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Students don’t arrive as blank slates.
They have pre-existing mental frameworks and models.
It is very easy to make minor adjustments to a model, so it is relatively easy for good students to extend their knowledge and understanding.
It is very difficult to replace an incorrect model with a correct one (regardless of whether or not the student wants to). This is why misconceptions (such as errors in basic algebra) can be so hard to fix.
To fix misconceptions:
- Doing lots of example problems doesn’t work. (This just entrenches either the wrong model, or the idea that the student can’t do maths.)
- Telling someone what they should think doesn’t work. (Consider how few debates on politics or religion actually change someone’s deeply held beliefs, regardless of the quality of the arguments.)
- Not only do these two approaches not work to correct wrong mental models, but they actually cause damage. If a student has a mental model for something, then they believe they understand it. If they get all the questions wrong, or get told they are doing it wrong, they can begin to doubt their mathematical reasoning skills in general, leading to maths anxiety and resulting in the student hating, fearing and being frustrated by maths.
It seems that really the only way to effect a model change is to:
- get the person to predict something with their model …
- and discover for themself that it is completely wrong, …
- then show them the correct model, and allow them to demonstrate that it solves the problem.