When Good Teaching Leads to Bad Results

I found the article "When Good Teaching Leads to Bad Results: The Disasters of 'Well Taught' Mathematics Courses" by Alan Schoenfeld very interesting and he has definitely outlined a critical issue in mathematics. I always find myself sitting on the fence when it comes to issues similar to this one. On the one hand, I believe children learn best when they explore and there are activities and they find math fun. On the other hand, I believe there is a place for old school mathematics with your proof and your algorithms.

Schoenfeld (1988) describes of an instant where a math problem would have only taken the students five minutes to do, but because they had to outline their process and follow the correct format, it took them 15 minutes. I would like to compare this instant with that of writing an essay. Any child can throw a few ideas on a piece of paper and call it an essay. However, in the higher grades, it would not be accepted as a proper essay unless it had proper paragraphs, an introduction, a conclusion and so on. Why should math be any different?

Jumping back on the other side of the fence, I agree that children have trouble connecting the bits of pieces of math that they learn together. They choose to memorize instead of understand. I like how Shoenfeld does not blame this on teachers, he blames it on the curriculum. I long for the day when children are grouped based on ability, not age, and everything they learn they will find relevance in the world around them. Until a curriculum is designed for us that allows this and standardized testing is not used to criticize us, we will do the best we can with what we have.

Why Teach Mathematics?

Growing up I have always loved mathematics; I found it interesting, it came to me naturally and I really enjoyed numbers. (I had a student ask me yesterday: “Miss you really like numbers don’t you?” Yes, yes I do!)

I first decided to teach mathematics because I was very passionate about it and I wanted to share that passion with students. I also knew that there was a lot of negativity about math in schools and I wanted to make math fun and enjoyable again.

I believe that mathematics that young children learn should be relevant to them: math they would encounter every day. As they grow older, math should begin to prepare them for an adult mathematical world.

I came into the teaching profession just four years ago, thinking that I should teach math the way it was taught to me (notes on the board, questions from textbook). After all, I had learned perfectly that way. I realized in that first year, that I need to change that mindset, because not all children learn that way. As each year has gone by, I have experienced more and more with discovery and allowing students to develop their own methods and ways to remember things.

So why should we teach mathematics? Even after reading the article, I am not entirely sure of that answer. My first thought was because students will need it as adults. My second thought was to teach children to be thinkers. And as I write this, more and more reasons pop into my head: prepare them for university, etc.

Given that thought, I think the math curriculum is too jam packed full of topics that try to satisfy all these reasons why we should teach mathematics, that it is too much for students. Students are lacking basic skills and mental math which are so important to every adult, because we spend very little time teaching (or going back to) these topics. The high school general math curriculum (it is in the process of a much needed change) was teaching children about abstract math that they would never use. Instead they should have been taught how to do their income taxes, balance their chequebook, etc.

I am always torn when I have to write something like this, because I love the difficult math that children always complain they will never use again and I can see all the benefits of teaching it, but on the other hand, I see children lacking so much in the areas that they will 100% use again.

What Kind of Thing is a Number?

When I first started reading this article, I was thinking to myself that this guy way thinking WAY too much into mathematics and why this was that and why things were how they were. I often find it hard to sit back and be philosophical about things. After I continued reading on for a bit, the article reminded me of the philosophical thought of "If a tree falls in a forest and no one is around to hear it, does it make a sound?" that raises questions about observations and knowledge of reality (Wikipedia, Oct.12, 2011). Hersh says "There's no math without people. Many people think that ellipses and numbers and so on are there whether or not any people know about them; I think that's a confusion" (Brockman, 1997, p. 1). I'm one of those people who would say that yes the tree makes a noise because if there was someone there they would hear it. The same goes for the math, I believe the math still would exist if people were not around, it just would not have a name. All humans have done is name it. However, it certainly makes me think more about the philosophy of math, that's for sure.

There are three philosophical attitudes towards mathematics: Platonism - some abstract entities; Formalism - calculations only, there is no meaning; and Humanism - mathematics is part of human culture (Brockman, 1997, p. 4). Hersh feels that humanism is the only educational friendly philosophical attitude of education and that it "brings mathematics down to earth" (p. 4). In other words, it is just what we have been calling making math real for children. We have been saying for years that we have to relate math to the real world for children in order for them to have a connection to it and find it meaningful to them. Having only been teaching for 4 years, I am finding myself more and more heading in this direction. My first year out, brand new to the whole thing, I resorted to the way I was taught with notes and questions, maybe a scattered activity. As each year went by, I felt more comfortable allowing children to discover formulae and why certain things happened the way they did. The children enjoy the math more this way and then they are much more likely to remember it.

The Romance of Mathematics

I believe the following about mathematics

·         Mathematics is an objective feature of the universe; mathematical objects are real; mathematical truth is universal, absolute, and certain.

·         What human beings believe about mathematics therefore has no effect on what mathematics really is. Mathematics would be the same even if there were no human beings, or beings of any sort. Though mathematics is abstract and disembodied, it is real.

·         The mathematics of physics resides in physical phenomena themselves - there are ellipses in elliptical orbits of the planets, fractals in the fractal shapes of leaves and branches, logarithms in the logarithmic spirals of snails. This means that "the book of nature is written in mathematics," which implies that the language of mathematics is the language of  nature and that only those who know mathematics can truly understand nature.

I think this affects how I teach mathematics in that I am constantly trying to show students how real mathematics is and where they can find mathematics in places they never thought. When a child thinks of nature, they do not think of mathematics, so I like to show them how different things can follow different mathematical patterns. Some develop a greater appreciation for mathematics because of that.