The order of the problems has got to be a key part of the planning process.

And, of course, the actual implementation must take cues from students, as it currently does.

Reflection - the curriculum

2010-11-23T23:57:00.007-05:00

I use the same geometry textbook as everyone else.
... Sort of. We have books, same for everyone. We don't reference them often, only as needed. We reference the online supports -- which does NOT include THE book.

The book is not online. It is, but we purchased supports that did not include the book. Our supports include video tutorials and practice test problems, both of which I use.

My book...
I prepared a book of 65 pages, ~10 problems per page. These are "nice" problems. They are differentiated problems! They teach the same curriculum as everyone else... perhaps a little more depth. A lot more understanding is required of the students to do these problems (Term 2 already and we're on page... 13). Well... I mean... They have to actually read... (huh?) ... and understand what it is they are reading... (you expect them to what?!) For instance, if a problem says, "Show that the triangles formed by the intersecting lines.... is isosceles..." a student has to recognize that this problem is about triangles, know what isosceles means, and know how to find the length of the sides. ... Oh well...

My book has interesting problems. It has challenging mathematical tasks. I recognize that some students cannot handle the higher order thinking questions. They must have problems at a very fundamental (easy) level. I supply those in my book, too. But am I to cater to the low fliers at the expense of the more advanced student?" Again, I differentiate.

I recently discussed the rules of the Exterior Angles Theorem. I even put it to a song (as a memory aid... basic fundamentals about triangles):

"The measure of an exterior angle is the sum of its remote interior angles."

We did a number of the basic problems. Then, I threw an open response question at them at them:

This is a very short example of the exterior angles theorem. If you are not familiar with the exterior angles theorem, and even if you are, you may not see it at first. When I first saw this, I think it took 30 minutes before I grasped all the relationships. (I went about it the hard way.) So now, I gave students the hint of "exterior angles theorem" (something I didn't have). If 10% of the kids figure this out on their own, great. A nice problem. Move on. So you didn't get it. That's OK.

I post the pages to my book on my website. I also post helps to each individual problem. There are also the tutorials (mostly from the textbook, but some are of my own making).

By the way... my book is ... free.

How would you, as a parent feel, to know of parents who complain about my teaching too many ways to solve a problem? Look. I've been trained. I am an expert in my field. I know what I am doing. Further, I follow the frameworks. We are to teach, for instance, linear systems using graphing, substitution, and elimination. I do that. All math teachers do that. It is what we're supposed to do. If a kid gets confused about several ways of doing this, I would suggest she learn elimination, for it works in all cases... I would suggest it if the person asked. I'm still teaching the whole population, however. So I am doing my job. To not teach the other methods would not be doing my job.

Besides, aside from a hammer in our toolbox, don't we want the awl, the bevel edge chisel, the vernier calipers ...

I think that when a parent or administrator questions that I teach more than one method, they are really questioning my integrity as a teacher.

On administrator recently said to my colleague, "Your assessments don't line up with what it is you teach." We have smart boards. We save all our lessons to *.pdf files. The evidence showed otherwise. What this teacher showed that had been taught was also on the test. (In fact, some were the SAME QUESTIONS!). What evidence would bring one to make such an accusation? Was it comments of a parent? Student? Why was the teacher not consulted (or supported)?

This pattern of putting into question teacher integrity appears to be systemic. It does not evoke trust. It does not bring out the best in me or my peers. Demotivating.

Top Game for Direct Instruction

2010-11-22T20:23:00.002-05:00

Today, I taught triangle congruence. I believe this is among the better of my direct instruction lessons - a rarity. I give an overview of the entire topic, in one sitting.

It starts with this question.
Q. How many unique 3-letter combinations can you make using the letters S and A? (Example: SSS)

I write the list students give, making 2 columns:

SSS AAA
SAS ASS-SSA
ASA
AAS-SAA

I give students a handout of a graphic organizer - a booklet showing all 5 possible shortcuts for proving triangles congruent, the two invalid shortcuts, and CPCTC. They mark the two invalid methods. We fold it into a booklet, which I instruct will be used throughout the rest of the term.

Q. How many parts does a triangle have?
A. Six. 3 sides and 3 angles.

If two triangles are congruent, all six of the corresponding parts must be congruent.
I draw, label and mark two triangles, such that ABC corresponds to FDE (not DEF).

Q. Can you write the congruence statements for each of the six pairs of congruent parts?
Q. ... and for the two triangles?

