# Should we assess ability to use specific problem solving methods, or general ability to solve problems?

This is my first semester being an instructor of record for a college algebra course. One of the sections we cover is "methods of solving quadratic equations", where we discuss factoring and using the zero product property, the square root method, completing the square, and using the quadratic formula.

One of the questions I pondered while creating their exam was: "Should I assess their ability to use a particular method to solve a quadratic equation, or should I assess their ability to solve a quadratic equation in general?" In other words, should I give the quadratic equation and specify which method they need to use in order to get full credit, or should I give a quadratic equation and allow them to use any method they choose to find the solution set?

It seems that the benefit of the former is that it will encourage them to study all of the methods we've learned and practice using them. If I allow them to use any method they choose, they may decide to just learn the quadratic formula and disregard the other three altogether.

On the other hand, I think that being able to successfully figure out which method to use to solve a particular problem is a skill that is worthy of assessing. This is the point that makes me lean toward giving them a quadratic equation and allowing them to use any method they choose. It also seems like posing a question in this way is more "authentic" in a sense. If they get to calculus and attempt to find the critical points of a polynomial function, they aren't going to be told whether they should use the zero product property or complete the square to solve the resulting quadratic.

So I am torn, which is why I thought it would be beneficial to get some feedback from some more experienced instructors. Do you assess your students ability to use a particular problem solving method, or do you assess your students general ability to solve problems?

Another context that this could come up is with series convergence.

• I've not used completing the square and other methods in school aside of quadratic formula. Sure, there was a whole section on polynomials before starting with linear and quadratic equations and there was a chapter explaining where the quadratic formula comes from. There were formulas for polynomial expansion/factorization. After that it was pretty much the same quadratic formula method all the time unless the equation was very simple. Just used the hammer for every nail big and small. Jul 14, 2021 at 22:36
• Maybe you are thinking about things appropriate for an instructor in a much more advanced course. At your level, use questions just like the ones in the textbook. Jul 14, 2021 at 23:28
• I would do both: Pose problems tied to specific methods, and add a question or two where no approach is suggested. Jul 14, 2021 at 23:32
• When teaching College Algebra, I let my students use whatever technique they want, but I make sure to include questions like "solve $(x+2)(x-3) = 0$" to test to make sure they know the zero factor property as well. (although some of my students will FOIL and try to solve it that way). Be aware that your students may be coming into your course knowing other ways to factor quadratics, and may be upset if they can't use them. This is how I learned about the slip and slide method: valenciacollege.edu/locations/lake-nona/documents/… Jul 15, 2021 at 1:36
• @JochenGlueck It's solving an equation like $x^2=16$ by taking the square root of both sides. It's like training wheels for completing the square, since the LHS is already a perfect square. Jul 17, 2021 at 6:49

As a fellow college algebra (et. al.) instructor, I think that knowing specific methods is a necessary thing to confirm. That's what I do on my tests. If you want and have time to add yet another general question (so as to assess both), then that's commendable.

The point is that the specific methods will pop up in future courses in different contexts, and your students need to be proficient enough with them to follow the development. This could be in the context of more advanced definitions, proofs, algorithms, etc. Here are some use-cases off the top of my head (maybe others can come up with better ones):

• Method of factoring: You don't mention that the Fundamental Theorem of Algebra is part of your course. At some point math students should digest this, and factoring is critical to their awareness. Ultimately it leads to the fact that any polynomial of degree $$N$$ can be factored into exactly $$N$$ factors, a wonderfully clear observation. Factoring will be used to reduce lots of stuff in calculus, so getting more practice there is good.

• Solve by square roots: Students should be aware that solving an equation by square roots gives two solutions. This prepares them for higher radicals in the complex plane; cube-roots of unity have three values, etc. And it's needed as part of completing the square. (Also personally I feel it's by far the simplest method, and if a technical person can't do this mentally they'll seem slow.)

• Completing the square: The essential thing is that this is the method by which you'll prove the validity of the quadratic formula. (You do that in class, right?) With this in hand, I find it's actually a fun class moment to see how much the students can help drive that symbolic proof. This activity paves the way for more proof-based courses in the future; you should verify that students have the tools to follow that proof, say. Another application: manipulating general circle equations so as to easily graph them.

In some sense, the quadratic formula is actually the most limited of all the techniques you've listed, because -- while formulas exist for 3rd and 4th degree equations -- a major achievement in algebra was proving that no such formula can exist for equations of the 5th degree in higher. So solving by a formula is ultimately a dead-end. In contrast, at least hypothetically, any such equation still has a predictable number of roots, and can always be factored.

IMO: It's easy to lose sight of the fact that almost everything we do in a math course like this is an exercise, not really a problem. (And I'm careful to use that former terminology.) Does anyone in the real world need to literally solve a quadratic equation by hand? Well, no: you can just use some technology (Wolfram Alpha comes to mind.)

The real point of our math classes is to understand concepts; like, any nonlinear polynomial is factorable (in complex numbers), that $$n$$th roots have (you know) $$n$$ roots, and that we need to verify our truths through proof (such as the techniques of completing the square vis-a-vis the quadratic formula). The test questions we present are merely somewhat-regrettable verifications that the students have indeed absorbed the mathematical facts.

So again on my test at this point I ask specific questions directing that students show me they know how factoring, completing the square, and the quadratic formula work. Separately, a homework quiz has questions on solving by roots, the key step in completing the square, and recreating one step in the proof of the quadratic formula.

If you add another general question so that you've also assessed their general ability, great -- but you will probably find they just use the quadratic formula, so you're not getting much extra information from that. Personally I might lean more towards a written question a la, "Given the following equation, which method would be simplest and why?", but you might get a bunch of pushback on that (and I don't have space on my own tests for it).

• For what it's worth, completing the square also turns up when finding certain inverse Laplace transforms by algebraic means. (I teach that to electricial engineers, by the way.)
– J W
Jul 15, 2021 at 8:58
• "It's easy to lose sight of the fact that almost everything we do in a math course like this is an exercise, not really a problem." Thanks for raising this distinction—I hadn't thought to explicitly make it before Jul 17, 2021 at 14:43
• Students often lose motivation when the fake problems (in writing, maths, etc. class) that they have to tackle feel like meaningless activities, so candidly referring to them as "exercises" instead feels more candid, and is a good reminder that they are primarily for honing skills that have upcoming direct relevance. Jul 30, 2021 at 13:59

Assessment is the measure of progress and decides a student's capability or competence o get into the next level.

General ability to solve problems is a broad term.

But, as you said - "Should I assess their ability to use a particular method to solve a quadratic equation, or should I assess their ability to solve a quadratic equation in general?"

The term general in the above statement is only allowing flexibility to a student in whatever way he/she can solve a problem to score high. But, does it really assess anything as regards to "general ability to solve problems".

In our experience and opinion, " General ability to solve problems" gets enhanced when a student understands and appreciates the need of solving problem using different techniques. Only then, he truly finds beauty and essence of the different methods and automatically generalized in his/her mind the right applicability of a method to solve problems. When he becomes capable in generalizing different methods or finds connection between them, their problem solving ability gets enhanced.