Meyer and Land [8] suggest that the crossing of a threshold will incorporate an enhanced and extended use of language.

While I’m sure careful attention to language is not the only way to help students “cross” a threshold, I am particularly struck by the examples Steve mentioned and the idea that we may unintentionally be confusing students by using the one word to mean more than one thing.

]]>http://www.ee.ucl.ac.uk/~mflanaga/thresholds.html

http://www.doceo.co.uk/tools/threshold.htm

http://www.etl.tla.ed.ac.uk/docs/ETLreport4.pdf

Meyer and Land seem to be the main researchers originating this use of the concept.

Perhaps one way to identify threshold concepts is to think about ideas that students have difficulty with, particularly over extended periods of time, partly because they keep using their prior ways of thinking in ineffective ways. Then we try to name the key concepts that they have to grasp that reorder their understanding.

To what extent do difficulties with multiplying negative numbers benefit from a changed understanding of multiplication rather than a better understanding of “negative numbers”, the new arrival on the scene. And, this leads to the idea of operations defined on sets of numbers, and the idea of certain numbers resulting from particular operations on particular sets.

One of the topics in the morning conversation was the idea that the word “same” was a dangerous word in K-12 math classes. Are these two expressions the same? No, but they might be equivalent under certain conditions. Are the 2nd differences in a table of exponentially related values the same? Yes, they follow the same pattern as the first differences (a key to recognizing it as an exponential relationship). No, the second differences are not all the same constant (a key to determining it is not a quadratic relationship). How do I add 2/3 and 3/5? Can I add 2 apples and 3 oranges? Yes, they make five. Five what? Oh. Five fruit? That would make them all the same, right? Yep, you have just turned apples and oranges into the common denominator of fruit. — Is there a threshold concept or two lurking in those examples? Operations on quantities can introduce new types? How operations behave depends on the types of quantities they are used with? Numbers and expressions can be of different types, depending on the operations?

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