Three sets of three
questions:
Generating Inquiry in Calculus
Class
Brent Ferguson
The Lawrenceville School, NJ
[email protected]
Purposes of today’s talk
• To make the case that our students’ deep cognitive
engagement is at the heart of learning.
• To explore some ways in which we can stimulate
student inquiry by modeling good questions in class.
• To provide specific moves that assist our work,
specifically with the generative, creative work of
proofs and problem-solving.
Punchline unveiled:
The “three sets of three”
• During proofs and problem-solving
(1) How is that legal?
(2) How is that helpful?
(3) How is that intuitive?
• When generating/initiating creative action
(A) What do you see?
(B) What do you want?
(C) What tool do you have to get there?
• When motivating Taylor series
(i) Can we model this?
(ii) How good is the model?
(iii) For what values is the model ‘good?’
Empowered learners:
what do they look like?
Motivated, Resourceful,
Self-Regulated Learners are…
• self-appraising (who am I?)
• self-directed (who do I want to become?)
• self-assessing (how am I doing?)
• self-advocating (what is my voice?)
Yes, it has
unlimited gas
mileage (so far
it’s traveled
+
12,000 miles
on 0 gallons of
fuel).
The Lawrenceville
School has a large
solar farm, which
generates over 90%
of our energy needs,
so the energy for our
grid is clean!
The Students are Watching…
• Do we want students to ask questions?
• Do we want them to grapple with difficulty?
• Do we want them to have some templates/ scripts
with which they can approach difficulty?
…then we need to provide these models in our classrooms.
Let us then begin with this end in mind.
The first questions we ask…
• Who are my students?
• What is my subject matter?
• Who am I?
Our moves should match the answers to these questions.
Mathematics…
• μ𝛼𝜃𝜂𝜏𝜂𝜍 – means ‘one who is training, learning a discipline,’ (a disciple, actually!)
• What patterns do I see in the world?
observation
• Can I prove it, in general?
proof
• What good is it?
application
Proofs…
We prove most results along the way (as we need them) including:
𝑠𝑖𝑛 𝜃
𝜃→0 𝜃
lim
𝑑
𝑑𝑥
𝑠𝑖𝑛 𝜃
𝜃→∞ 𝜃
=1
lim
𝑑
𝑑𝑥
𝑥 𝑛 = 𝑛 ∙ 𝑥 𝑛−1
Derivatives of other function models:
𝑒𝑥 ,
…and those of various function operations:
𝑑 𝑓 𝑥
𝑑𝑥 𝑔 𝑥
,
𝑑
𝑑𝑥
𝑓 𝑔 𝑥
,
𝑑
𝑑𝑥
𝑓 −1 𝑥 ,
lim
=
𝑎0
𝑏0
= 𝑐𝑜𝑠 𝑥 , and five other trig functions’ derivatives
𝑠𝑖𝑛 𝑥
𝑑
𝑑𝑥
𝑎0 ∙𝑥 𝑛 +𝑎1 𝑥 𝑛−1 +𝑎2 𝑥 𝑛−2 +⋯+𝑘
𝑥→∞ 𝑏0 ∙𝑥 𝑛 +𝑏1 𝑥 𝑛−1 +𝑏2 𝑥 𝑛−2 +⋯+𝑘
=0
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑎𝑥 ,
𝑑
𝑑𝑥
𝑙𝑛 𝑥 ,
𝑘∙𝑓 𝑥 ,
𝑑
𝑑𝑥
𝑓 𝑥 +𝑔 𝑥 ,
𝑑 𝑥
𝑓
𝑑𝑥 𝑎
𝑡 𝑑𝑡 = 𝑓 𝑥 ,
𝑑
𝑑𝑥
𝑏
𝑓′
𝑎
𝑙𝑜𝑔𝑎 𝑥
𝑑
𝑑𝑥
𝑓 𝑥 ∙𝑔 𝑥
𝑡 𝑑𝑡 = 𝑓 𝑏 − 𝑓 𝑎
…endeavoring at all times to use the language of function operations
(composites, products, etc.) so as to enhance students’ capacity to
connect this work to previous learning.
The “three sets of three”
• During proofs and problem-solving
(1) How is that legal?
(2) How is that helpful?
(3) How is that intuitive?
• When generating/initiating creative action
(A) What do you see?
(B) What do you want?
(C) What tool do you have to get there?
• When motivating Taylor series
(i) Can we model this?
(ii) How good is the model?
(iii) For what values is the model ‘good?’
