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Let them discover $\;\ell=2\pi R\;$ by their own, at least the invariance of $\ell/R$

Do you think that the following method works well in a mathematics class of elementary school? What do you think about the idea that math education system would remove the dictation of the math formulas, instead, it would encourage the students to observe some part or the whole formula themeselves? In the following post I mainly focuse on "The perimeter of circle" in elementary schools.

As a generalization of this question, at the last lines of this post, I will extend this question to high school geometry, for example sphere area.

Do you have any ideas or similar suggestions or some situations you already experienced? When you were student, how did you feel about memorizing and be dictated of mathematical formula?

Here is my suggestion about discovering processess of circle perimeter formula in elementary school:

"Let's distribute a few circles, with different radii, among some different groups of students of elementary school. (Circles that are made of flexible wires and easily form a line segment. Or we can use some pulley and thread or-string. The radius of pulley is easilly measurable. The perimeter of circle is the length of thread ) Then let the students calculate the ratio of the perimeter of the circle to the length of the radius and they compare the result of each other. We do this in a situation where they are not yet familiar with the formula for the circumference of a circle. But they would observe that the ratio of perimeter by radius is independent of our chosen circle.They would observe that the ratio is the same for measurments of all groups. Perhaps they would be surprised.

We do not expect them to discover the number$\pi$ but hopefully they observe and discover that $\ell/R$ does not depend on datas so is a universal constant."

Generalization to high school geometric formulas: How can one extend this idea to high school problems for example observing that the surface area of sphere divided by $R^2$ is independent of chosen sphere?

Your answers and comments are very appreciated.

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    $\begingroup$ You could have the students cut circles into "wedges" and arrange them into near-rectangles, then get the area of those rectangles using LxW. $\endgroup$
    – Opal E
    Jun 3 at 21:19
  • $\begingroup$ @OpalE Thank you for your comment. I am asking about perimeter. Any way how can they discover the invariance of $\ell/r$?I think the triangulation you suggest destroy the accuracy. Right? $\endgroup$ Jun 3 at 21:24
  • $\begingroup$ @OpalE We assume they are neither aware of the standard formula of perimeter nor of $\pi$. they need to discover $\ell/r$ is an invariant! So they need to have very accurate measurement $\endgroup$ Jun 3 at 21:28
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    $\begingroup$ What do you want in an answer? A yes/no question doesn't take us very far. My answer is yes. My college students can tell me that C=2πr, but they (often) don't really get it. Lots of circle measurement in elementary school would be great. The sphere thing doesn't make sense to me. How would you measure that? And it's a different sort of question entirely. C=2πr comes from the definition of π (C/D). The rest come from building up the mathematics (theorems, etc). $\endgroup$
    – Sue VanHattum
    Jun 4 at 0:27
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    $\begingroup$ I am confused about what your question is. The title suggests that your question is asking whether you should use this method with elementary students to discover the that ratio of the circumference to the radius is a constant. But upon reading your question, you seem to be asking how to extend this method to high school students for a sphere, to discover that ratio of the surface area to the radius squared is a constant. Which question are you asking? $\endgroup$
    – Amy B
    Jun 4 at 9:22
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Yes, it is always good when students independently discover mathematical facts, in contrast to being told that some old Greek dude figured it all out 2,500 years ago. It makes students feel useful and participatory, it prepares them for a future where they will be investigating phenomena where formulas aren't as well established, and the experience forms mental pathways that you don't get from reading a formula in a box on a worksheet.

If anything, I think your exercise is rushing the students towards quantification. Have the students take a piece of string that is the length of the circumference of their circle. (Dental floss works well as an inexpensive, disposable, and easily cuttable string.) Then, using their floss, they can measure how many times they could go straight end-to-end across the middle of the circle. Then ask the students to compare their results with their neighbor and they will find that they all came up with "a little more than three". From there, you can dive into calculations, but that will verify the students' conjecture instead of forming it.

As far as the surface area of a sphere goes, it's a formula that you never forget after doing the orange peeling activity. My tip is to have a knife to cut the oranges in half so you can have a good (albeit slightly runny) template for your circles, and to make 6 circles instead of 4 to add some uncertainty of what the final result will be.

