# Should one justify formulae in middle school?

Consider two possible lesson outlines:

1. Check homework.
2. Show a visual demonstration for the area of a circle, e.g. https://tube.geogebra.org/student/m279
3. Calculate the area of a circle as an example.
4. Students do exercises.

-

1. Check homework.
2. Calculate the area of a circle as an example.
3. Students do exercises.

Also, at some point, write the/a formula for the area somewhere where the students can see it.

Suppose that due to various constraints these are the only two possibilities - we can't, for example, have the students do the demonstration, or have an interactive demonstration. (Possible constraints: planning time, lesson duration, coming up with interesting exercises for several courses is overwhelming for a new teacher, students don't have the equipment to use an online version, etc.)

Particular question: Is step number 2 in the upper lesson plan useful enough to justify the reduced time that students have to complete exercises, during which the teacher gives personal feedback and helps at least some of them?

General question: How useful is it to have the teacher demonstrate (at middle school level and above) or even prove (at, say, high school level and above) things, when the demonstrations are unlikely to be grasped by (all or most of) the students, given that the demonstrations have an opportunity cost: there's less time for exercises and personal guidance?

Edit: Thus far, the responses say that of course one should justify formulae, since doing that is exactly the essence of mathematics, and keeping unjustified formulae in mind is much harder (in the long run).

1. Even if the area of a circle, let us say $A = \pi \cdot r \cdot r =\pi r^2$ (once powers are available), is taken as a black box, the exercises will include reasoning and inference. The simplest example is the case where only the diameter of a circle is known.
2. The answers thus far do not consider the trade-off at all. Certainly there is a benefit to doing more exercises (especially the more difficult ones). A good answer should, in addition to arguing why spending more time on justifying the formulae is good, also argue why it is better then time spent on exercises.

There was also a suggestion to use less time on homework and more on the justification, but since that can be applied to both scenarios above, it does not really answer the question.

• You have a responsibility to serve your top students, not just your average students. – Ben Crowell Feb 10 '16 at 15:46
• Very related, but no good answer there either: matheducators.stackexchange.com/q/10112 – Tommi Feb 14 '16 at 17:13

Yes, of course. As Benjamin Pierce famously put it:

Mathematics is the science that draws necessary conclusions.

If you don't justify your reasoning, then you're not really doing math at all.

Or as I put: then you're doing "faith-based mathematics", which is a much harder task. Most people can't remember a bunch of raw abstract formulas, and those who attempt to proceed in that way inevitably crash and burn at some later point.

If some prioritization must be made, then I would more highly recommend discarding item #1, checking homework in class. Either offload this to out-of-class, or prompt if students had specific inquiries on any items. Require justifications from students on all work to get them in the habit of thinking mathematically.

• General consensus among local teachers is that checking that homework is done significantly increases the amount of homework done. Pupils at this level are not used to saying if they want to have certain exercises checked (but not others) - though maybe they can be taught to do so. – Tommi Feb 10 '16 at 19:42
• @TommiBrander: Then: By the teacher out-of-class. Work enforcement can't be at the expense of actual mathematics discussion. – Daniel R. Collins Feb 10 '16 at 20:35
• Unfortunately, homework can't be part of a grade at my school, so unless homework is checked it simply doesn't get done. I check homework and make it a part of the requirement for retake exams (which are also required). If I don't, the students don't practice independently and fail (even with me reminding them). My suggestion: Have students complete an entry task WHILE you check homework, hopefully one that foreshadows the math discussion you will have later. – Opal E Feb 10 '16 at 21:16

I'd like to answer this question by quoting from something I wrote a couple of years ago as part of the "Teaching Philosophy" that I submitted with my job search:

I look back on my time teaching Nira, and many students like her, with mixed emotions. On the one hand I am proud to have played a role in teaching her that mathematics is, at its core, the science of pattern, reason, and logical argument. On the other hand I am saddened that Nira did not make this discovery until her final year of high school. All of the mathematics she had learned prior to that point had failed to impress her as anything more than a collection of techniques to be memorized, practiced, and deployed when necessary. Even the proofs she wrote in her 9th grade Geometry class had been disconnected from the making of reasoned arguments: a proof, for her, was a particular kind of exercise, requiring a solution produced according to an idiosyncratic and not particularly sensible set of rules. Proofs were never about anything — other than “doing proof” itself.

All mathematics teaching, regardless of its level or the specific course content, ought to be largely about the art of sense-making. The teacher’s job is to ask questions, and in so doing to model for students how to ask their own questions: not only how to solve a particular kind of problem, but also why the problem is worth solving, why a given method works, whether there are other methods that might work as well or better, and so on.

If all you are doing is showing students "a collection of techniques to be memorized, practiced, and deployed when necessary", without any investigation into why those techniques work, then you are teaching your students that, fundamentally, mathematics does not need to make sense. In the long run that does much more harm than any gain that might come from having your students get more practice in the use of those techniques.

The standard of mathematics is rigor, and the standard of science is experiment. Between these two there is one fundamental commonality: clarity and exactness. Without a demand or a demonstration of clarity and exactness there is nothing to gain from an education in mathematics or science. The elimination of vagueness together with doubt is the cornerstone of any quality education in the basic methods of math. Outside of mathematics, education at any level is more a matter of character than content. We teach to give knowledge to students that allows them to live a "good life" as one "inspired by love and guided by knowledge." To teach is to reveal the benefit of guiding one's love inspired actions by knowledge. If your goal as a math teacher is to empower students to develop that personal character needed to live a good life, then clarity and exactness are the standards to which you should hold yourself and others.