The opportunity cost of digging deeper ⛏️
Why pushing for depth is not always optimal
This newsletter is made possible because of Eedi. Check out our brand-new set of diagnostic quizzes, videos, and practice questions for every single maths topic, ready to use in the classroom, and all for free, here.
Once upon a time…
Recently, I was watching a teacher deliver a Do Now. As many maths teachers do, this teacher was using a 5-a-day, from the fantastic Corbett Maths. This particular 5-a-day looked like this:
As we circulated the room, it became clear that students were having particular difficulty completing the table of values for Question 4. Some students had left it out:
Others had made mistakes::
When it was time to go through the answers, the teacher employed the Book-to-Board approach that we had discussed the previous day. So, when instructed, the students copied their final answer to Question 1 onto their mini-whiteboards and showed the teacher.
It was clear that every single child had the correct answer of -1. But the teacher wanted more. He wanted to ensure his students could articulate why this was the correct answer. So, he chose a student at random and asked them to explain their answer. The student struggled, stumbling over their words and giving an unclear explanation. So, the teacher stuck with them, offering prompts and hints, but holding out for excellence.
The teacher then moved on to Question 2, where it was much the same story. The students showed the correct answer on their board, and the teacher then asked two students to explain their reasoning.
The same happened with Question 3.
I timed this exchange, starting my timer at the point students held up their answers to Question 1, and stopping it at the point students were asked to copy their answers to the troublesome Question 4 on the boards.
8 minutes had passed.
The opportunity cost of going deeper
Nothing the teacher did during those 8 minutes was bad practice or a waste of time. Indeed, pushing for excellence as he did, ensured he had extra information about the depth of his students’ understanding, and conveyed the message that in this classroom language and reasoning are important.
But the question I put to him during our subsequent coaching session was: was this the best use of those 8 minutes?
Every decision a teacher makes has an opportunity cost, as the time we spend doing one thing is time we cannot spend doing something else.
The 8 minutes spent probing deeper into Questions 1 to 3 meant that the teacher was unable to spend as much time as the students needed when addressing Question 4. The mini-whiteboards revealed - as we had suspected - that students’ understanding was not secure. The teacher then, very smartly, offered an explanation and rechecked for understanding. But, with one eye on the clock, and almost 20 minutes of the lesson gone, only felt he had time to do this for positive values of x.
And so, all misunderstandings with negative numbers were not addressed in that lesson.
A better use of time?
Let’s imagine that instead, having seen that all students had the answer correct to Question 1, the teacher confirmed the correct answer and moved on. And then did the same with Questions 2 and 3.
The price he would pay for this is that he would not be able to probe the depths of his students’ understanding. But what he gains is 6 or 7 minutes that could be spent on the question that students most need his support on. I feel that would be a better use of our finite class time.
So what?
I think the lesson here is that if we have evidence to suggest our students’ understanding is secure - mini-whiteboards are fantastic for this - and also data to suggest that understanding may not be quite so secure for a later question - circulation is key here - then we need to prioritise our finite time and allocate it where it is needed most. This is not ideal, but it is sensible.
Then, if it turns out that understanding is secure for all questions, we are free to probe further in the exact way the teacher did.
Can you relate to this?
What do you agree with, and what have I missed?
Let me know in the comments below!
🏃🏻♂️ Before you go, have you…🏃🏻♂️
… checked out our incredible, brand-new, free resources from Eedi?
… read my latest Tips for Teachers newsletter about stepping away from the board when modelling?
… listened to my most recent podcast about surviving and thriving an Ofsted inspection?
… considered booking some CPD, coaching, or maths departmental support?
… read my Tips for Teachers book?
Thanks so much for reading and have a great week!
Craig
I usually have students who are reluctant to share their answers on a mini white board. I encourage them the growth mindset of course, but it is not easy, and some of them are always stressed to show their answer fearing that they have incorrect answer.