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  • $\begingroup$ "Dental floss works well as an inexpensive, disposable, and easily cuttable string." I've always preferred a strip of paper, maybe folded once for added rigidity, and held on its long edge. A bit easier to bend into a circular shape and mark with a pen for measuring its length. [Also, there's a ton of scrap paper I can pull from the recycle bin, so no buying floss from the store.] $\endgroup$
    – Nick C
    Jun 4 at 18:29
  • $\begingroup$ Thank you very much and +1 for your very interesting answer. $\endgroup$ Jun 11 at 15:08
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I'm NOT so sure that one should emphasize discovery learning for the FIRST learning of a topic. Humans are imperfect computing machines. You have to consider the ma...person in the loop, as the military would say.

Also, young children have high capability and even enjoyment of recognition and memorization. Consider the game "Go Fish" for instance, or even the decades long affinity they have for memorizing state capitals. There is time. When kids are in calculus, the will learn a different reason, intuition for geometric formulas.

Note: I am not 100% opposed to discovery learning, even at young ages. But I would just be cautious about thinking this is the only path to Heaven.

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Over all I think the idea, of hands on for something like C/D is an invariant is laudable, but there are a few caveats.

First, students at that age are very aware of precision. The answer "the ratio is 3 and a bit" is fine, but they will not accept it is a constant if one group measures 3.1, the second group 3.2, a third group 3.15. Indeed, in that case you end up asking them to put aside their concrete observations and accept the teacher's assertion "it's your measurements that weren't accurate enough", which will be counterproductive - it says "you're not good enough" and "were you a better student you would have measured this right." Both false, and counterproductive to encouraging an interest in the mathematical arts.

You also need circles over several orders of magnitudes. If all students have circles about the same size, say 1 to 5 inches, that is not ideal; again the variability in measurements will make them skeptical that what you state as a truth is in fact a truth they have observed. Better to also go to the playground and measure some HUGE circles too, say some with 10 or 20 foot diameters. Switching to metric: a centimeter, 10 centimeter, 1 meter, 10 meter diameters. After seeing that circles of very different sizes all have SIMILAR (and with the sloppiness of elementary student measurement, NOT IDENTICAL) C/D ratios, you can PROPOSE that the C/D ratio is a constant/invariant. If you are pursuasive, you MIGHT convince them that the ratio COULD BE invariant. If you are lucky the students will begin (and just begin) to believe you that the ratio of C/D is invariant. So then what?

Trying to figure out the exact number, pi - 3.14159... is beyond elementary school. Honestly, beyond high school. But you can get them to believe it is a number that is bigger than 3, bigger than 3 1/4, smaller than 3 1/2 or 3 1/7. And you can use it to tease, foreshadow, anticipate ideas in middle school and high school math (the existence of irrationals, transcendentals, etc)

And then you CAN bring up that it's hard question, and that very very clever people, like Archimedes, Zu Chongzhi, Liu Hu, Madhava, and Aryabhata did a huge amount of work to get the value more accurate - and importantly, between upper and lower bounds. And that getting the value of pi to better than 1% (two digits) precision took on the order of a thousand years.

(note, one of my pet peeves in teaching is how often ideas that took decades to millennia for the brightest minds our earth has ever produced to come to grips with are insultingly and demeaningly presented to students as ideas so simple and obvious that the student must be "stupid" if they don't grasp the concept as the instant it comes from the teacher's lips.)

All this can tie into conversations about how good is good enough a measurement. The idea of precision. And the difference between an exact answer, which in some ways can't be had because the number is "irrational" - and conversations on what that means, and "good enough" for - building something, and that different precisions are needed for different tasks. And that a mathematician (as opposed to a carpenter, baker, engineer) wants to know it EXACTLY.

Anyways, just some quick notes about, sorry not more coherent.