The first set
“What do engaged students ask when being taught?”
• The “How to tutor a friend” conversation
(1) Did you get problem #17? (2) Can I see your solution? (3) Oh, I get how this one works…!
(They need our help to debunk myths about learning and studying!)
• During proofs and problem-solving
(1) How is that legal? (what’s the mathematics behind that action?)
(2) How is that helpful? (did that get us closer to our solution/goal?)
(3) How is that intuitive? (how would I have [in a million years] thought to do that…?)
The second set:
“How would I have thought of that?”
(A) What do you see?
(B) What do you want?
(C) What tool do you have to get there?
Let’s look at this in an example (or two…or better still, THREE).
Let’s prove…(on paper)
𝑑
𝑑𝑥
𝑓 𝑥 +𝑔 𝑥
= …?
𝑑
𝑓 𝑥 ∙𝑔 𝑥
𝑑𝑥
= …?
𝑑
𝑠𝑖𝑛 𝑥
𝑑𝑥
= …?
(We’ll need to refer to earlier results for this one):
𝑠𝑖𝑛 𝐴 + 𝐵 = 𝑠𝑖𝑛 𝐴 𝑐𝑜𝑠 𝐵 + 𝑐𝑜𝑠 𝐴 𝑠𝑖𝑛 𝐵
and
𝑠𝑖𝑛 𝜃
𝜃→0 𝜃
lim
=1
The third set:
Why study Taylor series?
(i) Can we model this function with an easier function?
We crave a simpler function, rather than this one. What’s our favorite kind? Polynomials...
(ii) How good is the model?
Once we’ve built this function, we need to know, at least roughly, how close we are to the actual values.
(iii) For what values is the model ‘good?’
If there are boundaries on the usefulness of this model, we need to know those. Nice motivator: 𝑦 =
1
1−𝑥
Here’s your chance…in 5 minutes!
• I have two wonderful gems to share with you: Vieta’s
formulas and Euler’s solution to the Basel Problem
(which use the Veita formulas, along with other results, to create a real thing of
beauty).
1 + 𝑎𝑥 1 + 𝑏𝑥 1 + 𝑐𝑥 ∙ ⋯ 1 + 𝑘𝑥 = ? ? ? in standard form
∞
1
1
1
1
1
=
+
+
+
+ ⋯ = ???
𝑛 2 12 22 32 4 2
𝑛=1
• But if you’d like to leave, I have (wait for it…)
THREE short slides to share with you, and the
evaluation instructions/details.
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 work.
• Strong exhortation to use study groups and TALK
• Music, videos, and bad jokes distributed liberally.
• Use of tau, the appropriate circle constant. It helps.
Read Michael Hartl’s TAU MANIFESTO, please.
Resources/authors for
further study and reflection:
• Brown, Roedinger, and McDaniel, Make It Stick: The
Science of Successful Learning
• Daniel Willingham, Why Don’t Students Like School? A Cognitive
Scientist Answers Questions About How the Mind Works and What It Means
• Carol Dweck, Mindset: The New Psychology of Success
– http://www.youtube.com/watch?v=ICILzbB1Obg
• Malcolm Gladwell, Outliers: The Story of Success
• Dan Pink, Drive: The Surprising Truth About What Motivates Us
• Claude Steele, Whistling Vivaldi: How Stereotypes Affect Us, and
What We Can Do
• Angela Duckworth’s extensive work on grit development
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
Strongly
Disagree
Disagree
Agree
Strongly
Agree
0
1
2
3
Send your text message to this Phone Number: 37607
poll code
for this session
_______
Speaker was engaging
and an effective
presenter (0-3)
(1 space) ___
___ ___
(no spaces)
Speaker was wellprepared and
knowledgeable (0-3)
Other comments,
suggestions, or
feedback (words)
(1 space)
___________
Session matched title
and description in
program book (0-3)
Example: 38102 323 Inspiring, good content
Non-Example: 38102 3 2 3 Inspiring, good content
Non-Example: 38102 3-2-3Inspiring, good content
Thank you for your interest and
for participating in this CMC-South session.
Please give feedback for session #694: code 19310
Text to 37607 the following: 19310, then a space,
then the three digits, followed by comments!
