I am currently a PhD student in mathematics and I have done a lot of tutoring/teaching math over my college career. One of the biggest issues that I’ve noticed in my classmates and students that struggle with math is this: They view math as memorizing a bunch of formulas and applying them. This is often how they were taught, and it’s horrible! To show the difference, let me give an example of the two ways that a simple concept can be taught:
The teacher states area formulas for the following: equilateral triangles, squares, rectangles, right triangles, regular hexagons. These are all pretty basic shapes, but their area formulas are decently different at first glance.
The teacher shows visually why a rectangle’s area is equal to its width times its height, by showing a diagram like this
The students can clearly see that there are 15 squares, which is the area. By doing a couple more examples, students will realize that the area can be found by multiplying the side lengths.
From here, the teacher can have students come up with the formula for a square based on the rectangle area formula, since squares are just special rectangles.
At this point, the students are also capable of finding the area formula for right triangles; by noticing that two of the same right triangle put together make a rectangle. Thus the area of the triangle is one half the area of the rectangle. That is, Area(right triangle)=(1/2)base*height.
Next, let’s memorize one formula (the teacher could go more into depth explaining how to obtain this, but for now it may be easier to memorize).
The area of an equilateral triangle (a triangle with all side lengths the same) is where is the side length.
From here, the students can construct the area formula for regular hexagons (a 6 sided shape with every side the same length and angle the same), as a regular hexagon with side length is just 6 equilateral triangles put together (each with side length ).
Thus the area of the regular hexagon is six times the area of each equilateral triangle, i.e. , or when reduced.
The difference between these lessons is that, in the first lesson, students are given 5 seemingly different equations to memorize. If they forget one of the equations, they will have NO way of doing related problems! This can be extremely frustrating, especially for younger students. Students who are taught like this get bored with math because there doesn’t seem to be any point to what they’re learning- each fact seems isolated and uninteresting.
Students who learn from lessons like the second one, however, are learning to THINK not memorize. Students in the second lesson only have to memorize one formula! Any of the other formulas, they can recreate themselves during a test or on homework. They will also be able to come up with formulas for other shapes, such as octagons or parallelograms. Students in lessons like these will build connections between the seemingly different branches of math and learn to see some of the cool intricacies within math.
Of course, it also helps if teachers connect math to the real world; simply learning the area formulas for different shapes may seem pointless to some students. But if the teacher explains how this could be used, it might stick in a kid’s head better! For example, students could consider what the perimeter (side lengths) of different shapes were with the same area (ie, if a square and a circle have the same area, what are the perimeters of each?). The teacher could then explain how this comes up in nature sometimes; for example, bubbles take on a sphere shape because it minimizes the surface area (analogous to perimeter) per volume (analogous to area). The teacher could bring in bubbles and a variety of tools (like hangers folded into different shapes) to show that, no matter what shape the bubble wand is, the bubbles always snap back into a sphere. This relates to what they’re learning in class because they’ll see that circles (the 2d versions of spheres) also minimize perimeters.