Provoking Learning in Calculus: Activities for

Problematizing Pedagogy Reem Jaafar

Associate Professor Department of Mathematics, Engineering and Computer Science

rjaafar@lagcc.cuny.edu

CUNY 2014 Mathematics Conference: Effective Instructional Strategies Graduate Center of CUNY

May 9th, 2014

… Let’s use Writing to Learn Activities

Common Student Challenges

in a first Calculus Course

• Weak algebraic skills

• Perception of mathematical learning (math = apply-the-

algorithm)

– No training in connecting mathematical definitions, notations

or formulae to conceptual understanding

– Difficulty in understanding/applying theorems (seen as

problem-solving steps)

– Difficulty “visualizing”, translating, interpreting graphs

CUNY-Wide…. • While Calculus is the most important course

among all disciplines, it ranks first in terms of

fail rate among all courses across CUNY.

• Recently, CUNY has launched a Calculus

Summer Boot Camp to better prepare students

exiting precalculus for calculus.

Who are Our Students?

Words used by students in describing

a writing-Intensive Math Course at the

Beginning of the semester

Assignment #1: A “Simple” Definition?

• Give the definition of the derivative of the function f at a

So What is Notation??

• Research shows that Many students use notation as sort of “whatever” that gets them from one point to the other, not as a language used to communicate ideas or translate thoughts (Gopen&Smith, Porter and Masingila, Morrel).

• A detailed in-class activity was given as an intervention to address some of these issues.

• Porter, M.K, J. O. Masingila 2000. EXAMINING THE EFFECTS OF WRITING ON CONCEPTUAL AND PROCEDURAL KNOWLEDGE IN CALCULUS. Educational Studies in Mathemat-ics . 42: 165{177).

• Morrel, J. H. 2006. USING PROBLEM SETS IN CALCULUS.PRIMUS: Problems, Resources, and Issues in Mathematics Under-graduate Studies . 16(4): 376-384.

• Gopen, G., D. Smith 1990. What's an Assignment Like You Doing in a Course Like This?: Writing to Learn Mathematics. The College Mathematics Journal . 21(1): 2{19.

Assignment #2: Low-stakes writing Task

• “If the velocity of an object at time t is 0, its acceleration then is also 0”—True or False? Please justify your claim.

Sample Work• When v(t) = 1, we get v’(t) = 0, but when v(t) = t,

we get v’(t) = 1.

Sample Work• When v(t) = 1, we get v’(t) = 0, but when v(t) = t,

we get v’(t) = 1.

• Zero acceleration means no change in velocity.

Therefore v(t) ≠ a(t).

Sample Work

• When v(t) = 1, we get v’(t) = 0, but when v(t) = t, we

get v’(t) = 1.

• Zero acceleration means no change in velocity.

Therefore v(t) ≠ a(t).

• When the motorcycle at startup, it can be stopped in

place but its wheels is in turn, this proves that it has

the acceleration but its speed is 0!

How to Initiate Mathematical Knowledge-Building?

Definition Notation as sort of “whatever”

Low-Stakes Writing Lack of Conceptual Understanding

An algorithmic Context of Learning

• Is a natural one for students to adopt, as documented

by Pettersson. So one should use it as a springboard

towards conceptual understanding.

• Students' natural inclination to algorithmic learning

can be used to stimulate “problematizing” activity, in

other words, to encourage students to encounter and

articulate questions they must then seek answers to. Pettersson, K. 2008. Algorithmic contexts and learning potentiality: a case study of students understanding of calculus. International Journal of Mathematical Education in Science and Technology .39(6):767-784.

Problematizing Some Activities…

Create the appearance of “paradox”

Options: Turn it to a Low-Stakes or High-Stakes Assignment

A Must: Provide students with checklist to make sure all necessary points are

addressed.

Dear Dr. Littlestone,

Yesterday I got into an argument with my boyfriend over a question on velocity and acceleration,

and I am hoping you can settle it for us.

(...) Even if the velocity at that moment is 0 it could be changing, so wouldn’t the acceleration

reflect that change and be different than 0? He said I was ridiculous, if the velocity is zero the

object is not moving so its acceleration is also zero.

(...) Who is right? If I am correct, could you please provide an example of a velocity function v(t)

where v(t1) = 0 but the acceleration v’(t1) ≠ 0?

  Thank you so much in advance,

Disconcerted by Derivatives

PS: pictures are welcome because I find them easy to understand, but could you please also

include an explanation that uses function notations? My boyfriend likes these explanations better.

Checklist: Does your response clearly…

• Say who was right and who was wrong?

• Correctly define velocity and acceleration?

• Correctly identify an error?

• Explain what the error is? (i.e. do you provide an example to illustrate why a

claim is wrong?)

