What is an effective means to get senior high school students to write their complete working out as part of their answer.

Question

In Australia and in the International Baccalaureate (2 systems I have worked in), for better or worse, mathematics is assessed by criteria. This increases the importance of students to express their mathematical processing and problem solving strategies in their working out - as this is explicitly assessed as part of the criteria.

This is also a source of an infrequent, yet prevalent pedagogical-headache, which is the basis of my question - what strategies are there to effectively train senior high school students to compose legible, coherent and logical mathematical sequences leading to their solution?

Answer 1

This is a very common issue with students. There are several main concerns :

1. Students do not like writing up their solutions.
2. Students do not benefit from writing up their solutions.
3. Students hate to be wrong.

What you can do, is to change the motivations for them to write up their answers.

On Brilliant, I get a 75% submission rate when asking students who got a numerically correct answer to explain their full solutions. Note that this is not 'homework', nor are they required to submit answers in order to proceed on the site. When compared to math circles where this percentage is often close to 0, it is remarkable. [When I was training the singapore IMO team, we had to force the younger students to submit at least 1 solution as an entrance ticket to next week's session. That was the only way we were getting write-ups from them.]

The student solutions I receive on Brilliant are often well crafted, and heavily thought through. Several even start to learn Latex (though it is not required), in order to improve their presentation. Their solutions are then graded on a scale of incorrect, incomplete, correct, excellent. The best solution (up to my interpretation) is presented the following week for others to learn from. What I believe we are doing right, is to influence the motivations for students to write their answers.

1. It should not be seen as a chore, but rather a privilege. This can arise by saying "If you can solve the problem, there is a random chance that you will be asked to submit your solution."
2. Students want to know how much they are improving, and in what areas they can improve. Give them an easy way to see what a good solution is and why. Don't simply point out "You made this mistake", but instead say that "Several of the solutions made the following mistakes".
3. Students want to seek validation from their peers. The fact that their solutions are presented to the community to read, provides strong motivation for them to learn how to improve their answer. I often deal with the lament of "Why isn't my solution chosen to be featured?", to why I'd point out the good aspects of the solution (be it choice of argument, style of writing, , etc.)

I would suggest that you try adopting this method, and see if it works for your class. I would love to hear feedback about it.

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Of course, we also offer them points to submit solutions. However, if bribery is so effective, I'd suggest that you simply bring cookies into class.

Answer 2

The one thing I can suggest, though it's not a guarantee (students can be stubborn in their habits) is to ensure you "model" again and again exactly what a "good" solution looks like. And say so explicitly: "A great solution to [present problem] would look like this...[model "how to" proceed and "what" needs to be stated, "what" needs to be established, and "how"]. This can be done in lectures (you doing the modeling), displayed on a website, handed out after solutions have been, etc. It's great if you can start a collection of samples of the excellent work of "real-live" students (even if they become former students, and with their permission, of course.) This will spare you some of the work of creating model solutions.

Too often, students are told to "show more work", "please explain", etc, in feedback they receive, and end up learning best what they don't do right, rather than how to do it right. Reinforce good solutions, or good parts of incomplete solution, in the work you do get.

When students are given the criteria on which their work is assessed, include some examples of "good solutions, not-so-good solutions" to help students acquire (or begin to develop) some internal sense of what they need to do.

In the end, though..."you can lead a horse to water, but..."

Answer 3

I will start by expressing the limitations of my experience. It has been almost 30 years since I was an Australian High School student. My teaching experience consists of 2 semesters of statistics to MBA students (who have widely varying backgrounds and therefore widely varying levels of numeracy)

There are 2 general approaches that I have seen used with varying degrees of success.

1. Actually breaking the question into parts (that each require working) with each part generating an intermediate result, and each part being worth marks. This is a good approach for younger students.

2. Being very specific with the marking criteria. This can follow the break it into parts step but does not have to exclusively. For example a marking criteria on a 10 mark question my be:

   • Correctly restating the question in algebraic terms (3 marks)

   • Correctly demonstrating the method of solving the problem (3 marks)

   • Intermediate calculations (3 marks)

   • Getting the right answer (1 mark)

It may also help to point out to your students that there is very little difference between constructing a good argument in mathematics and constructing a good essay in English or History; viz:

1. They state a position

   • Maths Prove Pythagoras Theorem

   • English Compare and contrast the journeys undertaken by the title character in "Othello" and Albert Facey in "A Fortunate Life"

   • History "The emergence and growth of Hitler and the Nazi Party were a direct consequence of the Treaty of Versailles." Discuss.

2. They require the statement of facts

   • Maths Axioms, arguments by others (i.e. well-established theorems)

   • English The primary texts, the definition of "journey", arguments by others

   • History Primary & secondary sources, arguments by others

3. They present an argument

   • Maths Calculations

   • English Testing of the facts

   • History Testing of the facts

4. They draw a conclusion

This may help those students who tend to think more in a "humanistic" way than a "scientific" way.

Answer 4

Looking at this from the perspective of a student, I never wanted to write out all of my working because to me there were steps that seemed obvious, and all that was needed was the answer. The most effective incentive in getting me to show my full working was always the same: an answer alone, if incorrect, gets zero. Not just no marks, but no help either. An incorrect answer with full working shown might get you partial marks for the bits that are correct, and the teacher can point out where you went wrong along the way.

I remember many questions that made this explicit by presenting an initial question, and then saying something like this:

a) Find the value of x

b) Use your answer to a) to prove this other thing.

c) Using your answer to c) (or otherwise) show that y = 2 in this other, related equation.

As you progress, exams stop phrasing it like this and start lumping them together into "Find the value of y in this equation" and expect you to work backwards, using the other thing from b) above in order to get a simpler equation, solving that for x and then substituting back into the first equation to solve for y. Having seen questions like this broken into parts before, students tend to spot that this question is worth a lot of marks and realise that there are likely to be marks for intermediate steps.

One way to help here is to let the students see the mark scheme, or at least an example one, so that they can see the breakdown. When students see that a 15-mark question only gives 2 or 3 marks for the number at the end, and the other 12 or 13 marks are spread around various key stages of the working, they'll start showing the working.

Essentially what this boils down to is this: if you want students to show their working, make it worth doing. People (students especially) are much more likely to do something if there's an actual benefit to them in doing it. This may be a better shot at getting the marks, even when they go wrong partway through, but even when the mark scheme isn't yours to control, you should stress that showing their working also lets you point out where students went wrong so that they can identify and improve on anything they don't understand fully. For me, that was often a stronger incentive than getting the marks.