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Doing maths without good mental models
To experience what it is like to do maths without good mental models, try the following experiments.
- Make $1.70 using only 10c, 20c and 50c coins. (Assume you have unlimited supplies of each type of coin.)
This is very easy because you have a mental model in your head for how money works. It is associated with your model for place value, and your model for addition, and your model for the number line (to measure how close you are to $1.70).
- Now make $1.58 using only 7c, 12c and 20c coins.
This is difficult, because you don’t have a model in your head that lets you get to the value.
To solve this second problem, you probably either painstakingly worked out a pattern that gets you to the value you need, or you used trial and error.
If you don’t have a pattern or model, then you are forced to use trial-and-error on each and every case, and then memorize the combinations to get to each value.
This is what it is like to do maths without an operational mental model. It is what many students are faced with when they do maths, and it makes doing maths prohibitively difficult.