We explain steps in great detail to students but often omit the underlying concept. The topic of adding or subtracting fractions with unlike denominators is an example of this.
The example above right is a short cut for what is shown above left. These short cuts, which math teachers love to use, add to the student’s confusion because these rules require the student to use rote memorization which does is not readily retained in the brain.
I suggest using what I call a meaning making approach. I present the student 2 slices of pizza (images courtesy of Pizza Fractions Game) and explain the following setting. “You and I both paid for pizza and this (below) is what we have left. You can have the pizza slice on the left and I will have the pizza slice on the right. Is that OK?” The student intuitively understands that it is not because the slices are different sizes. I then explain that when we add fractions we are adding pizza slices so the slices need to be the same.
I then cut the half slice into fourths and explain that all the slices are the same size so we can now add them. Then the multiplying the top and bottom by 2 makes more sense.
Below is a model for information processing (retention and retrieval). Here are a couple key points I want to highlight:
A lot of information is filtered out so what gets through? Information that is interesting or relevant.
Information that is connected to prior knowledge, is relevant or that is organized has a better change of being stored effectively for retrieval.
Working memory has a limited capacity. Consider what happens to your computer when you have a lot of apps open. Your computer may start to buffer which is basically what happens to our kiddos if instruction involves opening too many apps in their brains.
Long term memory is basically retrieval of information. Think a student’s book bag with a ton of papers crammed in it. How well can he or she find homework? Compare this to a well maintained file cabinet that has a folder labeled homework with the homework assignment in question stored in this folder. That paper is much easier to retrieve. This is analogous to long-term memory. If the information is relevant or meaningful it will be stored in the file cabinet folder and more easily retrieved. In contrast, rote memorization like the rules teachers present students are papers crammed into an overflowing bookbag.
It involves numbers and not just words that we use on a regular basis to communicate.
It is a language all of its own so students have to learn the language as well as the concepts.
We teachers sometimes make learning the math more challenging.
I want to elaborate on this last reason. The photo above speaks to this. We present a topic. If the students struggle there is a tendency to “dumb down” the topic to rote memorization of a meaningless set of steps. Below is 1 of dozens of examples of memorization tricks we use. I call these Math Rulz.
Despite Common Core for State Standards, a multitude of initiatives, best practices, differentiation etc. math is challenging for most students. A major reason, from what I have encountered, is that math is often taught in piecemeal fashion with an attempt at rote memorization of an overabundance of rules. I compiled many of these rules into a single document.
Imagine our kiddos receiving special education services who already have trouble grasping various topics having to memorize all these rules and mnemonics.