Exploratory Practice

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].)

exploratory practice notes_opt

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?)

conditional probability

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.

Five Superb Maths Lesson Ideas #2

1. Pythagoras and Trigonometry Revision Activity

I love activities that get students out of their seats. This task (designed by Steel1989) asks students to distinguish between Pythagoras and trig questions. Yet instead of a worksheet, the questions are designed to be printed out and stuck around the room on sheets of paper. Students get one to work on, answer it (in their book or on a mini whiteboard) and then write the answer on the back of the sheet. Then they put the sheet back up on the wall. When another student answers the same question, they check their answer with the one already written there. If the answers differ, they students need to talk to each other to discover which is correct.

pythagoras or trig

2. Polygraph Desmos Activity

Oh, wow, I’ve discovered a great one here and maybe you’ve heard the hype already. Desmos has introduced a teacher section that allows you to run class-side activities. I tired out the Polygraph: Lines activity with one of my classes. Have a look at the teacher guidance to learn more. Only you as the teacher needs to create an account; you give students a code to join the game. One student chooses a linear graph and their assigned partner has to ask yes/no questions to guess which graph it is. Meanwhile, as a teacher you can see all the questions and answer being given, who has been successful with the task (or not). I called one of my students over when I saw that she had typed “Does your graph go through the point y = 2x?”. I was able to clear up a misconception I didn’t even know she had until then.

The student’s view is shown in the screenshot below. Desmos is adding to the collection of class activities and I’m sure I’ll use them all in time!

polygraph lines.PNG

3. Tree Diagrams Challenge

A few of my year 11 students are ready to take on the challenge of those nasty tree diagrams questions that lead to quadratics. Fortunately, tonycarter45 has created this lovely sheet with probability extension questions. The sheet includes the answers.

Tony (who works at my school) has produced quite a few nice worksheets and you can see them on TES Resources. He specialises in thought-provoking questions. I like that his investigative worksheets often remove scaffolding parts as the questions progress.

tree diagrams


4. Two is the Magic Number worksheets

Three activities called “Two is the Magic Number” from Just Maths. Each one is a collection of cards solving a short problem, only two of which are done correctly. The rest show common errors and misconceptions. The cards generally cover number and algebra skills such as simplifying terms, using indices, and calculating with fractions. Depending on what you have taught your students, there may be a few topics that they haven’t learned, so check first. (My bottom set year 8 need to practice like terms, but they can’t do a conversion between meters squared and millimeters squared.) These sheets are great for checking students’ misconceptions.


5. IB DP Maths Resource Collection

I have a former colleague, Andrew Clarke, who is a brilliant resource collector. He has now started three curated collections of maths teaching ideas for IB teachers. The one that is most relevant to me is Teaching Diploma Program Mathematics. He has collected all kinds of teaching ideas for Maths HL, SL, and Studies SL. One item that caught my eye is an investigation about using calculus to describe concavity, which is one topic I have never found a good way of introducing.

Andrew’s other two sites may interest you: Teaching MYP Maths and Teaching PYP Mathematics.

What superb lesson resources have you seen or used recently? Comment below or tweet me @mathsfeedback.

Five Superb Maths Lesson Ideas #1

1. Areas of Flags

Areas of Flags (from Owen134866 on TES Resources). One of my colleagues introduced me to this brilliant series of worksheets (and powerpoints) that use flags as a context for finding areas of rectangles, triangles, parallelograms, and trapeziums. There is also a further activity with circles.

areas of flags

2. BC Numeracy Tasks

I was browsing on the website of Peter Liljedahl from Simon Fraser University, Canada. (I was reading a paper of his about task design.) I discovered that he was on a team to develop tasks to assess students’ numeracy in British Columbia. They look as though they are lovely, well thought out tasks. However, there aren’t any solutions that I can see, likely because these are in use as assessment tasks in BC. I note that some of them are too Canadian, though! “Last week I went out crabbing with a friend. We took my canoe and paddled out to a point just off Belcarra Park and threw in our trap.” I’m not sure my city-dwelling, mostly expat students would know what to make of this. However, there are lots of great tasks here and I reckon I will try some of them out soon.

crab trap.PNG

3. GCSE Five a Day Sheets

These GCSE starter sheets, Five a Day, by Corbett Maths. Each sheet has five questions. They are available for numeracy, Foundation, and Higher, and answers are provided. One sheet for every day of the year. I have asked some of my students to use them at home on weekends, too.


