## Unlearning mathematics

I was reading the comment thread in an old post of mine when I hit this gem by Bert Speelpenning:

Here is a short list of things that kids in math class routinely unlearn in their journey from K through 12:
* when you add something, it gets bigger
* when you see the symbol “+” you are supposed to add the numbers and come up with the answer
* the answer is the number written right after the “=” symbol
* you subtract from the bigger number
* a fraction is when you don’t have enough to make a whole
* a percentage can only go up to 100
* the axes on a graph look like an L
* straight lines fit the equation y=mx+b
* the values (labels) on the axes must be evenly spaced
* putting a “-” in front of something makes it negative
* a reciprocal is a fraction that has 1 on top.

What are some other things our students unlearn?

Which things are acceptable to teach initially in a way that will later be changed? When is unlearning problematic?

Which things are impossible to avoid having the unlearning effect? (For instance, even if the teacher avoids saying it explicitly, it’s hard for students to avoid assuming “when you add something, it gets bigger” before negative numbers get introduced.)

### 12 Responses

1. They are not unlearning, they are refining. It’s like learning Newtonian physics and then Relativistic physics. You start with simple models that then grown in sophistication, and models, simple or sophisticated, are much more than a few statements.

• Your comment reminds me of Asimov’s essay “The Relativity of Wrong”.

I’m not 100% behind saying for the students are “refining”, because sometimes they don’t. It seems like there should be a different word for that.

One of the obvious ones that comes up a lot is “when you divide you make something smaller”. Mathematically you can say they were going from natural numbers in real numbers and refine prior knowledge, but a lot students get stuck and somehow never grok that dividing by 1/2 will make a number bigger.

• What you are describing is a student who isn’t learning to being with.

Bob Hansen

• Well yes, they aren’t refining. What’s that called?

It’s just usually I equate refining with being an expert and getting it more polished. This is more like a redo of past preconceptions which may or may not make the mark.

The terminology isn’t that important though.

2. In that same topic thread, I shared that there is “desirable difficulty” and “undesirable difficulty” in learning. Bert’s list above is what I think of as undesirable difficulty because they are not ever pulled into an overarching framework. If it did, that framework would provide the context to add to our understanding rather than unlearning something.

Here is a snippet:

“I agree that the easiest path is not always the best path when it comes to learning. What we think of as intuitive does not always pan out in practice. Two great examples of this are the Kaminski study that determined that kids do not need concrete instantiation to learn abstract concepts, and Kornell’s findings that spacing is better than massing for memory and induction.

I am not doing these studies justice by oversimplifying, but the point is that we can be led astray by our intuitive thinking about learning. We always need to remember that results are what count. And we know that context and framework produce better results (learning) than a lack thereof. The reason is that for learning there is difficulty, and then there is desirable difficulty.

Spacing is an example of desirable difficulty. Unlearning is difficulty that is not desirable. Fits and starts are OK, but deep learning does not need to be based on learning something, and then just when we thought we were starting to understand it, unlearning it and replacing it with something else.

Here is an example of what I mean. This is just a story example but it is a good illustration of how people can learn systems of knowledge. In the story of the Karate Kid, Mr Miagi taught Daniel-san how to wax cars and paint fences. Later, he showed Daniel-san how those techniques could be used to perform Karate. Daniel-san did not have to unlearn anything, he added to his understanding, expanded it, and had several “aha” moments. Unlearning does not produce those “aha” moments, but sudden understanding does.

So unless the results show otherwise, let’s teach in a way that provides opportunities to expand understanding rather than in a way that requires relearning.”

Here is the thread:

3. Your post is very timely as I’m working right now on planning a PD for teachers in my district about replacing math “tricks” with understanding. I was inspired by the NCTM article, “13 Rules that Expire” and the online book “Nix the Tricks.” Both of these resources outline more misconceptions that students have to “unlearn.”

Others, that I would add to the list, are what you could call imprecise language. These would be when “opposite” is used to describe inverses (Add/Subtract, etc) or “similar” to describe what two things have “in common” but aren’t mathematically similar.

• Here’s a challenge for you then: even though teachers may not mention explicitly, some of those “rules that expire” will have students come up with them anyway (for instance, #4 on the list, addition & multiplication make numbers bigger, is sort of unavoidable).

What should we do as teachers when the contradictions come up? I know students who never can seem to shake their old models in some topics.

It seems a lot of what I’ve read just says “don’t teach the bad rules!” but the students are learning them anyway without explicit teacher instruction, and it’s really hard to tell a 1st grader “by the way, in the future numbers won’t always get bigger when you add”.

4. I am going to take these one at a time
* the answer is the number written right after the “=” symbol
This is a direct consequence of the fake “Algebraic Thinking” stuff and the fascination with equations and number sentences.
More of the 10 = 3 + 7 is in order here, and play witht the equations.
At least when I was at school we were correct in writing “Answer = “, even though the teachers hated it!

