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Imagine that you are teaching a high school class in the last years of high school, an undergraduate class in university, or you are a tutor of a small group at university.

Should one provide examples of concrete calculations in front of the class, i.e., what is a good balance of "telling what to do and just stating the result of an example" and "really calculating everything in the given example including calculations with numbers"?

I have in mind the following concrete situations:

  • You want to calculate a minimum value of a function and have to take the derivative of it, solve an equation and evaluate the function for a given concrete value.
  • In the above example, you want to find the minimal value by evaluating the function at the minimum and want to perform the number calulations.
  • You want to use the inverse function theorem for a concrete function and have to calculate the determinant of the Jacobian.
  • You have explained and derived the abstract algorithm for the LU-decomposition and want to perform this for a concrete example.
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I will answer you from my own experience: I am a lecturer in a Maths Learning Centre at a University. We provide a drop-in centre, and also revision lectures and other resources.

Students' problems with lecturers skipping the "simple" calculations

Students often talk to me about how frustrating it is when a lecturer skips steps, or at least when the lecturer does not indicate that they have skipped steps. This is not just the struggling students, but the ones who are very clever with most of the rest of the course. They stare at it for 20 minutes trying to figure out what happened, when what really happened was some expanding and simplifying.

I have other students who got through their high school maths, or worse didn't and the university let them do the course anyway (economics students are the classic example), for whom it is precisely the basics that they don't know. In particular, they have a shaky hold on the rules for algebraic manipulation of powers and fractions. Now we can work with them with the general rules, and give them exercises to do, but no matter how much you do this, they still need to see these operations in context with the new stuff. They don't know it well enough to embed it in something else, or to have the instinct of when an algebratic operation might be useful. They need to see full examples with the old stuff fully worked out so they can see where it fits with the new stuff.

Even some very high achievers have this issue -- there is some particular algebraic trick they just have never happened to see, and which only appears as useful while doing something new, such as simplifying the result of a quotient rule. These students will be able to see it as algebra and transfer it to other areas. (Others might need to be told that this is a possibility.)

We are in danger here of thinking that algebra is the only problem, but for many students it's actually the arithmetic that's the problem! Don't forget that many of your students actually have very poor arithmetic skills, having done all their maths on a calculator from a young age. They can often do a lot of high-level maths stuff, but have very few strategies to draw on with the low-level stuff. Students often tell me they really appreciate me doing the most basic whole-number calculations because they learn tricks they had never seen before.

These ideas are not restricted to strugglers with prerequisite knowledge, but at many points throughout the course. Often we teach X in one week, then assume they have it down pat next week and skip all the parts involving X when we teach Y. But of course, they are still trying to assimilate X. The classic example is eigenvalues. Many of my students simply don't have good strategies for calculating determinants, so when they try to find the characteristic polynomial, they struggle. Showing the whole process of actually expanding it out / using row operations cements the previous learning.

Finally, on a purely practical level, the students will have to be able to do all these calculations actually by themselves at some point (ie the exam). There is a danger that if they have never seen an example of doing the whole thing including all the little bits, that they will never realise that it actually is all necessary. They will think they know how the process works, but struggle to actually do it themselves, often failing simply because of the calculations rather than the understanding. To never show the whole thing is to give a false picture of your expectations.

Choosing when to do full calculations

As you have probably guessed, my general rule is to do full calculations whenever you can. Choosing examples where the numbers are easy to work with is one strategy, but you must include harder ones too at some point!

If your purpose is to show the structure of the new solution, I recommend going back through the solution you have done and highlighting the overall structure, noting when you were using theory and when you were doing calculations. I find drawing boxes around the different parts of the solution really helps students to see the parts. So does actually putting headings in that say what you are doing eg: "calculate determinant", "solve equation". Indeed, if they learn to put in helpful signposts like this in their own working then you will find it easier to mark!

For processes that do involve a lot of calculation, these headings can allow you to skip parts of the working without it seeming like magic. The notes say what to do and the students are not staring at it wondering what you did.