• Provide a variety of different examples for your reader? (function example

v(t) = ?, pictorial/graphical example, real-life situation…)

• Summarize mathematical conclusions in English everyone can understand?• Present all the above in clear, organized format accessible to student readers• Provide follow-up questions or solved problems for reader to think about?

Advantages… • It initiates a dialogue with students in class

• It Creates and the unique opportunity to understand some of students' difficulties and how to address them.

• Students are more aware of what makes sense, as reflected in their improved performance.

The benefits of turning some

textbook problems into writing tasks

…..Distance or Displacement?

• The velocity function (in meters per second) is given for

a particle moving along a line. Find(a) the displacement

and (b) the distance traveled by the particle during the

given time interval.

• Results:

70% of students answered (a) correctly but only 50%

answered (b)correctly.

Students did not understand the difference between

distance and displacement.

High-stakes writing tasks

…..Distance or Displacement?

Dear Dr. Littlestone,

(...) If I do what my professor did, I should be able to find the distance traveled by the ball from

the moment I threw it until it landed... but I don’t understand the result I get below!!??

2.5 2.5

0 v(t) dt = h(t) ]0 = h(2.5) – h(0) = 0 – 0 = 0

 

Why am I getting an answer of 0? Is this telling me the ball has traveled 0 feet in all this time??

I know that can’t be true, so what am I doing wrong?

 

(...) And if my definite integral isn’t telling me the distance traveled by the ball, what

information is it actually giving me?

Many thanks in advance,

Insaning from Integrating

Checklist: Does your response clearly…

• Check each claim and computation made by the student (is each right or wrong)?• Correctly graph the student’s velocity function v(t) = -32t + 40?

2.5

• Correctly interpret the integral 0 v(t) dt in terms of areas?

1.25 2.5

• Compare the integrals 0 v(t) dt and 1.25 v(t) dt, and explain how they are similar and how they are different? 2.5

• Compare the integral 0 v(t) dt to the total area between the curve y = v(t) and the x-axis?

• Explain to the student how to find the total distance traveled by the ball?

2.5

• Explain to the reader what the integral 0 v(t) dt represents?

2.5

• Explain to the reader what the integral 0 |v(t)| dt represents?

• Before this Assignment; Only 50% of students understood the difference between displacement and distance.

• Before this Assignment; Only 50% of students understood the difference between displacement and distance.

• Students’ essay responses to this assignment showed that a vast majority (over 90%) had a clear understanding of the difference between displacement and distance.

• Before this Assignment; Only 50% of students understood the difference between displacement and distance.

• Students’ essay responses to this assignment were surprising. A vast majority (over 90%) showed a clear understanding of the difference between displacement and distance.

• With guidance of the checklist, they graphed the velocity function and were able to interpret the area under the curve.

• Before this Assignment; Only 50% of students understood the difference between displacement and distance.

• Students’ essay responses to this assignment were surprising. A vast majority (over 90%) showed a clear understanding of the difference between displacement and distance.

• With the guidance of the checklist, Students graphed the velocity function and were able to interpret the area under the curve.

• Essays also reflect Engagement: The average number of words used by students in composing the essay is around 500.

Words used by students in describing

a writing-Intensive Math Course at the

Beginning of the semester

…At the End of the Semester

Student Feedback

• To write about mathematics you have to grasp the

topic quite well. When doing writing assignments it

helps grasp the idea.

Student Feedback• To write about mathematics you have to grasp the

topic quite well. When doing writing assignments it

helps grasp the idea.

• I think my communication skills remain the same;

however my ability to interpret math improved.

Student Feedback• To write about mathematics you have to grasp the topic

quite well. When doing writing assignments it helps grasp

the idea.

• I think my communication skills remain the same; however

my ability to interpret math improved

• I think my understanding about some graphs and formulae

has been improving since we tried the writing assignments

Student Feedback• To write about mathematics you have to grasp the topic quite

well. When doing writing assignments it helps grasp the idea.

• I think my communication skills remain the same; however

my ability to interpret math improved.

• I think my understanding about some graphs and formulae has

been improving since we tried the writing assignments.

• It is useful because it enhances the critical thinking level.

Student Feedback• To write about mathematics you have to grasp the topic quite well.

When doing writing assignments it helps grasp the idea.

• I think my communication skills remain the same; however my

ability to interpret math improved.

• I think my understanding about some graphs and formulae has been

improving since we tried the writing assignments.

• It is useful because it enhances the critical thinking level

• They are useful because they encourage us to think in “real-life”

terms.

Implications• No such thing as “too much” guidance (checklist, pre-

reading, class discussions, homework).

• To move beyond a comfort zone, start from the familiar

(allow computational work).

• Create appearance of “paradox” to motivate engagement

(find hidden error).

• Follow-up is as crucial as feedback (next problems, exam

questions).

• Grading should recognize the “extra” burden on students.

Thank you!