Learner-Generated Examples are when – surprise, surprise – the students come up with the examples given certain constraints, as opposed to the teacher setting questions for them to answer.
Diagnosis
- Do you challenge students to come up with their own examples?
- If so, how do you structure this?
Evidence
For most of my teaching career, I have focused on improving the quality of the examples and questions I give my students. I can use my knowledge and experience to design examples and questions that progress in difficulty, avoid erroneous generalisations, confront misconceptions, or are most like the exam questions students will encounter, and so on.
However, this means my students’ experiences are confined to what I present. And that is very unlikely to be complete or optimal, both for me as their teacher as I attempt to paint an accurate picture of their understanding, and for my students as learners as they strive to demonstrate what they can do.
Research suggests that an effective companion to teacher-generated examples are Leaner-Generated Examples (LGEs). LGEs can be a valuable formative assessment tool for teachers, providing insights into student understanding, promoting mathematical thinking, and supporting the development of mathematical writing skills
Solution steps
We could simply ask students to come up with their own examples.
However, when we do this, students often gravitate to the same “obvious” example:
Or they generate examples that are not all that useful:
The keys to effective LGEs are:
- Multiple examples
- Constraints
Thinkers – a 2004 ATM publication by Chris Bills, Liz Bills, Anne Watson and John Mason – contains a set of prompts (or what I will refer to as frameworks) that challenge students to construct effective LGEs. Each framework requires multiple examples amidst constraints and is designed to draw out students’ natural power to generalise.
Here I will share my take on two of the frameworks from the book.
Framework 1. Give an example of…
Have a go!
You will need a piece of A4 paper, or a mini-whiteboard, turned landscape, and divided up into quadrants:
In the top-left box, please write a fraction and its decimal equivalent.
Have you got one?
- Okay, in the top-right box, please write another fraction and its decimal equivalent.
- We are on a roll now, so in the bottom-left box, please write another fraction and its decimal equivalent that no one else will think of
- Finally, in the bottom-right box, please write a fraction and a decimal that are not equivalent, but that someone might think are
Here are the responses of a Year 7 student to these prompts:
The purpose of each example
Each example has a different purpose:
- Top-left: This is the initial example to kick-start the process
- Top-right: Asking for a second example tests the depth of students’ understanding and starts them thinknig about generalisation
- Bottom-left: Next we want a non-obvious positive example
- Bottom-right: And finally, a non-obvious negative example
Those final two boxes prompt students to generate boundary examples. Boundary examples are those right on the edge of a concept. Asking for an example no-one else will think of challenges students to push a concept to its limits. Asking for an example someone might think was correct but is not – in other words, an interesting non-example – tests whether students can take a small step over the boundary of the concept – as opposed to thiking of any old non-example – as well as challenging them to appreciate where others might struggle. Sure, I could come up with a selection of boundary examples myself for us to discuss, but I know of no better way to check my students’ understanding of a concept than to set them up to think of these boundary examples themselves. Some examples will inevitably lie on the wrong side of the boundary, but this draws out misconceptions that we can confront as a whole class.
How Give an example of… plays out in the classroom
Students respond on mini-whiteboards instead of paper, setting their boards out as described above. Pause after each prompt to give students sufficient time to think and write. Students work silently and independently, keeping their eyes on their own boards.
Here is an overview of how this activity progresses:
See all students’ responses
After sufficient thinking time – I can get a better sense of this by circulating the room – I ask students to hover their mini-whiteboards face down to show me they are ready. They then show me their boards on a count of 3, 2, 1…
Seeing all responses allows me to get a sense of my students’ understanding, as well as to ensure all students are participating. I pay particular attention to the responses of my weaker students, looking in particular at their top two examples. I make a physical note of any examples I wish to discuss with the class later – interesting correct or incorrect answers.
Partner check
Next, I ask my students to swap mini-whiteboards with each other and check each other’s work. They do so by ticking any examples they agree with, and putting a question-mark next to any they don’t. Students thus have an opportunity to practice checking for equivalence as well as generating their own, with the added twist that they could well be presented with challenging examples to check for their partner’s third and fourth ones. I say to my students: If you disagree with any particular answer, have a conversation about it with your partner, and if you still disagree, let me know. This always produces interesting examples that we can discuss as a class in the next stage of the process.