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  • $\begingroup$ Thank you very much for your attention to my question and your very interesting answer.Yes I agree the accuration on computation of $\pi$ is a serious problem in particular in that age. To be honnest my objective was the following: they would realize that this formula implies that C/D is independent of the circle we choose. Please see the comment by Madam Sue Wanhattum. In fact there is no guarantee that people get the geometric meaning of this formula. You pointed out to "that age". Is it a good idea they learn this formula 2 years later?Is this formula a prerequisite fo their subsequent $\endgroup$ Jun 13 at 12:22
  • $\begingroup$ subsequent study? $\endgroup$ Jun 13 at 12:22
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    $\begingroup$ My experience is that the concept "C/D is an invariant" is too abstract for many elementary school students. There is a story (from Feynman)? about teaching soldiers, possibly without high school education, the rudiments of electronics. It was necessary to teach Ohm's THREE laws; V= IR; I =V/R, and R = V/I. It's an abstract jump from learning a given formula like V=IR to understanding Ohm's Law describes the relationship between the three quantities. Similarly, students can feel comfortable calculating with C = 2πR, but realizing C/D is invariant is a higher level of abstraction. $\endgroup$
    – Michael G
    Jun 15 at 18:42
  • $\begingroup$ Yes I see that is an abstract subject. myself I learn the formula at age 11, but with $\pi=3.14$ not moredigits. I completly belived this lie at that agae. But i remember the teacher call me to blackboard and said me this question: A wire of length , (say 1), is formed into a circle find the diameter of cicrcle. This is a part of mtivation of my current post. But I agree with you that the invariance of C/D is a very abstract subject. $\endgroup$ Jun 15 at 19:07
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The question of when and why less guidance works or doesn't work is far from obvious. Instead of providing my personal opinion, as many have already done in other answers, I will try to collect some pointers to research that has already tried to answer this question, which you can use to critically review your own approach.

Andrew Blair in his article Inquiry is not discovery learning distinguishes between discovery and inquiry learning, as a response to critics of discovery learning. He makes the following distinction:

  • discovery learning, which he defines as when "students are expected to derive a procedure or concept from an activity devised by the teacher"
  • inquiry, where the procedure or concept that appears at the end of the discovery process is incorporated into the course of an inquiry

In discovery learning, the teacher attempts to preserve the pretence of discovery, even to the extent of withholding knowledge; in inquiry, the teacher, as a participant in the classroom activity, aims to introduce subject-specific knowledge when it is most relevant and meaningful to her students.

Clark et. al on the other hand, in Putting Students on the Path to Learning advocates for fully guided instruction:

Decades of research clearly demonstrate that for novices (comprising virtually all students), direct, explicit instruction is more effective and more efficient than partial guidance. So, when teaching new content and skills to novices, teachers are more effective when they provide explicit guidance accompanied by practice and feedback, not when they require students to discover many aspects of what they must learn.

Clark et. al. does not advocate using explicit instruction all the time, and recognize the need for independent inquiry, but suggest to use it to practice newly gained knowledge rather than to discover it.

Minimally guided instruction can increase the achievement gap. [..] more-skilled learners tend to learn more with less-guided instruction, but less-skilled learners tend to learn more with more-guided instruction. For these relatively weak students, the failure to provide strong instructional support produced a measurable loss of learning. The implication of these results is that teachers should provide explicit instruction when introducing a new topic, but gradually fade it out as knowledge and skill increase.

TL;DR the details of how you design such activity are important. You have to carefully consider how and when you provide explanation and instruction along the way of the inquiry.

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  • $\begingroup$ Thank you very much and +1 for your very interesting answer $\endgroup$ Jun 14 at 13:09
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A good think is to students pick up several rounded objects (wheels, cans, glasses, etc.) and mesure with rule the length and the diameter and simply divide. And observe if something happens. You can see this idea here (in Catalan - automatic English translation). It is original from Anton Aubanell.

Just a note: for measuring the length I think the better is to pick up a rope and surround the object you want to mesure to. Put a mark in the rope and mesure from the beginning to the mark of the rope.

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    $\begingroup$ Thank you very much for your very interesting answer. $\endgroup$ Jun 11 at 15:04

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