Finally, if you have the ability to record your lectures, those who are a bit slower can catch up on the details later.

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You can also plug in your computer and show the automated approach, e.g. in Mathematica:

$\mathtt{f[x\_] := x^4 + x^3}$

$\mathtt{Plot[f[x],\ \{x, -1, 1\}]}$

$\mathtt{df = D[f[x],\ x]}$

$\mathtt{Solve[df == 0,\ x]}$

$\mathtt{f[-3/4]}$

$\mathtt{Minimize[f,x]}$

$\mathtt{NMinimize[f,x]}$

$\mathtt{jac = D[Sin[x\ y],\ \{\{x, y\}, 2\}\ ]}$

$\mathtt{Det[jac]}$

This allows you to show several examples, including ones too hard to do by hand.

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  • $\begingroup$ Have you utilized a plan like this in the classroom? I'd be curious to hear how it went. $\endgroup$ – Chris Cunningham Mar 27 '14 at 21:21
  • $\begingroup$ @ChrisCunningham, not this exactly, but a plan like this, yes. As an undergrad I took a class from George Boolos, who showed us software for automating Turing machines. I taught in the Wolfram Science summer school, where we spent a lot of class time doing live Mathematica, and once had a student get a publication out of it. And when teaching linear algebra or second-year calculus for economists, I never used a computer in lecture, but sometimes had twenty or so students in a computer lab doing similar things on their own computers. $\endgroup$ – user173 Mar 27 '14 at 22:16
  • $\begingroup$ I down-voted as this does not answer the question. $\endgroup$ – Benjamin Dickman Mar 8 '18 at 21:30
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I find that too much detail distracts poor students, in particular when calculating numbers.

Almost every student in a lecture wants to understand something in the lecture. Some students may not follow complicated proofs or discussions, and may pay more attention to examples. If you perform an example with concrete numbers, then a lot of students will try to calculate with you and reproduce the same number values. But the speed of their calculations varies. If you as lecturer are too fast, the students will try to follow anyway and then probably miss the purpose of doing the example.

As a teacher, I try to minimize big calculations (when the topic is NOT the calculation itself), especially when calculating concrete numbers. For example, a minimal example can show the concept, e.g., performing an LU decomposition only for a 3x3 matrix with a lot of zeros. Students can work on more sophisticated examples as homework.

As a tutor, you can have a student perform a calculation on the board and then instruct that student. This has the huge advantage that the explanation and the concept come from one person and the calculation comes from a different one. The other students can then recognize from the voices if they have to pay attention to the concepts or if they want to reproduce the numbers.

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    $\begingroup$ Although I've answered my perspective on the question myself, I am very much interested on different opinions or other arguments. $\endgroup$ – Markus Klein Mar 17 '14 at 7:22
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    $\begingroup$ It's OK to ask and answer your own questions ....see there, especially the part about it being "explicitly encouraged" ;) $\endgroup$ – Tutor Jun 20 '14 at 22:07
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I teach high-school Calculus. When I teach a new concept, I almost always give a specific example, then I choose a student at random and ask him or her how to begin solving the problem. After he/she gives one step, I write that down then ask another random student for the next step.

In this way I see whether or not this kind of problem is truly easy for the students. I then adjust my teaching. This means I do not have a standard pace, but then the students really do understand what is being taught.

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When I want to present a problem ("if a man can row so fast and walk so fast, then what is the quickest path from A to B?") in a lower-level math class that is important, but that requires detailed calculation, I generally display a set of slides for it, which I then make available on the class web page. (This is similar to, but I think also meaningfully different from, Matt F. (https://matheducators.stackexchange.com/a/881/2070)'s answer.)

This allows me to present every detail, but removes the problems of requiring the students to wait for me to write the details, and of requiring me to wait for them to copy those details. It also allows for limited interactivity, because I can pause at crucial steps and allow students to fill in details. (It would be fun to have 'choose-your-own-adventure' slides to see the result of expected common mistakes!)

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