Typical Teaching = Typical Results

When I finished reading Shoenfeld’s article, my very first thought was of good teaching, what is good teaching, and who defines it? In his article Schoenfeld cites researchers Romberg and Carpenter to point to the fact that much of the instruction in schools at that time (and currently if I had to guess) was based on the “absorbtion theory of learning”. This theory is based on the premise that good teaching is defined by having many ways to say the same thing so that students will eventually “get it”.


Absorbtion theory, chalk and talk, kill and drill, call it what you like, students need to make sense of mathematical concepts on their own if they are to develop a lasting understanding of an idea.

Case in point, I had a tutoring student this week, who, although I have stopped tutoring her now that she is in Gr 7, I agreed to see her once more to help her get ready for her first test of the year. The topic being tested was divisibility rules and patterns, a skill that, if understood, can certainly be useful in completing more challenging mathematical tasks. We took some time working with the other rules, which she had no trouble with, she could write numbers that were divisible by a given number, check to see if a number was divisible by a given number and list all of the divisibility rules that applied to a number........she could apply all of the rules....except for the rule for dividing by 8. When it came to the divisibility rule of 8, “A number is divisible by 8 if its last three digits is divisible by 8” this student was having real difficulty......and with good reason I thought....I mean really, who knows their multiplication tables up to 3 digit products, yet this is the “rule” that she was supposed to use (and incidentally is the rule in the text book). I asked her what she knew about the number 8, what was it divisible by.....she quickly came up with 4 and 2, I asked her if she thought this could be of any help to her in determining if a number was divisible by 8. She didn’t see a connection right away, but when she did some more work with the divisibility rule for 6 ( a number is divisible by 6 if it is divisible by both 2 and 3), she was immediately able to make the connection that a number that is divisible by 8 is divisible by  4 and then by 2, for example if 328 is divided by 4 the answer is 82, 82 is divisible by 2, therefore 328 is divisible by 8. 292 is divisible by 4, but the resulting answer is 73, which is not divisible by2, therefore 292 is not divisible by 8. It took some time, and a little trial and error, but eventually she was able to use one rule to help her come up with another.
 She used her understanding of one rule to develop her own understanding of another, she made sense of it on her own, and has a more reliable way to find a solution as opposed to applying a rule that she doesn’t understand and doesn’t have the skills to use. Had she been forced to rely on this rule she had been “taught” it is likely that she would not be able to apply it and would not demonstrate an understanding of something that she does in fact understand quite well......when she’s given the time to think about it.

Another question that came to mind while I was reading this article is whether or not we are meeting the long term needs of our students with instruction that is intended to produce short term results? In the school in which Schoenfeld conducted his 1983-84 study, “Learning was operationally defined as performance on achievement tests” (Schoenfeld p. 4), “the primary goal of instruction was to have students do well on the exam” (Schoenfeld p.9.)

In placing so much emphasis on these tests we are influencing the very activities which take place in our classrooms. The phrase “teaching to the test” is one that is echoed by teachers everywhere, often with contempt, but is one that many adhere to even though they may be in disagreement. Our school districts are adopting new curriculums and new instructional materials that emphasize learning for understanding, yet we are still using materials for testing that emphasize memorization. I think that our assessment methods have to come in line with our instructional methods before we are able to see any real change in producing students who are able to apply their mathematical understanding in different and new situations.

On page 16 Schoenfeld states that what he observed in his study was “typical, if not better than average, instruction—with typical results” , as long as standardized test scores remains one of the driving forces of math instruction in our schools I feel it is likely that we will continue to see this “typical situation.

Melanie