I was struck by this passage:

In terms of implementing this in practice, I think that college is way too late, and also quite difficult because college math (and STEM) courses tend to be mostly about transmitting massive amounts of boring technical content and technical skills, leaving little to no room for actual ideas or ways of thinking. Nevertheless, I do think it would be an interesting experiment to have students keep something akin to a "vocabulary notebook" where they record the meaning (as opposed to the formal definition) of the various kinds of expressions they run in to. For example, a fraction ab is supposed to mean "a number which when multiplied by b gives a "; it is short and illuminating work to figure out from this (using distributivity of multiplication over addition, which we definitely want numbers to satisfy) that ab+cd=ad+bcbd , that there is no number meant by a0 , and that 00 can mean any number). This of course, presupposes that somebody takes the time and makes sure that the language in which these meanings are explained is coherent, so it would be a lot of work to design a course around this method.

I did in fact once successfully disabuse a(n Honors Calculus) student of "the Law of Universal Linearity" using these ideas. The particular instance concerned manipulating the Fibonacci sequence, and the student had made the error of writing something like Fx+Fx=F2x . What I did is explain the stuff above and had the student apply them by analyzing the meaning of the various expressions he had written down was, and then ask whether that equality was justified based on what he knew the expressions meant. That seemed to make an impression on the student, but I personally believe it was an impression made ten years too late...

## 18 comments:

"The law of universal linearity"

Why is it that when I Google this, I only come up with this thread and its link?

I'll agree with this:

"I think that college is way too late..."

But not the other blather.

"... vocabulary notebook" where they record the meaning (as opposed to the formal definition) of the various kinds of expressions they run in to.."

Ugh!

What's wrong with learning the formal definitions? What's wrong with homework sets that require students to DO problems that cover all variations?

I think there is a lot that can be done to improve this understanding, however, but I see nothing of that here.

For example, what is 'a' or 'b'? It's a factor. What is a factor? Can a student look at an equation and circle the "terms"? Can they circle the factors in each term? Can they see terms as rational fractions? Do they see that every factor has an assumed exponent?

For 1/a+b, they have to be able to see it as (1)/(a+b) to show the factors in the rational expression. How can student ever move from Algebra I to Algebra II without knowing that 'a' could be (2-3x)^6? When students learn that:

a/1 = a

do they know that (backwards) this could mean that

(2-3x)^6 = (2-3x)^6/1

How about

a^-1 = 1/a

Do they know that this also means that

(2-3x)^6 = 1/(2-3x)^-6

This thread's discussion is extraordinarily simplistic in understanding the huge problem of going from simple Algebra I rules to being able to manipulate any algebraic expression and knowing why you can do what you are doing. I had a teacher in high school who converted an equation to end up showing that 1 = 2. Of course, you had to find the subtle mistake. My son got the same thing.

This is not a "vocabulary" problem. It has to do with knowing how to apply the basic identities in all situations. In the link, however, the problems are so simple that this is about basic teaching and school competence.

College STEM courses "tend to be mostly about transmitting massive amounts of boring technical content and technical skills" ??????

Wow, what a statement. In computer science, the courses, especially those first year programming courses, are all about learning to think - learning to think in a way that has never been taught to the poor students before. That is why the first year courses are so hard for most students.

In high school algebra you have to really work with the axioms of arithmetic. I don't think this is commonly done, or not done well, but to understand 1/(a+b) you have to first understand that this is 1/x for some form of x. Since x could be anything, not just an integer, this is best thought of as the multiplicative inverse of x. The multiplicative inverse of x is the number which, when multiplied against x, yields 1 (the multiplicative identity). So what do I multiply against 1/(a+b) to get 1? Answer: a+b. If I multiply a+b against 1/a + 1/b I will not get identically 1, so it cannot be that 1/(a+b) is the same as 1/a + 1/b. To make sense of this you need in high school algebra you need plenty of time, patience, and insistence on using the correct terminology. To clear up misconceptions, which naturally occur, you have to drill down to the real basics. Even smart kids who are good at math with make these kinds of mistakes.

Another way to think about it is that 1/(a+b) = (a+b)^(-1) (this is just a notational change, nothing really significant). Now what rules for exponents do we know? Only rules that deal with products and quotients of bases, not sums. There is no rule that says you can distribute an exponent across a sum (although you can across a multiplication). This fact does not mean that it isn't true, but that you should be skeptical of such any easy formulation -- if it were true for exponent of -1 wouldn't there be some general rule for it?

