Why is it that I would not dream of giving my students a problem like:
The problem appears in a section called “Logarithmic Evaluation without Tables” in Smith and Fagan’s “Mathematics Review Exercises” (Ginn and Co., 1968). This is an American book, written as a supplementary text for high school students. If I were to assign this problem, (a) most students would decide just by looking at it that it is too hard and they would give up and (b) even if they did try it, they would not be able to do it, most likely because of the fractions and the relationship between (4/9) and (27/8). So the problem is too hard, right? Then how come it appeared in the middle of other similar problems in an American book written for American students of 44 years ago?
To my mind, the answer is clear – we have dumbed down the curriculum and our expectations. In particular, middle school math (even when the kids complete Algebra 1 in 8th grade) does not prepare the kids for any kind of higher-level thinking. From what I have observed, middle school math teaching, even when it claims that it is rigorous, graduates kids that have only a sketchy idea of math and spoon feeds the kids at every step of the way. One of the reasons for the poor performance of these kids in high school is “the system”– often middle school kids can not be held back if they fail a subject – it is deemed too damaging to their self-esteem. Another possible reason is that middle school teachers are driven by the state testing to cover all the topics in the book and there is no time to teach any concept in depth.
The result is a process where the middle school teachers I observed give the same kind of simple problems 50 times in a row, with the aim of having the kids master the procedures by ad nauseam repetition. The (unintended) consequences are kids who are bored, kids who do not see connections between the various math topics, kids who are not forced by the material to focus on a problem more than for about 4 minutes and kids who are not asked to use the procedures in a creative, higher level thinking way. These kids are lost to math – indeed, I would argue that they are lost to critical thinking – and the situation only gets worse as these kids “progress” through high school and college. Statewide, in California, roughly 30% of the students entering junior colleges have to take remedial math – essentially Algebra II. Locally, where I teach, that figure is over 40%. What use is instilling a “college going culture”, if the kids end up failing in college?
When I look through teachers’ blogs, such as the ones I follow, I am struck by the amount of effort and ingenuity that goes into procedures for teaching and grading and how little is said about what we ought to teach. Of course, one can not divorce what we teach from how we teach it. However, I would argue that what we teach should take precedence.
George Cobb, a highly esteemed statistics teacher and author, said: “Judge a book by its exercises and you cannot go far wrong”. I would paraphrase this as “good math teaching should be judged by the problems we set out for our kids”. Thinking, “hard” problems should start in middle school – we should abandon the idea that these are still children that need constant spoon-feeding. When are they supposed to grow up? If we start to demand and stick to our demands for critical thinking, we will get it despite all the grumbling that teenagers are prone to.
The resources are there: the Exeter problems, in Geometry Weeks and Adkins, in Algebra Dressler and Rich, the Dolciani books, Weeks and Adkins Algebra 2 and Smith and Fagan. Singapore books, such as New Additional Mathematics and the series on New Syllabus Mathematics, provide exercises far superior to those in American textbooks.
We teachers, especially the middle school teachers, need to be math teachers first and not just teachers of math – the priority should be on the subject and only then truly good teaching will follow.