Math for Teaching

Math Knowledge for Teaching

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The mathematics that engineers, accountants, etc and teachers of mathematics know are different. They should be. There are some engineers, accountants, chemists, etc who become very good mathematics teachers but I’m sure it is not because they have ‘math knowledge for engineering’ for example but because they were able to convert that knowledge to ‘math knowledge for teaching’.

What is math knowledge for teaching?

It includes knowledge of mathematics but on top of that according to Salman Usiskin, it should also include knowledge of:

ways of explaining and representing ideas new to students;
alternate definition of math concepts as well as the consequences of each of these definitions;
wide range of application of mathematical ideas being taught;
alternate ways of approaching problems with and without calculator and computer technology;
extensions and generalizations of problems and proofs;
how ideas studied in school relate to ideas students may encounter in later mathematics study; and,
responses to questions that learners have about what they are learning.

I don’t know why some people especially politicians think teaching is easy. Surely college preparation is not enough to learn all these. You certainly need to be a practicing teacher to even start knowing #1 and #7. Teachers need more support in acquiring these knowledge when they are already in the field than when they are still in training.

I started this blog to contribute towards helping teachers to acquire the seven listed by Mr. Usiskin. After 250 posts, it looks like I have not even scratched the surface 🙂

More posts: teaching mathematics and levels of teaching mathematics

Algebra

A Challenging Way of Presenting Math Patterns Problems

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Mathematics is the science of patterns. Part of the math skill students need to learn in mathematics is to see regularities. The usual way of introducing pattern searching activity is by showing a sequence of figures or numbers and then asking the students to find or draw the next one. Here’s a more exciting way of presenting math patterns problems. Bernard Murphy of MEI (Innovators of Mathematics Education) shared this with us. MEI is an independent charity committed to improving mathematics education and is also an independent UK curriculum development body.

The figure below is the third in a sequence of pattern.

How does the first, second, fourth, fifth figure look like?

Here are three of the patterns I produced and the questions you could ask the learners after they produced the sequence of patterns. Note that the task is open-ended. There are other patterns learners can make.

1. How many unit squares will there be in Figure 50?

2. How will you count the number of unit squares in Figure n in this pattern?

3. This is my favorite pattern. How many unit squares will there be in Figure n?

Note that in all the sequences, Figure 3 looks the same. Note also that for each of these sequences, you can have several expressions depending on how you will count the squares. Of course the different algebraic expressions for a particular sequence will simplify to the same expressions.

You can use this activity to teach sequences, linear function, and quadratic function. But this is not just what makes this activity a mathematical one. To be able to see regularity is already a mathematical skill and much more of course if they can generalize them as well in algebraic form.

I am so tempted to just give you the equations but that would mean depriving you of the fun. Anyway, here are two examples on how you can think about counting the squares in Figure n: Counting Hexagons and Counting Smileys. Have fun.

Quotes

Churchill on mathematics and mathematics education

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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.

$Winston Churchill on mathematics education$

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.

Algebra

Making Sense of Power Function

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The power function, $ax^n$ , n = positive integral exponents is actually the ‘basic’ polynomial function.They are the first terms in the polynomial function.

With graphing utility, it is no longer as much fun to graph function. What has become more challenging is interpreting them. Here’s are a set of tasks you can ask your learners as review for function. You can give it as homework as well.

Consider the sets of power function in the diagrams below. Answer the following based on the diagram

What are the coordinates of the points of intersection?
Why do all the graphs intersect at those points?
When is $x^4 < x^2$ ?
When is $x^7 > x^3$ ?
Why is it that as the degree or exponent of x that defines the function increases, the graph becomes flatter for the interval -1<x<1 and steeper for x > 1 or x >-1 ?
Sketch the following in the graphs below: $t(x) = x^{10}$ , $l(x)=x^9$
Why is it that power function with even exponents are in Quadrants I and II while power function with odd exponents are in Quadrants I and III? Why are they not in Quadrant IV?