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When I teach math classes, one goal I have in mind is to help students develop the cluster of thinking skills usually called analytical skills or critical thinking skills. And I think that math classes can help develop analytical and critical thinking.

These skills include being able to apply mathematical reasoning, but also other skills: to be able to take big problems and break them down in to smaller one, to be able to combine two ideas to create a third idea, to be able to generalize a concept to apply it to similar situations.

Are there currently accepted measures of how good learners are at these skills?

While I really think math and science courses help develop these skills, I would love to not only learn more about them, but to be able to point to research that says these skills are developed by quality math and science instruction. Can I get some leads on that research?

My own Google Scholar searches come up with mostly noise level results.

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    $\begingroup$ I would suggest Eric Mazur's papers; they are not specifically focused on what you are asking but I think you would found them interesting (also see his speeches on youtube). $\endgroup$ – onurcanbkts May 8 at 14:05
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    $\begingroup$ Eric Mazur's ideas looks pretty interesting. youtube.com/watch?v=WwslBPj8GgI $\endgroup$ – David Elm May 8 at 18:35
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    $\begingroup$ Eric Mazur's talk was amazing! $\endgroup$ – David Elm May 8 at 20:37
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You may be interested in Academically Adrift: Limited Learning on College Campuses, Arum and Roksa, 2011. Also summarized in http://www.newyorker.com/arts/critics/atlarge/2011/06/06/110606crat_atlarge_menand . They have a lot of discussion of something called the Collegiate Learning Assessment, which is a standardized test of critical thinking. They find that students in the liberal arts do significantly improve their critical thinking abilities over the course of their college education, while the gains for students in most other majors, such as business and engineering, are smaller. Science majors tend to be more similar to liberal arts majors, which might support your claim, but I don't have anything in my notes about math majors specifically.

They find that greater learning is correlated with faculty's high expectations, high standards, and approachability.

It's possible that the differences between majors really just measure preexisting differences between the student populations that go into those majors. Also, many people who are initially STEM majors in college end up switching to easier majors.

Another book with useful material on this topic is Valen E. Johnson, Grade Inflation: A Crisis in College Education. His main topic is grade inflation and the corrosive effects of student evaluations of teaching, but he also has some carefully constructed statistical studies that show huge differences between the level of difficulty of different majors (a point that is also supported by Arum and Roksa). He rules out interpretations saying that the different groups of students are equally smart, but just at different things. Math is one of the majors that does come out to be objectively harder according to his data, and therefore one would expect math students to be more able and to improve their critical thinking skills more while in college.

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    $\begingroup$ That New Yorker article was really interesting. $\endgroup$ – David Elm May 8 at 0:40
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The idea that math-study imparts certain fundamental intellectual skills was taken for granted for most of the history of Western education, and reaches back at least to Plato, who (reportedly) refused to accept students who had not mastered geometry. "Let no one ignorant of geometry enter here" was supposedly inscribed over the entrance to his Academy.

In late Antiquity and the Medieval period the term 'liberal arts' designated an elementary course of studies in language arts, logic, and math. The majority of the traditional liberal arts (four of the seven) were mathematical, and mastery of all seven arts was considered an essential pre-requisite to higher studies in philosophy, theology, or one of the professions.

You were probably asking for a different sort of literature than what I'm about to suggest, but it wouldn't hurt to become familiar with the older tradition, if you're not already.

Some ancient and medieval texts that discuss the educational theory are Aristotle's Nicomachean Ethics, Boethius's De Trinitate, Aquinas's commentaries on both of the former as well as a few questions in his Summa Theologiae, and Hugh of St. Victor's Didascaliae. And since the "Great Books" revival of the mid-20th century there has been an enormous amount written about these topics. A heavily-footnoted survey of all of this is Clark & Jain, The Liberal Arts Tradition.

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