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I ran across the following sentence in 4th grade mathematics teaching materials:

垂直である2本の直線は、十字になっているものだけではないことを、定義の中の例示に触れながら、ア、エ、オをもとに帰納的におさえる。

What meaning of おさえる is being used here? I assume it is the following meaning:

  • 大切なところをしっかり理解する。把握する。

I would like to be sure this is right and not specific vocabulary/terminology from the teaching realm. If I am right on this, does the last part mean "to inductively grasp... based on ア, エ and オ" (those are examples from the book)?

Also, would one expect to choose 抑える or 押さえる here? Is it the same or is there a subtle difference?

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Yes, that definition of おさえる is correct. It's used for something that is important but easily goes away from memory. You need to consciously keep it in mind, hence "to press". It can be used both by teachers in the sense of "stressing the point", and by students in the sense of "consciously keeping the point in mind".

As for kanji, 抑える and 押さえる are both very common, but I personally prefer 押さえる because 抑える tends to mean "to suppress".

Additional notes:

  • So is this about skew lines (ねじれの位置) in 3D geometry? I thought this was a topic for middle school students, not 4th graders.
  • This use of 帰納的 sounds inappropriate to me. Using a few examples to convince elementary school students should be called a 直感的な説明 (intuitive explanation), not a 帰納的な説明 (inductive explanation).
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  • Very specifically, this is about perpendicularity (垂直) and parallelism (平行) in planes, i.e. 2D. The examples are presenting different situations. Here, ア represents two lines crossing as in 十, and the examples エ and オ represent situations where the lines aren't crossing over but are still perpendicular, for instance adjacent sides in a square (口) where the lines simply touch without "going over". The textbook then goes on to define various figures, such as 平行四辺形 (parallelograms), 台形 (trapezoids) and finally ひし形 (rhombuses).
    – Saegusa
    Commented Sep 3 at 5:27
  • While situations where the lines don't meet at all is also in the scope of the textbook, the specific examples chosen here are the ones mentioned above. I should add, I double checked and it is indeed 帰納的 which is being used, but I agree that 直感的 makes more sense, and is used in other places in the book.
    – Saegusa
    Commented Sep 3 at 5:37
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    @Saegusa Oh, I see. A line (直線) extends infinitely in higher education, but it refers to line segments (線分) in this context!
    – naruto
    Commented Sep 3 at 8:11
  • Yes, this confused me as well initially, since they were explicitly using 直線, and not 線分, but I am assuming this is because the notion of 線分 is not yet fully crystallized in the pupils' mind. Interestingly enough, one property of parallelism they consider is that どこまでのばしても交わらない, which seems to indicate they can imagine infinitely extending a line (segment), but they do not yet seem to define what a line segment is.
    – Saegusa
    Commented Sep 3 at 8:46
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    小学校の算数の学習指導要領の中に、「帰納的に考え説明したり、演繹的に考え説明したりする活動」とか「帰納的に考えたり推論したり」みたいに、「帰納的・演繹的」って結構何回も出てくるんです。大学で教免とるとき（算数じゃなかったですが）にも帰納的・演繹的ってやらされました
    – chocolate ♦
    Commented Sep 3 at 10:46
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I think recursively is a better way of understanding 帰納的 here. It sounds to me like something is being recursively built up from ア, エ, and オ.

I would take

ア、エ、オをもとに帰納的におさえる。

to mean

obtain by recursion upon ア、エ、オ

where "upon" is my rendering here of をもとに.

This sounds like instructions for a geometric construction of some sort. I would feel a bit more confident in my answer if I knew a bit more about what ア、エ、オ are. Is there more context you can provide?

These are instructions from 4th grade teaching material? Wow! I'm impressed.

Technically, induction is a kind of recursion within mathematics. But usually, when we say something like "inductive" in English what comes to mind is "inductive reasoning". However, this is not the sort of reasoning mathematicians have in mind, and in this context I think saying "inductive" could be a bit confusing. Though, at least in the States, I would be very surprised if a 4th grade teacher understood either "inductive" or "recursive". (That might seem like extreme prejudice on my part, but I've spent a good part of my career instructing K-6 teachers in mathematics.)

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    Here, ア, エ and オ are different examples of perpendicular lines. Recursively really doesn't work here, I really do believe the meaning is inductive, as in inductively building comprehension of perpendicularity through examples showing that a line can be perpendicular to another even if they don't cross. Also I am actually studying mathematics education, so if you happen to have more resources on this on the US system, let me know! If you're interested in some papers comparing it with the Japanese system, I might have some as well. :)
    – Saegusa
    Commented Sep 2 at 17:24
  • I should add, these are meant for the teacher, not for the pupils!
    – Saegusa
    Commented Sep 2 at 17:33
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    I would be very surprised if a 4th grade teacher understood either "inductive" or "recursive" 小学校の先生だし「帰納的」「演繹的」の意味くらい知ってないとだめだと思うのだけど・・少なくとも大学で教免とる時に必須の科目（「教科教育法」とかなんとか）で勉強してるはず・・
    – chocolate ♦
    Commented Sep 3 at 10:36
  • @chocolate I agree with you. Unfortunately, such is the state of education in the States.
    – A.Ellett
    Commented Sep 3 at 13:20
  • @Saegusa I would be interested in see such papers. Unfortunately, I don't have much to share with you other than my experience teaching prospective K-12 teachers and also working with K-12 teachers in the classroom.
    – A.Ellett
    Commented Sep 3 at 13:27

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