4. “Think of a Number” Lesson for HCF and LCM

I’m planning to use this lovely lesson about highest common factor (HCF) and lowest common multiple (LCM) from the Mathematics Assessment Project. I like that it provides a pre-test (which could be used as homework) to help me plan the lesson. The main tasks are really well explained in the teacher notes and include a whole class discussion with mini whiteboard responses, and a card sorting activity. Then there’s a post-test to see what students have learned. All 100 of the lessons in this series are designed with a pre-test and a post-test; I love that it makes it easy to see how students have improved.

The only downside of this lesson is its American vocabulary. I am going to need to use white-out to correct greatest common factor (GCF) to HCF throughout!

hcf lcm shell map

5. Shakespeare and Numbers

Our Head of English has started talking about upcoming celebrations for the 400th anniversary of Shakespeare’s death (23 April 2016). I have been thinking about what we might do in maths to celebrate. So far I found this Numberphile video about the numbers in Shakespeare’s sonnets. I will continue hunting for some other things to use in lessons but this video (duration 4:36) will be a nice ender for lessons on that day.


What superb lesson resources have you seen or used recently? Comment below or tweet me @mathsfeedback.

Numerical and Algebraic Integration Cards

I made this set of eight cards about areas found by integration for my IB Standard Level students. The graphs are taken from a textbook exercise. (Screenshot below. Follow the link to get the cards.)

numerical integrals cards 1

I wanted them to use several methods for finding the areas, including numeric and algebraic integration, so I presented these instructions.

numerical integrals cards

With the answers displayed on the board, students could feel confident as they went through the cards.

a mathematics lesson that worked

This lesson really worked for my students. It was less boring than a textbook exercise, and allowed them to discriminate between methods for finding areas. It also provided good practice of integration and GDC skills.

How I Organise My Lesson Ideas

My favourite discussion with other teachers is to share lesson ideas. But remembering them at the right time is hard work. I think, “I’m sure someone told me a good idea recently about quadratic equations…!” I have probably forgotten more good ideas than I have remembered.

A few years ago I started a document to save good ideas. Below is a screenshot from the document, and you can see my whole Resources Listing document here.

resources listing screenshot 2

Every time someone tells me a good idea or I see one on the web, I write it down here. Then later I can look back through the doc to find it again.

how to get stuff done as a teacher

You can make a document like this, too. Recently I made a blank copy of this document so I could share it with others. Here it is – download a copy and enjoy! Please let me know if it’s useful to you.

Concept-Based Teaching in Mathematics

Some members of my team and I were recently meeting with the IB subject manager for Maths HL assessment. She mentioned that one possible development for the next round of curriculum updates (pdf, IB OCC login required) is a greater focus on inquiry learning and concept-based teaching. When we started discussing this I realised I don’t really know what concept-based teaching looks like in maths. Do you?

I am just beginning my learning journey about concept-based teaching in maths. Here are my stops so far.

Jennifer Wathall has written a book about concept-based maths – it will be published in 2016. I knew Jennifer when I worked in Hong Kong and I attended workshops given by her. I think it is likely to be a very practical and useful book.

This brief blog post by Jeff Hadad argues that the idea of concept-based learning is a good one for maths teachers.

The blog post above links to a book called How Children Learn: Mathematics in the Classroom by the US National Research Council (free download).

Here is a short article by an MYP teacher that gives an example of concept-based learning using simultaneous equations.

What can you share about concept-based mathematics?

What Went Well Bookmark for Peer Assessment

Peer assessment can be a bit bland if students don’t know what to look for in their friend’s maths work. Today @tesmaths tweeted a resource to help with this: these What Went Well bookmarks (free, sign-in required). I adapted them slightly since I’m more interested in students’ mathematical communication than in their neatness. Here is my version of the What Went Well bookmarks.

www bookmarkebi bookmark

These are in a PowerPoint file and are double sided, five to a sheet.

Reflection Time

Last week we had a professional development session with Andy Hind (@andyhind_es4s) about deep learning. One thing that stuck out to me was the value of reflection time in order to deepen learning.

