I give my Grade 11 Mathematics learners homework almost every day. The work they do during class time is simply not enough to develop the fluency they need, so each day I send them home with additional problems that they should do on their own. When the learners arrive in class we review the homework from the previous lesson before beginning on the next section of the work. “We review the homework” sounds so simple, but it’s not!

In this post I am going to avoid “flipped classroom” ideas and the entire “Is homework necessary?” debate. I may deal with those in later posts (which I will link here). For now I am writing about a traditional teaching arrangement, and accepting that homework is a necessary part of learning for senior math learners. So: homework from a teacher’s point of view, where my main concern is to identify the best use of the learners’ (limited) time in class, while remembering that learners are by no means all equal. This, the first post in the series, analyses homework outcomes: what can the learner have done?

(Image above from Flying Colours Maths,  where Colin has written a rather nice post about what parents’ attitudes to math homework should be!)

Homework outcomes

Let us consider just one of the problems they were given to complete – for any learner we may find:

• He didn’t attempt the problem (this actually includes “He copied the solution from a friend”, “He Googled the solution” and “His dad did his homework”)
• He tried, but didn’t reach any solution, or his answer was wrong
• He correctly solved the problem

We can break these categories into various subsets. Without claiming to be exhaustive, I came up with the following.

Problem not attempted

• Learner is completely intimidated by the extent to which he is out of his depth and after walking out of class today does his best not to even think
  of math until he arrives at class tomorrow and realises that oh shit we had homework
• Learner looked over the problems, realised he knew how to do them all and decided not to waste his time doing stuff he already knew
• Learner had an important hockey match or operetta practice, or was attending extra physics classes and by the time she arrived home at 7:00 in the evening was simply too tuckered out to even think about homework. Variations on this theme can include going to a party, spending the afternoon looking after younger children, visiting the mall, etc.

Problem attempted and solution incorrect

• Learner didn’t know what to do, wrote down a few algebraic manipulations or computations and came up with some kind of solution. No effort was made to analyse the problem or review class notes and examples. (Didn’t know, didn’t try)
• Learner didn’t know what to do, struggled to understand the problem, made a few steps in the right direction but didn’t obtain the correct solution. This may have been due to misconceptions, procedural or computational errors, or simply giving up. (Didn’t know, tried, didn’t succeed)
• Learner knew what to do before beginning the problem, went through the motions but made a procedural or computational error (typically a sign error!) along the way. (Knew how, slipped up)

Problem solution correct

• Learner knew what he was doing before beginning and produced a textbook perfect solution. (Knew how, did it)
• Learner didn’t really know how to do it, picked a few random algebraic manipulations or computations, applied these and by luck or coincidence obtained the correct solution. (Didn’t know, got lucky)
• Learner approached the problem not knowing how to do it, but by doggedly following the example was able to emulate the steps and obtain a correct solution. (Didn’t know, followed steps)
• Learner approached the problem not knowing how to do it, but by connecting ideas and through insight and reference to notes from class, gained a good understanding of the solution process. (Learned!)

What is actually important here?

When you look over this list, one thing that is important to notice is that whether the solution is correct or incorrect is not actually the important thing here. There are better ways to classify the homework. One stab at this is the table below, where we look at the situation when the learners arrive back at school with complete or incomplete homework.

Quadrant model of homework responses

Just an aside here about “cognitive effort”: what I am referring here to is the level of thinking required, the mental effort needed to figure things out, to connect new ideas to pre-existing ideas, to developing understanding (for more on this see my earlier post on Schemas and Learning).

We can use this table as the basis for a different classification of homework outcomes:

• The learners in Quadrant 1 (Low Effort/Cannot Do) quadrant gained nothing from the problem.
• The Quadrant 2 (High Effort/Cannot Do) learners may have benefited, depending on whether their efforts developed new connections between ideas, gave them insight into mathematical relationships, etc.
• The learners in Quadrant 3 (Low Effort/Can Do) should gain speed and fluency from doing the homework. Their ability will be bedded in, mathematical connections reinforced and so on.
• In Quadrant 4 (High Effort/Can Do) we have the learner who, by completing the homework has made progress toward understanding.

Back to homework review

Now that we have a classification of homework outcomes, we can begin to analyse the usefulness of the process of homework review.

Quadrant 1

Those who didn’t do the homework at all or didn’t put in any cognitive effort are not going to benefit from seeing the solution except for being able to copy it down as an additional example. Most of them are going to copy without engaging with the content and may never return to review the problem. Without engagement and cognitive effort these kids are not going to learn; I need to induce an attitude change to move them to another quadrant.

Quadrant 2

The “Didn’t know, followed steps, succeeded” learners may benefit from a solution on the board if they are actually engaging with solution process in a way which will improve their understanding – in other words, depending on the cognitive effort they are expending in reviewing the correct solution. And finally the “Didn’t know, tried, didn’t succeed” learners need to see the steps of the correct solution in order to see where they went wrong and this is potentially an important learning opportunity.

Quadrant 3

For the low effort learners who knew how to do the problem, the correct solutions are in the back of the textbook (just the answers, not the steps). The learners who knew how to do the problems can confirm their answers. For those who slipped up, the key to benefitting is that they should self-diagnose their error so that they don’t keep making the same mistake. Although it is time-consuming from the learner’s point of view, self-diagnosis is a far more useful learning opportunity than using the teachers’ solution to identify the error. The learners in this quadrant should not need to see the teacher’s solution (which may, in any case, be just one of a few different possible methods).

Quadrant 4

The high effort learners who succeeded in getting the correct solution can see it in the back of the book, punch the air in triumph and move on. The learning experience occurred the previous afternoon when they struggled through developing understanding. They also should not need to see the teacher’s solution.

We need to review homework in a way which will benefit the last two categories of learners, while at the same time not disadvantaging the learners who by now are able to successfully solve this problem. And (rather unforgivingly) without having the whole process brought to a virtual standstill by the Low-Effort-Can’t-Do-It learners.

Part 2 of this series will be linked here.