# Has a List of Fundamental Mathematical Skills been compiled?

... in the literature.

I an wondering is there a (considered) list of "fundamental mathematical skills".

I am not sure can I give a solid definition of "fundamental mathematical skill". What I mean by a fundamental skill would certainly include "pattern spotting", "visualisation", and what I would call "substitution".

A skill that would perhaps also be a fundamental skill would be "estimation".

However something like "algebraic manipulation" would not be a fundamental skill.

A soft definition might be to say that things that belong to the realm of "mathematical thinking" might be fundamental skills, while things that belong to the realm of "mathematical knowledge" would not. I appreciate this is very far from a clear demarcation.

Context: Colleagues and I are thinking about developing a diagnostic test for new students. We are thinking around including questions that might test these fundamental skills.

• Estimation, e.g. en.wikipedia.org/wiki/Fermi_problem , is a highly sophisticated skill that has many intricate parts. To me it seems similar to algebraic manipulation in that respect. Colleagues and I are thinking about developing a diagnostic test for new students. We are thinking around including questions that might test these fundamental skills. Constructing such a thing with good face validity, predictive power, and statistical reliability is a very complicated project requiring a lot of specialized knowledge. How do you know your a priori list correlates with success? – Ben Crowell Apr 22 '20 at 16:14
• @BenCrowell thank you for your comment. We don't know that any such list correlates with "success". Our tentative plan is to design as best we can a diagnostic test, and analyse the results together with students' grades and use e.g. PCA to see which questions or subsets of questions predict-well later student performance. – JP McCarthy Apr 22 '20 at 16:39

Common Core includes Standards for Mathematical Practice. Those may be a good description of what you're looking for. I have seen a few other good lists of mathematical thinking skills. I enjoyed thinking about how I might use this one, put together by Avery Pickford, in my teaching. (You might find more good thinking in the comments.)

• This is excellent thank you very much. – JP McCarthy May 2 '20 at 14:06

This is not really an answer, but is too long for a comment. BTW, the downvote wasn't mine.

I tried constructing something similar to this about 25 years ago, with some guidance from my mother, who worked in standardized testing. The details of the list of skills were different in my case, but it was a similar situation. The problem I would expect you to run into is the following (which is what happened for me).

All questions were correlated with one another, and all were correlated with students' grades. However, these correlations were all relatively weak -- some weaker than others. And there was no real evidence that the discrete list of skills we had in mind were actually separate skills. If they really were separate skills, then you would expect that when you did a covariance of question $$m$$ with question $$k$$, there would be a strong correlation when $$m$$ and $$k$$ were designed to measure the same skill, and a weak correlation when $$m$$ and $$k$$ were designed to measure different skills. This didn't happen.

So I can't predict what your results would actually be, but by default I would kind of expect results such as the ones I found. And I think then what's happening is that you're simply measuring some over-all measure of intelligence, academic preparation, and reading comprehension. This is the kind of experience that led Spearman to originally postulate the existence of some kind of general intelligence or "$$g$$" parameter.

Although IQ testing is basically pseudoscience, there is a kernel of truth in it, which is that basically all intellectual abilities tend to be correlated with one another, and it's difficult to disentangle separate factors.

One possible way to approach your problem might be to look around at what well-constructed tests already exist, and give those tests to your students. Then if you have a hypothesis that there is some well-defined skill like estimation, look for questions that seem to you to prove estimation, and see if they really do correlate strongly with each other, weakly with other questions, and strongly with students' grades.

The advantage of starting with other people's published tests is that you get the advantage of all their previous work in making sure the test works well. For example, they will have already eliminated questions that are not useful because they're too easy (everybody gets them right), too hard (nobody does better than random guessing), or not valid (don't correlate with anything external). For a multiple-choice test, there is a huge amount of work that can go into finding good distraction answers that probe actual student misunderstandings.

• Ben. Thank you for your comment/answer. We are not planning to design a test based just on such fundamental skills, or even majorly on such fundamental skills. The test will also test mathematical knowledge, etc. We are just hoping to include some such questions. – JP McCarthy Apr 23 '20 at 8:54