Hello!

I hope you are all well.

Time for another round of Coaching Case Studies where I reflect on some of the things I have learned from working with fantastic teachers around the country. You can catch up on the previous issues in this series by clicking the links below:

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### Case study 1

I arrived early on in a lesson about the highest common factor and lowest common multiple of two numbers. The Do Now involved checking students’ understanding of factors and multiples. Questions were projected individually on the board, like this:

This is the third time I have been lucky enough to work with this teacher, and his progress has been phenomenal. Previously, he would have asked for volunteers or Cold Called individual students. But he now habitually uses tools of mass participation in order to check both the effort and understanding of all students, instead of hearing from just one or two. Indeed, eliciting mass participation is now so ingrained in his practice that when I asked him in the coaching session to list the things he did well in the lesson, he didn’t even mention this!

Anyway, students were asked to respond to this question on their mini-whiteboards, and all students were able to write down a factor of 21.

The teacher then did something really smart which I never used to do enough: he asked for a non-example.

Students again responded on mini-whiteboards, and the teacher grabbed a few boards off students to share their responses and to further probe how they knew the number they had written down was not a factor of 21:

This was fantastic. I am a huge believer that in order to really understand something we must know both what it is and what it is not, hence I love a non-example.

In the coaching session that followed, we discussed how the use of non-examples could be made even better. The non-examples students came up with were things like 24 (as in the picture), 37 and 100. In order words, they were *obvious* non-examples.

I made two suggestions.

The first is to change the question slightly: Write down a non-factor of 21* that someone might think is a factor.*

Now students have to think a bit deeper. I hypothesise that had students been given that prompt we might have seen things like:

42 and 210 (multiples)

3.5 and 10.5 (numbers that “go into 21 a whole number of times” but are not factors)

-3 and -7 (negative numbers)

This may also have revealed some misconceptions. If the teacher showcased some of these answers as he did with 24, there may well have been students who disagreed and thought they were factors, and this could have been addressed.

My second (related) suggestion was to use my favourite framework for checking students’ understanding: *Give an example of…*

On their mini-whiteboards, students could have been challenged to give an example of…

The first example is an opportunity for students to play it safe, often coming up with the most obvious example. The requirement to then come up with a second example increases the challenge and tests students’ understanding further. But the third example - an “interesting” example - is where it really kicks off. This is an attempt to elicit a *boundary example* from students - in other words, how close can they get to the edge of this concept without tipping over into non-examples? And finally, asking students to come up with a non-example someone might think is an example is a challenge to see if they can hop over to the other side of that boundary line.

Students fill in all four quadrants independently, then when instructed show their mini-whiteboards to the teacher as a check for effort (there is too much data for a comprehensive check for understanding). Students then swap boards with their partner and mark each other’s work: a tick for examples they agree with, a question mark for anything they don’t. The teacher then asks for examples with a question mark and shares these with the class, because these will invariably be the very boundary examples or misconceptions that the teacher wants to draw out and discuss. Often a student will have an example in the bottom-left quadrant that another student has in the bottom-right, and a productive discussion can ensue.

I discuss the *Give an example* framework in my Tips for Teachers book. I use it to check students’ understanding across many areas of mathematics. For example:

Give an example of… a ratio equivalent to 2:3

Give an example of… a polygon

Give an example of… a linear equation with a solution of 6

With either approach - asking for a non-example someone might think is an example, or going all the way with the *Give an example* structure - it is always worth having a few interesting examples and non-examples up your sleeve that you can discuss in case students do not come up with them themselves.

The teacher is excited to give this a go!

### Case study 2

I saw two lessons in two different schools recently. In both lessons, I was interested in how the teachers circulated around the room to check the effort and understanding of their students whilst they were working independently.

The first classroom was laid out like this:

The second classroom was laid out like this:

Would you like to predict which desks the two teachers spent most of their time near during circulation, and which got the least attention?

Using a seating plan, I tallied up the seconds the teachers spent in different locations during circulation. I then created two rough heat maps, with green signifying lots of time spent at these desks, red signifying much less time, and black signifying no time at all:

Many students’ work was hardly seen during circulation, and some students’ work was not seen at all!

This is clearly an issue because circulation has several benefits. It allows us to check students’ effort, pick up on any misunderstandings, offer words of praise and encouragement, and reduces the chance of poor behaviour.

In both coaching sessions, we discussed two possible solutions.

The first solution is to ensure the layout of the room does not prohibit us from getting to any student. Desks at the end of rows where there is a wall, or desks in the middle of a row where there is a tight squeeze to get through are both problematic. This is not an argument for groups, by the way. I am a big fan of desks all facing the front, but considering altering the layout to eliminate black spots is key.

The second solution is to be strategic with our circulation. It is a good idea to plot a route around your classroom that you know takes in every desk. Perhaps you could be strategic, going first to the desks of stronger students who are likely to have more for you to feedback on early in a task, then later to weaker students, and perhaps circulate twice past the desks of students who you know need extra attention. Plotting the route in advance and practising it until it becomes routine means we have one less thing we need to think about when teaching.

Have a think about the layout of your room.

Do you have any red spots?

How about any black spots?

What could you do to combat this?

**Three final things from Craig**

My calendar is full up for this academic year, but I am now taking bookings for September onwards. So, if you are interested in a workshop, departmental support, or coaching, please check out this page

Last week my Mr Barton Maths podcast had its 2 millionth listen! To celebrate, I have revamped my podcast website with a brand new player, better search, and better show notes. Take a look and perhaps reminisce over some favourite conversations from years gone by.

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Once again, I the teacher have become the "padawan" to the Jedi Maths-ter Barton. Thank you exponentially for the idea of a Non-answer as well as a non-answer that somebody else will think is correct. May I "add" just a single dose of my 3 favourite words, topic dependent of course......What Happens IF...... Example What Happens IF I multiply/divide 3 negative numbers ??? Etc

As always really helpful and interesting. I almost wish I were still teaching in a school so I could use these techniques.