Solutions to exercises

One of the things I have come to discover since becoming more interested in the research behind teaching and learning is that what students like is often at odds with what helps students learn.

(This has huge implications for the correlation between student evaluations and student learning, and how this impacts career progression in academia, but that’s a topic for another day…)

At the end of each semester that my textbook has been used to teach a course at Carnegie Mellon, I have sent a feedback questionnaire to the students in order to find out how it can be improved. Every semester, by far the most common request from students is to include solutions to the (many) exercises.

To illustrate, here are a couple of quotes:

Could you have the answers to the exercises somewhere? It was hard to know if you did it right when there were no answers.

I understand the point of exercises, but to me without solutions they don’t really help. Going to the professors and TAs all the time isn’t very appealing to know if we have done the problem correctly.

My goal in writing this post is to justify my decision to withhold solutions to exercises, by way of introducing inquiry-based learning and discussing some of the research surrounding its effectiveness for teaching mathematics.

But first, in keeping with the principle of ‘if it ain’t broke, don’t fix it’, I should motivate the need to transition away from the traditional teaching model.

Tradition, tradition!

In a traditional undergraduate mathematics course, new material is delivered to students during class time by means of lecture—some courses include additional sessions where an instructor or teaching assistant demonstrates some examples or reviews some concepts. There might or might not be an accompanying textbook, where students can look up definitions, precise statements of theorems and fully fleshed-out proofs.

Assessment is typically by means of regular problem sheets,  a small number (0 to 3) of midterm examinations and a final exam; there might also be quizzes during class time. A student’s grade in the course is usually computed by taking some kind of weighted average of their scores on individual assessments (e.g. 20% homework average, 10% quiz average, 20% midterm 1 score, 20% midterm 2 score, 30% final exam score).

This formula for teaching is so entrenched in undergraduate mathematics education that it is difficult to imagine any other way of doing it. It is so widely accepted as the norm that no-one even things to ask whether there are improvements that can be made.

If you take a step back for a moment, things appear less than ideal.

Under this model, for example, problem sheets are the first opportunity students have to practise using their new knowledge, and yet they must turn in all their answers for credit. Whilst they may have seen examples in class or problem sessions, they have not had any chance to work independently and get things wrong without it hurting their grade.

I would argue that this is unfair.

Let me elaborate. A student’s final course grade should be a reflection of their performance of the course’s learning objectives upon completion of the course. Thus if a student achieves a B grade in a course, you should expect them to have a better grasp of the concepts and skills taught in the course than a student who achieves a C grade. This much I don’t think is too controversial.

If we begin assessing a skill for credit before providing opportunities to practise that skill, then we end up punishing students for acquiring a skill after it is first introduced but before the end of the course. This means that two students might end a course having met the same learning objectives, yet they could achieve different grades because one found it harder at first than the other to grasp the ideas or skills involved.

One way of resolving this problem is to build in opportunities for students to practise a skill or apply a concept before assessing their ability to do so for credit.

This is where inquiry-based learning (IBL) comes in.

Seek, and ye shall find

The main underlying principle of IBL is this: people learn more when they discover something for themselves than they do if someone tells them about it.

There are many flavours of IBL and many possible course design models on the spectrum of ‘traditional lecture’ to ‘fully fledged IBL classroom’.

A fully fledged IBL approach changes the teacher-student dynamic from giver and receiver (of knowledge) to facilitator and inquirer. Rather than copy down notes for later reading, a student in an IBL classroom is presented by the teacher with some kind of task, and is challenged to identify what assumptions must be made to complete the task, to ask questions, to make conjectures and to discover useful facts. The teacher’s role in this process is to guide the student through this process.

Of course, this is at the other extreme of the spectrum; inquiry-based strategies can be incorporated on a small scale, too; indeed, IBL activities could be interspersed throughout a predominantly lecture-based class.

At a very basic level, this could look something like stating a theorem and asking students to find a proof; or indicating what a new mathematical concept will need to do and then asking students to write down a proposal for a definition for that concept; or stating a problem and having students work out a solution.

By having students think in this way about the mathematical content being delivered, they stand to gain better conceptual knowledge. Furthermore, it helps students to practise skills in an environment where they can be wrong and learn from being wrong before being assessed on these skills for credit.

See here and here for just two studies demonstrating the effectiveness of IBL for teaching undergraduate mathematics.

Why there are no solutions in my textbook

Returning to the quotes from students at the beginning of this post, it is evident that the reason why students want solutions is because they want to know if they are correct.

When I was deciding whether to include exercises, I had to answer the following two questions:

  1. Does knowing when you’re correct improve your learning?
  2. If so, what is the best means of finding out if you’re correct?

After reviewing the research on IBL, I couldn’t quite convince myself either way on the first question. If the answer were to be ‘no’, then I would feel entirely justified in omitting solutions without further argument. So, for the sake of playing devil’s advocate against myself, I assumed the answer was ‘yes’ and moved to the second question.

I firmly believe that an important part of mathematical education is learning how to communicate mathematics. Being able to articulate your approach to solving a problem is a skill in its own right, and acquiring this skill is of fundamental importance to succeeding as a mathematician.

(Can you see where I’m going with this?)

By forcing students to communicate with others in order to determine whether their approach to a problem is correct, I am covertly incorporating an IBL strategy into the design of my textbook.

Each time a student approaches another student, their course instructor, their teaching assistant, or even their pet guinea pig, and articulates their approach to a problem, they are strengthening their mathematical communication skills and are performing mathematical inquiry.

To include solutions would be to deny students, particularly those lacking the necessary self-control to avoid looking at the solutions, the opportunity for learning a key mathematical skill and, at that, a skill that is not often explicitly targeted in undergraduate mathematics curricula.

And this is why I did not include solutions in my textbook.

A compromise

Having given my argument that providing solutions to exercises is antithetical to inquiry-based learning, I am now ready to announce a compromise that I recently made. Namely, there are now hints to selected exercises in the appendices of the textbook.

These hints are not really intended for students who are using the textbook; instead, they are intended for teachers: in order for a teacher and a discussion to have a fruitful solution about the student’s approach to a problem, it is helpful (but certainly not necessary) if the teacher has some idea of how to solve it!

Whilst students may find it helpful to look at a hint in order to solve an exercise, it is likely that their long-term conceptual knowledge of pure mathematics will reap greater rewards from speaking to others about it.