I point out that segment AB can be written as segment BA without altering the meaning.
When we write the triangle congruence statement, I ask why triangle ABC is not congruent to triangle DEF. I point out how order matters here. When I write it, I align the letters ABC and FDE right over each other, instead of in a row.

I show two triangles with sides listed 3-4-5, and 3-4-5, but one is a mirror image of the other.
Q. Are these two triangles congruent? If so, explain why and write a triangle congruence statement. If not, explain why not.

Q. Are angles C and E congruent? How do you know? Nothing was told you that they are.
This is where I introduce CPCTC.

Q.
Given: ABCD, AB=CD, BC=AD. Draw the figure. Prove that triangle ABD is congruent to triangle CDB.
This is where I introduce the first formal proof, writing it step by step on the board.

Q. If I wanted to prove that angles A and C were congruent, how would I do this?
A. First prove the triangles are congruent, SSS. Then, all the corresponding parts must be. CPCTC.

Next, introduce SAS, by showing two triangles with two congruent sides.

Q. Is there enough information to prove these two triangles are congruent?
We will explore this in the lab tomorrow.

Show the triangles with the included angles marked, and ask,
Q. How about now? Why do we call this the "included" angle?

Show ASA, same thing.

Q. If you know two angles of a triangle, can you find the third?
The first three shortcuts: SSS, SAS, ASA, for proving triangles congruent are postulates - we accept them as true without further proof. This fourth method, AAS-SAA is a theorem, because we can prove it.

Next show two triangles like this, and ask:
Q. Is there enough information to show these two triangles are congruent? Why, or why not?

At this point, I offer three proofs of the Isosceles Triangle Theorem (base angles of an isosceles triangle are congruent.) First, given isosceles triangle ABC, AB = AC. I draw this. Then add the midpoint, M, of segment BC...

Next, given isosceles triangle ABC, AB = AC. I add auxillary segment AM, which bisects angle BAC...

Finally, given isosceles triangle ABC, AB = AC. I add segment AM, an altitude from vertex A... This should work. But I keep getting ASS-SSA, the invalid shortcut.

Q. What kind of triangle is this?
A. Right.
Q. What do we call this side?
A. Hypotenuse.
Q. What do we call this side?
A. Leg.

This brings us to the fifth shortcut, the hypotenuse leg, valid only for right triangles.

Q. Are these two triangles congruent?

Q. But I have SAS, don't I?

A. No. (Point out the "included" angle... would need the other side...) This is a prime example of ASS-SSA, better known as the "Donkey" theorem. Not a valid shortcut.

Q. Are these two triangles congruent?

A. No. AAA makes good auto insurance, but is not a valid test for congruence. It IS a valid test for SIMILARITY.

At this point, I review the entire lesson, adding HL to the original slide.

This is only an overview of what we will be discussing the rest of the term. The booklet will serve as a guide.

Would You Rate Some Questions Below Others in Significance?

2010-11-17T16:59:00.004-05:00

PROBLEM: Show that the triangle formed by the lines y = 2x – 7, x + 2y = 16, and 3x + y = 13 is isosceles. Show also that the lengths of the sides of this triangle ﬁt the Pythagorean equation.

How can you identify the right angle just by looking at the equations?

This problem is the culmination of what we've been doing in geometry since the beginning of the year. In this little question is linear systems, distance between points, slope of lines, slope of perpendicular lines, and simplifying radicals.

So a student asked, "Why are we moving away from triangles to do these problems?"

I tried not to find that incredulous. "Well... this is a triangle problem."

He came back, "But, why are we moving away from triangle problems to do this kind of problem?"

Huh? Persistent he is...

V: "That's just a dumb question."

"Well I guess I'm stupid."

V: "I didn't say that. I don't believe that. Let's re-read the question."

"I don't care anymore."

V: "You asked why we're moving away from triangle problems. The question says..."

"I don't care."

V: "The question IS a triangle problem. 'Show that the TRIANGLE...is isosceles .... and is right.'"

I took the better part of the period in helping students get through this one. It makes a good test question, because it covers so much of what we've done in class. If you know what is going on, you'll be able to do this. Most students were able to work through the linear systems.

When we got to the vertices of the triangle, I had students plot them. I plotted them on the board, and pointed out, "This looks like a right angle. But I'm not sure. How can I be sure?"

One student said, "Just put your paper up next to it."

V: "Is there a way mathematically to do this?"

A different student, "Could we use the y = mx + b thing?"

V: "I'm not sure. It wouldn't help me. But you may find a way to use it. y=mx+b is the equation of a line. I'm not sure how knowing the equation of a line will help you to show two lines intersect at right angles."