Brent Ferguson,
The Lawrenceville School
[email protected]
– Taylor’s Theorem: a proof with a
synthesis of FTC, product rule, & substitution
Appendix
Begin with:
𝑓 𝑥 =𝑓 𝑎 +
𝑥
𝑥
𝑓′
𝑎
𝑡 𝑑𝑡
(NOTE: This means 𝑥 ∙ 𝑓′ 𝑥 = 𝑥 ∙ 𝑓′ 𝑎 + 𝑥 ∙ 𝑎 𝑓′′ 𝑡 𝑑𝑡 ; we’ll use this in red below…)
𝑡=𝑥
= 𝑓 𝑎 + 𝑡 ∙ 𝑓′ 𝑡
𝑡=𝑎
−
𝑥
𝑡
𝑎
∙ 𝑓 ′′ 𝑡 𝑑𝑡
𝑥
= 𝑓 𝑎 + 𝑥 ∙ 𝑓 𝑥 − 𝑎 ∙ 𝑓′ 𝑎 − 𝑎 𝑡 ∙ 𝑓′′ 𝑡 𝑑𝑡
𝒙
𝑥
= 𝑓 𝑎 + 𝑥 ∙ 𝑓′ 𝑎 + 𝒙 ∙ 𝒂 𝒇′′ 𝒕 𝒅𝒕 − 𝒂 ∙ 𝒇′ 𝒂 − 𝑎 𝑡 ∙ 𝑓′′ 𝑡 𝑑𝑡
′
=𝑓 𝑎 +𝑥∙
𝑓′
𝑎 −𝒂∙
=𝑓 𝑎 + 𝑥−𝑎 ∙
𝑓′
𝒇′
𝑎 +
𝒂 +
𝑥
𝑎
𝒙
𝒙
𝒂
∙ 𝒇′′ 𝒕 𝒅𝒕 −
𝑥 − 𝑡 ∙ 𝑓′′ 𝑡 𝑑𝑡
𝑥
𝑡
𝑎
∙ 𝑓′′ 𝑡 𝑑𝑡
Taylor’s Theorem: an ‘integrative’ proof
𝑥
𝑓′
𝑎
Recall that starting from: 𝑓 𝑥 = 𝑓 𝑎 +
𝑡 𝑑𝑡, we generated , from the previous page:
𝑥
𝑓 𝑥 = 𝑓 𝑎 + 𝑥 − 𝑎 ∙ 𝑓′ 𝑎 +
𝑥 − 𝑡 ∙ 𝑓′′ 𝑡 𝑑𝑡
𝑎
𝑥
𝑓′′
𝑎
= 𝑓 𝑎 + 𝑓′ 𝑎 ∙ 𝑥 − 𝑎 +
=𝑓 𝑎 +
𝑓′
−𝒇′′
𝑎 ∙ 𝑥−𝑎 +
′
=𝑓 𝑎 + 𝑓 𝑎 ∙ 𝑥−𝑎 +
=𝑓
𝑎
=𝑓 𝑎
+ 𝑓′ 𝑎 ∙ 𝑥 − 𝑎
+
+ 𝑓′ 𝑎 ∙ 𝑥 − 𝑎
𝑓′′ 𝑎
2
+
=…and so on: 𝒇 𝒙 =
𝒕 ∙
𝒇′′ 𝒂
𝟐
𝑥−𝑎
𝑓 ′′ 𝑎
2
2
𝒙−𝒕
𝒙−𝒂
𝒏 𝒇 𝒂
𝒊=𝟎 𝒊!
𝟐
2
+∙
𝒕=𝒙
𝟐
𝒕=𝒂
𝑥
𝑓′′′
𝑎
+
−𝒇′′′ 𝒕 ∙
+
𝑥−𝑎
(𝒊)
𝟏
𝟐
𝑡 ∙ 𝑥 − 𝑡 𝑑𝑡
𝟏𝟏
𝒙−𝒕
𝟐𝟑
𝒇′′′ 𝒂
𝟑!
𝒙−𝒂
𝒙−𝒂 𝒊+
𝑥1
𝑎 2
+
2
∙ 𝑓′′′ 𝑡 𝑑𝑡
1
𝑡 ∙ 2 𝑥 − 𝑡 2 𝑑𝑡
𝒕=𝒙
𝟑
𝒕=𝒂
𝟑
𝑥−𝑡
+
𝒙 (𝒏+𝟏)
𝒇
𝒂
+
𝑥1
𝑎 3!
𝑥 (4)
𝑓
𝑎
𝒕 ∙
𝟏
𝒏!
𝑥−𝑡
𝑡 ∙
1
3!
3
∙ 𝑓 (4) 𝑡 𝑑𝑡
𝑥 − 𝑡 3 𝑑𝑡
𝒙 − 𝒕 𝒏 𝒅𝒕