Our 13 year old is getting a taste of functions in her 8th grade algebra class. She's finding their abstractness very strange. She's been doing concrete, real-world problems all her life, and is just getting her first taste of the less-tangible stuff.

I think the whole education experience feels like learning lots of "rules" to most kids, and most of those rules are at least partially arbitrary: spell it "principle" in these cases, "principal" in those; 20% off per day late; commas here but not there; in H2O, the 2 applies to the H, not the O, etc.

Even in math: fractions must be in "simplest form" or lose points (simplest is whose opinion?); show your work (which parts? what's the rule? says who?); write the "0" in 0.2 or lose points (.2 is NOT correct claims the teacher); if you have a remainder of 2, which is correct: r2? R2? r 2? 2r?

In such a world, you can hardly blame kids for thinking that math is about learning a bunch of rules and applying their intuition directly to the notation.

I try very hard to teach my kids that math (at their level) is about real things, and that math notation is a somewhat flawed and arbitrary attempt to describe those things. Learning math is about learning about those real things, not about learning "rules" for manipulating the description (notation). If you keep reminding yourself of the real situation being described, the "rules" for manipulating the notation can be seen as just what they have to be to describe the situation sensibly.

This Law of Universal Linearity is a result of kids applying their intuition directly to the notation. I think the solution is to keep showing them how to read the notation and picture the reality it describes. They need to have lots of experience taking notation and converting it to real examples and vice-versa. With that, they can learn to focus their intuition on the math, not on the math notation.

If the notes you take for a class in college don't include the meaning of the thing you're writing down, then you do not know how to take notes.

You don't need a separate "vocabulary notebook", you need to learn the habit of thinking about what the teacher is saying, and writing it down in your notes. Similarly, you need to read books and think about what the meaning is of their terms and equations, and write that down, too.

Students who are successful in physics learn that to understand an equation, look at the boundary conditions--what that equation of x means at say, x=0 or x=infty.: does the result make sense?

When you are a novice and unsure of the sense yet, you learn what happens in physics from examination of boundary cases (equations says XXX happens--remember that!); as you grow, you test your derivations against the intuition of what should happen at them.

Most STEM students make poor assumptions because they aren't asked what things mean, and have learned to turn the crank without thought.

I know a group of student who are in grade 8, supposedly having already finished a year of alg 1 through quadratics.

We just found out that they can't comprehend volume except as an equation V=lwh.

For them, Volume *is* length times width times height. No sense that Volume is a quantity telling you how much stuff fits in something. No sense that the equation to compute a value for volume isn't volume. And sadly, no sense at all that not all objects' volumes can be calculated using that formula.

An Honors Calc student who made it through AB Calc and maybe BC with high scores may never have been asked what a derivative *is*, what is implied by linear functions, what nonlinear functions are telling you, what closure means, etc. They were taught to turn the crank, not to think.

which means we have solved the Chinese room paradox....successful

symbol manipulation does not mean the entity manipulating understands, not even if you draw the system to include "the room".

LOL. This is the result of poor math placement procedures in middle and high school. See it here all the time...instead of placing students who take the time to understand the math & make a few errors in doing so in honors, those seats go to students who have phenomenal memories and ace quizzes and tests on that strength. Of course it helps that NY doesn't do too much in the way of formal axiom definitions....I had to pull out the Dolciani to keep my kid out of that trap.

LOL. This is the result of poor math placement procedures in middle and high school. See it here all the time...instead of placing students who take the time to understand the math & make a few errors in doing so in honors, those seats go to students who have phenomenal memories and ace quizzes and tests on that strength. Of course it helps that NY doesn't do too much in the way of formal axiom definitions....I had to pull out the Dolciani to keep my kid out of that trap.

LOL. This is the result of poor math placement procedures in middle and high school. See it here all the time...instead of placing students who take the time to understand the math & make a few errors in doing so in honors, those seats go to students who have phenomenal memories and ace quizzes and tests on that strength. Of course it helps that NY doesn't do too much in the way of formal axiom definitions....I had to pull out the Dolciani to keep my kid out of that trap.

Apologies. Blogger seems to be having difficulties.

@Allison

"For them, Volume *is* length times width times height. No sense that Volume is a quantity telling you how much stuff fits in something."

