## Computer Based Math Redux

I’ve posted before about Conrad Wolfram’s efforts to remove calculation from the curriculum and make everything computer based. There is now a website devoted to the initiative (http://computerbasedmath.org) and Conrad Wolfram’s blog recently announced their first country interested in taking up the curriculum: Estonia.

Estonia isn’t too surprising a choice; they recently put programming in the standard curriculum starting at first grade.

However, they’re not diving into axing algebraic manipulation from the curriculum yet; rather Computer Based Math (abbreviated CBM) is planning to “rewrite key years of school probability and statistics from scratch”. This is a reasonable first step given statistics is often taught computer based or at least calculator based these days (my colleague who teaches AP Statistics next door does so) and it does feel very silly to work through a passel of “figure out the standard deviation” problems by hand.

However, I’m going to play devil’s advocate again with a thought experiment. Since algebraic manipulation is not being removed at this time, these questions aren’t going to be applicable to Estonia yet, but presuming Computer Based Math continues working with them it should come up soon.

Suppose you are in a curriculum where you are used to algebraic manipulations being done by a CAS system. You are learning about statistics and come across these formulas:

Mean for a probability distribution
$\mu = \Sigma[x \cdot P(x)]$

Variance for a probability distribution (easier to understand)
$\sigma^2=\Sigma[(x-\mu)^2 \cdot P(x)]$

Variance for a probability distribution (easier computations)
$\sigma^2=\Sigma[x^2 \cdot P(x)] - \mu^2$

Standard deviation for a probability distribution
$\sigma=\sqrt{\Sigma[x^2 \cdot P(x)] - \mu^2}$

[These are incidentally off page 208 of Triola's Elementary Statistics, 11th Edition.]

What is necessary to use the formulas conceptually? What understandings might someone lack by not having experienced the algebra directly? Is it possible to understand the progressive nature and relations with these formulas just by looking at them? Is it necessary (to be well-educated in statistics) to do so? If it is necessary, what specific errors could somebody potentially make in a statistics calculation? Could this be mitigated by the text? Could this be mitigated by steps taking during the CAS portion of the education that while not leading to lengthy practice in “manipulate the algebra” problems will still allow understanding of the text above?

Is it possible to explain something too well? That is, something appears very clear to students after it is explained, so they don’t practice (or at least pay attention to their practice because they assume they already understand the topic), and then the lack of practice means they forget what was explained? I’m not meaning “they never learned it in the first place” but rather “they learned it so well that they forgot it because they assumed the memory was permanent”. (This is a slightly different issue than students who assume they learn something but really just keep their misconceptions.)

Are there circumstances where practice can actually lessen understanding; for example, when a student who learns a “trick” that works for an entire worksheet may attempt the same trick in circumstances where it doesn’t work? Thus it may be a bad idea at times to have a student practice a topic without all the special cases? (Specific example: suppose a student practices integer addition using only a positive with a negative number, but doesn’t attempting adding negative numbers with negative numbers until later. Will their earlier practice hinder their learning in the new situation?)

## TED-Ed attempts logarithms

Each of the TED-Ed videos is meticulously animated and represents, I am sure, many many (many) hours of effort. Knowing this made the TED-Ed take on logarithms rather painful to watch:

Oof. Let me attempt to sort my thoughts:

1. The hook baffled me.

A hook should, optimally, be incorporated into the topic being learned. This hook was simply a preview of a future part of the video, and didn’t carry much interest on its own. The “red eyes” made me think it was referring to the eye-bleedingly long numbers being presented.

While my own logarithm video isn’t perfect (also not entirely comparable since it’s about the addition property in particular) I do at least manage a hook that’s useful in the explanation of the topic.

2. “…small numbers and in some cases extremely large numbers leading us to the concept of logarithms.”

Logarithms come out of the inverse concept of an exponential. The numbers don’t have to be large or small. (If you want to get historical, they were often used as a method to multiply quickly by turning the operation into addition.) While a logarithmic scale can be used to handle large or small numbers, I don’t see how that leads to the statement in the video.

3. “the exponent p is said to be the logarithm of the number n”

Math videos often are on the glacially slow side, but this part was presented so fast parts of my brain melted.

Look: Logarithms represent, in essence, the first new mathematical operation students have had to reckon with since grade school. They cause intuitions to fail. I have seen students who have never had problems with mathematics before have them for the first time with logarithms.

It’s worthwhile, then, to spend a little more than five seconds on your definition.

The definition is confusing, anyway; a logarithm is a function. It applies from one number to another number in a specific way. It is not simply an extract from an exponential equation. While the video mentions that (sort of) it waffles on the implications of introducing a new mathematical operation.

