Common Sense Strategies to Enable Mathematical Thinking

This 2020 interview with Craig Barton, author of Reflect, Expect, Check, Explain, covers Barton’s view of variation theory and how sequences of questions can help students think mathematically. The section below the original explains variation theory in plain terms and sets out what researchers have and have not shown about it.

In this interview with Craig Barton, author of Reflect, Expect, Check, Explain: Sequences and behavior to enable mathematical thinking in the classroom (John Catt Educational), find out about his interpretation of the variation theory. It’s an idea of mathematics that holds essential parts of concepts, ideas, and examples constant while changing only specific parts. Then, in the classroom, the teacher draws student attention to those changes, impacting the outcome.

A few years ago, Barton began reading all he could on variation theory, and started to write sequences of questions, launched a website, and started talking about the concept. Much to his fascination, he experienced an initial backlash as some teachers found his theories “too reliant on common sense” and unhappy with some of his interpretations. Instead of trying to convince educators that he was right, Barton concentrated on the clarity of the message and concept that eventually led to his recent book.

In this interview, Barton examines the motivation behind math learning and the surprising realization that teaching kids in a way that makes them feel successful can be the most motivating of all. While listening to the interview, think of teaching methods that you have found useful and then consider the following questions:

• What do you think of the variety of motivation tactics used in the classroom? Is there an over-emphasis on the knowledgeable and “highly able” students that naturally thrive in mathematics? 

• Are engaging activities as effective as they can be? Do you notice certain teaching methods that make students feel successful and thus drive learning outside their comfort zones?

• Are you instituting key student practicing or routines, in a way that enables them to think mathematically while building competency and fluency, without becoming dull and uninspiring? Instead of unrelated mathematical questions, are there practice methods used that string together a sequence of questions for a broader relational concept?

• How can we draw out the beauty of relationships inside mathematics so that every child has the opportunity to appreciate them? How can teachers help the mathematically reluctant become more inspired?

Learn more about Craig Barton’s book, Reflect, Expect, Check, Explain, and more great John Catt Educational publications, by visiting us.johncattbookshop.com

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Variation theory in maths teaching: what it is and what the evidence shows

Variation theory is a way of planning teaching that focuses on what changes and what stays the same across the examples students see. Craig Barton describes in the interview above how Barton came to use it, and how some teachers first found the ideas too reliant on common sense. This section explains the idea with a small example and looks at what the research literature says.

The idea in one example

Suppose a class is learning to solve equations such as 3x + 2 = 11. A teacher using variation might give a short run of questions in which only one thing changes at a time: 3x + 2 = 11, then 3x + 2 = 14, then 3x + 5 = 14, then 4x + 5 = 21. Students are asked what is the same and what is different between neighboring questions, and what that does to the answer. Because only one feature moves, the students’ attention is pulled toward it. A set of unrelated questions of the same difficulty would give the same amount of practice but would not draw attention to any single feature.

A May 2019 summary from Cambridge Mathematics describes the approach as a way of analysing and planning learning activities that focuses on what changes and what stays the same, and notes that researchers recommend students see difference before sameness, which includes counter-examples and non-examples. This corresponds to what Barton describes as sequences of questions that build toward a broader relational concept.

What the research literature says

The sources reviewed for this section describe a theory with a good deal of classroom use and a smaller body of controlled evidence. The Cambridge Mathematics summary draws several cautions.

• Researchers disagree on exactly what to vary and what to keep fixed, and the use of structured variation differs considerably between countries and between textbooks.
• Well-designed tasks are not enough on their own. The teacher must also direct students’ attention to the feature that is meant to be noticed.
• Teachers often describe variation as something they already do, and the summary says the theory may simply describe formally what they do already. It also notes that studies have found teachers are often unaware of differences in quality among the examples they choose, or choose them at random.
• Professional development that builds teachers’ skill in choosing examples and problems may be particularly beneficial.

The National Centre for Excellence in Teaching Mathematics (NCETM) in England, on its research and evidence page, calls variation theory the component thought to be key to the success of the East Asian system, and says varying practice questions in this way is also known as intelligent practice, which aims to avoid mechanical repetition and promote connections between ideas. Its reading guide points to a 2017 paper by Kullberg, Runesson Kempe and Marton, which it describes as illustrating the principles with one teacher’s practice before and after lesson studies. That is a descriptive case and not outcome data. The NCETM also states, on a page it says was originally published in August 2020, that teaching for mastery is in the early stages of adoption in England and that it will be monitoring research and evidence of its effectiveness and adding to the page.

The fair summary is that variation theory has a coherent explanation and wide use, that case studies show how it can be taught, and that the sources opened for this section do not report a large trial isolating its effect on student results. The criticism Barton mentions, that the ideas lean on common sense, is therefore partly right: they are intuitive, and the contribution of the theory is to make the choice of examples deliberate.

Using it in a department

1. Pick one concept and write the sequence together. A department meeting is a natural place to agree which feature each pair of questions should highlight.
2. Plan the question you will ask students. The prompt “what changed and what did it do?” is the teaching move; the sequence only prepares the ground.
3. Include a counter-example. A question that looks the same but behaves differently shows students where a rule stops working.
4. Watch what students say. The Cambridge summary points out that planning includes anticipating how students will react, so a short note of likely answers belongs in the plan.
5. Compare results over a unit. Because the evidence is limited, a department that adopts the approach can compare its own quiz results or student explanations before and after.

Barton’s reflective questions at the end of the original, about motivation, engaging activities and practice routines that do not become dull, fit with this approach. A sequence of related questions gives practice, as Barton asks, in a form that builds a connected idea. Whether the approach suits a given class is for teachers to judge by what their own students show.

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