  1. Reflection time for students.
  2. Reflection time for me.

Students need time to reflect on their learning in order to embed it and connect it to their existing knowledge. One strategy Andy used which I will use in lessons was a small picture of a nutshell that popped up about twenty minutes into a session. Andy said, “Tell your partner everything that has happened so far, in a nutshell.”

He said that students should reflect at four points in a one hour lesson. At the beginning (thinking back to the last lesson), after twenty minutes, after forty minutes, and at the end. I scribbled down this time line.

reflection times

Tomorrow I’m incorporating reflection time into my lessons twice. At the beginning of one of my lessons we are going to recap the last session with the instructions on this slide.

recap last lesson

In another lesson I want to ask students to reflect at the end of the lesson, and I’m using the slide below. It’s a feedback structure I have used since even before I was a school teacher. I learned about it from my colleague Richard Hoshino while I was a lecturer at university.

3 min feedback

The “3 Minute Feedback” questions always follow the same pattern. The first question is related to today’s lesson and allows me to see if students have succeeded with the objective of the lesson. The second question relates to my teaching. The third one is always worded exactly as above and gives students a chance to share anything on their minds. I like to respond to these via Edmodo after the lesson by giving the class an idea of the proportion of responses of each type and by answering the questions.

I need reflection time in order to become a better teacher. I used to blog more regularly and this was a good method of reflection for me. But Andy also suggested a private reflection journal and I’ve started one this week. I set an alarm for half an hour before I want to go home. I use 15 minutes for writing reflection and 15 minutes for tidying up my desk. I haven’t managed to do it every day this week, but I’m pleased that I have done it three out of the last six workdays. I’m going to either write in my journal or on this blog during my afternoon reflection times this year.

How do you include student reflection into your lessons?

Stereotypes of Mathematicians

I was reading on the NCETM website about films that have mathematicians as main characters. Four films are mentioned:

  1. A Beautiful Mind
  2. Pi
  3. Good Will Hunting
  4. Enigma

One commentator says that these films contribute to stereotypes about maths since they are all about men, men who are unsociable, and that they are uncomfortable in their roles. Films are not helping maths break away from a “nerdy” stereotype.

Then another commentator goes on to lambaste this idea by saying:

the first three films are about mental illness, not mathematics: the characters happen to be mathematicians, their profession is incidental to the drama that arises from their malfunctioning brain chemistry. The negative, frightening “nutter” stereotype they perpetrate is far more reprehensible, and dangerous, than any “nerd” stereotype.”

The article ends by asking what schools are doing to counteract these stereotypes. For our part, we have named ten of our classrooms after mathematicians. Nine are men and one is a woman: Sophie Germain. They are not all dysfunctional; though Georg Cantor did go insane.

2015-02-12 17.33.09

My classroom is named after Paul Erdös but I have yet to capitalise on this with my students. There are so many good stories about Uncle Paul and his love of maths. He was a bit nutty though, so I am not sure he refutes any stereotypes. The truth is, a lot of mathematicians are men and a lot of them are a little odd. (I say this as a proud nerd.) I think this is especially true among academics, and perhaps less so among mathematicians in industry.

Does your school do anything to counter stereotypes of mathematicians?

A Nice Activity for Statistics and Data Representation: Estimating One Minute


My year 8 students were learning about working with grouped data. I used an activity that I pull out regularly when I need some data from the class to work with.

Students worked in pairs and one partner timed the other while they were estimating one minute. I asked the estimating student to close their eyes, say start, and then sit quietly until they think one minute has passed. Their partner used a stopwatch (on their iPad this time) to time the duration.

We recorded on the board all the times from the class. In my year 8s we took several tries each since we are a small group and I wanted to have about 30 pieces of data.


Then we went on with our lesson about grouped data. We got to consider all the authentic data questions–for example, how broad should the classes be to record this data? Also our outliers seemed very far from the bulk of the data. Students naturally asked about these. I like it when there is less need for me to point out these things. Because it is their own data they are intensely interested.

a mathematics lesson that worked

We had a bit of a giggle because one of the biggest outliers was my own estimate for one minute. Once I had my eyes closed I vastly overestimated how long one minute was. My teaching & learning assistant said I was having a nap for 114 seconds!

What data collection activities do you like to use?