Fractions of an Amount: Intelligent, varied practice
One of the most significant (and I believe beneficial) changes to my teaching has been in my selection of the examples and practice questions I use with my students. I endeavor to make full use of the principles of Variation Theory, holding everything constant from one example to the next apart from one key element. This serves two key purposes. Firstly, it allows students to attribute any change in the answer to the change in the question, thus drawing their attention towards the underlying mathematical structure. Secondly, it allows students to form expectations about the next answer in a way that is simply not possible if the examples are not connected. When the expectations are realised, it is a confidence booster, and when they are not realised it is a key learning opportunity as students seek to understand why. I describe my process for using intelligent, varied practice and minimally different examples in Chapter 7 of my book, How I wish I'd taught maths.
Contents
Example: Fractions of an amount keyboard_arrow_up
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1. Example-Problem Pair
- I model the worked example first in silence
- I then narrate and annotate that example, explaining why I did each step
- I then show my students the problem I want them to try (Your turn)
- I circulate the class and choose a couple of interesting answers or approaches to show to the rest of the class using Show-Call
2. Intelligent Practice
- Once I am happy most students have understood the example-problem pair, I give out/project up the series of Intelligent Practice questions
- I remind students of the importance of pausing after each question, looking at the next question, noticing what has changed and what has stayed the same, and forming a prediction about what they expect the answer to the next question to be.
- Until I am confident students are carrying out this process, I may ask students to write down their prediction to make it explicit
- Once students have worked out an answer, they know to reflect on their prediction. Was it right, or was it wrong? If it was wrong, why? How is the answer to this question related to the answer to the previous question?
- If any students are still struggling with the example-problem pair, or the basics of carrying out this process, it is at this point I go to help them
3. Answers
- I project these up and ask students to mark their own
- I ask if there are any they don’t understand, or if there are any that they expected a certain answer but got a different answer
- We then have a class discussion about these