* you subtract from the bigger number
Subtraction is an action only meaningful on unsigned numbers (the counting ones and the measuring of quantity ones)
Subtraction has no real meaning in the signed numbers. The problem is that negative numbers are introduced as an extension

to the “numbers we have already”, overlooking the fact that signed number were created to deal with position and not

quantity. Even with voltage we say “the voltage is high” and not “how many volts have we got?”.
In the signed number system 3 – 7 means “starting at 3, move 7 steps down or to the left”, and of couse in the unsigned system

it is a garbage expression.

* a fraction is when you don’t have enough to make a whole
Problem here is the way fractions are introduced, as “parts of a whole”. This makes statements like “3 is half of 6” only meaningful if we then say “6 is the whole”. I thought 6 was a number (or at least a representation of a number!)
Parts of a whole is an application of fractions. Many difficulties will then fade away.

* a percentage can only go up to 100
Similar problems with this. 100 is treated as “the whole”

* the axes on a graph look like an L
Well, they do, when used with unsigned numbers.

* straight lines fit the equation y=mx+b
I thought they did
* the values (labels) on the axes must be evenly spaced
This is not really a problem. Graphs with different spacing on the two axes is a step in the right direction.
By the time one wishes to introduce logarithmic plots the majority of the students are so lost anyway that it doesn’t matter too

much!!!!!

* putting a “-” in front of something makes it negative
Where they get this from I cannot imagine. Could try “So putting a + in front of something makes it positive” and sit back

until the dust settles.

* a reciprocal is a fraction that has 1 on top.
Technically this is ok, since the reciprocal of 2/3 is 1/(2/3), but can be usefully rewritten.

Math teachers are either so familiar with the stuff that they don’t see that some of the statements are oversimplifications or have limited application, or they are one chapter ahead in a textbook of indifferent quality.

5. I went to the “old post of mine” to find an almost endless discussion on “multiplication is or is not repeated addition”, particularly the Keith Sherwood/Bert Speelpenning conversation. I got interested in this one when reading the Keith Devlin post.
None of the discussion takes into consideration the nature of “number”.

Numbers are first of all whole and for counting and measuring “how many”. So 3 bags of 4 sweets in each is 3 times 4, or 3 lots of 4, and the simple way of seeing this is as 4 and 4 and 4 (I am deliberately avoiding the + sign here). This of course can be visualized as
@ @ @ @
@ @ @ @
@ @ @ @
or as @ @ @ @ and @ @ @ @ and @ @ @ @

As soon as numbers are required for measuring attributes of things we get the rational or fractional numbers, which are NOT the same as the natural numbers, even though 5/1 and 5 can be paired.
This leads to two types of question:
1: London is 210 miles from here. How long is the return trip?
This leads to 2 lots of 210 .. and repeated addition
2: My car does 25 miles to the gallon. How far will it go on 3.2 gallons?
Here we have a rate problem, which is less easily seen as repeated addition, and the argument starts to rage. Out comes the infamous “area model”, but even here repeated addition is sensible:
1 gall 25 miles
1 gal 25 miles
1 gall 25 miles
0.2 gall 5 miles
total miles 25 and 25 and 25 and the extra bit, 5 : result 80
I can see the area model as counting squares and parts of squares, or as adding up rows of squares, a row of part squares, a column of part squares and an annoying little bit in the top right hand corner !

The real problems of explanation come with electricity.
power = voltage x current
but by that time the kids just get out their calculators.

6. one big lie. “all you need to know is….”
many so-called teachers.
the clueful get it despite ’em.
it was ever thus.

7. /*At least when I was at school we were correct in writing “Answer = “, even though the teachers hated it!*/— howardat58, upthread.

i’m more likely to’ve encouraged this behavior
than to’ve “hated” it.

but “Answer” is a pretty awkward variable-name
so, given a chance, i’m also likely to’ve made it
as plain i could find a way to do that what i’d
*really* like to see is a clear

A = …. messy expression to be simplified

right at the beginning and then the

bit at the end. which gives a presentation
clearer than one is likely to find on the
blackboards unerased by the previous
class. alas.

because “define variables (with units) precisely”
is a *major* sticking point for *many* students
and i’m not just talking about Remedial Algebra.
one of my favorite-ever calculus tutees
refused my excellent advice on this subject
*many* times.

but without it, we simply *cannot* organize
our presentations coherently.

she finally… same calc ii student here…
couldn’t endure my continual insistence
on keeping equations balanced as she
wrote out her calculations. we broke up
over it.

the attitude seems to be “it’s all just
ritual-process calculation anyway
until i can get the Answer”, whereas
of course one seeks to instill instead
something like “the Answer is itself
a collection of equivalent statements
(leading to the value of a variable)”.

“scratch” work is *obviously* the enemy of clarity
once one is made to *grade* the work.
and not just clarity of *presentation*.
having calculated out some expression,
let’s say correctly, one is in the position
of having to *do something* with the result.

but without the whole A = Answer format…
a “proof”, if you will… one is left with a
bunch of area-on-the-page with certain
code-strings (and scattered english)
bearing no particular *stated* relation
to one another at all.

and if Answer = “the thing i want to see”
i’m very likely to give ’em full credit.
but that won’t make it good work.