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In answering another question What is the justification to teach the (redundant) use of parentheses in multiplications? I was left wondering what we actually know about students' progression in terms of how they see algebraic expressions.

Is there any research work that has been done into the progression of students' ability to parse math expressions?

My impression is that as students progress in their mathematical careers, the way that they mentally process mathematical expressions changes from seeing them as a sequence of characters (and having to consciously think about order of operations) to automatically seeing their structure of nested subexpressions.

In other words, for early students (5 x 10) + (5 x 8) and 5 x 10 + 5 x 8 are processed differently, but for later students they are processed nearly the same.

But at what time scale does this happen? My impression is that students make substantial growth in their parsing skills during calculus.

This is analogous to going from reading a sentence in a foreign language and having to puzzle out the grammar to this process happening subconsciously during the reading stage.

I suspect if there is any research at all along these lines it would be eye-tracking research, but I couldn't find eye-tracking research in math focused on formulas specifically.

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    $\begingroup$ @AdamRubinson This can be viewed as a part of a more general question: how do students read mathematical texts. I don't remember how I learned the order of algebraic operations in the early childhood, but I still remember my student years and reading textbook passages. In principle any mathematician should have some recollections about that too and it may shed some light on the original question, though, of course, it cannot count as a hard research evidence for anything. Or just look at how children read English and when "Once upon a time" will be read correctly if spelled "1ce uppon a tme" $\endgroup$
    – fedja
    Jan 30, 2023 at 1:24
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    $\begingroup$ "My impression is that students make substantial growth in their parsing skills during calc 1." Could you please use an age instead of a class name? I have no idea what age "class 1" corresponds to. In another context I would have guessed "calc 1" was a university course, but in this context I'm guessing it's in primary school? Although giving such a specific name to a primary school math course seems a bit odd to me. $\endgroup$
    – Stef
    Jan 30, 2023 at 16:28
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    $\begingroup$ I mean differential calculus! There's a lot of parsing practice in determining what order to apply derivative rules. $\endgroup$
    – TomKern
    Jan 30, 2023 at 16:53
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    $\begingroup$ I'm aware of one study that used eye-tracking software to compare how students and expert mathematicians read mathematical proofs. I won't post this as an answer, though, because you're curious about parsing algebraic expressions, and I'm not sure if there's any work on that specific topic. Here's a blog post summarizing the study I mentioned, with a link to the study itself in the post: blog.oup.com/2016/01/reading-mathematics-proofs $\endgroup$ Jan 30, 2023 at 22:00
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    $\begingroup$ I have been having my students in intro-proof write abstract syntax trees to make sure they are able to parse sentences in first order logic correctly. I think they could help with order of operations as well. Leaving this as a comment because it is just a hunch: I have 0 evidence to support this. $\endgroup$ Jan 31, 2023 at 22:17

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So to know what some expression will evaluate to you will need to look into the operators in other words what you referred to in your post as characters.

Q1: (and having to consciously think about order of operations) to automatically seeing their structure of nested subexpressions?

Due to use of terms in algebra it fairly to understand mathematical equation.

(a x b) + (a x c) is a little bit hard to read.

But if we instead write

ab + ac it is a lot easier.

To summarise the example you have given is not really encountered but if it did you will need think about the operations i.e. 5 x 10 + 5 x 8. at least a little bit otherwise you will end up 440. So I don;t think you can answer a question subconsciously.

Q2: But at what time scale does this happen?

Assumption is being made by the question about subconsciousness.

Very quickly I will presume. Order of operations is taught in year 5-7. Though I don't think subconsciousness will occur as mention in Q1.

Regarding eye tracking. A mathematical expression is read left to right by convention. I don;t any stats my guess would be most people would look at the left first to see if anything can be done. If not they will look at the other terms.

e.g.

8 / 2(2+2)

Since 2(2+2) is one term it needs to be evaluate first before division.

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