Categories

## Plan for the Unexpected!

Whatever you teach (or learn), you should look for the links, the more unexpected, the better.  This post is the story of one of them, simple but lovely and strong.

I have a class with a group of adult students who never had a positive experience with mathematics (or as they would say, “they hate mathematics”), have very limited knowledge of mathematics, and yet, likely to work in nurseries and primary schools where they have to teach mathematics somehow or other. My aim is to help them not to hate mathematics (if not like it) and have some positive experience of doing mathematics. They refer to my aim as “great expectation” 🙂 But, last week something  changed in them; something that made me so excited that the whole class burst into laughter.

Previously in the class we had worked on the idea of subitizing and the ways we might help children to do so conceptually. We had learned how the “shape” or “structure” of a group of objects could help us (children) to realize “how many they are without counting”  (that is the essence of subitizing). For example, children should learn to see the number of dots in the following figure is five without counting.

Also, We had played with Mathlink Cubes to experience different shapes for numbers, and to discover similarities and differences between those shapes. In particular, we learned numbers 3, 6, and 10 can be represented by triangular-looking figures like the following.

So far, it was just the story of what the class knew before facing with this question:

In how many ways can we represent number two by using the fingers of one hand?

Here are two of them:

Using just one hand, there was no need to be that much systematics; somehow or the other we could find the answer. However, the problem became more difficult when we were allowed to use the fingers of both hands. Here, the unexpected link of the story appeared. It is amazing and worthy of being discovered if it is the first time that you are counting the twos of your fingers. Please try it before continuing reading.

Categories

## Plan for the Mess!

Plan for the mess” is one of my favourite teaching ideas. The idea is to bring students to a point where after a heavy messy work (most of the time calculations and symbol pushing) they say “why I didn’t see that earlier!” Here is how recently I used the idea in my linear algebra class.

The students had just solved a couple of two equations with two unknowns by whatever tools they had from highschool. Many of them had no idea about any geometric interpretation of such equations (i.e., solution, if there is one, is at the meeting point of two lines), and of course, none knew the vector interpretation in which, say  ​$$\begin {cases} 2x + 3y = 4 \\x + 2y = 3\end{cases}$$​, can be thought as ​$$x\begin {bmatrix}2\\1\end{bmatrix}+y\begin {bmatrix}3\\2\end{bmatrix}=\begin {bmatrix}4\\3\end{bmatrix}$$​. Of course, one should be too naive to assume that students immediately appreciate the importance of having several  interpretations for the same thing. Here is where you plan for the mess.  Thus, I asked them to solve a system of three equations with three unknowns such as ​$$\begin {cases} 2x + 3y+4z = 4 \\x + 2y-z = -1\\3x-y+2z=2\end{cases}$$​. It was “fun” watching students messing around with the equations, substituting this into that, sometimes eliminating this unknown, the other time eliminating that unknown until they got the answer: ​$$x=0, y=0, z=1$$​.

I am sure you can imagine how their faces look like when they realize that they could get the answer just by “looking” at  the equation as

$x\begin {bmatrix}2\\1\\3\end{bmatrix}+y\begin {bmatrix}3\\2\\-1\end{bmatrix}+z\begin {bmatrix}4\\-1\\2\end{bmatrix}=\begin {bmatrix}4\\-1\\2\end{bmatrix}$.

Learning from a hard way is unforgettable!

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## Proof-Generated Definitions!

Okay, for a long time, I didn’t know what to blog about. Now, I have decided to write about my teaching ideas that take ages to turn into a piece of research. That is why I have started with this strange title for my first blog. The title comes from the exact same phrase coined by Lakatos in his book Proofs and Refutations.  Here is how I used the idea recently.

Last week, I was going to teach the definition of a convergent sequence. Previously, we had played a lot with sequences geometrically: how they are represented on a number line and how they are graphed as a function on the coordinate plane. We hadn’t done any calculations; just we drew a lot of figures to see how different sequences (i.e., convergent, divergent, bounded, increasing, decreasing) behave graphically.  The question I (as the lecturer) had to answer was which “theorem” could justify and motivate the definition of a convergent sequence. Indeed, at that stage of students’ knowledge, there was perhaps only one choice; but, surprisingly that choice seemed to be a perfect fit:  if a sequence converges, then its limit is unique. There was only one small problem, students have no reason to think otherwise: how on earth a convergent sequence could have two limits? Thus, I needed to raise the question and also play devil’s advocate. Here is how the plan goes in the class.

I: How do we know that this sequence has only one limit (I drew the following figure)?

Students: What? Obviously, it has only one; see, it is getting closer and closer to the red line.

I: But, what if there is another line very close to the red line that the sequence is getting close to it too (I didn’t draw that other line on the graph).

Students: It (the sequence) is going to pass that line; see (one of the students drew the following (blue) line):

I: What if the line I thought of is closer to the red line.

Students: After a few other terms, it is going to pass that line as well.

Here, suddenly one of the students said: “What if the sequence also passes the red line?” As a result, I changed my role from someone who was “against” uniqueness to someone who was defending it.

I: But, it can’t go that far from the red line, can it?

Then, I cheated and I told the rest of the story! After all, it was the first time that I tried to generate the definition of convergnce of a sequence via a theorem in the class. I more motivated the definition and less generated it, though, we were very close to hit the target and generated a number of key ideas that were used in the definition. Next time that I teach sequences, I’ll try to go all the way and report it in another post.