The Dangers of Math Magic

In the first paragraph of his paper, Erlwanger states that his purpose is to show the IPI system to be is unsuccessful because of the way in which it presents Mathematics to the learner. The main point is that the answer to the questions “What is the subject of Mathematics?” and perhaps more importantly “What is not Mathematics?” (according to Erlwanger himself) is in disjoint with the subject being taught to students such as Benny. The paper is, in a sense, simply Erlwanger’s own version of our own first blogs, albeit a little longer. For example, Erlwagner finds that Benny believes in a kind a Math “Magic” – if he applies the correct rules, rules which some guy devoted his life to creating, he will be able to produce the correct answer. The task is simply to find the rule that works for that particular problem, and as such, Benny has no trust in the consistency of Math. He has no idea whether his answer is right until the mysterious answer key hanging over his head deems to respond with a “Yes” or “No”. In contrast, Erlwagner sees Math as a rational subject, one in which separate concepts, such as fractions and decimals, can be related and answers can be verified according to common mathematical sense.

I think that Erwanger’s definition of Mathematics is still extremely applicable to learners today. In my own experiences in the schools, I find that teachers hand out Math worksheets and explicitly state “Don’t show any of your messy working on the paper”. How can a child conclude anything from this other than that a correct answer is all that is required from them? This encourages the kind of rule application favoured by Benny: Use any method that seems to work, regardless of whether or not it makes any kind of Mathematical sense! Teachers need to try and gain an understanding of how a child produces answers, and why he thinks the way he does.

Instant vs Delayed Gratification

According to Skemp, when we speak of understanding a Mathematical concept, there are two different types of understanding that we could possibly be referring to. The first, termed Instrumental understanding, is knowing what to do in order to get a correct answer. The second, Relational understanding, expands on this: it is knowing what to do, and the reasoning behind why this produces a correct answer. Both provide methods for solving Mathematical problems and both produce correct answers. The benefit of learning Math instrumentally is all to do with ease and speed. Given the right set of rules, a child can quickly, and correctly, produce a set of correct answers. It is easier for the child to understand what she or he has to do. However, there are drawbacks, many of which can be avoided by using relational teaching and learning. For every new set of problems, an instrumental learner must learn a new set of rules. On the other hand, a relational learner can adapt his knowledge of previous concepts and derive a new method. He is also more likely to remember something that he understands. Consider the 2 words "Happy" and "T8grP". Which are you more likely to remember tomorrow? The one that makes sense to you, that you can imagine using correctly in situations, that you feel confident adapting for a variety of uses. The learner will enjoy mathematics all the more for being able to solve problems without being dictated to at every step.

I have too much fun with numbers....

What is Mathematics? For me, Mathematics is the way we express our trust in the universe. Every time we use numbers to express the happenings in the world around us, we are declaring “Yes universe, I believe that you make sense.”

I’m going to talk a bit about what I call “Mathematical sense”. I see so many students in schools that simply don’t have it. They add together 2 numbers and get a number smaller than either. They count …97, 98, 99,100, 200, 300… For them, and the numbers have no meaning to them. They represent nothing, except a lot of wasted time, trying to reach some unknown goal along this dark unknown path that their teacher is trying to push them along. Mathematical sense is the art of knowing, of feeling, that when we are solving mathematical problems, we are not just shifting around numbers because the teacher told us to, but that something deeper is coming into play, something powerful, something that is fundamentally true and right.

Personally, I like working through lots of examples in Mathematics. The more I play with numbers, the more develop my mathematical model of the world – my knowledge of how numbers are supposed to work. I think too often we push students straight to the more difficult questions, the ones that will challenge them and “stretch” their knowledge or understanding, without allowing them proper time to get really comfortable with the material.

I think that also, too many of our public school teachers hated math in school and consequently hate to teach it. Our children are taught from a very young age that Math is boring, difficult, nerdy and completely detached from real life. Far too often I have heard statements such as “I’m sorry class, you’re not behaving, so we’ll just have to do Math sheets instead of our colouring”. An effective bribe, but one that places Math firmly in the undesirable category. We have developed a culture where Math is not respected as it should be.

I cannot think of one person who would stand out in public, give a small burst of apologetic laughter and say “Sorry, I’m just not any good at reading”. So then do we hear the same thing about Math every day? Why are people so proud of their mathematical inability? Why is it socially acceptable, desirable even, to declare a complete revulsion of numbers! Rise up Flatlanders! Rise out of obscurity and the shout to the world “Hurrah! I Love Math and my world is beautiful!”. My dear Flatlanders, lets go out and teach them.

Much love,

Emily the circle