A boy student, "We could look at the slopes."

V: "How will that help you?"

"Well, if the slopes are opposite reciprocals, you'll know the lines are perpendicular, so they intersect at a right angle, so it must be 90 degrees."

His logic was near perfect!

He got it! At last. Someone gets it! Hurrah!

I've heard some folks say that there is not such thing as a dumb question. I would like to put forth that some questions are... better than others. (Meaning some are...)

If you look at the books in the Bible, there is a lot of difference in the value of Isaiah, versus Song of Solomon!

Praise in the Classroom

2010-11-16T21:12:00.003-05:00

Problem: A plane is flying at 36000 feet when above Lincoln Nebraska. A little while later, when above Des Moines, 160 miles from Lincoln, the plane is at 28000 feet. If it continues at the same rate of descent, how far from Des Moines will it be when it lands?

Students are presenting their homework at the board all week. A girl came up today when I asked (recruited) her to do this problem. She stared at it a while. Circled parts, underlined parts... She drew a picture, which was pretty good. It showed that the plane descended 8000 feet in 160 miles. She marked off sections of 160 miles length for 8000 feet in height... until she got to zero.

In the recent past, my remarks have been to issue hints. I sometimes get frustrated, visibly so, when students have very weak algebra skills, or when they fail to recall specific concepts we'd discussed previously (such as how to factor, or how to simplify a radical etc...). I have to learn to take a deep breath and let go of the frustrating parts... My expression of frustration has a not so positive affect on the classroom climate. Same, with the feedback I give that is more about the problem than about my relationship with the student.

Now, logically, I recognize that if I can get the students to feel that I am concerned about their success, individually, that they will perform better for me... But the engineer in me has this drive to solve the problems, and to not complement work unless it meets expectations. (Which expectations are good, but maybe too high)... so I really have to make a practice at giving sincere, discrete compliments.

Today, after this young lady presented, I made a conscious effort to give real, positive feedback. I said,

"I see that you drew a rather effective picture. You correctly show the distance versus the drop in height of the plane.... and used this information to deduce the solution, how far from Des Moines the plane would be... Are you satisfied with your work?... Kudos to you. This is excellent. There are other ways, but getting your hands dirty and going through this is an effective strategy.... Did you do this on your homework?... Why not?..."

It turns out, she hadn't done this kind of thinking on her homework, though she is certainly capable. Sigh.

On the up side, the complement / praise / wording of my comments seemed to have a positive effect on the student, and on the tenor of the class.

I showed an alternate strategy: writing the equation of the line.
(which happens to be what we've been doing in class up to now...)

I notice that this year is worse than previous years. Student attention spans are about 10 - 60 seconds. They can't stay focused on a problem long enough to get themselves to the point where the learning will take place. I can't do the thinking for them. I need them to work up to staying focused 5 minutes.... or even an hour...

This is rough on me. I have a curriculum to teach. The book of questions I've created will do that and much more, but students must put forth the effort. If students are given a chance to ask questions about the problems, they don't because they haven't given enough thought to them to know what questions to ask!

It would be very easy to go back to direct instruction. I could say, "I taught them, but they didn't learn it." (This comment from Post and Weingarten's book, "Teaching as a Subversive Activity" is said tongue in cheek... It is what some teachers say. The traditional method lends itself to that. You teach the standard, and folks either fall in line or miss the boat.) Inquiry-based learning, which I truly believe in as the most effective way to teach and to learn, is harder. It requires students to do the work. We have to live through a period of students getting used to ... not getting the right answer ... getting stuck ... and finding a lot of confidence and inner drive when they are right.

What I am interested in is not the traditional... answer! I'm OK with students getting the wrong answer. I'm looking for the approach... and how far does one get (what all did you do) before getting stuck. The PROCESS is what is important.

So, if there are ordered pairs in the problem, and students don't even plot the points, they are lazy. I don't like labeling students as "lazy." But that is what it is. Same with not using graph paper to plot the points. Lazy. A lesson on the law of the harvest is in order. You reap what you sow. The great football coach, DX Bible, said it this way, "You've got to pay the premiums in order to reap the dividends."

Unless there is change soon, it promises to be a long school year.

Meaning - Seekers => Meaning - Makers

2010-11-15T21:10:00.004-05:00

PROBLEM: A plane is at 36000 feet above Lincoln, NE. A little later, it is at 28000 feet above Des Moines, IA, which is 160 miles from Lincoln. If it continues at this same rate of descent, how far from Des Moines will it be when it touches the ground.

