Tuesday, October 30, 2012

There are a wide variety of mathematical errors, and it's worthwhile to try to find patterns and themes that stand behind the particular mistakes.

The following quote is from Thinking Fast and Thinking Slow:
"I propose a simple account of how we generate intuitive opinions on complex matters. If a satisfactory answer to a hard question is not found quickly, [the intuitive capacity] will find a related question that is easier and will answer it. I call the operation of answering one question in place of another substitution. I also adopt the following terms: the target question is the assessment you intend to produce; the heuristic question is the simpler question that you answer instead."

Not all errors involve some sort of substitution. Sometimes we avoid jumping to a conclusion, we grapple with the difficult problem, but we still make a substantive error. This is just one type of mathematical error.

Questions for homework:
1. Do you agree with the author of this post?
2. Are there other categories of mathematical error that you can identify?
3. What can teachers do to effect the usage of heuristic questions by their students?
4. Is this the sort of question the domain of psychologists? Of teachers? Of both?

1. Is Kahneman talking about errors? It isn't clear from the quote, and I haven't read the book.

I ask because answering a simpler, but related, question, when done mindfully, is a powerful problem solving strategy. It helps identify the essence of a problem, and it allows the solver to develop some intuition about the larger problem.

I don't associate this with "mistakes" at all.

2. Agreed. Fuller quote:

"Substituting one question for another can be a good strategy for solving difficult problems, and George Polya included substitution in his How to Solve It...But the heuristics that I discuss in this chapter are not chosen; they are a consequence of the mental shotgun, the imprecise control we have over targeting our responses to questions."

1. Over 125 words, and I still don't understand what Kahneman is talking about. This is one reason I avoid reading economists.

This "mental shotgun" sounds like mindless guessing. Is there anything of value to be gleaned from inspecting that?

2. I think so. It's not mindless guessing. The idea is that the mind is a pattern-recognizing, resource-skimping, jumping to conclusions machine. The mind uses the useful substitution heuristic as a problem-solving mechanism, but it can backfire if the "easier" heuristic question isn't a good one.

3. I suggest that it's not a matter of choosing an easier question, just a more familiar/expected question. This would explain why students sometimes cannot solve the easiest question using pythagoras rule, if the question is embedded in a more tricky exam on trigonometric functions and they therefore expect to use trig. It also explains why students keep trying to set up an equation and solve for x in question that only require them to factorize or expand an expression. I also think that there is a fundamental difference between looking at simpler cases as a heuristic method, and simplifying the question by accident, without meta-cognitive awareness.

This kind of changing the question into something more expected fits right in with schema theory, which Bransford (one of the "sheep and goats"-problem authors above) has done pioneering work in.

4. In all my carrier, my life, my social relations, the fast answer is more likely to be the wrong answer. Inspecting a work of art, of music, or even of a student, I found that the first impression yields usually a very incomplete picture. The same certainly holds for economy people assessing some complicated business model. Intuition might be a good way to start, but it can lead into mud if not controlled by careful thinking.

I also disagree with the "answer an easier question" strategy. While this may help to get a start, the effort of the complicate question is still ahead.

1. "Expert performance entailed (a) a large knowledge base of domain-specific patterns; (b) rapid recognition of situations where these patterns apply; and (c) reasoning that moves from such recognition directly toward a solution by working with the patterns, often called forward reasoning."
(Are Cognitive Skills Context-Bound?

Expertise just is the ability to take reliable shortcuts. But, sometimes, the shortcuts are unreliable. If we understand how the shortcuts are formed then we have a better chance of helping kids avoid the wrong shortcuts. In general, I think that teachers want kids to work slowly and carefully through problems. But that's not how expertise works. We focus too much on the slow, careful processes without helping kids form reliable, quick shortcuts (i.e. intuitions)

5. Related to comments above - the brain does not limit itself to mathematical patterns, so a new problem may be similar to something the student understands, but not similar in a mathematical way. This is what I think is usually going on with my students. Not really guesses, but non-mathematical connections to other problems/ideas.

5^0 looks like 5*0, (stuff + other stuff)^5 looks like (stuff + other stuff)5, (3 + x)/x looks like 3x/x so I will "cancel" the x's, etc.

The step that is often missing is to put a filter on your brains pattern finding/use. Find a pattern/notice something. Great! Now slow down and reason out the stucture supporting that pattern or observation.

Obviously, finding a less complicated problem that is mathematically similar can be a very useful strategy. Finding a more complicated problem that is similar can also be useful at times.