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There is the so-called FCI test. It contains a list of questions such that anyone who can speak will have an opinion. Based on the answers one can determine if the answerer knows elementary mechanics. I learned about this test from an interesting lecture from Eric Mazur describing an application of such a list.

It seems that there are no analogous tests in mathematics. Did I miss something?

Sergei Tabachnikov shared a couple of questions that might work:

  • One needs to peel potatoes. What is faster and why: peeling a pound of large or small potatoes?

  • Why does a reflection in a mirror interchange left and right, but not up and down?

  • Two people are walking down an escalator (that is moving down), counting their steps. Who will count more steps, the one who is moving faster or slower?

Could you share more questions of this type?

Please make one question per answer. The questions should not use any professional jargon; no graphs no formulas; say, they could be asked in a kindergarten. I think it is possible to find such question for Euclidean geometry; I am not sure about other branches of mathematics.

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    $\begingroup$ The title mentions Euclidean geometry, but the question in the body is about "mathematics". Which do you mean? $\endgroup$
    – Dan
    Commented Dec 25, 2023 at 8:55
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    $\begingroup$ @Dan I changed the question a bit. $\endgroup$
    – Anton Petrunin
    Commented Dec 25, 2023 at 18:51
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    $\begingroup$ The question about potatoes is an example of (IMHO) bad question, because it's too real. A kid with experience might answer "large ones take longer because they're harder to grasp", and an adult conversely "small potatoes are hard to grasp", or even "how small do you mean?". Or "when peeling with knife or with peeler". And so on. Neither having anything to do with surface. $\endgroup$
    – Pablo H
    Commented Dec 26, 2023 at 13:28
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    $\begingroup$ Are you aware of the Mathematics Educators Stack Exchange site? It would seem to me an appropriate place to ask your question. $\endgroup$
    – J W
    Commented Dec 27, 2023 at 11:25
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    $\begingroup$ Whether your assumption regarding needing to be a professional mathematician to come up with such questions is true or not, please note that Mathematics Educators certainly contains professional mathematicians as regular participants, albeit fewer than on MO. $\endgroup$
    – J W
    Commented Dec 27, 2023 at 22:13

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The FCI is a "concept inventory" test for classical mechanics. There exist several such tests for mathematics. Some may be implemented as "peer instruction" (as Eric Mazur advocates), but they can also be taken individually.

  • One example is the Function Concept Inventory test, designed to investigate undergraduate students’ understanding of the concept of function.
  • Another is the Precalculus Concept Assessment to "assess essential knowledge that mathematics education research has revealed to be foundational for students’ learning and understanding of the central ideas of beginning calculus."
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    $\begingroup$ Unlike the test discussed by Eric Mazur, these questions cannot be asked in a kindergarten. $\endgroup$
    – Anton Petrunin
    Commented Dec 25, 2023 at 0:30
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    $\begingroup$ These are sorely lacking from conventionally used tests to ascertain whether those who have grades of $A{+}{+}{+}{+}$ in the prerequisites to calculus actually understand anything. Perfect grades are given to those who have worked hard and meticulously followed all instructions but have not understood anything. $\endgroup$
    – Michael Hardy
    Commented Dec 25, 2023 at 4:20
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This question is slightly easier than your questions but seems thematically of the sort you are aiming for.

Pat walks around the edges of a triangular park. After Pat has walked two edges, they go to finish their walk by walking the third edge. Assuming that Pat walks the same speed for their entire walk which of these must be true:

A) The remaining part of the walk will take more time to complete than their walk has taken so far.

B) The remaining part of the walk will take less time to complete than their walk has taken so far.

C) The remaining part of the walk will take the same amount of time to complete than their walk has taken so far.

D) There is not enough information to compare how much time it will take their remaining walk compared to the how much time their walk has taken so far.

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Since no kindergarten level answers have yet been proposed, I'll start a post with just one:

Standing in the middle a large flat field with a straight fence along one side, say to the child Run to the fence as fast as you can! The test is to see whether they run along the perpendicular segment between their starting point and the fence.

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  • $\begingroup$ Another option "Draw the shortest curve that divides the area of pentagon into equal parts", but maybe it is too much. $\endgroup$
    – Anton Petrunin
    Commented Dec 26, 2023 at 20:09
  • $\begingroup$ @AntonPetrunin Do you mean a regular pentagon? I'd suspect it's a perpendicular to an angle bisector, but I wouldn't have a clue how to determine the offset without brute force calculations, and wouldn't be sure without confirming its length with more brute force calculations. What am I missing? $\endgroup$
    – user519702
    Commented Dec 26, 2023 at 22:28
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    $\begingroup$ @user479223 I'm assuming 'equal parts' means 'two regions of equal areas'. Then it is not an angle bisector. Or does it mean 'two congruent regions'? $\endgroup$
    – user519702
    Commented Dec 26, 2023 at 23:18
  • $\begingroup$ @Servaes Oh, I see what you mean now. $\endgroup$
    – user479223
    Commented Dec 26, 2023 at 23:19
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This blog post points out psychometric issues with the Calculus Concept Inventory:

This NSF project report, "Year 1 (Expanded) Annual Report for DUE-1625678 Project DIRACC: Developing and Investigating a Rigorous Approach to Conceptual Calculus" discusses a "Calculus 1 Concept Inventory" (C1CI) https://www.researchgate.net/publication/319875221_Year_1_Expanded_Annual_Report_for_DUE-1625678_Project_DIRACC_Developing_and_Investigating_a_Rigorous_Approach_to_Conceptual_Calculus; also see https://www.nsf.gov/awardsearch/showAward?AWD_ID=1625678 which says more about ongoing work.

I also came across a reference to "a validated Statistics Concept Assessment, the Benchmark Assessment for Statistics Introductory Concepts (BASIC)" in an email to a statistics mailing list from Michael Pearson of the MAA dated Nov 30, 2023.

This website lists a lot of physics concept inventories, but also has a page about math concept inventories: https://www.physport.org/recommendations/Entry.cfm?ID=124939

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