Multiple Solutions Methods vs. Encouraging a Particular Approach

It happens frequently in math that problems have multiple possible solutions. This might become troublesome, e.g. when students use some other approach, hence, not learning the current topic.

One could say that it is the task that is wrong, but in many cases examples which require the more complex approach are too hard to use for introduction. This leaves us the "solve using X and without using Y," but it seems artificial.

There are also problems with grading - one would like to check the understanding of the advanced techniques, but appropriate problem might be to hard for a short test.

Are there methods for encouraging the use of a particular approach?

Edit 2: For example when teaching logical connectives, students learn the "table approach" first, and then get to know some other laws, like De Morgan's, distributivity, etc. Then, many times, instead of using a series of reductions, they create a big table (e.g. if there are multiple variables) and do not practice the transformations enough. And then it happens again, when they learn sets and use repeatedly element-chasing instead of algebraic methods.

Another example could be from computer-science. It happens that during abstract algebra courses students learn about matroids and perhaps that some graph-theory concepts are expressible using those structures. Then, in some algorithm-design class the intended proof might be to create an algorithm which constructs the solution (and in the process prove the solution exists), but students might use the properties of matroids, frequently just remembering the facts and not understanding them.

Edit 1: One possible example is when you have to find the extrema of the function $$f(x,y)=x^2+y^2-2x-4y+9.$$ You can solve this by calculus techniques with derivatives, but also by writing $$f(x,y)=(x-1)^2+(y-2)^2+4.$$

• Alternative solutions are always possible, and I encourage my students for example to hand in alternative solutions to exams or examples, even after the official solutions have been posted. An interesting alternative approach in an exam or homework is worth some extra grade. – vonbrand Mar 14 '14 at 12:25
• @vonbrand: To my understanding this is about the rare cases where the alternative ways are not a sign of a creative, out-of-the-box approach to the solution (and are thus not worthy of a reward), but rather of laziness since they do not require any learing effort and are faster for every problem that is feasible for a test. – Wrzlprmft Mar 14 '14 at 16:55
• @Wrzlprmft, then either ask for a specific way to solve the problem (i.e., "use of a truth table here will get no points"), set the problem up so that the "easy," lazy way leads to a quagmire, or just so that the advanced technique is natural. – vonbrand Mar 16 '14 at 16:52

In the example you mention of computing truth tables rather than use algebraic manipulations to answer questions about boolean expressions/sets, it's actually a wonderful situation where the students chooses the long and tedious way, not noticing a much more convenient method works much more nicely. In such cases, I let the student waste time getting the answer in long way, and then I show the short and sweet way. This usually has the effect of burning into the student's brain since they just worked unnecessarily hard to do something and here is a really simple way of achieving the same end-result. Same holds for the example of finding minima/maxima.

I'm of the opinion that if you want the students to use a particular method, then there needs to be a reason to use that method and not any other they prefer to use. If that is the case, then you should be able to convince them, by demonstration, that it's worth their while to learn that other method. We all prefer the comfort of the methods he had already mastered. But we all appreciate learning a new trick if it can help us. Show them the trick.

• It's not always possible. Sometimes the new method is for solving problems, which they can safely assume that won't be on exams (and with homework, you can ask for help). Sometimes the new method is just a different perspective, which will be useful later, but right now is not. And even if the students chooses the long and tedious way, he/she might be so proficient in it, that it's shorter for them (e.g. you don't need to think). – dtldarek Mar 16 '14 at 10:50
• @dtldarek, we learn (and retain!) new tricks if they are (relatively) immediately useful. "You have to learn this [total nonsense] because you'll need it later" won't cut it. By when they need it, it'll have been long forgotten. Use the Just In Time idea in pedagogy too ;-) – vonbrand Mar 16 '14 at 16:56
• @vonbrand I cannot agree with you. For example there are multiple techniques that students learn in linear algebra course, which come to use much, much later. – dtldarek Mar 16 '14 at 19:48
• @dtldarek can you point to any specific such technique that simply can't be shown immediately to the students to be useful? I can't think of a single such situation. – Ittay Weiss Mar 16 '14 at 20:39
• @IttayWeiss Sure, the only thing that is immediately useful is solving linear equations (and related tools and techniques). The rest isn't particularly interesting until multidimensional analysis, and eigenvalues become interesting even later, e.g. with functional analysis, differential equations, differential geometry, spectral theory, etc. Of course, you could find some particular uses, but are not necessary as in "you cannot progress further if you don't know this one". – dtldarek Mar 16 '14 at 21:10

This only works in some cases:

Ask not only for the final result but also some intermediate results which are only produced by the new method. This way the new method becomes more feasible, since the students are required to use it anyway to produce the intemediate results. In your example on finding extrema, you could, e.g., ask your students to also calculate the gradient of the function.

