Is the 'start with the desired result and work backwards' problem solving method discouraged? I'm now in my 3rd university year and every physics problem has been solved using the 'given values, manipulate until you find the final answer' (let's call it 'working forwards', unless there is an actual name).

I think working forwards is a good method for derivation questions, when formulas for the final answer are not known. However, for application problems this often makes the solution unclear until the last step, when it all comes together. In my limited experience working backwards always yields a more logical process. I'd like to know if there is a reason that it isn't used more frequently in solutions, at least when it comes to physics problems.

To illustrate

Given some initial values for variables, and the name of the desired variable, you could approach it in (at least) 2 ways:

Method 1: Take the variables with given values and manipulate them to find a formula for the desired variable. E.g.

Given a, b, c. Find d.

One relation with a is x = f(a,b). One with c is y = f(c). It is also known that d = f(x,y). Substituting, we can now write d = f(a,b,c).

Method 2: Take formulas for the final variable, and manipulate other variables until you get it in terms of the given values. E.g.

Given a, b, c. Find d.

One solution for d is d = f(x,y). It's also known that x = f(a,b) and y = f(c). So we can now write d = f(a,b,c).

• You wont always know what the answer is to work backwards from, so why get in the habit? Working backwards is good for checking, but working forwards is the only way to push the boundary of what is known. Commented Dec 7, 2014 at 17:52
• @ChrisC Like I said, working forwards is good for derivation questions. When either formulas are unknown, or you're trying to find some new relation between two variables, totally work forwards. But for questions outside a research context (e.g. applied problems one might expect in an industry setting), working backwards allows for a clear, logical route to a solution. Working forwards amounts to 'feeling around' until you get an answer. Again, great for discovery, but feels inefficient for typical problems. Commented Dec 7, 2014 at 18:08
• While I agree with Chris C, there is one caveat: a number of problems can sometimes be more easily solved by working backwards. I know that I wouldn't come up with certain formulas just by working forwards. It takes insight and experience to reach them, and working backwards has learning value just as understanding a difficult solution to a difficult problem. Commented Dec 7, 2014 at 18:09
• Here's the argument for working forward: You are mostly learning tools that have proved effective for various cases in the past. The purpose of the problems you are facing now is (hopefully) to acquire facility in applying those tools and understanding their workings. This will enable you to adapt them for new situations that may arise or combine them with other methods. Much of this can only be obtained by working forward, reaching deadends and getting frustrated. Frustration is a not a bug; it's a feature! :) Commented Dec 7, 2014 at 18:12

In both my high school Physics and Calculus classes, I emphasize that problems can be approached both ways: forwards or backwards. A student should try one method, and if that doesn't work he or she should try the other method.

With the multiplicity of formulas at his command, a student may not be sure which information that can be derived from the known facts is useful. Working backwards may be more likely to stick to useful information.

Here is one problem that my Physics class almost always only succeeds with working backwards. Given a mass, with a distance and a time to raise it, find the needed power of a motor to raise the mass (assuming constant power). The way the students get is: $$P=\frac Wt=\frac{Fd}t=\frac{mgd}t$$

Working forward is difficult as they do not see the need to include $g$ (the free-fall acceleration due to gravity at the surface of the earth).

It should also be noted that the classic book How to Solve It by George Polya emphasizes the backward approach. Polya devotes little space to the forward approach, and some of that space concerns combining it with the backward approach. This shows that at least one master of problem solving considered the backward approach to be more important and useful.

My own philosophy is to teach and show both approaches. Use whatever works!

One point I think is worth emphasising is that working forward is not just a good method for derivation questions. When formulas for the final answer are not known, it is an essential method. Consider the following toy example as an illustration of this. Say a confused student has a crazy new theory that 1=2. Working backwards, they multiply both sides by 0 to get 0=0. Seeing this is a true statement, they may think they have shown that 1=2.

I recently had a conflict about this with a pupil. He had answered a question in which you were supposed to determine a parameter such that the equation that contained it had a specific solution. He guessed a value for the parameter and then confirmed it by calculating out the equation.

I took points off because it was possible, and part of the tested material, to derive a value for the parameter by reasoning forward in logical steps. Very often I am interested in testing in whether the student has used logical steps rather than guesswork.

If the problem makes it possible to work backwards from the answer in logical steps I have no problem with that strategy. If it involves luck or guesswork I would be against it.

• This might depend on the subject and level, but I wonder about punishing students for well-chosen guesses (assuming those guesses are then confirmed, of course). I find that my students have the opposite problem: they only want algorithmic procedures, and get uncomfortable as soon as they have to make educated guesses. Having to make choices is already intimidating enough---"what if I choose wrong and waste a bunch of time"---without adding "what if I solve this problem correctly, but didn't read the professor's mind about what method to use". Commented Dec 8, 2014 at 21:58
• The final behaviour we are looking for is of course both: hypothesis and then rigour. I am happy with any mathematical procedure that delivers the result: I recently told my class that if they solved a problem with the cosine rule that was fine, but if they used pythagoras van basic trig in stead that was also fine. Commented Feb 19, 2015 at 10:46
• @HenryTowsner: Tell them about undecidable problems, ad hoc answers, the nonexistence of a magic bullet, and that there is no shortage of things in life that are hard enough to deal with as they stand to say nothing of the absurdly general problems they can be embedded into as instances of. It's sad but at the same time part of our wretched condition. Commented Nov 16, 2015 at 4:40
• If I was your student, this would make me lose interest in the topic you teach. You asked a question, the student solved it correctly and formally, and you took points off because you expected another method. So now the student learned that for your tests they are supposed, not only to know several methods to solve problems, but also to read your mind to guess which method you prefer, and possibly use a method which is not the fastest just because they're guessing that it's the one you want.
– Stef
Commented Nov 9, 2023 at 17:43
• Plus, tests usually have a time constraint; if the student guessed the parameter (which shows good intuition on their part), and they can then very quickly prove formally that this value of the parameter is correct by simply calculating out the equation, why would they use a slower method instead to rederive the parameter which they have already found? This sounds extremely suboptimal
– Stef
Commented Nov 9, 2023 at 17:47