Classroom Assessment Techniques Mathematical Thinking (Screen 1 of 5)

Malcolm Swan
Mathematics Education, University of Nottingham
Malcolm.Swan@nottingham.ac.uk

Jim Ridgway
School of Education, University of Durham
Jim.Ridgway@durham.ac.uk

"Most assessment practices seem to emphasise the reproduction of imitative, standardised techniques. I want something different for my students. I want them to become mathematicians - not rehearse and reproduce bits of mathematics....Assessment doesn't have to be an isolated, threatening business; it's not just answers, but approaches..." -Malcolm Swan

"Thinking mathematically is about developing habits of mind that are always there when you need them - not in a book you can look up later. For me, a big part of education is about helping students develop uncommon common sense. I want students to develop ways of thinking that cross boundaries - between courses, and between mathematics and daily life...People should be able to tackle new problems with some confidence..." -Jim Ridgway

WHY USE THE MATH CATs?
The Mathematical Thinking Classroom Assessment Techniques (Math CATs) are designed to promote and assess thinking skills in mathematics.

WHAT ARE THE MATH CATs?
The Mathematical Thinking Classroom Assessment Techniques (Math CATs) are designed to promote and assess thinking skills in mathematics. Few faculty have difficulty finding or developing tools which assess the specific mathematical techniques which they teach; a challenge which faculty do face is to find ways to promote and assess the development of mathematical thinking - notably to help students know what to do when faced with problems which are not identical to the technical exercises commonly encountered in mathematics classes.

Here we define mathematical thinking as

...the development of a mathematical point of view - valuing the process of mathematization and abstraction and having the predilection to apply them; and the development of competence with the tools of the trade, and using those tools in the service of the goal of understanding structure. (Schoenfeld, 1992)

The Math CATs are designed to address this challenge by offering ways to assess and instill a broad range of the mathematical thinking skills important for students in the science, mathematics, engineering, and technology disciplines. These skills include:

• checking results and correcting mistakes (Fault finding and fixing)
• making plausible estimates of quantities which are not known (Plausible estimation)
• modeling and defining new concepts (Creating measures)
• judging statements and creating proofs (Convincing and proving)
• organize unsorted data and draw conclusions (Reasoning from evidence)

WHAT IS INVOLVED?

Instructor Preparation Time:	Minimal if use existing tasks.
Preparing Your Students:	Students will need some coaching on their first task.
Class Time:	Some tasks take 5 minutes; others as much as 45 minutes.
Disciplines:	Varies with specific CAT.
Class Size:	Any.
Special Classroom/Technical Requirements:	None except for Reasoning from Evidence CAT.
Individual or Group Involvement:	Either.
Analyzing Results:	Varies with specific CAT.
Other Things to Consider:	Fairly demanding task for students who are unfamiliar with open-ended problems.

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Tell me more about this technique:

Introduction
Description of the 5 Math CATs
Goals, and Theory and Research
Links, Sources, and References
Malcolm Swan, Jim Ridgway, and MARS
View Entire 'Mathematical Thinking' Technique
Download 'Mathematical Thinking' Technique
Tools

Mathematical Thinking CATs || Fault Finding and Fixing || Plausible Estimation
Creating Measures || Convincing and Proving || Reasoning from Evidence