Don’t round up

Let’s take a look at an important component of checking for understanding and responsive teaching.

Three examples from the classroom

Example 1

Example 2

Teacher: What is the first step to solve 6𝑥 + 5 = 29?

Molly: You move the add 5 to the other side, and it becomes a takeaway 5

Teacher: Exactly, you subtract 5 from both sides of the equation because subtraction is the inverse of addition

Example 3

Teacher: How do we simplify 14/21?

Dan: You cancel 7 from the top and bottom

Teacher: Yes, you divide the numerator and denominator by 7 as 7 is the highest common factor of 14 and 21

What is going on here?

In each of the examples above, the teacher is Rounding Up. In Teach like a Champion, Doug Lemov explains:

Rounding Up involves a teacher responding to a partially or nearly correct answer by affirming it, and in so doing, adding critical detail (perhaps the most insightful or challenging detail) to make the answer fully correct.

Is Rounding Up a problem?

I think so. For three reasons:

1. The student thinks they have understood, when really they haven’t
2. The standards in the class fall, because students realise they can get away with half-baked answers
3. The message conveyed is that, in this classroom, the teacher does the hard work

What is the solution?

The solution is to push for excellence. So, we can rework Example 2 as follows:

Teacher: What is a good first step when trying to solve 6𝑥 + 5 = 29… Molly?

Molly: You move the add 5 to the other side, and it becomes a takeaway 5

Teacher: Let’s improve this answer together. The add 5 does not move, but it does seem to disappear from the left-hand side of the equation. What happens to it?

Molly: I’m not sure

Teacher: How did you know that taking away 5 would be involved, because that is correct?

Molly: Because taking away 5 is the opposite of adding 5?

Teacher: Yes. Can you remember the proper words we use for taking away and opposite?

Molly: Subtracting and inverse?

Teacher: Love it. So, what do we do to make the add 5 look like it disappears?

Molly: Subtract 5?

Teacher: Yep, and if we do something to one side of the equation…

Molly: We do the same to the other side

Teacher: Exactly. Right, Molly, let’s put this all together. What is a good first step when trying to solve 6𝑥 + 5 = 29?

Molly: Subtract 5 from both sides of the equation

Teacher: Because…

Molly: Because subtracting 5 is the opp… I mean the inverse of adding 5

Teacher: Superstar

Sure, this takes longer, but it is more genuine, and addresses each of the three problems outlined above.

Check for listening

But what about the other students in the class? Because of the increased length of the interaction, there is a danger their attention wanders.

We can counteract this by adding in some checks for listening.

So, Example 3 could play out as follows:

Teacher: How do we simplify 14/21… Dan?

Dan: You cancel 7 from the top and bottom

Teacher: We need to be more precise with our language, Dan. What is a better word than cancel?

Dan: Divide?

Teacher: What did Dan say?… Sophie?… Yes, we divide. And why did you pick 7?

Dan: Because it is a factor of 14 and 21

Teacher: Tell me what Dan said, and if you agree with it… Mo… Dan, you are on a roll now. And the proper name for the top and bottom of a fraction

Dan: Numerator and denominator

Teacher: What did Dan say the names are for the top and bottom of a fraction… Sam… Okay, so let’s put it all together.  How do we simplify 14/21?

Dan: We divide the numerator and denominator by 7, because 7 is a factor of 14 and 21

Teacher: Loving your work.

What about motivation?

When I spoke to English teacher, Mark Roberts, his advice was the opposite of mine! He said we should round up as it keeps students motivated:

Rephrase to Amaze is a way to boost students’ confidence by taking a correct, but basic, answer and adding in technical terminology and detail. It makes their answer sound really impressive, motivates students, and makes them feel successful.

I respectfully disagree. I think Molly and Dan feel far more motivated by the end of the reworked interactions than they do when we round up their answers.

A good motto I try to live by comes from maths teacher Michael Pershan: End every conversation with the student saying something smart.

Another type of rounding up?

Here is another interaction from a recent classroom visit:

Teacher: I want to share £20 in the ratio 3 : 2. How do I start… Flynn?

Flynn: No idea

Teacher: Well, first, I need to know the number of parts. How many are there?

Flynn: No idea

Teacher: There are 3 parts here, and 2 parts here. What is 3 plus 2?

Flynn: Five

Teacher: Exactly! Okay, so what do I do next?

Flynn: No idea

Teacher: Okay, so I am going to share £20 equally between these 5 parts, so 20 divided by 5 is…

Flynn: 4

Take a moment to consider what is happening here?

The student is clearly contributing to the solution, and what they are saying is correct, but the teacher is once again providing the critical insight. Like in the three previous examples, there is a real danger that the student (and maybe the teacher) concludes the interaction thinking they understand how to share in a ratio.

This one is more difficult to solve. Responding to the initial No idea with some of these responses might help:

1. What question did I ask? (to check the student has actually been listening)
2. What do you know? (to get things moving)
3. I am going to ask three students what they think, and then come back to you to see which answer you think is the best (to provide support, but not let them off the hook)
4. Okay, listen to me explain, then repeat it back to me, and then I am going to ask you a related question (again, this provides support, but does not let them off the hook)

Once again, this takes more time. But I think it is time well spent in creating a classroom environment where thinking hard is the norm.