by Michelle Lefolii
We often hear two questions about the way that we teach math:
- why is it important for students to show their work?
- why are students sometimes given test questions requiring them to apply their skills to unfamiliar situations?
As you know, at Abelard we have our own philosophy of education. We have carefully developed our own pedagogical techniques over the years; these are based on what we have learned works best to prepare our students for the demands of university and life beyond.
What we are attempting to do is to train our students to think, and to be able to apply the knowledge and skills they acquire in creative and productive ways.
To know something is not necessarily evidence that you have thought about it, or even that you understand it. We can know things through memorization, and through intuition or instinct, and these are valid and important first steps. However, when we have come to terms with understanding the underlying structure of a concept, this knowledge becomes deeper, more useful and more reliable.
If we both know something and understand it, this helps us to communicate our knowledge to others, and to apply it in many different contexts. And these skills are invaluable both within the classroom and in the world at large.
It is also important for our students to be prepared for university, and how better can we prepare them than by training them to perform the same types of tasks in the same fashion as the universities will require? This is the aim of our curriculum in general, and of course is true of our mathematics courses.
Allow me to reinforce what I’ve just said by quoting from two texts written by university professors as guides to their first year students: the advice they give is equally pertinent to our own.
The following passages are extracted from “A Guide to Writing Mathematics”1 by Dr. Kevin P. Lee, which is given to first year students at UC Davis.
“The Greek word mathemas, from which we derive the word mathematics, embodies the notions of knowledge, cognition, understanding and perception. In the end, mathematics is about ideas. In math classes at the university level, the ideas and concepts encountered are more complex and sophisticated… (and) will include concepts which cannot be expressed using just equations and formulas.”
“If a mathematician wants to contribute to the greater body of mathematical knowledge, she must be able to communicate her ideas in a way which is comprehensible to others… When you use your mathematical knowledge in the future, you may be required to explain your thinking process to another person (like your boss, a co-worker, or an elected official), and it will be quite likely that this other person will know less math than you do. Learning how to communicate mathematical ideas clearly can help you advance in your career.”
“Putting an idea on paper requires careful thought and attention. Hence, mathematics which is written clearly and carefully is more likely to be correct. The process of writing will help you learn and retain the concepts which you will be exploring in your math class.”
“As you learn more math, being able to express mathematical ideas will become more important. It will no longer be sufficient just to be able to write down some final “answer”. There is a good reason why Herman Melville wrote Moby Dick as a novel and not as the single sentence:
The whale wins.
For this same reason, just writing down your final conclusions in an assignment will not be enough for a university math class… You will not be writing papers to demonstrate that you have done your homework. Rather, you will be writing to demonstrate how well you understand mathematical ideas and concepts. A list of calculations without any context or explanation demonstrates that you’ve spent some time doing computations; however, a list of calculations without any explanations omits ideas. The ideas are the mathematics. So a page without any writing or explanation contains no math.”
In a document titled “Math Survival Skills for First Year Students”2 compiled and edited by Geanina Tudose for the University of Toronto at Scarborough Math and Stats Help Centre, we find the following advice regarding problem solving for homework and tests:
“The higher the math class, the more types of problems… increasingly, you will tackle problems which require several steps to solve them. Break these problems down into smaller pieces and solve each piece.
- Problems testing memorization (drill)
- Problems testing skills (drill)
- Problems requiring application of skills to familiar situations (“template” problems)
- Problems requiring application of skills to unfamiliar situations (you develop a strategy for a new problem type)
- Problems requiring that you extend the skills or theory you know before applying them to an unfamiliar situation.
When you work problems on homework, write out complete solutions, as if you were taking a test. Don’t just scratch out a few lines and check the answer in the back of the book. If your answer is not right, rework the problem; don’t just do some mental gymnastics to convince yourself that you could get the correct answer. If you can’t get the answer, get help.
The practice you get doing homework and reviewing will make test problems easier to tackle.”
At Abelard, we do ask our students to work hard. We do expect them to get high marks. But these are not the end goals. We will consider our job well done only if we have encouraged our students to learn, to think, and to apply the fruits of their learning and thinking to a greater context.
Lee, Kevin P. “A Guide to Writing Mathematics.” Nina Amenta Homepage, http://web.cs.ucdavis.edu/~amenta/w10/writingman.pdf.
Tudose, Geanina. “Math Survival Skills for First Year Students.” Teaching and Learning Services, Math & Stats Help Centre, University of Scarborough, 2004, https://www.utsc.utoronto.ca/mslc/sites/utsc.utoronto.ca.mslc/files/resource-files/survival_guide.doc.