"Above all else, do no harm" is the sound advice given (by Hippocrates?) to new physicians, which suggests to me the following mathematicians' analogue: "Thou shall not inflict irreversible mathematical brain damage on thy students." Unfortunately, we teachers sometimes do exactly that when we gloss over a question posed by a student that we don't know how to answer at the age-appropriate level. One of the main purposes of this webpage is give examples, advice, and references that can help us deal with these pedagogical problems. Readers of this web page are invited to submit their own examples. A particularly useful reference on this topic is the book Knowing and Teaching Elementary Mathematics [Lawrence Erlbaum Associates, Mahwah, NJ, 1999, ISBN: 0-8058-2909-1 (paperback)] by Liping Ma. In it, she poses the following interesting pedagogical questions:
"Multi-digit Number Multiplication: Dealing with Students' Mistakes Some sixth-grade teachers noticed that several of their students were making the same mistake in multiplying large numbers. In trying to calculate
the students seemed to be forgetting to "move the numbers" (i.e., the partial products) over on each line. They were doing this:
instead of this:
While these teachers agreed that this was a problem, they did not agree on what to do about it. What would you do if you were teaching sixth grade and you noticed that several of your students were doing this? "
"Division by Fractions"
People seem to have different approaches to solving problems involving division with fractions. How do you solve a problem like this one?
Imagine that you are teaching division with fractions. To make this meaningful for kids, something that many teachers try to do is relate mathematics to other things. Sometimes they try to come up with real-world situations or story problems to show the application of some particular piece of content. What would you say would be a good story or model for ?"