Below are links to the other posts in this series. Scroll past them to read the article.
- Atomisation
- Overtisation
- Expansion and Context Shaping
- Cognitive load
- Review
- Lesson structure and schemes of work
- Speed principle
- Difficulty and Motivation
- Defining range/scope
- Categories for different types of knowledge
- Instruction for basic types of knowledge
- Instruction for linked types of knowledge
- Instruction for routines
- Instruction for problem solving techniques
- "Real world" maths
- Prompts and scaffolding
- Correcting mistakes
- My take on the strengths and weaknesses of Direct Instruction
Expansion:
Expansion takes place after initial teaching, where things have been simplified to the only the essential features (see cognitive load). Expansion then introduces more variety, showing how questions can be worded differently and still mean the same operation
One type of expansion activity would be 'implied conclusion' tasks. These are tasks that you use the current skill (possibly alongside other skills which are already firm) to solve new problems. An example of this is ordering fractions after being taught how to compare fractions.
Matching ("circle all the fractions greater than two thirds") and "here is the answer, make up the question") are other types of expansion activities.
One type of expansion activity would be 'implied conclusion' tasks. These are tasks that you use the current skill (possibly alongside other skills which are already firm) to solve new problems. An example of this is ordering fractions after being taught how to compare fractions.
Matching ("circle all the fractions greater than two thirds") and "here is the answer, make up the question") are other types of expansion activities.
Context Shaping
Context shaping is the name given in Direct Instruction for demonstrating the range of contexts where a skill can be used.
This may be covered in a single activity or multiple activities, depending on the number of different possible contexts and the amount of difficulty added with each new context.
DI recommends juxtaposing highly different contexts one after the other (before interleaved practice). This maximises the chances that students will interpolate and infer any contexts in between those shown.
DI recommends juxtaposing highly different contexts one after the other (before interleaved practice). This maximises the chances that students will interpolate and infer any contexts in between those shown.
Both expansion and context shaping activities should happen as soon as possible after students are firm on the initial teaching. This is so that students do not stipulate by viewing the skill in too narrow a way.
Example:
These questions from https://www.map.mathshell.org show a variety of contexts for using direct proportional reasoning:
*Note, there is also some question discrimination here as students need to decide if the numbers are in direct proportion or not.
I would likely split this in to several activities in order to reduce the difficulty of individual sessions. In order to juxtapose highly different contexts I might show them in this order (though other orders would also work):
- Speed/cycling and driving
- Lines
- Similar shapes/triangles
- Maps
- Money and Petrol
Reflections
Textbooks generally do this in their exercises. However, I find the structure that DI gives to these activities helpful for phasing in the wider activities. So for a high ability set, I may just give them the textbook and let them get on with it and, for a low ability set, I can expand their understanding in small stages over several days. This allows students of all abilities to feel successful and to deepen their understanding of a topic, something I definitely struggled to get right in the past!
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