Whole class discussion
Following the partner check, we discuss any examples that students disagreed on, together with any that I picked out when students showed their boards, or interesting examples I have prepared in advance. I record each of these examples on my whiteboard so they are visible. For each example, I ask students to think on their own if it is correct, vote with thumbs up or down, then discuss with their partner, and then I can Cold Call students to hear what they have to say.
With a selection of interesting examples and non-examples on the board, the whole class discussion might be an opportunity to generalise: How can we quickly spot if a fraction and decimal are equivalent? How can we quickly spot if they are not?
Other examples
When to use Give an example of…
- After fluency practice
- End of the lesson
- Assess prerequisite knowledge
- In an assessment or homework
Framework 2. Additional constraints
Additional constraints is certainly more challenging as a teacher to write, but it can lead to some superb thinking and discussion amongst students.
Have a go!
This time, turn your piece of paper or mini-whiteboard so it is in portrait orientation, and divide it up into five roughly equal rows.
In the top row, please write a pair of fractions that add together to give 1.
Have you got a pair?
In the next row, please write a pair of fractions that add together to give 1 AND have different denominators.
Then, in the next row, write a pair of fractions that add together to give 1 AND have different denominators AND have the same numerator
If at any point you have already written an example that matches the new constraint, think of a second example.
Okay, here is an overview of how this activity progresses:
See all students’ responses
With the first constraint announced (the two fractions must add to 1), I give students thinking and writing time, and then ask them to hover their mini-whiteboards face down to let me know they are ready. I then ask students to show me their mini-whiteboards. I do this at each stage of Additional constraints, whereas I wait until students have all three examples when using Give an example of. The reason for the difference is that Additional constraints generates a sequence of examples, and an unchecked error early on messes up everything that follows.
As before, I make a physical note of any interesting responses.
Partner check
I then prompt students to swap boards with their partners so they can check each other’s example, marking it with a tick or a question mark. Once again, I tell my students that if there is a disagreement that they cannot resolve between them, let me know and we can discuss it as a class. As we get further through the sequence, the Partner check is also an opportunity for students to discuss how they constructed their examples.
Whole-class discussion
At any stage, I’m free to instigate a whole class discussion. I will do this if I’ve spotted any interesting answers, or if a pair has a disagreement from their discussion that needs resolving. If, however, all seems in order, I will skip this step and move on to the next constraint.
Additional constraint
Then we loop back to the beginning as we add in an additional constraint.
- In your second row, I want a pair of fractions that add to one, and the two fractions must have different denominators.
Can you think of a pair of fractions that work?
Some students may have gotten lucky and find their first example also matches this additional constraint. In that instance I ask them to check that is definitely the case, and if so can they come up with another example that also matches the constraint?
What follows is another round of showing me their mini-whiteboards, then partner check, and then whole class discussion if needed.
At this stage, a student’s mini-whiteboards might look something like this:
This continues as we work through the complete sequence of constraints, each building on the previous. So, all together we have:
Think of a pair of fractions that:
- add to 1
- …and have different denominators
- …and the same numerator
- … and one fraction is improper
- … and the numerator is 13
Can you think of an example for each?
Here are a few ideas for helping the Additional constraints framework run as smoothly as possible:
Utilise Variation Theory
I often encourage students to look closely at their first example and see if they can modify it in any way to match the second constraint. This is another example of how I like to use a key idea from Variation Theory – that meaning is found as things change amidst a background of sameness. This offers support to students who are struggling as they have a starting point, but also can provide challenge to other students as they seek to explain the relationship between the adjacent examples.
Speed demons
I use the hover your board technique as described in Tip 12, so I can easily tell which students have their examples ready to show me. If I have one or two speed demons, hovering their boards like mad men ready to show me their response, I will challenge them to think of the most interesting additional example they can that also matches the constraints.