4. “…log base 10 is used so frequently in the sciences that it has the honor of having its own button.”

First off, no: the sciences often use base e (given how much continuous growth and decay happens in real life). Base 10 logarithms do still get used for logarithmic scales, but the statement as given in the video is just confusing.

Also, that’s a TI graphing calculator? Which one of has a logarithm button but not a natural logarithm button? Even the TI-81 has one.

5. “If the calculator will figure out logarithms for you, why study them?”

The answer the video gives … is so you can figure out a logarithm base 2.

That’s a terribly weak answer, given a.) yes there are many applications of logarithms where understanding the mathematics is both good and necessary, and the video even goes into one application immediately after making this statement b.) the answer doesn’t really answer the question (since it doesn’t explain where the computer science-related equation came from) c.) with the current operating system, Texas Instruments calculators are perfectly capable of putting in alternate bases without a change of base formula (The video incidentally doesn’t mention the change of base formula even though one of the questions in the post assessment asks what it is.) and d.) The statement presumes the use of a calculator in the first place (computer-based systems are also perfectly capable of doing logarithms with alternate bases).

6. The video then wants to show how useful logarithms are by giving a formula from science.

Based on the post-test, I’m guessing this part is here merely to show how logarithms are used in “real life”.

In the master catalog of Ways to Convince Students Why Something is Useful, “look, a formula that shows up in science!” ranks somewhere between “because math is good for you” and “so you can get into a good college”.

Is it really that bad? Am I just being grouchy here?

## Cooperative learning tasks in mathematics

By cooperative learning tasks I mean giving particular “jobs” to students during group work; here’s a sampling from this website:

Checker: Checks team members for understanding and agreement
Datakeeper: Keeps track of information generated by group
Helper: Gives help in reading, spelling, problem solving, or using materials
Questioner: Asks questions of instructor or other groups
Reporter: Gives oral reports to the total group
Summarizer: Sums up what the group did or the conclusions the group came to
Validator: Paraphrases what is said for clarity
Writer/Recorder: Writes down ideas and records the task

I tried experimenting with them last year (based on the urging of several people) but I’ve been distinctly unhappy. It feels like the jobs segment up the work in a rote sort of way which gives a student permission to “shut down” when they aren’t needed for something in particular. I’ve still had some luck with engineering-like projects which involve building, but this sort of thing fails for me in general. For example, today I’m having my Algebra I students work on these questions in groups:

You have an
a row by b column matrix
and want to multiply by a
x row by y column matrix.
1. When is this multiplication impossible?
2. If the multiplication is possible, what is the size of the new matrix?

3. When multiplying 2×2 matrices, there is an identity operation (just like multiplying by 1 is an identity operation in arithmetic). What is it?
4. What about for nxn matrices?

5. Give an example (with all work) that shows that multiplying 2×2 matrices is in general not a commutative operation.
6. Even though the commutative law doesn’t apply in general there are specific cases where it works. Give an example of a matrix A and B such that AB = BA.

How would working out answers here divide neatly into jobs?

[EXTRA NOTE: I still think matrices shouldn't be taught before vectors. I'm making my best go at the curriculum I need to do, though. Bert Speelpenning's series on matrices (especially this post) is helping me get some motivation out there.]

## What I’ve been up to (redux)

My time this summer was mostly occupied with a secret project which should be revealed very soon. I can say it has to do with Common Core.

I also have been working on my book project. I’m currently calling it How Algebra Works and it is totally crazy.

It is targeted at adults who had algebra class in the past but it’s now a tangled muddle of memories they barely understand. Every problem given is a puzzle — that is, it wouldn’t be out of sorts in a Martin Gardner book or the World Puzzle Championship. Given I am not fussing over standard schoolbook curriculum, I am doing my best to rethink everything from scratch with the goal of explaining how it works rather than providing technical proficiency.

So, yeah — easy to crash and burn. Hence I won’t talk about it any more until I’m ready for beta testers (both for the text and for the puzzles).

Finally I’m teaching at a new school now.

## Thomas Aquinas on teaching

From Josef Pieper’s Guide to Thomas Aquinas:

. . . all knowledge of any depth, not only philosophizing, begins with amazement. If that is true, then everything depends upon leading the learner to recognize the amazing qualities, the mirandum, the “novelty” of the subject under discussion. If the teacher succeeds in doing this, he has done something more important than and quite different from making knowledge entertaining and interesting. He has, rather, put the learner on the road to genuine questioning.

## IMPROVEMENT IN ADDING-REGISTERS FOR PENCILS

(Following from Guess the purpose of this patent)

CHAELES C. FIELDS et al
Patent number: 198934
Filing date: Jun 21, 1877
Issue date: Jan 8, 1878