I can understand your point about "derivative", but isn't "volume" an ordinary, everyday word? If they haven't understood the common meaning of "volume" by eighth grade I wonder what they're doing.

I can't even understand the derivative problem. How is calc taught these days?

Measurement is now an optional unit in K-2. So, if they aren't in the kitchen cooking, or out helping with the vehicle or lawnmower maintenance , volume isn't something in their practical experience beyond the idea of a gallon of milk or a 2 liter of soda.

In Calc, my son's DE teacher didn't use the differential notation (dy/dx as opposed to y') soon enough -- it needs to be there early on for conceptual understanding. Download a copy of "Calculus Made Easy" by Thompson if the text isn't sufficient to make up for the inclass omissions.

It sounds like the students are confusing volume (the formula) with volume (the concept). I wonder if they'd get it if you picked up the nearest irregular object and starting measuring its length, width, and height to find the volume.

What I don't understand about not understanding a derivative, is how the subject is introduced in modern textbooks. When I learned, you were shown a curve, shown how it could be estimated by short segments, then shown how that estimation became more exact as the segments became shorter and shorter. In the end, with the concept of a limit, we were shown that the derivative was the slope of the line at any given point. Has that way of presenting it changed? I audited calc in 1985 in high school, then took it for real in college the following year.

We asked immediately "what is the length, width, and height of this sphere?". They didn't know and it didn't immediately rouse them from their cognitive slumber. They defaulted back to lwh when we removed the immediate pressure.

There is a large continuum between no-thinking fill-in-the-blank processsing and the step-by-step process that an expert takes. However, one cannot do well in math with "just" a rote understanding. That is not possible unless the course is taught very badly. Change problems by a tiny bit and they will fail. That does not mean that those who do get good grades understand everything very well. There are layers of understanding. It also does not mean (as many educators like to think) that the process of teaching has to be flipped around to come at everything from an "understanding" direction. Invariably, they define this understanding using real world problem solving ideas.

That's the big fallacy. They see some sort of roteness and use that to change the whole process of teaching to match their philosophy. Rather than work harder, set higher expectations, and work more on mathematical understanding, they see only what they want to see to confirm what they want to do. It's a classic case of confirmation bias.

Back when my son was in pre-school I thought about all of the things I didn't like (understanding-wise) about my traditional math education. Then I found out that our school used MathLand. I was horrified. They got it completely wrong and were going in the wrong direction. On one hand, they talked about critical thinking and understanding, but on the other, they set very low expectations. Their understanding was at the conceptual and trivial understanding level.

If students do not know what volume means, then there is either a misunderstanding or something very bad (competence-wise) is happening. Likewise for derivitives. If students don't (at least) know that the derivitive is the slope of the tangent at a point, then there is a teaching competence issue. It's not confirmation that somehow the direction of the teaching process is backwards.

This doesn't mean that there isn't a problem with the teaching of understanding. This is what I was thinking about when my son was in pre-school. I want the understanding that needs to accompany a botton-up, skills-based approach to math. Those two things must go hand-in-hand. However, a bottom-up, skills-based approached to math (with missing understanding) can be fixed, but a top-down, understanding approach with fewer skills, is nowhere. Understanding does not drive mastery of skills automatically. Doing math is not some sort of general Polya-thinking process. I see it as the best application of a toolbox of mastered skills.

A trivial analysis of rote skills in math is a major problem in math education. Many don't want to see it as a call for working harder. They see it as a way to change the direction (and level of expectations) of teaching. Many don't want reality to get in the way of a good philosophy.

It took me a long time to get to the point where I really "knew" algebra and not fall into any number of misunderstanding traps. I had to develop lots of ways of "seeing" equations. I had to really understand what the basic identites meant as equations got more complex. This is what I see lacking in a lot of math (traditional, discovery, or otherwise). A teacher told me that Algebra II is the most difficult class for a lot of students. I can understand why. I struggled. The understanding I needed was related to understanding why I couldn't solve a slight variation of an equation. Understanding the forest would have helped, but not so much when I struggled with one tree.

In 8th grade, I had trouble understanding why 'X' had to be one value. At that level, everything is confusing. There are no simple words or in-class discovery projects that will cover all of the things one needs to know. That can only be done with homework sets. General concepts might be helpful, but one needs much more low level understanding help. One needs a process the ensures that homework is individually done and mastered. That is where all of the important understanding is hidden.

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