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1

A compromise approach could be to give a problem (with parts), as you might on a guided worksheet. Such as: A function $f$ has domain $(2,4)$. We define $g$ by $g(x) = f(x-2)$. a) Is $g(3)$ defined? b) Is $g(5)$ defined? c) What is the domain of $g$? You might even give a follow up problem (perhaps as extra credit) along the lines of: A function $f$ ...


9

I think the given example is highly appropriate. You cannot cover every possible combination of ideas in class. Students display understanding of a concept (rather than "recipe following") by showing the ability to adapt at least a little bit to novel conditions. I think the problem you gave is a great homework problem. I personally like homework to be a ...


9

I would frame this issue a little differently than you have. I think it's unreasonable, at least in the context of courses which aren't well into a math major, to ask students to do something they have not been taught to do. That is, the problems on an assessment should be the same as problems they've seen already. The catch is that "the same" is actually ...


7

My advice is to minimize the amount of such synthesis required. Don't make it a large fraction of your tests, if at all. Teach the students the methods you expect them to display on the exam. Not something requiring some spark of creativity. Program for success. Creativity is tough in general and even tougher under test conditions. If you push too ...


4

What follows is a critique of the paper cited in another answer here: Setren, et. al., "Effects of the Flipped Classroom: Evidence from a Randomized Trial", Annenberg Institute at Brown University (2019). That paper doesn't, in fact, refute the value of the flipped classroom in any way and is an example of poor research design. In fact, it says nothing about ...


5

I would like to echo Peter Saveliev's point in the comments: You are asking about "higher results" but you do not specify what qualities you are interested in measuring. Do you care about grades? Performance on standardized tests of content knowledge? Student success in courses which depend on the given course as a prerequisite? Student perception of ...


13

Consider a paper from this year: Setren, et. al., "Effects of the Flipped Classroom: Evidence from a Randomized Trial", Annenberg Institute at Brown University (2019). In their introduction, the authors write: Despite the proliferation of the flipped classroom, little well-identified evidence exists on its impact on student learning. In this study a ...


9

The Education Endowment Foundation has a 'Teaching and Learning Toolkit' which aims to assess the effectiveness of various teaching methods or types of intervention, including collaborative learning, altering teaching for individual learning styles, and small group learning. It is focused on teaching 5–16 year olds, and not specific to mathematics education, ...


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