Purposeful Practice - Tips for Teachers by Craig Barton

Purposeful Practice can provide a nice bridge between Consolidation and out-right Problem-Solving

Contents

Diagnosis

Do your students ever find the jump from consolidation practice to problem-solving too big?
What are some of your favourite activities and strategies to help bridge this gap?

Evidence

In the ATM publication, Practice, authors Dave Hewitt and Tom Francome identify three types of practice activities often given to students:

1. Mindless Practice: The kind of mimicking of the method explained by a teacher where learners have not understood the mathematics behind a process but appear to be able to get correct answers as long as the questions do not deviate too much

Dave and Tom give the example of a Pythagoras worksheet where every question requires the students to square a side, square the other side, add them together, and then square root. Essentially, the students stop thinking about Pythagoras’ theorem and instead just regurgitates the same algorithm. Life is good, until they encounter a problem like this:

2. Unconnected Practice: This involves practice of the mechanics of answering questions but where more understanding of the underlying mathematics behind the methods is required.

Here, The questions are not necessarily connected in any way other than
they have been designed to practise a particular skill or idea. This is the Consolidation type exercises we discussed in the previous section.

I think I value this type of practice more than the authors. However, I agree that if this is the sole type of practice students experience, they will find their levels of interest and flexibility to deal with unfamiliar problems quickly diminish.

3. Open problems: When we think about an ‘open’ problem we tend to imagine situations where learners ask their own questions about a particular situation

For example, students might be told to join three points to make a triangle, and then asked to consider what questions a mathematician might ask:

The main issue with problems like this is it is difficult to know exactly how students will approach them and what they will think about, which means they may ot end up practising the idea we need them to practice. Assessing students’ understanding can also be tricky.

Purposeful Practice is a special type of activity that is a middle ground in between Unconnected practice and Open Problems, or what I would call Conbsolidation and Problem-Solving. Purposeful Practice activities can provide students with practice of a key method or procedure whilst at the same time providing opportunities for them to think deeper.

Solution steps

Where does Purposeful Practice fit into a lesson?

In a previous section, I shared three possible structures for a Learning Episode.

Learning Episode structure #1

Learning Episode structure #2

Learning Episode structure #3

Purposeful Practice is the bridge between Consolidation and Problem-Solving.

Whilst good Purposeful Practice activities contain lots of opportunities to consolidate, I feel there is a need for a small dose of Consolidation beforehand. This sets studetns up to think harder about the connections present within Purposeful Practice.

Likewise, it is thinking hard about these connections in the relatively structured ennvirmoiments and cintraints f a purposeful practice activity that then allows studetns to get the most out of the relatively iunstrucutured, unconstrained problem-solving tasks that follow.

High-value Purposeful Practice Structures: Introduction

In the past, I had a favourite activity for each topic I taught. A fill-in-the-blanks for Pytharoas, a mystery for sequences, a card sort for cumulative frequency diagrams, and so on. The issue was that I had to spend precious minutes explaining the structure of each activity. Moreover, my students had to exert precious attention.

Students have limited attention. When presented with an activity, they must divide their attention between the structure and the content. The more unfamiliar the activity, the more attention is diverted towards the structure. As we know from Dan Willingham’s work, students remember what they pay attention to.

Hence, I have developed a set of high-value Purposeful Practice structures. These structures can be used across different topics and classes. Students get used to the structure, so they can dedicate more attention to the content and thus learn more. I also improve at using the structure to support my students’ learning.

High-value Purposeful Practice Structure #1: Completion tables

Example of the activity for you to try

Can you fill in each of the boxes?

Why this is a good activity structure

To students, it doesn’t look like lots of maths… but it is!
There are lots of opportunities for Consolidation of key procedures
There are of opportunities for reasoning and problem-solving as students grapple with how to work forward and backward, given different information

Tips to make these activities effective

1. Do a We Do like the top row

I like to make up a different example than the one in the top row, but of the same difficulty. So, 2/3 + 4/5 would be good for the above activity. As students will have experienced consolidation practice of all the procedures by the time I give them this activity, I don’t need to do an I Do. Instead, I will point to a bo and ask students to work out their answer on their mini-whiteboards, hover, and show me in 3, 2, 1…

The data I get back helps me check students are secure with the structure of the completion table and gives me insight into their understanding. I can respond accordingly, letting students who are secure crack on and supporting those who are struggling.

2. Print out the Completion Table

I am all for saving trees, but projecting the Completion Table on the board and asking students to copy it out into their boos wastes precious time and effort that could be used for thinking and learning.

3. Give students a mini-whiteboard to try ideas on

Mini-whiteboards are also useful when students are working on the activity. They can try ideas out without fearing the permanence and messy crossings that working on paper brings. They can transfer their answer to their books when they are happy.

