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In Australian math curriculum, four strands of mathematical proficiency are defined: :

  • Understanding
  • Fluency
  • Problem Solving
  • Reasoning

Are these items hierarchical - I mean, for example, is reasoning a "higher" proficiency than understanding?

P.S. In Australian curriculum,

  • Understanding refers to a deep understanding of the mathematical principles and patterns that underpin classroom learning as well as the connections between concepts.

  • Fluency refers to building students’ content, basic skills, speed and accuracy in routine questions.

  • Problem-Solving refers to having students attempt never-before tried problems.

  • Reasoning is the proficiency strand that requires students to prove that their thinking is mathematically valid or that someone else’s thinking is not mathematically valid.

Reference: https://www.backtofrontmaths.com.au/uncategorized/what-do-the-proficiency-strands-mean

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    $\begingroup$ I don't know, sometimes I can reason things I don't understand. I think I may have once published a paper like that. That said, I'd hope there are larger definitions than these single term slogans. I'd need more to understand. $\endgroup$ – James S. Cook Aug 21 '17 at 19:14
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    $\begingroup$ I would assume that from the term "strand", the implication is that they are not hierarchical -- like intrinsic windings in a rope, repeated throughout the curriculum. $\endgroup$ – Daniel R. Collins Aug 22 '17 at 2:16
  • $\begingroup$ Are these terms defined or otherwise explained in the curriculum or elsewhere? I have found that mathematicians and educators mean different things when they discuss students "understanding" something. $\endgroup$ – Tommi Aug 22 '17 at 3:24
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    $\begingroup$ @TommiBrander You can find their definitions here: v7-5.australiancurriculum.edu.au/mathematics/content-structure $\endgroup$ – Behzad Aug 22 '17 at 5:48
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    $\begingroup$ Suggest that you include the definitions in your question to make the question clearer. You could include the link and copy and paste the definitions. $\endgroup$ – Amy B Aug 22 '17 at 14:38
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Based on the fuller descriptions in the links, I think the first 3 are hierarchical and the last one is separate.

PS1. fluency-basic test problems

PS2. understanding-word problems (same math concept as before, but able to decode and model)

PS3. problem solving-able to extend previous concepts to next step

W1. reasoning-able to describe the concepts

I would argue the first three, while having some differences, are all about problem solving and it is unlikely to show skill in the higher strand and not in the lower. The last strand is really separate from problem solving. You probably need to train on it specifically.

Obviously, you do need at least some modicum of ability to explain something, but probably not even very good "fluency". There is a big difference in description versus execution. (But on the other hand, in some cases you need to comprehend the concept much more particularly to describe it, than what is needed to execute any problems.) IOW, to a certain extent it is possible to be a good problem solver (by which I mean the general meaning, not the strand name) and a poor explainer and visa versa. Thus the need for separate practice in both problems and description. [If you buy that description is important...many people would think it is more minor than execution ability.]

FWIW1, I do think "problem solving" (the strand specifically) and to an extent "understanding" (the strand) is more about student intelligence than training. IOW, yes, you need to have the basic concepts down to extend them. Or to weed though the chaff of a word problem. But it's kind of hard to practice the ability to take the next step.

FWIW2: I think the strands are poorly named. Even writing here it is hard as problem solving and understanding are really more general concepts in the English language than what the Aussie edubabble labels describe. Something more like the following would be more helpful:

  1. Basic problem solving fluency
  2. Complicated problem solving

  3. Ability to extend concepts

  4. Descriptive ability

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