Student: "No clue."

V: "What do you know?... What do you see?"

Student: "No clue."

V: "You could draw a picture."

Students have not had success with problem solving. It takes perseverance, confidence, and creativity. Each of these feed off each other. So, if your attention span is 10 seconds, as it is with many of the young people when they arrive in my class, confidence will not develop easily. Trying to get a 1 minute attention span is good. 5 minutes better. How about an hour? ... a week?!!! (I won't EVEN ask for a lifetime... of life-long learning...) But that's what it will take. Some pondering time. Some thinking time.

I modify Ralph Waldo Emerson's quote, "Can there be any greater reproach than idle learning? Split wood at least." I say,

"Plot the points, at least."

If one has coordinates listed, doesn't plot the points, yet says, "No clue..." This, to me, wreaks of laziness. Call it like it is.

I grew up in an era when TV took hold of our lives. We became passive observers. In 1986, commentary was made, "What will you now do about..." ... the government corruption?... starvation?... etc... And then the commentator answered for you, "Nothing."

While that was true then.

It is not necessarily so, now. Check this out:

We can be meaning seekers. We can be meaning makers. One bird at a time.

Math Scores Drop

2010-11-11T05:24:00.002-05:00

National news is hyped. They're selling entertainment:

This report emphasizes how US math scores have fallen. Look back, and every year there is a report like this. You might think we're OK because we're in MA. Well, hold up a bit.

There is good news, and there is bad.

You can take data and twist it to mean different things. For instance, in a recent meeting with the math department, it was brought up that our school's MCAS average scores were below the state's average on all multiple choice questions except 1.

While that may be true, it is useless data. It does not tell me what I need to do to improve my instruction, nor where I fall deficient. If you look at the scores of kids where were taking Algebra 2 that took the MCAS (the brightest of the bright, the ones who took geometry in 9th grade...) their average scores were all correct on over 2/3 of the questions. In my honors geometry class, last year, same thing.

Then, look at the regular class scores. My class averages... well, you get the picture. To be useful, that is, to use the data to improve instruction, I must know where students fell short, on which of the types of questions, from which class, more specifics... This is not something about which blanket statements can be helpful. Nor does it help lump what my students did with the rest of the math department. To be helpful to me, it is my kids' scores...

Back to the national news hype. What do I learn?

Not to change the topic...

The results in all my classes on yesterday's Linear Systems test were very low. (Yet some students scored 90% or better). We reviewed this algebra 1 topic for 2 weeks in geometry, for a month in the algebra 1b class.

"Are you going to count that grade?"
Well, yes, I am.

"But you can't fail everybody."
Well, yes, I can.
However, I want you to succeed. Test Score Recovery questions, if completed and at least 80% correct - the first time, will raise your grade. It is not a test retake. Yet it is an opportunity to get a better score.

The fact is, I teach to a standard. If you don't meet the standard, it is you who needs to improve. I cannot do the work for you.

"But I didn't have time to finish."
The test was designed to be taken in 30 minutes, with 20 minutes to check your work. If you knew the material, you should have been able to finish. You need to know this like the back of your hand. Perhaps you should do more practice. You can find extra problems posted on the class notes.

Problems in the problem sets

2010-10-15T19:39:00.000-04:00

This past week, I showed students the mind map of where we've been. It nicely organizes for them what we've discussed, giving a broad picture of how things fit together.

I thought I could / would see how well they'd done. I gave two points A (-4, 5), B (4, -1), and asked students to find the length of segment AB, the midpoint, the slope, the y-intercept, the equation of the line, the slope of a line parallel, and the slope of a line perpendicular.

I had no idea the ruckus this would cause. Many still struggle in making ends meet (figuratively speaking).

Today, I gave as bell work this problem:

PROBLEM: How would you plot the line: 3x - 4y = 5 without writing it in slope-intercept form first.

This algebra problem gave remarkable insight into student's number sense. At first, no one volunteered a possible way to approach it. After a few minutes, I offered this hint: "Hint: How many points determine a line?" The chimed in unison, "Two." But it meant nothing to them.

My next clue was to draw an x | y table.

One student in each class came up with the idea of finding the intercepts. I am always careful to help students make connections. For instance, I said, "The y-intercept occurs where the x value is zero." I motioned on a grid how we went neither to the right or left of the origin. "It's coordinates would be (0, b), where we have to solve for the b, the y-intercept." And I wrote (0, b) on the grid.

