Cutting Off The Path To Answer-Getting

John Benson at Angles of Reflection has a post–a very important one–that’s been sitting at the top of the blog for a long time. Every so often I re-read that and think about it. It offers a telling vignette of how students are taught a rich technique for addition (“near doubles”) but do not use the technique. The students still revert to standard procedures of addition to “get the answer.” Then they tack on the new technique, which means they aren’t really learning about the structure of addition.

This explains something that has troubled me for some time. I frequently hear from parents that their child is not challenged by the curriculum that is being offered in their school. When I have looked at the curriculum I find that it is often very rich and full of many challenges. Perhaps the reason for this disconnect is that when the student looks at the material, the only thing the student is thinking about is how can I get this done and get a right answer, while I am looking at how many different things a student can learn from the variety of approaches taken.

I think the key to cutting off answer-getting in students is to notice that what they want is a good grade for just having the answer, so the teacher strategy is to give a middling grade for just having the answer, and a better grade for employing new strategies.


Schools are the Hearts of Communities

Although Anthony Cody gives us much to think about in his questions and suggestions about Bernie Sanders’ run for President as a Democrat, I want to address one point that speaks to my experience:

School closures are devastating to communities, and have been focused in African American and Latino neighborhoods. This results in community decay and spurs gentrification. Research shows federal efforts to “turn around” high poverty schools have not succeeded. This policy should be halted, and schools in these communities should be supported with wraparound social services to directly address poverty, not destroyed.

I grew up in a rural, poor community and went to small public schools with few resources. Looking back, it is clear to me that the school was the heart of the community, and that is true for schools in the city as well as the country. From the sports teams, to the Pancake Suppers, to the local votes on millage increases to pay for repairs and new buildings, to the Quiz Bowl teams that travel to neighboring schools, schools provide identity, purpose, and a sense of the future to a community. To lay off half the staff or shut down a school is for America to apply a scorched earth policy against its own citizens…this might make sense if the Wehrmacht were rolling in.

But there ain’t no Wehrmacht today. There’s just elites tearing apart American communities… and it is even more shameful they are focusing this on communities of historically oppressed minorities.


So, it is AP Calculus prep time, my least favorite time of the year.

This year I am lucky to have one student, just one, who will NOT take the AP exam. So, I asked him, “Do you think it would be a boring waste of time to do the old AP exam your classmates are doing?”

“Yes,” he answered. So, I pulled down the old Calculus textbook by Paul Foerster–whose explorations are essential to my class–and thumbed through it for a good problem that would take him some time to do…

The factorial function and the Gamma function!

He set to it, doing some improper integrals, and some integration by parts and then he used his calculator to integrate the factorial function to get 0.5! = 0.886.

The book said, find a simple expression involving pi that equals 0.5! and after a little time, he found.

Moreoever, he learned that (-1)! and (-2)! were infinite but (-0.5)! and (-1.5)! were finite.

And I learned these facts, too!

Back in college, I saw the gamma function in my textbooks but spent zero minutes using it, much less investigating its definition. So having a student who was not constrained by the AP test prep (and I’m only spending a week and a half on prep) gave me the opportunity to broaden and deepen my knowledge of mathematics… as well as this one student, and after the exam is over and done with, I’ll use this experience to push the rest of the students to learn (0.5!). It will fit in with the “Integrals and Statistics” unit I’ve composed for the month & a half after May 6.

This is what studying math brings you: new discoveries, new connections, a sense of wonder at how the world of numbers makes sense.

Thank you, Paul Foerster, for your wonderful books!

“Ceterum censeo Synthetic Divisionem esse delendam”

…and polynomial division, too.

After teaching AP Calculus BC for almost seven years, I wonder whether there is any value in teaching Polynomial Division and Synthetic Division in PreCalculus.

So, in Calculus, you might have to integrate (or differentiate) a rational expression. You might want to use algebra to make an equivalent form of that so it is easier to differentiate. My school’s textbook recommends polynomial division for a problem like this but I think it’s better to use algebra:  UseAlgebra

I feel it is much better to simply give students exercises wherein a little creative addition of forms of zero can be used to create the factors you want in the numerator. It keeps them on their toes as far as “chunking” and thinking in terms of factors.

*my apologies to people who know Latin for the appropriation of Cato the elder’s quote. I probably used the wrong case in there.

Synthetic Division and what’s wrong with Education

Iota Delta

Today one of my coworkers came to me and asked if I could help her out with synthetic division.  She’s taking a College Algebra Course in her spare time, as part of her efforts to earn a degree in Computer Science.  I of course said I could help her, mistaking synthetic division for polynomial long division, then once we started talking about her problem, I realized I didn’t know how to do synthetic division.

It’s a shortcut for the process which I do know how to do, Polynomial Long Division.  My teachers tried to teach me how to do it back in high school, but I couldn’t grasp the concept.  I’m sure in college someone even tried to impart the knowledge upon me but it never stuck.  I began to wonder why it was that I never learned this simplistic process which you’re first taught in Algebra II courses, when…

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Synthetic Division: to teach or not to teach?

One person asks, Why teach synthetic division? and I’ll see that question and raise you a “Why teach polynomial division?”

Tree in a Forest

I really can’t think of a good reason why synthetic division is still in the Algebra 2/PreCalculus curriculum.  When I was in high school (at my HS, I was in the last class not to use graphing calculators), synthetic division was a handy shortcut because we had to do tons (often enough for a whole page for one problem) of divisions to test potential zeros for higher-order polynomials.  However, with technology as it is, I can think of better ways to spend the 2-3 days it would take to cover synthetic division and finding rational zeros by hand.

I do still teach polynomial division, because it is useful for finding polynomial asymptotes of “improper” rational functions.  But for the synthetic division shortcut, unless I’m doing tons and tons of polynomial division problems (and only the case where we’re dividing by linear factors at that), I really can’t think of a good…

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