The power of asking "What if?..."

Why this is my all-time favourite question, and how to use it

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I don’t know of a more powerful, versatile question to ask students during a mathematics lesson than: What if?.

• We have worked out that 0.8 is equivalent to 80%… What if the decimal changed to 0.08?

• We have worked out the rule for this sequence: 2, 3, 5, 7, 9… What if the sequence changed to: 102, 103, 105, 109…?

• We have worked out the position of the image following an enlargement of scale factor 2. What if the scale factor is -2?

• We know the mean of this set of numbers (5, 6, 7, 10, 12) is 8. What if the set of numbers changes to 5, 6, 7, 10, 12, 8?

Here we take a key principle from variation theory: meanings are acquired from experiencing differences against a background of sameness, rather than from experiencing sameness against a background of difference

In other words, if we hold all else constant but change one critical feature of an example, students’ attention is drawn to the change and its effect on the outcome.

For many years, my favourite time to use questions like this was during the independent practice phase of the lesson. I called such sequences of questions Intelligent Practice, and built a website full of them.

Now, my favourite time to use such questions is during the We Do phase of a worked example. Once students have answered one question on their mini-whiteboards, I can ask them: what if I change this… and challenge them to change as little as they can in their working to answer the new question. I do lots of these examples, and find they help students make meaning to such an extent that I don’t have to do worked example after worked example. I will show such a sequence of questions in action below.

What If? in action!

During a coaching session in one of the schools I am supporting, I worked with a teacher on What If?. The lesson I had observed had consisted of disconnected examples on factorising into single brackets during the We Do, and as a result, students were not making the connections as quickly as the teacher hoped.

He really liked the idea of using What If at this stage of the lesson, having been a keen dabbler in Intelligent Practice in the past. We worked together to plan a few sequences for upcoming lessons, and the very next week, I saw it in action.

It was a Year 11 class who were recapping solving linear equations. The I Do and first We Do looked like this:

Students responded on mini-whiteboards, and everyone nailed the first example. The teacher then said: What if I do this?… and subsequently changed the + to a -

Students were challenged to change their working out as little as possible to answer this new question, noticing that they now needed to add 9 to both sides instead of subtracting.

The mini-whiteboards provided evidence that students had understood this, so the teacher introduced the next change:

This threw the students, so the teacher modelled the process and then rechecked students’ understanding by making up a couple of related examples on the board:

Students got this right, and so were back on track. Next, the teacher asked what would happen if the 21 became a 20?

This was really nice because students could shift attention away from the structure of the equation and instead focus on how to deal with a division that did not give an integer answer.

Again, the whiteboards revealed some students struggling, so again, the teacher modelled and then rechecked:

Back to the original sequence of examples, next up the 3 became an a:

Without them realising, and without the teacher needing to do a separate worked example, students had gone from solving equations to rearranging equations. Students’ attention was drawn to the impact the unknown had on their solution, and everyone took it in their stride.

The teacher gambled slightly at this point and introduced two changes:

But this was a fairly safe gamble, because he knew he could always jump over to the other board to help students get back on track.

Here are the final three examples:

Each time a new change was made, some students said: too far, sir! But 30 seconds later they were so proud of themselves when they nailed it.

The full sequence took around 25 minutes. Notice how we went from solving a simple two-step linear equation to rearranging a fairly complicated formula, all in a coherent sequence that made total sense to the students and gave them lots of moments of success along the way.

This occurred on mini-whiteboards, and some readers may be concerned about the lack of “evidence” in students’ books. but the medium was entirely appropriate for this style of questioning, allowing students to make changes to their working out easily and allowing the teacher to keep a constant check on his students’ understanding at each stage.

Independent fluency practice and problem-solving followed.

Another example

The Head of Department had also been trialling What Ifs during the We Do phase, and I was lucky enough to see it in action during a lesson on probability:

Improvements

In the coaching session that followed, we discussed some improvements:

• The teacher could ensure that the previous question remains visible at all times, so students can more easily notice the change made. The best way to do this is to copy out the previous question, slowly and deliberately rub out the critical feature in the copy, and then make the change.

• To ensure every student has the best chance of noticing this, the teacher could make sure they have empty hands and eyes on the board and then Cold-Call a few students to ask them to describe the change.

• Likewise, the teacher could keep the working out for the previous example visible so changes to that working out can be made clearly and deliberately.

• After the change has been made, students could be given, say, 10 seconds in silence, without doing any writing, to predict what impact the change will have on either the answer or the way they work the answer out. This stops students from diving straight in and ignoring any key relationships.

• For key questions, the teacher could ask students to share their predictions with their partner or the rest of the class. Again, this helps to draw attention to the relationship between examples, and also sets up the potential for confirmation or surprise.

• Students could use one side of their mini-whiteboards to answer the What If series of examples and the other side of their boards for the extra checks for understanding to save them from losing the working for the What If sequence.

• Occasionally, it is a good idea to throw in a re-set example that changes several things that have been covered in previous What Ifs to see if students can tie these ideas together

The two teachers will work on embedding these into their practice over the next two weeks before supporting their colleagues in building them into their modelling process.

A source for these types of questions

If you like this idea, you can use the sequences of questions from my Variation Theory website to get you started. Whilst they were written to be used during the practice phase of a lesson, they can work really well during the We Do phase as described in this post. You may need to reduce the number of questions, or make other adaptations to fit the needs of your students, but hopefully it is start!

How do you use What If questions?

Let me know in the comments below!

🏃🏻‍♂️ Before you go, have you…🏃🏻‍♂️

1. … checked out our incredible, brand-new, free resources from Eedi?

2. … read my latest Tips for Teachers newsletter about not using mini-whiteboards?

3. … listened to my most recent podcast with Kris Boulton about Atomisation?

4. … considered booking some CPD, coaching, or maths departmental support?

5. … read my Tips for Teachers book?

Thanks so much for reading and have a great week!

Craig