**…***of understanding***.**

Hello and welcome to the 71st edition of our fortnightly newsletter, *Things in Education.*

*Day 1 of a TPD session at a school*

*Things Education:** When writing learning outcomes (LOs), it is important to write verbs that are observable and measurable. So using verbs like ‘differentiate’, ‘describe’, and ‘list’ are better than verbs like ‘know’, ‘understand’ and so on. And your formative assessments should be such that they directly test the learning outcomes.*

*Day 2 of a TPD session at the school*

*Things Education:** We always need to check for student understanding. We need formative assessments for that. *

*Teacher:** I am a bit confused here. Earlier we had said that we should not be writing LOs with verbs like ‘understand’ and that we have to assess directly for the LOs. So do we assess student understanding or not?*

There are two things that are obvious from these interactions. One, this teacher was paying attention all through the sessions and was making connections between various aspects of lesson planning. This is very gratifying for us. Secondly, student understanding is a complex thing and even before we get to that, understanding what understanding is is pretty nuanced.

So let’s explore the nuances of what understanding really means with the use of a simple example—division. What does it mean when we say that the students have understood division? The topic of division has multiple layers to it. Students need to be able to understand the procedure of long division. They need to understand why the procedure of long division works. They need to be able to apply the procedure of long division in different circumstances. They need to understand division as repeated subtraction.

**Unpeeling the first layer: Students need to understand the procedure of division**

This is a procedural skill that students need to pick up and understand the different steps involved. Once they understand the procedure of long division, they need to practise it to master it as a skill. Mastery of the skill happens post understanding of the skill. Practising something builds mastery, but does it build understanding?

**Unpeeling the second layer: Students need to understand why long division works**

Once students understand how long division works, they practise the procedure of long division. During the practice, students can be made aware that when they make a mistake early while performing long division, their answer is going to be way off; but if the mistake happens closer to the end of the procedure, the answer is going to be closer to the real answer. This is just one of the examples of how to get students to understand the procedure of long division. Make them see the patterns and guide them through why differences in the erroneous answer and the actual answer are larger when the error is made earlier in the procedure than later. This should help students understand that when 10043 is divided by 5, in the first step you are dividing 10000 by 5, and so on. This understanding will help students understand why the procedure of long division works. It is repeatedly dividing!

**Unpeeling the third layer: Students are able to understand when to apply the procedure of long division in new circumstances**

This is essentially ensuring that students have a nuanced enough understanding of division (and the procedure of long division) that they are able to abstract this to new situations. Bloom called it “application” in his taxonomy. This comes from students’ understanding of where and when division can be used. This is a deeper level of understanding for students.

**Unpeeling the fourth layer: Students are able to understand that division is repeated subtraction**

The above are all different levels of understanding, but these are still restricted to the procedure of long division. If one is to get a deeper understanding of division, it is for students to make the connection between division and subtraction. By middle school, the connection between division and multiplication is made often, but not the connection that division is repeated subtraction. This is interesting because when the topic of division is first introduced in primary school, it is sometimes introduced as repeated subtractions. When a student is able to articulate why division is repeated subtraction, it means that they have truly understood what division, as a concept, is.

In the example here, we saw how nuanced understanding can be. Not only are there different things to understand within a relatively narrow topic, but there are levels or nuances to the understanding that students build. In the study of Physics, for example, the variety of things to understand within a topic is more than in Mathematics but less than in Biology. Do you think as the variety of things to understand increases, the nuances also increase? Or do you think all subjects, right from Mathematics to Physics to Biology to Geography to Political Science to History, would lie on a line as shown in the diagram below?

The point is that when we look at student understanding or even understanding, in general, there are different things to be focused on to ensure all understanding is recognised, articulated, and then taught by educators.

Going back to the conversation at the beginning, just by using the phrase, ‘*Students will be able to understand…*’ we are masking how nuanced understanding is. So the ‘understanding’ needs to be broken down into its different components, taught separately and assessed separately. We do ourselves a disservice when we use a broad brush to understand understanding. It reminds us of this anecdote about a school teacher who would take her students to watch birds. She would insist that her students observe the birds and not try to identify them. It was her opinion that once the bird name is identified, all observation ceases. The nuances are lost. In the same way, let’s not sweep all understanding away into the same bucket.

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*Edition: 3.19*

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