Orchestrating the FTC Conversation: Explore, Prove, Apply Brent Ferguson The Lawrenceville School, NJ

Purposes of today’s talk To initiate some reflection on one of the ‘big ideas’ in mathematics (FTC) and to suggest that multiple, rigorous approaches will enhance understanding. To show some ways in which we can stimulate interdisciplinary connections with FTC, and teach a few non-mathematical lessons along the way. To share some tasks that bring attention to the importance of the idiosyncratic but useful notation and conventions of our discipline. This will help make the case that doing math well requires careful reading.

Speaking of reading, here’s a gem …from over a century ago This excerpt (1911) precedes the cognitive science of today that has only served to verify the points Whitehead makes about working memory space, etc. Brilliant… From Introduction to Mathematics, A. N. Whitehead, which is vol. 15 of the “Home University Library of Modern Knowledge” series (1911)

Sequence Some language considerations A classical approach: – FTC-II generated by considering Euler’s linearization method for approximating functions, then a definition of the definite integral – proving FTC-I using that definition Making connections: Algebra I class, personal development, identity formation, and growth mindset

The Fundamental Theorem? Let’s look at a familiar item first…

…of Arithmetic: (unique factorization theorem) – every positive integer can be written uniquely in standard form as a product of primes. …of Algebra: A polynomial of degree n has n roots. …of Calculus: (1) integration and differentiation are inverse operations, and (2) antidifferentiation, evaluated at two inputs, can be used to calculate a definite integral. “The Fundamental Theorem”

Development of the Concept Context for consideration: Riemann Sums, using data from helmet speedometer. This leads us to ask questions… –How accurate was our answer? –Was it likely an under- or overestimate? –Can we do better? How (or why not)? Definition time: the definite integral…

FTC-II emerges…first!

FTC-I emerges next, by proof…

Observations… Differentiation and integration are inverse functions. (This is a big deal!) The independent variables t & x are a bit tricky. Order of operations matters. Things get a little funky if you switch it around (let’s look at some examples).

Fun with the F.T.C.

A Calculus ‘poem’

This got me thinking…

Linearization as an approximating model

* Euler’s method as a connection to FTC

Other practices for success in Calculus Math histories: students write 2-4 pages about their relationship with math over the years…a trove of information! Formative feedback, decoupled in time from delivery of grades. Required revisitation of all test problems, and modeling/practice in class: homework revision. Keen attention in the first weeks to the form of students’ written homework, and commentary on how and why to use the notation effectively. Strong exhortation to use study groups and TALK mathematics…again, this is modeled and practiced in class as a way to equip students. Music, videos, and bad jokes distributed liberally.

A Message on ‘Fit’ “Good teachers join self and subject and students in the fabric of life.” –Parker Palmer, p.11, The Courage to Teach

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Thank you for your interest and for participating in this CMC-South session. Please give feedback for session #195: code 4253 Text to the following: 4253, then a space, then the three digits, followed by comments! Brent Ferguson, The Lawrenceville School

Taylor’s Theorem (time permitting) : a proof with a synthesis of FTC, product rule, & substitution

Taylor’s Theorem: an ‘integrative’ proof