The perils of math pedagogy

It seems that the prevailing wisdom in teaching mathematics is to make abstract concepts as concrete as possible. The thinking is that if math can be related to everyday concepts or pictures then it will be more palatable and understandable. I happen to disagree. I think part of what makes math fun is the abstractness of it. You make up some rules and follow them to their logical conclusions. This is also how I see children play. They like to invent make-believe worlds and then play within them according to the rules of the world.

Usually these attempts at concreteness seem harmless enough but I have recently come up with an example where making things concrete is much more confusing than just teaching a rule. The example is in division with remainders. My third grader was asked to “draw” 7 divided by 2 in terms of items and groups. Her instinct was to draw 2 groups with three balls each with one ball remaining, like this

(x x x)

(x x x)

x

She then was supposed to write this as a mixed number, which looking at the diagram she wrote 3 1/3. When she asked me if this is correct I asked her to multiply this by 2 and see if it gets back 7 and when she got 6 2/3, she was extremely confused as to why she didn’t get the right answer. I tried to explain to her that the way she grouped things, the remainder was in terms of the fraction of the number of groups, which is very unintuitive and almost impossible to explain. It would have been even worse if the example was 8 divided by 3.

I then tried to tell her that a better way to think of division is not to ask how many elements would you get if you divided 7 into 2 groups because this amounts to begging the question (phrase used the correct way), since you need to know the answer before you can do the operation. Rather, what you really want to ask is how many groups would you have if you divided 7 items into groups of size 2 (which is a local rule), whereupon the diagram would be

( x x)

(x x)

(x x)

x

Now if you write down the mixed number you get 3 1/2, which is the correct answer. She then argued vehemently with me that this is not what the teacher taught her, which may or may not be true.

I think even most adults would get confused by this example and maybe working through it would give them a new appreciation of division. However, if you wanted children to learn to divide correctly than teaching them the rule is better. To divide 7 by 2 you find the largest integer that multiplied by 2 fits into 7 and what’s left over is divided by 2. Even better, which introduces and motivates fractions, is that you write 7 divided by 2 as 7/2 and this then becomes 3 1/2. If you learn the rule, you will never end up with 3 1/3.

Selection of the week

Leonard Bernstein conducts the “Orchestre National de France” in Hector Berlioz’s (psychedelic) Symphonie Fantastique 5th Movement:Larghetto, Allegro (Songe d’une nuit de Sabbat) in Paris, 1976.

If you have the time, here’s the whole piece played by the Chicago Symphony Orchestra conducted by Stéphane Denève.

New paper on path integrals

Carson C. Chow and Michael A. Buice. Path Integral Methods for Stochastic Differential Equations. The Journal of Mathematical Neuroscience,  5:8 2015.

Abstract: Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult. Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs. The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder.

This paper is a modified version of our arXiv paper of the same title.  We added an example of the stochastically forced FitzHugh-Nagumo equation and fixed the typos.

Selection of the week

Vivaldi’s Spring from the Four Seasons, with Julia Fischer and the Academy of St. Martin in the Fields.

Selection of the week

The first movement of the Cello (and Piano) Sonata No.1 Op. 38 by Johannes Brahms played by Alisa Weilerstein, cello and  Ilan Rechtman, piano.

 

Here is the whole thing with Yo Yo Ma and Emanuel Ax

Jobs at the Allen Institute

The Allen Institute for Brain Science is currently recruiting. The positions are listed on this link but in particular see below:

SCIENTIST I – MODELING, ANALYSIS AND THEORY The Modeling, Analysis and Theory team at the Allen Institute is seeking a candidate with strong mathematical and computational skills who will work closely with both the team as well as experimentalists in order to both maximize the potential of datasets as well as realize that potential via analysis and theory. The successful candidate will be expected to develop analysis for populations of neurons as well as establish theoretical results on cortical computation, object recognition, and related areas in order to aid the Institute in understanding the most complex piece of matter in the universe.

Contact: Michael Buice, Scientist II michaelbu@ alleninstitute.org

Selection of the week

Early Romantic composer Felix Mendelssohn’s Song Without Words Op 109 played by English cellist Jacqueline du Pré, whose career was tragically shortened by multiple sclerosis.

Michael Buice on the dress

Read former LBM fellow Michael Buice’s explanation of the dress colour illusion.

Huffington Post: …In the case of the dress, one’s assumptions about lighting have a strong impact on the perceived color. In particular, your perception will be affected by whether your visual system sees the dress as being in bright light or in shadow. Comic book coloristNathan Fairbairn put together the following in order to illustrate these two different potential hypotheses about light and color in the picture.

So what happens if we try to remove contextual information? It so happens that these average colors are close to being inverses of one another. Inverting them gives us:

Inverting the colors in the original photo should approximately “swap” the two colors on the dress, as well as remove contextual information (or perhaps render it nonsensical). The color inverted dress looks like:

I see white-and-gold here, and I saw white-and-gold in the original. My wife is a die hard Black-and-bluer, and she sees the inverted dress as light-blue-and-gold. Notice that the image now has artifacts that look (to me anyway) like damage in an old photograph. This is a sample size of one, so I’m curious to know if this inversion changes the perceptions of any other black-and-bluers out there.

We know that training can alter the “light-from-above” prior, and it seems plausible that people’s differing perceptions of the photo are due to their different experience, and in particular their experience with light, shading, material, and overexposed photographs.

Our brains have to make guesses, but they don’t always make the same guesses, even though we live in the same world. One of the hardest inference problems our brains have to solve is figuring out how everyone else sees the world. Perhaps with some very hard work, I can be a Black-and-bluer, too.

Michael Buice is a scientist at the Allen Institute for Brain Science. His research interests are in identifying and understanding the mechanisms and principles that the nervous system uses to perform the inferences which allow us to perceive the world.