The mathematicians concern themselves with doing mathematics at high level of abstraction. The mathematics educator concern themselves with what it is that one does when doing mathematics and what kind of experiences are propitious for a person’s later successes. – P. Thompson.
According to Thompson, the following quote is a paraphrase of what Winston Churchill said about mathematics and mathematics education.
There are mathematics educators who believe that mathematics and mathematics education are one and the same thing. Most mathematicians will hear none of this. But there are quite a number who now finds it part of their responsibility to educate as well.
As a mathematics educator, the most important part of teaching mathematics for me is to be able to provide learners with experiences to think mathematically and to develop their mathematical thinking habits even for those who would not be using hard-core mathematics in their lives. I believe that it should be one of every student’s inalienable rights – to learn to think mathematically. Sadly of course that most of the students, because of their traumatic experience with school mathematics, would rather go through life without math.
I have put together in this post some of the ideas behind the kind of mathematics teaching I promote. As I stated in the subheadings of this blog, the articles and lessons I write here are not about making mathematics easy because it isn’t but about making mathematics makes sense because it does. Before reading any of the articles below, I suggest read the About page first and what I think mathematics is. I hope I make sense in the following articles. Click here for the list of math lessons.
Among the ‘rules’ for working with negative numbers, the most counter intuitive is “negative times a negative is a positive”. It is easily forgotten especially if it was learned by rote. It is also not an easy ‘rule’ to make sense of so it needs to be learned with conceptual understanding. Here’s my proposed lesson for teaching multiplication of integers. This lesson takes from the lesson Subtracting integers using tables- Part 1 and Algebraic thinking and subtracting integers – Part 2. Note that this lesson like the rest of the lessons in this blog is not just about students learning the math but more about them engaging in mathematical thinking processes such as searching for patterns, making generalization, reasoning, making connections, etc.
Set the task
Fill up this table by multiplying the numbers in the first column to the number in the first row. Start filling up the rows or columns you think would be easier to do.
For discussion purposes divide the table into 4 quadrants. The top right quadrant is Quadrant 1, top left is Quadrant 2, bottom left is Quadrant 3, and bottom right is Quadrant 4. This is also one way of leading the students to consider filling-up the quadrants according to their number label.
Explore, Observe, Explain why
Students are more likely to fill-up Quadrant 1 because the numbers to be multiplied are both positive. The next quadrant they are more likely to fill-up is Quadrant 2 or 4. You may want to give the following questions to scaffold their thinking: What do you observe about the row of numbers in Quadrant 1? How can it help you fill up quadrant 2? Do the numbers you put in Quadrant 2 make sense? What does 3 x -2 mean? What about in Quadrant 4?
From Quadrant 2 students are more likely to fill up Quadrant 3 or Quadrant 4 by invoking the pattern. Questions for discussion: Do the numbers in Quadrant 4 make sense? What does -3 x 2 mean? This is one way of making the students be aware that commutativity holds in the set of integers. The problematic part are the numbers in Quadrant 3. None of the previous arguments are useful to justify why negative times negative is positive except by following the patterns. But this explanation will be enough for most students. You can also use the explanation below.
Revisit the rule when teaching another topic
We know that 8 x 8 = 64. This means that (10-2)(10-2)=64. By distributive property, (10-2)(10-2)= 100+-2(10) + – 2(10)+ ____ = 64. Previously students learned that -2(10)= 20. Hence, 100 + -40+___= 64. What should go in the blank must be 4. So (-2)(-2) = 4. This proof was first actually proposed by Maestro Dardi of Pisa in year 1334. In explaining this to students I suggest rewriting (10-2)(10-2) to (10+-2)(10+-2) to reinforce the distinction between the dash sign as minus and as symbol denoting ‘negative’.
Girolamo Cardano sometime in 1545 proposed a geometric interpretation of this operation. He argued that (10-2)(10-2) can be interpreted as cutting off 2 strips of 2 x 10 rectangles from the two sides of the 10 x 10 square. Cutting the rectangles like these meant cutting the 2 x 2 square twice so you need to return back the other square. The figure below shows this. This proof by Cardano is usually used to teach the identity square of a difference . This is a good opportunity to revisit the rule negative times a negative is positive.
Learning mathematics is not just about knowing, understanding, and applying its concepts, principles and all the associated mathematical procedures and algorithms. It’s not just even about acquiring the capacity to solve problem, to reason, and to communicate. It is about making these capacities part of students’ thinking habits. It is only then that one can be said to be mathematically literate.
The test for example that solving problem is no longer just a skill but has become part of students thinking habit is when students are doing it without the teachers still having to ask “Can you explain why you solve it that way?” or “Can you do it another way?” Those should be automatic to students.
“A habit is any activity that is so well established that it occurs without thought on the part of the individual.”
Here’s is a list of important mathematical habits of mind that I believe every teacher should aim for in any mathematics lesson.
Habit #1: Searching for Patterns
Students should develop the habit of
generating cases and generalizing patterns
looking-out for short-cuts that arise from patterns in calculations
investigating special cases, extreme cases from patterns observed
Habit #2: Reasoning
Students should develop the habit of
explaining the positions they take
providing mathematical evidence/justification for the conjectures or generalizations they make
testing conjectures by generating cases both special and extreme
justifying why a generalization will work for all cases or for some cases only
Habit #3: Solving and posing problems
Students should develop the habit of
always looking for alternative solutions to problems
extending problems and solutions to more general case
. This is a good opportunity to revisit the rule negative times a negative is positive.
