Hopefully not – because your choice of words inspired my last blog post. Mental calculus sounds better in the context of your article in the sense that students learn calculus as a calculus (i.e. mechanically) without putting their understanding, soul and feeling into it (subject of my last article). That they learn to shuffle symbols and formulas to get a “desired output” without developing an intuition about what’s going on.

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]]>Hopefully not – because your choice of words inspired my last blog post. Mental calculus sounds better in the context of your article in the sense that students learn calculus as a calculus (i.e. mechanically) without putting their understanding, soul and feeling into it (subject of my last article). That they learn to shuffle symbols and formulas to get a “desired output” without developing an intuition about what’s going on.

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]]>Yes indeed, blue collar. I wonder why I read ‘color’ as ‘collar’? I never saw the mistake until you pointed to it.

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]]>BTW – did you mean “blue collar neighborhood?” (although I confess that the wording you chose made it immensely more vivid, and I dare say… colorful?)

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]]>y = x with appropriate transformations from 0=<x<=1

y = x^2 with appropriate transformations from 1<x<2

y= ln (x) with appropriate transformations from 2<=x<=3

Or this?

Not possible. No transformation of the given functions will produce the graph for 0 <= x <=1

The first is a more literal (mis)reading of the prompt, the second is even more literal and overlooks the identity transformation as students often do. Yet each response reveals quite a bit of knowledge, and each would score poorly on an automated grading system. Fortunately, we're not yet automatedâ€¦

This happens to me ALL THE TIME. As carefully as I phrase an assessment prompt, as many drafts as I go through in response to unanticipated interpretations, it still happens that a student will answer a question I didn't intend to ask that is contained within my very words. I've come to view such responses as fundamentally unpredictable, and come to see that they often demonstrate competence that I would otherwise not have seen. It's not "partial credit," it's credit where credit is due. The response you cited from 1/3 of your students could be something like B+ quality. They partitioned the domain correctly, and correctly identified the linear, quadratic, and logarithmic segments. Even though it's wrong, I couldn't call that response an F.

As instructors, we're completely versed in the language and assumptions of our subject. The hard part, as you say, is to "put aside" what we've already learned and see our subject with fresh eyes. What is transparent to us is often opaque to our students. On good days, I get to translucent.

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