# How to teach if calculations and algebraic manipulations are off limits

This question may be just too broad in scope, but some form of it has been on my mind during this year of remote learning as I imagine a future cycle of educational upheaval.

Much of what we teach in mathematics involves algorithms: ways to calculate certain things, methods for solving equations, etc.

I am curious what (and possibly how) you would teach if it was assumed that everyone had access to calculating devices (e.g. Symbolab, Wolfram|Alpha) and could use them at any time, and that teaching students methods for calculating, simplifying and solving numerical and algebraic expressions/equations was decided to be outside the scope of what you could teach. [Note: an introduction to how a calculation works could be fine, but it would be pointless$${}^\star$$ to put these types of questions on an exam as the answers are just a few button clicks away.]

($${}^{\star}$$ Unless you were merely testing whether someone brought their calculator that day)

I ask this to determine what things (skills, understandings, habits) educators think remain at (or should be brought into) the core of this discipline in the age of technology that can perform all of the arithmetic and algebra we teach. Note that I am not suggesting that it would be the best scenario to disallow certain arithmetic or algebraic methods, but rather I am asking what, in their absence, we would choose to teach to students. Just what survives this hypothetical scenario?

I am interested mainly in first- or second-year topics in college, such as precalculus, trigonometry, calculus, differential equations and linear algebra.

Are there colleges that have officially adopted such a program, or do you imagine a curriculum or specific set of outcomes for any of these courses that eschews teaching/focusing-on/assessing calculation skills or algebraic manipulation? Would any of these courses cease to exist or might some be combined?

• When I was in high school (in Germany), there were two variants of math courses (depending on the school): With graphing calculator and with CAS (which was then allowed to be used everywhere including exams). I got the former, and even there it didn’t even remotely feel like we were taught algorithms for merely performing them instead of illustrating general principles and so on. In university, the focus was clearly on understanding concepts and being able to apply them. So, your scenario is reality since at least twenty years. May 21 at 7:10
• I use a number of conceptual problems on my tests, that I think are hard to cheat on. Sadly, the more conventional problems do point me sometimes to the students who are willing to cheat. A student this week did a u-sub problem (in Calc I) by partial fractions. They didn't include all of the work, so at first their solutions look like nonsense to me (nonsense with a right answer spells cheating). Of course photomath doesn't know the course. Partial fractions made sense for it. May 21 at 14:56
• For balance, a conceptual problem that was cheated on: I ask the definite integral on a (linear) absolute value function. I tell them to graph first, and then to show me how the graph gives the solution (by triangles). I don't say the key words for my students, which would be "use geometry". So a desperate student looked it up, and integrated the two lines separately. She would never have thought of that herself. May 21 at 14:59

I highly recommend looking at the Calculus exams from the University of Michigan.

https://dhsp.math.lsa.umich.edu/examshops.html

These problems tend to focus on extracting relevant information for solving problems from a variety of representations (symbolic, graphical, numerical, verbal), and on translating between these representations.

Example: Here is a problem where the student needs to translate from a verbal description of some quantity to a symbolic expression.

https://dhsp.math.lsa.umich.edu/exams/115exam1/f18/p5.pdf

Example: Here is a problem where the student needs to extract information about a function from its graph. Problem 4.b.ii is very nice, since it really requires the student to understand composition and one sided limits (even though one sided limits are not obviously indicated). Problem 4.b.iii requires the student to have really understood the graphical meaning of the standard difference quotient.

https://dhsp.math.lsa.umich.edu/exams/115exam1/f17/p4.pdf

The exams are full of superbly crafted questions of this form. Symbolic computation is kept to a minimum: a "gateway" exam which is mastery based ensures that students are developing calculation skill. The essential outcomes of the course center on reasoning and conceptual understanding.

• That is a really nice one-sided limit question. I've never thought to ask a question like that before! May 21 at 12:27
• @MatthewDaly I know! I have been mining these questions a lot, and the gems are frequent. May 21 at 13:20
• This is a nice resource of problems that tend toward translating from words <-> algebra (more so the algebra stuff than the calculus), and the topics are the traditional ones. For your take on my scenario, would math courses maintain the same topics, assessed as in your links, without much change to the content? How would your teaching change if these were now the main focus (e.g. translating a mathematical statement into words)? Could you fill an entire semester's worth of a class with this kind of thing only? May 21 at 15:11
• @NickC Once the words and pictures are translated into symbols (which human or digital computers can process), you can then have a computer do the computations, and further interpret the results. This could open up a host of interesting problems which are infeasible to hand calculate, but are feasible to calculate with digitabl computers. A recent example from my own life involved computing the forces needed to "planche": desmos.com/calculator/b7vj73cxmu May 21 at 15:17
• Be sure that if you're going to include these sorts of questions in your course that count towards students' grades that you teach students the necessary skills and intuitions to solve them! May 21 at 19:05