Recently I have been reading and learning about exploratory practice, thanks to a very interesting talk and a few articles.

**Movement in Maths and Exploratory Practice**

Last week I went to a talk at the Singapore National Institute of Education. Dragan Trninic was talking about how maths can be learned through bodily movements. He has done some work on proportional reasoning in which students raise their two arms to different heights above the desk while looking at a coloured screen. The screen turns green if the students raise their hands in a certain proportion, for example, if the right is double the height above the table as the left. (Read more here [pdf].)

Trninic was explaining that he designed his study to see if the physical experience would help students understand proportions better. (It’s such a hard thing to teach and understand.) He wondered if practicing in this way – he called it exploratory practice – would prove valuable. And it did.

By responding to questioning from the researchers, the students were able to voice their findings. This helped them develop an understanding about proportions.

Trninic linked this kind of exploratory practice to the way people learn dance or martial arts. In those disciplines, students learn through a collection of sequenced movements, making improvements as they go. Trninic was careful to distinguish this from the kind of repetitive practice of a single skill that we sometimes use in maths classrooms; that kind of practice doesn’t include any variation and hence students learn efficiency and speed for the atomic skill. Exploratory practice, on the other hand, is set up by the teacher in a way that students are asked to learn as they go by trying to generalise.

**Shanghai Maths and Procedural Variation**

This reminded me of some reading I have been doing about procedural variation. In the UK, there has recently been a two-year-long teacher exchange with Shanghai. Sue Pope and Mike Ollerton wrote in Mathematics Teaching 250 (ATM members’ access required) about their experiences with secondary maths teachers from Shanghai in the UK. They recount that they were asked to read an article in advance: “Teaching with Procedural Variation: A Chinese Way of Promoting Deep Understanding of Mathematics” by Lai and Murray (pdf, free).

The article by Lai and Murray quotes international maths comparisons that show that Chinese learners have a very secure understanding of the mathematics they have learned, and that they can apply it. Yet some Western onlookers say that mathematics education in China is characterised by rote learning or passive transmission. Without an explicit focus on conceptual understanding, why can Chinese students understand and apply maths so well? Lai and Murray aim to illuminate the “paradox of the Chinese learner” by describing the teaching as procedural variation.

**Carefully Constructed Exercises**

When teachers in China use procedural variation, they set up a carefully constructed set of examples or experiences for students, who are then expected to attend to essential features and notice connections. The tasks are chosen help learners to create generalisations, and are sequenced to help this occur.

The article includes an example about the teaching of division involving decimal numbers. This series addresses the misconception that “division makes smaller” and invites students to form a new conclusion about division.

Problem 1: There are 9L of apple juice and every 3L is put in a jar. How many jars are needed?

Problem 2: There are 9L of apple juice and every 1L is put in a jar. How many jars are needed?

Problem 3: There are 9L of apple juice and every 0.3L is put in a jar. How many jars are needed?

Problem 4: There are 9L of apple juice and every 0.1L is put in a jar. How many jars are needed?

Problem 5: There are 9L of apple juice and every 0.05L is put in a jar. How many jars are needed?

In this series of tasks, the total amount of apple juice was kept constant while the amount in a jar was varied from a whole litre to less than a litre. This exercise might be considered rote drilling if computing for a correct answer is the focus. However, an experienced mathematics teacher will organise this series of tasks hierarchically and provide scaffolding to illustrate and generalize… mathematical ideas.

This type of procedural variation involves varying the problem. The variation is created by changing a constraint or feature of the the problem while other parts remain the same. (This reminds me of the what-if-not technique that I first read about in the book *Adapting and Extending Secondary Mathematics Activities* by Pat Perks and Stephanie Prestage. It’s also something Watson and Mason talk about in an article called “Seeing an Exercise as a Single Mathematical Object: Using Variation to Structure Sense-Making“.) There are two other types of procedural variation: examining multiple methods of solving problems and using a single mathematical method in varied applications. Lai and Murray use examples to describe these also: first, simultaneous equations can be solved using multiple methods, which are then compared and contrasted; secondly, a combinatorial method can be applied to various problems including handshaking, crossroads, and diagonals.

This type of practice is what what I would describe as exploratory practice: the questions are designed to lead students to a greater understanding of an underlying idea or structure.

**How Does This Apply in My Classroom?**

A good question, and one which I have not fully answered yet. I am still reading about exploratory practice, thanks to the Watson & Mason article linked two paragraphs above.

First, teaching through movement. I have been encouraged by a recent Jo Boaler article to use movement and gestures more. For example, when talking about transformations of shapes, we can use our hands to show reflection from palms up to palms down. (I wonder if this also extends to use of manipulatives?)

Boaler’s article also encouraged me to talk to students about how they see their fingers or bodies moving in their mind’s eye. But Trninic’s research is more about using movement to expose mathematical structure directly. He mentioned that he wants to work on conditional probability next. I am looking forward to hearing about it.

Regarding exploratory practice, creating or sourcing the exercises will be an interesting task. For number-sense exercises, I have used Pamela Weber Harris’ book *Building Powerful Numeracy for Middle and High School Students*. She presents problem strings which are sets of questions that lead a learner to see patterns and make generalisations about number.

Do you use exploratory practice in the classroom and have some resources to share? Tweet me (@mathsfeedback) or comment below.