Saturday, January 12, 2013

Unreflectingly Applying Known Steps

Encouraged by the non-zero readership of the previous post, here is the second chapter in the adventure. Thanks for the comments, and thanks, CJH, for the title!

Hypothesis About One Cause of Misperception

I will use the blanket term “misperception” to refer to a hasty misinterpretation of a problem and applying a familiar — habitual — operation to reach an answer. A motivating example is Suhana’s response “8” to the problem “There were five birds in a tree and three flew away. How many are now present?”

I hazard a statistical argument for why this might happen. In a nutshell, I argue that this strategy of quickly judging a problem is quite efficient for the tasks the student typically faces.

The mind-numbing homogeneity in problems is nicely demonstrated by the Kumon’s 80-page magnum opus “My  Book of Simple Addition”. I urge you to briefly pause and guess what the 700 addition problems  might be. It turns out that the problems range from “1 + 1” all the way through “28 + 2”. That’s right: the second number is always “1” or “2”. You don’t even see “2 + 28”. Mechanical strategies are perfectly adequate here! I haven’t read “My Book of Simple Multiplication”, but hope that it’s problems do not range from 1 * 0 to 28 * 1.

From elementary education, let’s turn to something a bit higher — Grade X in Maharashtra, India, where I went to high school. There is a well-defined pattern for what questions are asked. Question I has eight parts of which you may answer any six. One subquestion usually concerns finding the determinant of a 2x2 matrix, another subquestion is to simplify a poynomial ratio (where the denominator has degree 2), and a third is to find the HCF or LCM of two small-degree polynomials. Recent question papers can be found here. Who needs to understand math to excel at such an exam?
The homogeneity also extends to certain word problems. It makes possible tricks for guessing what the operation to be done is — multiply or add or subtract? An example from a random website:

The third step is to look for “key” words. Certain words indicate certain mathematical operations. Below is a partial list. Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved
    Additionincreased by
    more than
    combined, together
    total of
    added to

Really? “increased by” means addition? How about “I had a few balls. My collection increased by 3. Now I have 7. How many did I have?”. The answer ain’t 10. I don’t even have to use such esoteric examples as apparent magnitudes of planets “increasing by 1” — magnitude is a log scale where smaller values represent brighter objects. Such triggers only work in a limited context, but if one is exposed primarily to that context the trigger words appear quite potent.

Sprinkling Variety

To the extent that homogeneity is the cause of these ills, variety is the cure. If there is enough variety, the argument goes, then no automatic maladaptive strategy will get sufficiently reinforced and habituated.

I have started doing word problems with Suhana. I print these out, and making use of her recently increased fluency in reading, let her read the problems. The problems are tailored to include things in her world — real and imagined. I have been focusing on asking problems that force her to pay attention to what words mean.

Here is an example: “Suhana had three balls. Reesha had two bats. How many balls are there?” Can’t get simpler than that! Other examples I tried initially included “Three princesses had to go to the ball. They had seven socks in the closet. Each princess wore two, so how many socks are left?” and “Baba had two cats. They had a few kittens. Now Baba has seven cats. How many kittens were born?”

Teething Troubles

I have nothing insightful to share about “attention spans are short, distractions are many, I don’t want to push” except to note that it appears to my biased eyes that excitement is contagious and that doing things with a parent is a draw.

Some Success

Today, Suhana solved, with no help, this: “Suhana took six toys to the park. Reesha took four toys. Sadly, the girls broke five toys and brought the rest back home. How many did they bring back home?” She did not use the words add or subtract — those words are not really needed.

That's all for tonight, folks!


  1. It will be interesting to see when the abstract concepts of addition and subtraction emerge from the word problems. Are there links related to that (from studies)?

    1. To be honest, I am not sure when or how those will emerge. Suhana already knows the word "add" and can add small numbers using fingers (including, for problems whose answer exceeds 10, "Baba, can I borrow your fingers?")My guess is that as she does more word problems, she'd realize that certain situations result in things that look like the operation of addition that she already does. We'll see.

      Of course, the very notion of addition as you understand it today is very abstract and complex --- if you include additions of fractions, complex numbers, polynomials, elements of fields, adding 3 hours to "11:53 PM", the overriden addition operator of a C++ class, additions with infinitely many terms (such as the "addition" that defines pi), not to mention seeing integration as a limiting case of addition. Addition is not a monolithic concept; the conceptual apparatus for this rich concept must co-develop with the many other concepts it depends on. Multiple leaps of abstraction are needed and the notion of addition as understood by different people is perhaps quite different --- most people and neither mathematicians nor programmer, and a notion of addition restricted to real numbers is sufficient. Is there a qualitative difference in understanding this "simple" concept? I don't know. I am quite curious to see how it develops :)