Teaching Theory

Changing the order of math application

When a student solves a math problem for class there is generally some amount prior knowledge application, and an application of the new knowledge the student is learning. What if they also did it in that order?

One way to learn and internalize a new method in mathematics is to see and work through an example that uses that new method. But even a simple $2+2=?$ requires prior knowledge application of number recognition and interpretation. So most mathematics problems start with applying the new method, then using old knowledge to complete and clean up the answer.

Take solving a problem applying logarithmic algebra, like:

The first step might be to apply the log properties and get:
then change the logarithmic form into an exponential form:
$$2x +6=8$$
From this point we are solving a simple algebraic equation (to get x=-1). Hopefully the student knows this from a previous course.

This method of solving requires applying the new concept first to the problem, then once the problem becomes more recognizable using previously learned methods. One potential problem with this order is that “once you have a hammer, everything looks like a nail”. This leaves students to always apply the newly learned method to every problem, and not properly adding it to their mathematics toolbox.

Of course one solution would be to ensure that students are solving a mix of problems covering the new and old material. This takes time, and space that the student may not have. One alternative would be to utilize the approach shown in our $2+2=4$ problem.

In the $2+2=?$ problem, the student must first recognize the number symbols, then apply the new method of addition. To go back to our logarithm example, a student could solve a problem like:
$$e^{x+1} =e^x +1$$
In this problem one approach is to first rearrange the problem algebraically to:
$$e^x e^1=e^x +1$$
Then divide through by $e^x$ to get:
$$e^1 = 1+\frac{1}{e^x}$$
Which with a little more algebra yields:
$$e^x = \frac{1}{e-1}$$
Now we can apply our logarithm properties (although not quite as advanced as the first problem), to finally solve the problem,

Instead of applying our newly learned logarithmic properties, then applying the old algebraic knowledge, the algebra comes first and the logarithm application only comes at the end. Logarithms are still applied and demonstrated in the problem, but the student has to apply another tool first.

Taking this approach in more practice and testing may help to break students out of simply applying the most recently learnt method to every problem on the test. They can instead appreciate all of the tools they already know, and learn when and where to apply them.