Most Standards (e.g. Common Core Standards in the U.S.) explicitly promote the teaching of “mathematical reasoning”, with a usually vague description of what exactly that is. Does anyone know of experimental evidence clearly linking such an intervention intended to teach mathematical reasoning (by any desired clear definition) to improved performance? (Preferably in algebra 1 and geometry, but I’ll take any K12 level.)

You may be confused by my not specifying a specific performance measure. However, I can't be specific about the performance measure because I am asking for a pointer to research that uses whatever performance measure it uses. If I asked for a specific one, it would preclude a wide range of other possible performance measures. Examples might be specific math tests, SAT scores, college entrance, etc. Since I'm asking about research papers with "real" experimental results, most of these will have their own instrumented performance measures.

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    $\begingroup$ @jackisquizzical I don’t have any examples but understand what you are saying… at the risk of redundancy, I’ll reiterate that you are looking for 1) experimental evidence 2) suggesting a correlation or a causal relationship 3) between “mathematical reasoning” and “scholastic achievement” 4) where the definitions of “mathematical reasoning” and “scholastic achievement” may vary between studies but will be made precise by the methodology within each individual study. (And you have given examples of suitable achievement measures). Right? I don’t know what the confusion is about either. $\endgroup$
    – Steve
    Commented Jul 20, 2022 at 19:39
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    $\begingroup$ OP, I think the pushback you're getting may be that for math specialists, mathematical reasoning is the ultimate point, and all other exercises and testing are just a means to that end. As a thought-experiment, most of us would be happy to abolish all performance testing if we knew that mathematical reasoning was being achieved by students. To the extent that a testing instrument fails to measure mathematical reasoning, it must be broken instrument. $\endgroup$ Commented Jul 21, 2022 at 17:07
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    $\begingroup$ On that theme, the question seems circular. How would you ever know that mathematical reasoning is happening without having any effect on a performance test? The performance test is the only thing you have to tell you that mathematical reasoning is occurring! $\endgroup$ Commented Jul 21, 2022 at 17:09
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    $\begingroup$ I don't know why this is so hard to understand. Suppose that we hypothesize that the More Raisins (MR) intervention improves mathematical performance. We choose a measure, say, a standardized test like the SAT math test. Then we could do either a controlled experiment, for instance giving MR to half of a group of students vs no MR, and seeing how the arms differ, or perhaps an observational study where we ask teachers how often they employed MR, and then regress that against the performance measure, or maybe against a bunch of performance measures. This is all totally obvious stuff. No? $\endgroup$ Commented Jul 21, 2022 at 17:40
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    $\begingroup$ Look, the Question is poorly worded and the elaborations merely raise more issues: vagueness, circularity, and conflating a quantity and how it's measured (to measure volume, I might use geometry/integration, submerge in water, use augmented reality, make a rough visual guess, etc.), then giving an illustration that contradicts the original request. $\endgroup$
    – ryang
    Commented Jul 22, 2022 at 6:57

2 Answers 2


If we are into anecdotal stories, I want to share one about my former adviser Vicktor Havin, who once made a public speech about one way in which mathematical thinking can be useful to a layman. His main point was that while we can make any list of properties defining an object, not all such lists will be consistent and if the list is self-contradictory, the object may merely fail to exist, so if we decide to put forth any such list and act upon it, we'd better make all terms completely unambiguous and verify its consistency first. It is easy to comprehend when one talks about "primes divisible by 27", but many old and modern society slogans from "Liberty, Equality, Fraternity" to "Diversity, Equity, Inclusion" would better undergo a thorough scrutiny for ambiguities and inconsistencies before being implemented. While such scrutiny may not show a total self-contradiction in the list or in its individual properties, in many cases it will show a necessity for certain clarifications, limitations, and balance that are usually absent when we launch a crusade for the latest political fad, whatever it is. What is Equity, for instance? It can be interpreted as giving everyone opportunities according to his/her abilities, inclinations, and willingness to achieve their goals, or it can be interpreted as giving no one a clear advantage over anyone else, and these two definitions brought to their extremes will result in two completely opposite policies, neither of which (pure meritocracy or cutting off every head that sticks out) will make most (if any) of us happy.

Another thing is that in mathematics you can call anything by any name, but every notion you introduced should be clearly defined and have a clear domain of applicability. Lack of such definitions results in various empty universal agreements like "We have to hire the best candidate" often followed by bitter fights over the meaning of the word best when everybody is at everybody else's throat in the end. Or, everybody seems to have some notions of "Good" and "Evil" but no 2 people will completely agree on what is in which category because both words require about 10 specifiers ("for what purpose", "under what circumstances", "morally/logically/pragmatically", etc., etc.) to acquire minimally concise meanings. One cannot easily change how people communicate in general, of course, but one thing I find worth trying at least occasionally when somebody says something that I strongly disagree with is to ask "What exactly do you mean by that?" before throwing myself into a heated argument (not that it always prevents the latter, but, at least, one can get a better idea of what the opponent is after this way and avoid fighting ghosts).

At last, mathematics is something where we have a clear and universally agreed upon criterion of truth, which, in most cases, allows everyone to run full verification of claims by him/herself. That facilitates the communication enormously. In the modern world, where the notion of "information" seems to be almost completely substituted by that of "information wars" (which range from toilet paper advertisements to COVID vaccination debates to Ukrainian war coverage), this is something worth keeping in mind too.

Of course, this doesn't answer the question as posed but rather clarifies what I mean by "mathematical reasoning" that I'd like to see being taught in schools, so feel free to downvote :-)

  • $\begingroup$ "feel free to downvote :-)" Are you kidding me?? $\endgroup$
    – ryang
    Commented Sep 12, 2023 at 6:35
  • $\begingroup$ @ryang: Are you kidding me?? -- And yet the strategy seems to have worked . . . $\endgroup$ Commented Sep 13, 2023 at 10:15
  • $\begingroup$ @ryang: By saying "feel free to downvote", the OP is preempting a reader's possible desire to cast a negative vote by a strategy of creating sympathy for the OP. And since there are currently no negative votes . . . $\endgroup$ Commented Sep 13, 2023 at 12:22
  • $\begingroup$ @DaveLRenfro My comment just means, "this preemption isn't necessary, in fact I am going to upvote". $\endgroup$
    – ryang
    Commented Sep 13, 2023 at 13:03

I don't know if this is something you count as mathematical reasoning but lately I've read a story about a mathematician who has saved numbreds of lives:
During the second World War, somebody had the brilliant idea to make a summary drawing of all bullet holes that were shot into airplanes. The idea was to use that drawing in order to decide which parts of the planes should be better protected.

That summary indeed showed some regions, showing more bullet holes than other regions, so the idea came up to foresee more protection on those regions.

... but luckily there was also a mathematician in the room, who said just the opposite: instead of adding protection to the regions with bullet holes, he proposed to add protection to the regions WITHOUT bullet holes.


Well, he said: there is absolutely no reason to believe that some regions receive more bullets than others. The fact that our summary indicates otherwise, just means that the regions without bullet holes are the regions where the plane is so vulnerable that a single bullet on that spot causes the plane to crash. And as that plane is crashed, it does not appear in our summary!


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