This problem is nice in that when we solved for b, we got -(5/4). While the point (0, -5/4) is completely correct, it is worthless to me in graphing. I need two LATTICE points (or points whose values are integers -- they lie ON the grid.) I could use those to find the distance, the slope etc... But fractions... bleh.

Again, in every class, a student offered that we try x= 1. We did so and came up with b = -1/2. Nicer than -5/4, but still not a lattice point.

Students tried again, usually by guessing, until they found a number. Some eventually saw a pattern, or ... a very few had the number sense to make the number you multiply by 3 and combine with the 5 to be divisible by 4...

What I am finding is that the sequence of the problems and the way they direct the pedagogy is of great importance. I make revisions every year. I think that this year it will come together very well (for the next year's class).

Power School and Grades

2010-10-10T20:57:00.000-04:00

Our school adopted Power School for recording student progress online.

None of the experts from our school were consulted about the selection of this tool. Helas.

The program we previously used is obsolete. But, it had greater flexibility and speed.

I find this program cumbersome and time consuming to use. It takes me about 2-3 hours longer per week to enter the grades in this than it did in the previous program. Oh well.

I created a composite quiz score for each term, which counts the same as a test. (All the quizzes together count as 1 test). So, if students miss and fail to make-up a quiz, it brings down the composite score. If that is the only test score in the database (because we haven't taken other tests yet), or that score plus one other real test are the only scores in the database, AND since tests count for 50% of the overall grade, missing one (or more) score(s) will have a tremendous effect on how the grade fluxuates.

Also, there is a homework composite score. I input the homework actual grades, such as 6 correct out of 47 questions... I want to see how the kids are actually performing. But most homework is scored based on effort, not correctness. The homework portion of the grade may be higher for the effort than their actual performance indicates. So I record a composite score.

Finally, there is one composite score for the open response questions. These count for 25% of the grade. When you only have 2 assignments, and a student did not turn in the one of them, ... well... Again, you can see the dramatic effect this will have on the student's score.

Since we teachers are being made to use this program and spend the time inputting the grades and not liking it... I input grades without emotion. In the past, I would be prone to show more mercy in modifying the scores. As it is, what you see is what you get. The computer does the calculations. I tell it like it is.

The research agrees on one thing, which is that grades do NOT motivate students to action. Yet, this school district has chosen to place great emphasis on grades. What is their expected outcome? We'll see. What will the parent reaction be to the actual scores? How will the kids respond?

If a student does not turn in a paper, or does not take a test or quiz, it goes down in the gradebook as zero until such time as it is made up. And there is not unlimited time to make things up or turn things in late! Oh well.

Alfie Kohn has written a great deal about grades. Good read.

Learn to Split Wood At Least

2010-10-06T23:48:00.004-04:00

PROBLEM: Let A = (2, 4), B = (4, 5), C = (6, 1), T = (7, 3), U = (9, 4), and V = (11, 0). Tiangles ABC and TUV are specially related to each other. Make calculations to verify whic pairs of seides are congruent. Write a few words to describe what you discover. How wold you write symbols (no words) to indicate which of the sides of the one triangle match up same as the sides of the other triangle?

This problem, from the Phillips Exeter Academy... my students are capable of doing it. Why do they say, "I had no clue." "I got stuck." I can't help but wonder what it is that is bothering them.

V: "Do you know how to plot points?"
"Sure."
V: "Then at least, you should have plotted the points. For goodness sake, learn to split wood, at least."

This last remark had to do with a quote from Henry David Thoreau:
"Can ther be any greater reproach than idle learning? Learn to split wood at least."

A good part of my geometry classes have decided that they needn't do any of the homework. Today, I told them to hold onto it until I called them up to present their work. If they said they didn't have it, I informed them, matter of factly, that it goes down in the gradebook as a zero. Not good. Come prepared.

The problem states: Make calculations to verify which pairs of sides are congruent.

One student said he didn't know how. After all we've been doing since the beginning of school. Page 1. Take two points, draw a third that makes a right triangle out of it. (Pythagorean theorem... over and over and over again.) OR Distance Formula, which we even spent time deriving in class.

Sigh.

When students come prepared, I thrive. When they don't, it is a drag on all of us. One can't study with inquiry based learning practices and simultaneously be passive. I get the sense a lot of kids want to be entertained. They want me to think for them.

Mind Map: What Have I Learned in Geometry?

2010-10-02T22:54:00.002-04:00

Here is a mind map of the major concepts we've been learning about in class. I present this in that students may be wont to say, "I don't know what I've been learning." The open response questions follow the topics pretty closely, and they give students a chance to reflect on what it is that they've learned.