A drawback of this is, however, that you guide your students along the way of solving the task to some extent.

• If you've taught how to find the extrema using a gradient, then saying "use the gradient to find the extrema" is not giving too much away. They either know how to find the gradient and then extrema, or they don't. If they know how to find it but simply blanked on the question, what's the harm in giving them a helpful nudge through the test? – David G Mar 15 '14 at 6:00
• @Skytso: The point of the whole thing is that you do not need to say something like “use the gradient to find the extrema”. Anyway, the nudge in the example is not so much of a problem, but I can imagine other situations, where finding what new technique you can use to find the solution is a slight part of the challenge. – Wrzlprmft Mar 15 '14 at 9:45
• @Wrzlprmft, if that is the case, perhaps the question should be more along the lines of "Select a technique to solve this problem [or even a collection of similar problems], and defend your selection" (without solving it, mind you). – vonbrand Mar 16 '14 at 16:58
• @Wrzlprmft, I totally understand, but it's important to define which technique to use so the student doesn't use a shortcut (unless you're ok with that). E.g. I solved a Calc III problem using a Physics technique. It got the right answer and was a valid technique, just not the one taught in class. – David G Mar 16 '14 at 18:17

Such a situation usually occurs when it is necessary to introduce technique of solving the general problem while the previously known methods are intended for particular cases.

In this case, I always try to explain the reasons that require the introduction of more complicated method. I describe the limitations that "old" methods have and try to give examples that require general approach.

Of course, due to time limitation, general examples sometimes can not be used for initial introduction. In this case it is possible to consider simple example for introduction and give more complex example for home assignment.

The main problems arise during testing. In this case one has to to give simple examples which may allow for an alternative solution. However if prior to the test to announce its purpose, e.g. "test knowledge of particular technique", it can partially solve such problems.

In other words, if the student will know the situations in which the "new" method would be the only possible technique, it will be a more serious approach to the study of this method.

• But "you should use X and should not use Y" can be often circumvented. X can be used in some trivial, unrelated manner, while Y might be used indirectly (e.g. one day during my student days I got a task to give an algorithm without using functions and keywords from some given list, however, it was possible to reimplement the necessary ingredients and use them nevertheless). Then it's hard to grade such a solution. – dtldarek Mar 14 '14 at 9:24
• Your questions has two different aspects: first is how to explain need of use of complex method for simple example, second is how to prevent use of 'illegal' method during tests. My approach is like 'fair-play rule': we studing general approach for problem but consider only some simplified examples 'for education purpose only'. Therefore due to 'educational' aim of task some non-native limitation (e.g. restricted selection of method) is used. Main idea that students should know such restrictions and agree to play the game. – Danil Asotsky Mar 14 '14 at 10:16
• If students don't grasp the advantages of the alternative method, I'd say that you should focus on getting that across. Perhaps by carefully selected examples solved side-by-side (or even given as homework to be done both ways). – vonbrand Mar 14 '14 at 12:23
• @dtldarek, I might have graded your answer with the highest grade. Building your own tools to simplify the answer is the hallmark of a great programmer... – vonbrand Mar 16 '14 at 17:02

All of the courses I remember taking (science and math) from grade school through graduate were very liberal about allowing any solution method to be OK. And I personally tend to be a bit (not extreme but a bit) of a reductionist. Why learn the derivative quotient rule when you can use the multiplication rule and a raised to the minus 1? Why learn the two point formula of a line, when y=mx+b is so much simpler to remember and you can still derive what you need? (These are real examples, even now.) But it is a bit of a pain in the butt when I find myself doing 2 point line derivations (in real life business analysis) and only remember y=mx+b. In converse, the quadratic equation is drilled into my headflesh and I don't need to complete the square! bsquared-4 ac is almost an icon. And if you do chemistry, physics, engineering, quadratics are all over the place. Ready facility is useful.

So as I get older (different learnings, but many physical tasks), I find myself more valuing deliberate practice of specific methods. That's fine that you like to block karate chops with "wax on, wax off" but you need to learn "paint the fence" too. You'll be better off having both in repertoire.

So even though it is artificial, I think deliberate practice in specific methods is good training.