I was sitting with a small group of math teachers at a meeting and I asked about their methods of "Testing for Understanding." It seems that for many (most?) the answer was a combination of "They can do the homework." or "They can pass the tests." Am I just looking for too much?

Anyway, while doing some serious research (playing around on Google search) I came across a page called Understanding Mathematics, a study guide by Peter Alfeld.

He writes," You understand a piece of mathematics if you can do all of the following:

Explain mathematical concepts and facts in terms of simpler concepts and facts.

Easily make logical connections between different facts and concepts.

Recognize the connection when you encounter something new (inside or outside of mathematics) that's close to the mathematics you understand.

Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.)

By contrast, understanding mathematics does not mean to memorize Recipes, Formulas, Definitions, or Theorems. "

He then goes on to give a few examples (powers, logarithms, quadratics) which he uses to amplify his explanation.

I think the student who can explain why 8

^{-4/3}= 1/16 without too much armwaving probably has a pretty deep understanding of the exponentiation process.

I'm not as sure about his quadratic solutions model. He talks about the quadratic formula and then states, "Forget about the formula, it's an example of clutter. (To this day I cannot remember it.) But since you understand these matters you know how the formula was derived."

I'm not sure I believe him. Ok, I know that sounds kind of harsh, but if you derive the quadratic formula a few times it sort of sits there looking at you all the way through, doesn't it... I see it and know what it is I'm working toward.

I agree that one ought to be

**able**to derive the quadratic formula by completing the square, or at least solve quadratics without the formula, but then I remember a quote I have from Euler in my notes on "Twenty Ways to Solve a Quadratic Equation": Even a great mathematician like Euler, after deriving the formula, suggests “it will be proper to commit it to memory”. from An Introduction to the Elements of Algebra . Euler understood math, and when crunch time comes, you gotta' really sell me to make me disagree with him.

I don't have a wide range of clear delineators of understanding, but a kid who looks at a quadratic with a positive leading coefficient and a negative constant term and can't tell me how many real solutions it has does NOT understand solving quadratics, no matter what he has memorized. And if I write some ugly equation on the board the kid has never seen before and ask, "What happens to the graph of this if I replace all the x's with x-2 and he says it moves to the right (or left, I'm easy) I think they understand something big. If I tell him that (2,3) and (3,5) are on the graph of f(x) and ask him what f

^{-1}(3) is and they have no clue, I think they have missed something big.

Here is a recent example, we were talking about H1N1 and working with the logistic curve and I was explaining that the maximum rate of growth occurred at half the maximum value, such as here.

I suggested that from this point they should be able to see that the ab

^{-t}= 1 and that would lead them to see that b

^{t}=a. But for many (most) of them sorting out that the exponential part = 1 was too many steps at once... in fact for a large number, it just did not make sense that 1+ab

^{-t}=100/50 (

*seeing past the clutter?*). Whether you think of this as a division property or the means and extremes property of proportions, it seems to me that if you don't see this "clumping" (I think that was Polya's term) you can't be very flexible in math patterns.

Ok, so send me your gut tests... If they can do THIS, they mostly "get it", or, conversely, if they can't do THIS, then they definitely don't "get it".

## 8 comments:

I'm the wrong person to ask. I think 6th graders can handle Calculus and 8th graders can handle Quantum Mechanics, so I think the understanding of these things should begin much earlier than when you get them, Pat.

Hey Pat, do your students have to grade you? Click here for the worst idea ever. We may be crazy here in the States but at least we're consistently crazy. Do they do that is Europe? It's bad enough in-laws grade parents but teachers too? Who gets to grade teachers? Oh yeah ... peers.

Also, if you want to kill someone's love of Math, be sure to give lots of word problems in the 4th grade, or give them a really dense textbook. Textbooks, in almost any subject, are subject-love killers. Three companies in Texas corner the textbook market, right? Somebody should look.

Fortunately in this wired age, we have the ability for each teacher to make their own textbook for each class, on-line. Here's a toast to the day textbooks go away, or the school/government awards textbooks to only the 20 % brightest students who bring the natural love anyway.

When I was in junior high I could not remember the quadratic formula while taking a test, so I derived it by completing the square. The QF is too complicated, and you have to remember what a, b and c refer to. Consider a student trying to solve a+bx+cx^2=0.

Here are some resources on "chunking" that may further explain your point. We want students to become experts, which can be measured by how large their 'chunks' are?

http://psychclassics.yorku.ca/Miller/

http://ocw.mit.edu/courses/chemistry/5-95j-teaching-college-level-science-and-engineering-spring-2009/video-discussions/lecture-2-teaching-equations/

Regarding Memory, have any of you read The Memory Book by Harry Lorayne and Jerry Lucas? Definitely one of the most amazing books I've ever read. It's all about mnemonics, or "tricks" we can play to help us remember a ridiculous number of things.

Not everyone has a photographic memory, and not everyone who has a photographic memory is a genius. They may have read the book, and don't feel like sharing where they got their "amazing" ability from.

For example, early on if you read the book, there's a simple trick for memorizing long 20-digit numbers, like a driver's license. It doesn't take long. At a party, have someone show you their license, look at it for all of 5 seconds, then hours later, you'll be able to tell them the number exactly. Be the hit at your next party!

Tim and all,

First thanks for your comments,, I love (and yet hate) the quote on the MIT video Tim sent, "Teaching is a wonderful way of transferring the notes of the teacher to the notes of the student without passing through the brain of either."

It was many years before I remembered the quadratic formula: it was less error-prone for me to re-derive it every time I needed it. I've always had a bad memory, which is part of the reason I went into math: hardly any memory work needed.

I'm glad to see that Steven Colyer will lat the top 20% have textbooks, since they are usually so bored in class that they aren't paying any attention. My son learns a lot from textbooks, much faster and more reliably than from classes.

I was totally bored in Math class in elementary school, because I was the top student and I, get this, would actually read a chapter ahead, so I could figure out what the teacher was talking about the next day. Word problems and memorizing times tables by rote bored me the most, and they both happened in the 4th grade. Fortunately, my 4th grade teacher was the best one I had in elementary school, so we banged though "State" regulations.

I can't complain about my high school or college Math teachers though, they were all excellent.

Pat, I've checked out your mathematics glossary, and what can I say, except: Awesome! That's what I'm talking about ... the world is changing so much with fantastic on-line resources for science and math, do we actually need textbooks anymore?

And don't get me started on college textbooks ... $200 per book is price-gouging, pure and simple, and don't think the authors see much of that money.

I end my politics of education rant here. It's tough for any currently employed teacher to get involved in that, they have to do what they have to do. I do hope Pat finds time after retirement to comment though, and doubly hopefully not sound like he was educated by a "schoolmarm" in a one-room schoolhouse (which I'm sure he won't, but may be accused of sounding ... we've seen LOTS 0f changes in Education in our lives), ... unless of course that schoolmarm was Miss Crabtree. ;-)

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