The homework assignments may seem disjointed, however there is a great semblance of order. The problems are not structured like a traditional course, where you get a worksheet with 50 problems that look just like the example. Heck, there are only 5-15 problems per page, and we're only on page 6, beginning week 6. These problems are packed with information and thinking. Students must draw upon all their prior knowledge and piece it together, just as they will have to in the "real world."

Let me illustrate with a recent incident how much the students have accomplished.

One of my students was sitting next to a friend of hers. I overheard their conversation. The friend, who is in the honors class, said, "Here's what we're learning in geometry." She showed my student the distance formula. My student said, "Yeah, the distance formula. That's really just the Pythagorean theorem." The friend looked puzzled. "Don't you see it? Here, let me show you." She still didn't see it. At this point my student looked at me and asked, "Don't they learn that in the other classes." I told her that it's pretty common to teach the formulas.

My approach and desire has been for students to understand the formula well enough to derive it, if need be, rather than memorize it. If they understand where it comes from, they will be less likely to mis-apply it (add when they should subtract, or vice versa, or some other misapplication of the algorithm.)

They Liked It

2010-09-29T23:22:00.001-04:00

PROBLEM: (POW 361 from mathforum)
The problem goes something like this: A saleswoman saw her potential client had three kids, and asked, "How old are they?" The lady, a math teacher, replied, "Their ages sum to 13." The saleswoman said, "That's not enough information." The lady replied, "The product of their ages is the same as my house number." The saleswoman looked at the house number and said, "Still not enough information." So the lady said, "Alright. My oldest daughter likes chocolate." Then the saleswoman said, "I know!"

For the past two weeks, I've instructed my algebra 1B classes using direct instruction methods. I hate it. The kids are bored. They are memorizing enough to pass the quiz or test, but I sense it is not going to stick with them.

Today, I gave them the problem above. It is an inquiry-based problem. The purpose was to practice problem solving strategies.

It was pretty quiet in the room. The kids read the problem.
"I can't do that."
V: "Do what."
"Read all that."

I read the problem aloud.

"You want me to do that?"
V: "Sure... Where will you begin."
"I don't know."
V: "What do you know? ... What do you see?"

More silence. Kids studying the board.

V: "Let's see. What is important here?"
Silence. Kids started to talk.
"You know their ages sum to 13..."
I circled the part about ages summing to 13.
"That could be million possibilities."
V: "List them."
"You want me to ... what?!"
V: "Sure. Why not. Try to list them..."

Kids writing feverishly.
V: "Here. I'll do one. 1 - 1 - 11. These sum to 13."
More writing.
V: "And when you're done, I'm going to ask, 'Did you get all the combinations?' and 'How do you know you got all the combinations?' and 'How can you be sure you didn't double-dip, and repeat one of the possibilities?'"

Quiet. Several kids working at this. After a little time.
"I can't do that. There's too many."
V: "Did you try it?... You should have 14."

More quiet, as people tried.

V: "Let's list them."
We listed the 14 possibilities:
1-1-11
1-2-10
1-3-9
1-4-8
1-5-7
1-6-6
2-2-9
2-3-8
2-4-7
2-5-6
3-3-7
3-4-6
3-5-5
4-4-5

More quiet... Staring at the board. I read more from the board.
"What was the lady's house number? If we knew that, we could solve this."
V: "I don't know. It doesn't say. Maybe it's not important. What do you think?"
"Then why did they mention it? Is this a trick question?"
V: "I don't know?"

"It says 'the product of their ages.' Let's find that."
V: "OK"
The kids did the calculations. I wrote the products on the board. When we got to 2-2-9, there were two products with 36.
"Oh, I see. That must be her number."
V: "Why?"
"Because it's not enough information."
I knew what he meant, even though what he said made no sense.
We finished the products.

"It has to be the 2-2-9."
V: "Why?"
"Because it says, 'oldest.' And with 1-6-6, there's not an oldest."
"What does the chocolate have to do with anything?"
V: "I guess the important part is like ___ said: 'oldest.'"

For about 15 minutes, the kids were very engaged, much moreso than usual. I think, if they would admit to it, they were actually having fun. It was fun for me to watch them decode the problem.

This problem, from Drexel University's math forum, did a number of things. It was not just a fun exercise. The real thing I wanted to teach by it was a math habit of the mind: perseverance. After working through the problem, we discussed it, why we did it, what they were to get out of having done it etc.

Then, back to the old way...