Metacognition is ‘thinking about thinking itself’ and, more broadly, refers to the cognitive aspect of self-regulated learning. Through metacognition students learn to plan, monitor, evaluate & regulate their approach to learning and the way they are thinking about a given problem or particular learning activity. Before we delve into the various approaches to metacognition in maths, you might want to check out of free downloads section: there you can find a few useful teaching resources to get you started!
Metacognition has been shown to enhance educational attainment, for a long time metacognition has been seen as a particular boon to the mathematics teaching and learning community. For example, the Singapore Mathematics Curriculum includes metacognition as one of its core components. Until recently, there have been very few resources to help teachers bring metacognition into their lessons.
Metacognition is typically task-specific. Nonetheless some teachers may wish to explore more general reflections on ‘how to learn best’ and how to boost learning power with their students. Task specific metacognitive reflections can be guided and assisted by appropriately designed worksheets such as these.
Maths teachers may also wish to role-model metacognitive thinking in front of students when solving mathematical problems: over time students internalise the metacognitive style of thinking that a teacher can display for them.
Likewise, teachers of mathematics can use questioning to trigger metacognitive reflection in students. It is useful to base questions around the central project of planning, monitoring, evaluating and regulating the learning-process. A useful (and free) set of metacognition questioning prompts can be downloaded here.
Teachers may wish to evidence metacognition work whilst enhancing the design of student exercise books by including metacognitive/self-regulated learning tracking sheets such as these. Such tracking sheets encourage regular metacognitive reflection at the start and end of lessons: very simple! Teachers may wish to include information and/or ‘metacognitive extension tasks’ about metacognition and reflective learning at the front or back of exercise books as a way to educate their students about metacognition.
The Singapore Model Method for Learning Mathematics outlines a set of metacognitive skills, Heuristics for Problem Solving, that students can be prompted to try as a way to approach a particular mathematical problem in a new way:
· Act it out
· Use a diagram or model
· Make a systematic list
· Look for patterns
· Work backwards
· Use before/ after concept
· Use guess and check
· Make suppositions
· Restate the problem in another way
· Simplify part of the problem
· Solve part of the problem
· Thinking of a related problem
· Use equations
This might serve as a useful basis for a ‘prompt worksheet’ that can be at the front or back of students’ exercise books for them to refer to when they are struck.
Heuristics are an important concept in pedagogical theory around metacognition in maths; in the context of problem-solving, are a set of practical strategies to help students solve mathematical problems. Heuristics are ‘rules of thumb for making progress on difficult problems’. Polya’s heuristic breaks problem-solving down into four stages for students to focus on:
Stage 1: Understanding the problem
Stage 2: Devising a plan
Stage 3: Carrying out the plan
Stage 4: Looking back.
Students do not follow a linear path through these stages when working on a mathematical problem; instead, they tend to alternate between stages. For example, a failed attempt to carry out a plan (Stage 3) may lead students to devise a new plan (Stage 2) or prompt a realisation that they may not have fully understood the problem, which would lead them back to Stage 1. In any case: exploring heuristics can be an important aspect of a mathematics department’s metacognition work.
Mathematics teachers who wish to bring metacognition should consider exploring the more philosophical issues, debates and questions around mathematics: “What is a number?”, “Is Mathematics invented or discovered?”, “How do we know that mathematical facts are really true?” etc. This can be a fun and engaging way to guide students towards deeper reflections on both the significance of mathematics and the nuances of how mathematical understanding is acquired.
Use of ‘Dedicated Improvement & Reflection Time’ (DIRT) is the mark of a metacognitive maths teacher! Setting aside time so that students can reflect deeply on how to improve, perhaps with the assistance of appropriate prompts or ‘DIRT Worksheets’. It’s an easy way to foster metacognitive awareness in your students.
Aside from this: there’s nothing wrong with simply teaching students about metacognition and self-regulated learning directly. Why not? We made these informative ‘Knowledge Hunt’ lessons that might help you deliver the key details effectively: click here to download them!
A useful acronym Mathematics teachers may wish to use with their students is IMPROVE. It’s an instructional method for teaching mathematics that involves training students to ask metacognitive questions has been found to produce significant improvement in students' learning:
· Introduce new concepts
· Metacognitive questioning
· Practise
· Review
· Obtain mastery on lower and higher cognitive processes
· Verify
· Enrich
There are four kinds of metacognitive questions the students are taught to ask:
1. Comprehension questions (e.g., What is this problem all about?)
2. Connection questions (e.g., How is this problem different from/ similar to problems that have already been solved?)
3. Strategy questions (e.g., What strategies are appropriate for solving this problem and why?)
4. Reflection questions (e.g., does this make sense? why am I stuck?)
Metacognition boosts learning power, it creates reflecting and self-regulating learners, and in the long-run it can significantly increase achievement for students of maths – if you are looking for a new direction in your own professional developments as a Maths teacher or if you’re hoping to enhance your school’s Maths provisions with innovative new metacognitive strategies you should consider joining The Global Metacognition Institute today: that way you can download all our metacognition teaching resources through The Member’s Area: click here to join us now!
Download our NEW Metacognition in Maths Toolkit!
