The ManicaPost
Friday Lessons with Uncle Jay
Many Maths lessons suffer from the following issues:
a. The teachers don’t know why they are teaching particular maths topics, and they often don’t know what else the students are learning in other subjects.
b. As a result, the students don’t know why they learn those maths topics, either. The common question, “Why do we have to learn this?”, is a reasonable one. Do you have a good answer, beyond “It’s in the exam” or worse, “Because it’s good for you”?
Some possible ways to fix this:
Find out where the students will use each math topic you teach (it may be in their science class, or some engineering subject). It’s great when you can use actual examples from those other subjects and let the students know that’s where they’ll use each math topic.
Help students make connections between the math topic and the “real world”. If you’re not sure how it’s used in the real world do a search!
Start with concrete examples — leave the abstract concepts to later
Maths is largely about abstraction. Mathematicians for centuries have thought about real problems and come up with practical ways to solve those problems. Later, they have generalised the process, usually presenting the solution using algebraic formulas.
When students have no idea what the original practical problems actually mean, how can they be expected to understand the abstractions of those problems (using the formulas)?
When students have no idea what the original practical problems actually mean, how can they be expected to understand the abstractions of those problems (using the formulas)?
Instead of starting each topic with a formula, start with concrete examples of the problems that were originally solved using that maths. Then, help the students see how the math theory can help to solve such problems by showing them the thinking behind the solution.
Another thing to consider is these days students spend less time playing outdoors than they used to, so they have less experience to draw on when it comes to concepts like velocity, acceleration, gears on a bike, and generally how things work.
So when we say “Imagine . . ”, students often find it hard to imagine what you are talking about due to lack of concrete experience with that concept.
Start with an interesting, real-world problem (preferably localised)
Most Maths teachers start with “Here’s the new formula for today, here’s how you plug in values, here’s the correct answer.”
Problem is, there’s no attempt to motivate the learners.
It is good to pique curiosity with a photograph, a short video, a diagram, a joke, or perhaps a graph. This trigger should outline an interesting problem in your local area (so students can relate to it better and feel more ownership).
Where you can, use computers to do the drudge work
Many Maths courses seem to be more about calculation rather than concepts.
These days, it doesn’t make sense for humans to spend hours learning how to calculate using complicated algebra.
To quote John Allen Paulos:
Mathematics is no more computation than typing is literature.
And then this one, usually attributed to Albert Einstein:
Computers are incredibly fast, accurate, and stupid.
Human beings are incredibly slow, inaccurate, and brilliant.
Together they are powerful beyond imagination.
For the vast majority of your students (who will not eventually become mathematicians), it’s more important they understand the concepts and which process to use when confronted with different real problems.
They then should learn how to use computer algebra systems (or graphics calculators) to solve such problems.
Some examples:
a. I observed a lesson recently where the math teacher got his students to calculate the standard deviation of a set of scores. He started with the formula then allowed the students to use Excel.
He wanted them to apply the formula (rather than let Excel do it directly using STDEV, which is what many students actually did).
The problem was, the students didn’t even know what “standard deviation” meant. They were just plugging in numbers with little idea of what it meant.
Finally someone remembered the standard normal curve and could explain what they were all doing
The teacher should have made sure the students had the general idea first, including the general meaning of “distance from the mean”.
a. When finding the length of a curve using calculus, it’s actually not possible to find the answer using ordinary integration (because the integral does not exist).
But using a computer algebra system, we can easily find the answer for such problems and spend more time understanding the problem and the solution given.
Creativity and ownership
Many Maths students feel very little ownership for what’s going on.
They have little say in what the topics are (that’s usual for most formal education) and the exact same assignments are given to everyone.
It’s not surprising there is little enthusiasm for such “one size fits all” approaches.
There are many ways we can encourage creativity in math.
Technology is one avenue — get students to use creative means to describe a mathematical concept (it could be a video, an animation, a diagram or perhaps a concept map).
Such individualised assignments get them thinking about the bigger picture, encourages creativity, and is more likely to generate feelings of ownership than the normal mass-produced assignment.
