This morning’s water-cooler topic here at the Math Forum: *threshold concepts. *I’m not sure if it’s a phrase we just coined or not, it’s definitely not original, I think Erik Meyer and Ray Land coined the term, but it led to really interesting conversations and storytelling, more of which I hope will continue in the comments.

Our working definition of a threshold concept: something that changes the way you understand mathematics, which after you understand it, you can’t go back to previous ways of thinking, and you find yourself interpreting and understanding math through the lens of that concept.

We have a hunch that threshold concepts might have been those that were controversial on being discovered (e.g. irrational numbers, negative numbers, infinitesimals…) though I’m not sure that’s a necessary or sufficient condition.

Here’s one example: knowing that the ratio of the circumference to the diameter of a circle is π is not necessarily a threshold concept (it can be as boring as knowing the first 20 digits of π, if you just think of it as a fact). But understanding that there is a constant relationship between the circumference and diameter of a circle that can be described by a number that cannot be measured, represented as a fraction, or written out completely (and yet can still be represented, calculated with, etc.) is a *big deal*. When you encounter numbers like *e* and √2, you might understand them in terms of your understanding of π. You might go out looking for more numbers like π. You might begin to wonder about the concept of limits. A new door in mathematics has been opened, and after going through it, you can’t go back…

What other concepts might be threshold concepts? Do you have a personal story of a new door in mathematics opening up? And, as educators, should threshold concepts inform our teaching? How?

I recently ran into the Threshold Concepts idea reading about work done in the UK:

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?

On this site: http://www.ee.ucl.ac.uk/~mflanaga/thresholds.html, the features of a threshold are spelled out, I found this idea useful:

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.

[...] as Max explains, are concepts that change a person’s way of understanding a discipline or subject area. After [...]

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