If students get stuck
As constraints are added and the difficulty ramps up, students will get stuck. Circulating the room as students are working helps you pick up on this. In these cases, I stop the class, ask if anyone thinks they have an example that works, and we check it together. I then ask that student to describe their strategy for coming up with the example. I can then ask the rest of the class to either use that strategy to come up with their own example, or – if they already had an example – see if they can come up with the most interesting answer they can think of.
Allow collaboration
Another option as the challenge increases and students start to get stuck, is to allow collaboration. I like to ensure my students have had silent thinking time themselves, and then allow them to work with their partner to see if they can find an example that works between them.
Extra challenge
A nice challenge to ask at the end of the sequence is if students can think of their own additional constraint to tag onto the end that still allows another example to be generated. Leaving the final constraint (and the numerator is 13) would allow for this in the above sequence.
Strategy and generarlisation
As we progress through the constraints, I’ll use the whole class discussion to ask students how they constructed their examples. Not only does this offer a great insight into their understanding, but it can also put us on the path to finding more examples that work as we move to generalisation.
Other examples
Other examples of Additional constraints that I have used in maths lessons include:
Think of a set of 6 numbers that:
- have a mean of 5
- …and a mode of 2
- …and a median of 4
- … and a range of 10
- … and two of the values are non-integers
Write down a sequence of numbers that is:
- linear
- …and is descending
- …and contains at least one non-integer term
- … and has a 3rd term of 15
- … and has an 8th term of 13
Can you think of a sequence of additional constraints for an upcoming topic?
Framework 3. What if?
When Michael Pershan came on the podcast he inspired an alternate framework for student-generated examples called What if?
Here is how it could play out for another fractions example. This time you will need your piece of paper split into two columns:
- On the left-hand side, write down a fraction of an amount problem in the form abof c, where a is an even number and the answer is an integer. Also write down the answer.
- As an example, I am going to go for 45of 30, which gives me an answer of 24.
- You cannot use any of the same numbers I have used.
I write my example and answer on the left side of my board so students can see. I will continue to write my examples and answers throughout the sequence that follows.
Students then write their example and answer in the left-hand column of their mini-whiteboards, hover, show me on cue, then check each other’s work.
I add the additional instruction that if a student has used the same numerator, denominator or amount as their partner, then the student closest to the classroom door must change their example, and their partner must help them come up with a new one. This is really important to ensure each pair has two different examples to compare during the sequence that follows, and that both examples are different to mine.
And then I introduce the first What if?
- What if you double b?
- So, looking at my example, I would now have 410of 30
- We write the new example in the right-hand column
My board looks like this:
Notice, I have written the question in the right-hand column, but not the answer
Here is an overview of how this activity progresses:
Self-explain
After announcing each What if?, I encourage students to take a moment to predict what they think might happen for their example and write this prediction on their mini-whiteboard.
- Will the answer increase, decrease or stay the same?
- Can they predict an actual value?
Students then work out the answer on their own, writing it below their new example in the right-hand column. I then prompt them to pause and reflect:
- Did the answer match their prediction?
- How would they explain what is going on?
This opportunity for self-explanation is key. Modelling how I as a teacher make predictions and reflect, and sharing examples from students that I see on mini-whiteboards around the room is important to promote good practice.
See all responses
Then I ask students to hover their boards and show me when called upon so I can make a note of any interesting responses.
At this stage student’s mini-whiteboards might look something like this:
Partner check
Next I ask students to discuss their answer with their partner. Here I tell them to do three things:
- Check each other’s work
- See if the relationship they noticed between their examples is the same for their partner’s
- See if between them they can come up with the best explanation as to why
Whole-class discussion
I then Cold Call a few pairs to see what relationship they uncovered. With a number of different examples to work with, including my own, we can try to explain what is going on.
The next What if?
We continue in the same way for each new What if?. Students write each new example in the right-hand column, and the rub that out when we have finished discussing it. Their first example always remans visible in the left column as that is always the example we compare against. I model this each time with my own example.
Here is the full sequence of What ifs:
- What if you double b?
- What if you halve a?
- What if you add b on to c to create a new c?
- What if you multiply both a and b by 7?
- What if you you swap a with c and c with a?
