I was watching (as a fellow teacher) a lecture on arithmetic progression. It so happened that a formula was rapidly introduced and used.

The formula was $a_i = a_1 + (i-1)r$ (I trust the terms meanings to be easily guessable, but they were, of course, explained in the lecture)

I could not shake the feeling that it would have been more productive for the students to do some exercises before this abstraction, to "get a feel" for the subject.

Does your expericence as a teacher agree with my intuition? Is there some research on the matter? (I don't mean just for progressions, but in general. Is it reasonable to do some exercises without the formalization to get a hang of things?)

  • $\begingroup$ If the terms were explained in the lecture, then it might have been fine to introduce the formula. It's hard to tell without having seen the lecture, and without knowing the level of mathematical maturity at which this lecture was aimed. $\endgroup$ – Mike Pierce Nov 6 '17 at 15:46
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    $\begingroup$ Fairly low maturity. Not all of them could solve a first degree equation. At any rate, my intuition is still that if the equation was not obvious to them at first glance, then it should have been introduced after some exercises, and if it was, then doing some exercises before would not have been a problem $\endgroup$ – josinalvo Nov 6 '17 at 15:56
  • $\begingroup$ But, of course, if your intuition/experience differs (or better yet, if research differs) I would want to know $\endgroup$ – josinalvo Nov 6 '17 at 15:59
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    $\begingroup$ Get students to build some informal understanding of a problem so they could solve (or at least reason through) it without an abstract formula? You are describing my intent in every single class I teach. So, in short: Yes, I agree with your intuition. $\endgroup$ – Nick C Nov 6 '17 at 16:41
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    $\begingroup$ The best way to get students to understand a formula is to make them derive it themselves. So, I'd start with some examples and let them try to calculate, so I don't give them the formula, they do. $\endgroup$ – Dominique Nov 7 '17 at 8:39

I think it varies. Advantage of the more intuitive approach is that you get a feel for the topic before the formula. Advantage of the formula is you see where things are headed. I would say if the formula is complicated (TO THE STUDENTS) than the intuitive approach is better. If the exercises are difficult and the formula reasonable than the formula firs approach is better. In this case, I would prefer the examples first.

I don't agree with "theorem proof is how everything is going to be". Don't think this applies to the vast majority of students that learn math to use in science, engineering, business, medicine, etc. (rather than abstract math majors.) IOW, that's a bug, not a feature...

  • $\begingroup$ I agree with James (comment in different answer) that a lot of times "motivation" can be confusing if the student doesn't at least know where we are heading. That could mean the formula or could even just mean mentioning the topic or saying that we will do a few examples and derive a rule. Sorry to be wishy washy but it's a balance. I don't think there is one right way all the time. $\endgroup$ – guest Nov 7 '17 at 18:30
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    $\begingroup$ If someone is introducing motivation without saying what they’re doing, that is of course wrong. This problem is obviated by following the usual rhetoric advice: “Tell ‘em what you’re going to tell ‘em, then tell ‘em, then tell ‘em what you told ‘em.” $\endgroup$ – G Tony Jacobs Nov 8 '17 at 0:11
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    $\begingroup$ Great point. But I have seen too much intro be confusing at times. For instance, it's probably easier to grok a 2nd order constant coef. diffyQ as just a problem to break up and solve analytically ("solve for y") than it is to have an example mechE problem with damper and spring and physical input force (jackhammer, landing gear, etc.) And I say this as someone who LOVES physical problems and physical insights. Maybe it is enough motivation to say at the beginning, that this helps with problems in quantum theory, controls, and EE (just as a remark, but no intro problem.) $\endgroup$ – guest Nov 8 '17 at 18:36
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    $\begingroup$ Ah, you're talking about a whole different kind of motivation than I'm talking about. Like, in the OP, before showing them a formula for a general arithmetic sequence, give them an example one, and ask them to work out the 100th term, without writing them all down. If they do that for a couple of examples, they'll have a sense of what qualities are essential to describing and working with sequences of this type. Then when we give them names such as $a_1$ and $r$, they have notions in mind to which they can attach the variables. I wasn't thinking of motivating by talking about applications. $\endgroup$ – G Tony Jacobs Nov 8 '17 at 18:58
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    $\begingroup$ For the example in discussion, I think showing the example first is useful as the equation looks complicated and the student is more used to seeing a sequence (IQ tests, etc.) than an abstraction to a common term. Sequences and series are just like that. Yes, I agree there are a lot of types of intro/motivation, etc. and different situations call for different things. $\endgroup$ – guest Nov 8 '17 at 20:04

My experience is exactly the opposite: that the abstract formula should be displayed as soon as possible, then justified, then exercised. My philosophy on this has been developed by the idea that I want the formula in the student's visual field for as much time as possible, in the hopes that it will sink in mentally. Note that this is sympatico with the requirement in many locations that the teacher clearly state the goal of the lesson at the start of a meeting. Plus: The cycle of theorem-proof is simply traditional mathematical presentation and writing style (for exactly this reason, I think), and students should get to experience and expect that style of presentation as a "real" math class.

Personally, I always get weirded out when I see instructors doing the opposite. They seem to take most of the presentation time doing these warm-up exercises, and wind up squeezing the ultimate goal in the last few minutes of class (and not actually exercising the formula itself). As both a teacher and a student myself, I feel that we all get confused about what the "point" of the lesson is, what the real take-away skill is, and how it will be assessed in the future.

I understand that many of us wish that we could lead all of our courses and students in "discovery" style lessons where they take personal ownership for all the new material. But as teachers this is simply infeasible granted the limited time in class; and particularly so from my perspective in the college classroom (even if much of my career is spent remediating topics from middle school).

(I was reading material by Hung-Hsi Wu recently and he does this always-concrete-warm-up cycle, and while his material is otherwise excellent, I find this distracting and it forces me to flip back-and-forth a lot to uncover the "real" proofs.)

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    $\begingroup$ In retrospect, much of what I did not understand as an undegraduate was probably professors attempting to "motivate" things. Motivation must be properly outlined, otherwise, as students it's easy to confuse the cumbersome motivational-mode thinking with the actual intended take-away. $\endgroup$ – James S. Cook Nov 7 '17 at 5:14
  • $\begingroup$ @JamesS.Cook: Exactly. Well put. $\endgroup$ – Daniel R. Collins Nov 7 '17 at 16:08

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