Of course, this means that students’ workings out are lost as soon as they clear their boards, but I am okay with that. I want to choose the most appropriate medium to support students, not compromise this in the quest for written evidence. However, some teachers prefer to ask students to do their working out in their exercise books.

4. Use the 4-2 approach

The 4-2 approach works well with Completion Tables. By the end of the 4 minutes, most students have enough entries to check with their partner, providing either confirmation or fertile ground for discussion.

5. One big Completion Table between two

You could print out one big A3 Completion Table between each pair to encourage further collaboration. Students can be encouraged to do their working out independently on their mini-whiteboards, compare their answers for a particular box, and, if they agree, enter it into the shared table.

6. Regular whole class checks for understanding

While you can circulate as students work on their Completion Table, every 5 minutes or so, it is worth getting a more reliable snapshot of the whole class’s understanding. The easiest way to do this is to do Book-to-Board: Quick check:

Get studetns’ attention.
Choose a box – perhaps one you have identified one or two issues with
Ask students to copy their final answer from their sheet to their mini-whiteboards, hover, and show you in 3, 2, 1
Respond accordingly

Where to get these activities

High-value Purposeful Practice Structure #2: Venn Diagrams

Example of the activity for you to try

Can you think of 3-digit integers that could go in each of the four regions?

Now can you think of numbers with 3 decimal places that could go in each of the eight regions?

If you cannot find an example that fits a particular region, can you convince me it is impossible?

Why this is a good activity structure

Venn Diagrams require students to generate their own examples instead of answering teacher-generated questions. Learner-generated examples are a fantastic test of student understanding as they can reveal misconceptions in a way that teacher-generated questions may miss
There’s lots of thinking required
The barrier to entry is low as every example belongs somewhere. But as the regions get filled up, the challenge level increases
As we will see, it is easy to offer both support and challenge when required

Tips to make these activities effective

1. Lower the content demands initially

Always start with a basic example to ensure students are secure with the structure of a Venn Diagram. We want our students to be as fluent as possible in terms of knowing exactly what each region of a Venn Diagram means, so that when we change the content to something more challenging, students can divert their attention to thinking about the content and not the Venn Diagram itself. Here is an example where each student had to put their initials in the correct region:

2. Always do a double Venn Diagram before a triple Venn Diagram

I always warm students up with a double Venn Diagram. This helps identify misconceptions or misunderstandings with the Venn Diagram structure, or the subject content. It also helps to build student confidence.

Dan Rose takes this one step further. He asks his students to complete a double Venn Diagram first, then reveals the third circle and its label, and challenges students to see which examples from their double Venn will need to move and which can stay!

3. Start each Venn Diagram with an example the class does together

Consider this Venn Diagram where students need to give the equation of a straight line in the for y = mx + c

Pick a student and ask them to choose their favourite equation of a straight line (everyone has one). Let’s say they go for y = 2x + 3. Write this on the board and challenge students to decide on their own which region they think it belongs.

Get them to write their choice of letter on their mini-whiteboards, hover them face down, and show you in 3, 2, 1… Then ask students to place their boards in between them and discuss their answers and reasoning. I

f the vast majority of students have the right answer, you can confirm and quickly move on, confident that students know what they are doing and have tasted some success. If there is an issue, you might need to pause, reteach, and pick up the example when understanding is secure.

4. Ask students to do their working out on their mini-whiteboards

As with Completion Tables, I always ask my students to do their working out on mini-whiteboards and then record their answers on a blank copy of the Venn Diagram that I have printed out on paper. Mini-whiteboards are a great vehicle for students to do the kind of messy thinking and experimentation often needed to crack a Venn Diagram. They also support paired discussions and collaboration more so than with books or paper.

5. Use the 4-2 approach

You should be spotting as theme here! As with Completion Tables, I find the 4-2 approach strikes the right balance between independent work and focused collaboration.

6. If students are struggling, have examples for them to sort

For example, using the Venn Diagram below, I could ask a struggling student where they think 3/5 belongs.

Sorting is easier than generating one’s own examples and should help students get back into the activity. Indeed, I have seen teachers begin with 8 examples for students to sort into regions before asking them to generate their own examples.

7. Ask students who finish to think of additional examples for each region

This may appear a bit of a throwaway extension, but generating multiple examples can help students move towards generalisation.

Building on this, I like to ask students who have finished to think of the most interesting example they can for a particular region, whatever they consider interesting to mean. This prompt compels students to consider boundary examples, and is a revealing insight into the strength of their understanding. Give it a go yourself. Can you come up with three examples for Region D in the fractions Venn Diagram above? What is the most interesting example you can think of?