- What if instead of abof c, it was 1abof c?
- What if instead of abof c, it was baof c?
The link to Intelligent Practice
This structure is closely related to my use of variation during Intelligent Practice (Tip 72). Indeed, you could prompt students to use the same Reflect, Expect, Check, Explain format for each modification:
- Reflect – what has changed? – the denominator has doubled
- Expect – what do you think will happen to your answer? – it will halve
- Check – work out the actual answer and compare to your prediction – it matches!
- Explain – can you explain the relationship between answers, first on your own and then with your partner? – we are now dividing by a number that is twice as big, so our answer will be half the size
But the power of the What if? structure comes from the fact that students are all working on different examples and yet noticing the same relationships. Often you hear exclamations such as: wow, that happened with mine too!, during the paired discussion. This allows us to move towards explanation and generalisation when we pick a few of these examples to share as part of the whole-class discussion.
Other examples
Two other examples of What if? that I have used are:
Start with 5 numbers, all different, that have an integer mean
What if…
- You double all the numbers?
- You subtract 1 from each number?
- You add 10 to one of the numbers?
- You include an extra number equal to the mean of your 5 numbers?
- You copy your 5 numbers and work out the mean of the set of 10?
- You double your 5 numbers and work out the mean of the set of 10?
Start with the first 5 terms of an ascending linear sequence, and work out the nth term rule
What if…
- You subtract 1 from each term?
- You double each term?
- You start with the same number but go up by 1 more than before?
- You reverse the sequence?
Can you think of a sequence of What ifs for an upcoming topic?
The holy trinity strikes again
In Tip 17 we discussed the holy trinity of participation: Mini-whiteboards → Partner Talk → Cold Call. We can see a version of it here in each of the three frameworks:
- First students have an opportunity to think independently, and I can see all their responses
- Then they are challenged to check each other’s work, discuss strategies and resolve disagreements
- Finally, I can call upon individual students to share their thoughts on interesting answers, and add clarity when needed
Let me end with a few questions I regularly get when discussing these three frameworks
Do I need mini-whiteboards to use these frameworks?
I tell you what, if a mini-whiteboard manufacturer fancies sponsoring this book, just let me know, because it seems to be all we ever talk about. In theory, you could use books or pieces of paper, but mini-whiteboards are really useful for all three of these frameworks for learner-generated examples for the following two reasons:
1. Mini-whiteboards make checking for understanding much easier
- I can more easily see student responses as I circulate the room.
- I can use hover your board to easily see when my students are ready far easier than if students are working in books and I say put your pens down.
- At any stage, I can ask my students to show me their mini-whiteboards so I can get a snapshot of whole-class understanding. This also allows me to make a physical note of any misconceptions or interesting examples that we can discuss as a class.
- I can grab a board and stick it under the visualiser or hold it up so everyone can see.
2. Mini-whiteboards make peer assessment and discussion much easier
Partner Talk is a key element of all three frameworks, and such discussion is made so much easier with mini-whiteboards. Students can swap boards much easier than they can swap books, and mistakes and messy annotations can be rubbed away.
When should I use these frameworks?
I use these three frameworks in three key phases of a learning episode:
- As a relevant prior knowledge check. These three frameworks allow us to revisit concepts students have met in the past in a more interesting and challenging way.
- As a supplement (or alternative) to fluency practice. Students gain valuable practice both in the generation of their own examples and in the checking of their partners’. The opportunity to generalise and describe their strategy also takes this above and beyond the type of fluency practice I would usually give my students.
- As an end-of-lesson check for understanding. Students may have been working on a concept all lesson, gaining valuable fluency practice. But what do they really know? The depth of challenge elicited by these three frameworks, together with their inherent mass participation, makes them an ideal end-of-lesson activity.
How long should these frameworks take?
Great question, terrible answer – it depends. If students’ knowledge is not secure and you need to do lots of discussion, questioning and modeling, then they may take up to 30 minutes. If students’ knowledge seems secure, then you might spend five minutes on a framework. However, in this case, you may take the opportunity to probe students’ understanding further, focussing on interesting examples and non-examples, student strategies, or explaining relationship.