8. Challenge students to create their own Venn Diagram with certain constraints

The ultimate challenge, of course, is to ask students to create their own Venn Diagram on the current topic. You could be even more prescriptive and ask that their Venn Diagram has, for example, exactly one region that is impossible to fill.

9. Challenge students to change as little as possible to move from one region to the next

An interesting strategy to suggest to students is to see if they can change each example as little as possible to generate an example for a different region.

For example, say a student comes up with 5, 5, 5, 10, 20 for region A in the Venn Diagram below.

Then what is the smallest change they can make to that example so they now have an example for region B, and so on? This taps into the power of variation theory, enabling students to focus on the critical aspects of examples.

10. Use peer assessment to check answers

When assessing students’ work, we have the inherent problem with all learner-generated examples: there are potentially an infinite number of correct and incorrect examples for each region, so banging the answers on the board and asking students to mark them themselves will not work.

Instead, ask students to swap their completed Venn Diagram with their partner. I give them the following instructions:

Tick each example you think is in the correct region
* any you think might be wrong
Discuss the *s with your partner
If you still disagree, put your hand up. You can guarantee that any remaining starred examples will make for a fruitful whole-class discussion.

Where to get these activities

High-value Purposeful Practice Structure #3: Open Middle Problems

Example of the activity for you to try

Open Middle Problems require students to fill in the gaps to fulfil a given condition under certain constraints. Each problem has:

A “closed beginning” – they all start with the same initial problem.
A “closed end” – they all end with the same answer.
An “open middle” – there are multiple ways to approach and ultimately solve the problem

Why this is a good activity structure

They look like a fun puzzle (and they are!)
They generally have multiple ways of solving them as opposed to a problem where you are told to solve it using a specific method
They may involve optimisation such that it is easy to get an answer but more challenging to get the best or optimal answer
They may appear to be simple and procedural in nature but turn out to be more challenging and complex when you start to solve it.
You can always ask: How do you know if you have got all the possible solutions?

Tips to make these activities effective

1. Try an example together

Ask a selection of students to choose some numbers to go into the boxes and then test this out. This will help ensure students are secure with the rules and structure of the problem, and may give struggling students a starting point.

2. Give students a mini-whiteboard to try ideas on

Open Middle problems quickly get messy, and the mini-whiteboard is the perfect vehicle to house that messy thinking. Studetns will want and need to try out numbers, rub certain numbers out, and then try new numbers. This is much harder to do in an exercise book.

3. Use the 4-2 approach

The 4-2 approach works really well with Open Middle Problems. In those 4 minutes of silent, independent work, students can play around and find their “best” answer. They can then share their answer and approach with their partner for the 2 minutes of collaboration, learning from each other.

4. Keep track of the class’ “best” answers

For Open Middle problems that require optimisation (find the biggest, smallest, etc), it can be a good idea to stop the class regularly and:

Ask who thinks they have the best answer so far
Get all students to check this answer
If it is correct, record it on the board
Set students the challenge of beating it

This can help motivate students, as well as offer support and guidance to students who may be struggling.

Where to get these activities

Where do I find the time in lessons for Purposeful Practice?

This may all sound good in theory, but where are you supposed to find the time to do this with the curriculum so full and time so tight? There are no easy answers, but the following may help:

1. Spend time perfecting the routines for each lesson phase

The time that could have been spent engaging in some lovely Purposeful Practice activity is usually lost earlier in the lesson due to poor routines:

The Do Now takes longer than 10 minutes
Atomisation doesn’t happen, so students do not have the prerequisite knowledge to learn the new procedure.
The I Do takes ages as it is littered with discussion and checks for understanding.
The We Do is not effective at eliciting data about student understanding, so students get stuck on the Consolidation activity.
The teacher spends too long on Consolidation because they have not consolidated enough during the We Do

You can see how problems earlier in the lesson transform into curtailing any opportunity for problem-solving. Get those routines in place using the solution steps throughout this website, and you should find you have more time for problem-solving

2. Set yourself a reminder

Where possible, we should aim to leave at least 10 minutes of a lesson to do the problem-solving activity. This stops problem-solving from being ignored completely, or shoved in during the last 30 seconds, preventing many students from engaging with it.

3. Purposeful Practice doesn’t need to happen in every lesson

Learning Episodes are designed to span over several lessons. Thus, we should not expect to use tasks like the above in every lesson. However, students should be exposed to activities like the ones in this section in every Learning Episode.

Want to know more?

The ATM book, Practicing Mathematics, is a good starting place
Colin Foster’s work on Mathematical Etudes is also worth checking out

Implementation planning

Choose a class you feel comfortable to try something with
Choose an upcoming topic that class is studying
Choose one of the activity structures we discussed
Find a suitable problem
Prepare to deliver that problem by first doing it as a mathematician, and then putting your teacher hat on