A Capstone Mathematics Course for
Future Secondary Teachers
UTeach Institute – NMSI Conference
May 20, 2014
John Quintanilla
Co-Director, Teach North Texas
Professor of Mathematics and University Distinguished Teaching Professor
Tress Kringen
Alumna, Teach North Texas
Mathematics Teacher, Jack E. Singley Academy (Irving ISD)
Alyssa Mendez
Student, Teach North Texas
Capstone Course
“Why do I have to take Real Analysis?”
“[T]he mathematics courses taken by
prospective high school teachers
[should] include at least a three-course
calculus sequence, an introductory
statistics course, an introductory linear
algebra course, and 18 additional
semester-hours of advanced
mathematics, including 9 semesterhours explicitly focused on high school
mathematics from an advanced
standpoint. It is desirable to have a
further 9 semester-hours of
mathematics…”
p. 55
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Courses for TNT Math Majors
• MATH 2100: Functions and Modeling for Secondary
Mathematics Instruction
• TNTX 3100: Conceptual Algebra and Geometry
• MATH 3850: Mathematical Modeling
• MATH 4050: Advanced Study of the Secondary
Mathematics Curriculum
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Course Outline
Component
Assessed by
Mon/Wed lectures
(traditional math course)
%
Three midterms + Final 54
Homework
11
Certification preparation
Weekly problems
11
Friday presentations
Peer review
11
Course Projects:
Engagement activities
3 short essays x 4
13
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Course Outline
Prime Directive:
Math majors should not be embarrassed about not
knowing what they have not yet been taught.
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Monday/Wednesday Lectures
“Similar topics [to previous
courses], but we covered
wildly new nuances, and
explained previous knowledge
more in depth.”
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Monday/Wednesday Lectures
I think that the textbook
was useful and appropriate
for this course.
History
Mon/Wed lectures
Certification review
I think the textbook will be
a good resource for me to
have five years from now.
Friday Q&A
Projects
Student perceptions
Monday/Wednesday Lectures
1. Number Theory for Classroom Teachers
2. Polynomials and Rational Functions
3. Exponents and Logarithms
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Monday/Wednesday Lectures
1. Number Theory for Classroom Teachers
• Prove the rule to check if a number is a multiple of 9.
D15C0
• Convert 857,536 into hexadecimal.
• Convert 𝟎. 𝟏𝟐𝟕𝟗 into a fraction.
• Find all
𝟏
𝒏
with a delay of 1 digit
and a block of 3 digits.
• Find the decimal representation of
History
Mon/Wed lectures
Certification review
Friday Q&A
𝟖
.
𝟏𝟕
Projects
Student perceptions
Monday/Wednesday Lectures
1. Number Theory for Classroom Teachers
• Prove that |𝒛|𝟐 = 𝒛 ∙ 𝒛
• Prove that log 𝟏𝟎 𝟐 is irrational.
• Prove that 𝟐 + 𝟑 is both irrational and algebraic.
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Monday/Wednesday Lectures
2. Polynomials and Rational Functions
• Prove the Rational Root Test.
• Prove the Conjugate Root Theorem.
• Explain why a sixth-degree polynomial has at most five
“turning points.”
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Monday/Wednesday Lectures
3. Exponents and Logarithms
• Find
𝟏𝟗
𝟐𝟓, 𝟕𝟐𝟗 without a calculator.
• Why are the Laws of Logarithms true?
• Why does lim 𝟏 +
𝒏→∞
• Why does
History
𝒙𝟏
𝒅𝒕
𝟏 𝒕
𝟏 𝒏
𝒏
= 𝒆?
= ln 𝒙?
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Student Perceptions
Before (After) taking this course, I understood the reasons
why I had take theoretical courses like Real Analysis.
Student Perceptions
Before taking this course, I
thought I understood the concepts
taught in secondary mathematics
well enough to teach them.
After taking this class, I now
realize that I didn’t understand
secondary mathematics as well
as I had initially thought.
Certification Review
• Long-range purpose: prepare students for the state
mathematics certification exam
• Discussed in office hours, not in class
• Hopefully a relatively painless way to remind (or teach)
students about the secondary mathematics curriculum
• Hopefully not a huge time sink
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Certification Review
• Precalculus (8 weeks)
–
–
–
–
–
–
Functions
Systems of equalities and inequalities
Exponential and logarithmic functions
Trigonometry (3 weeks)
Polar coordinates and conic sections
Sequences and series
 ~15 problems,
choose 5
 Pledge that all
problems were
attempted
• Calculus (5 weeks)
– Limits and continuity
– Differential calculus + applications (3 weeks)
– Integral calculus
 Do NOT appear
on exams or final
• Combinatorics (1 week)
• Basic probability and descriptive statistics (1 week)
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Student Perceptions
Should the reviews be
retained in future years?
History
Mon/Wed lectures
Certification review
What percentage of
problems were…
Friday Q&A
Projects
Student perceptions
Friday Question-and-Answer Sessions
 Inspired by example of Iowa’s Central College.
 Students are given a list of difficult questions.
 Students are called at random to the board to answer
the next question on the list
 Professor/MT plays the curious/obnoxious student
 Grade obtained by peer evaluation
 Reviews (or, in some cases, teaches) content without
boring students to death
 Addresses their own misconceptions in a low-key way
 Complements the design/delivery of 5E lesson plans
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Friday Question-and-Answer Sessions
A student asks, “Is
History
Mon/Wed lectures
Certification review
𝒂𝟐 + 𝒃𝟐 = 𝒂 + 𝒃?”
Friday Q&A
Projects
Student perceptions
Friday Question-and-Answer Sessions
A student asks, “My father was helping me with
my homework last night and he said that the
book is wrong. He said that 𝟒 = 𝟐 and 𝟒 = −𝟐
because 𝟐𝟐 = 𝟒 and (−𝟐)𝟐 = 𝟒. He wants to know
why we are using a book that has mistakes.”
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Friday Question-and-Answer Sessions
You ask a student to find 28% of 50.
His solution: 𝟐𝟖 × 𝟎. 𝟓 = 𝟏𝟒.
What would you tell the student?
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Student Perceptions
Should the Friday
presentations be
retained in future years?
What percentage of the
Friday problems were
___ for you to answer?
Student Perceptions
Most questions were
realistic that I should be
prepared to answer.
I feel better prepared to
anticipate and respond to my
future students’ questions.
Project: Finding Engagement Activities
Basic premise:
Authors of textbooks (and hence aspiring math teachers)
seriously believe that the following problem actually
piques student interest in mathematics.
“Farmer Jones has to
enclose a pen with 100
meters of fence. One side
of the pen is already in
place. What are the
dimensions of the pen
that has the maximum
possible area?”
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Project: Finding Engagement Activities
 Answer 3 questions (150-250 words) from each of
Algebra, Geometry, Precalculus, Prob/Stat
o
o
o
o
o
o
What interesting word problems can your students do now?
How can this topic be used in the future (extend from the past)?
How has this appeared in pop culture (high culture) (the news)?
Who contributed to the discovery (development) of this topic?
What are the contributions of various cultures to this topic?
How can technology be used to engage students with this topic?
 Grading: Accuracy, Engagement and Understandability
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Project: Finding Engagement Activities
 All (ungraded) essays are amalgamated into a
miniature reference book and returned to class.
 The best essays (with students’ permission) are placed
on my blog, http://www.meangreenmath.com.
 One memorable example: order of operations.
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Student Perceptions
I feel like I have more ideas
for capturing my future
students’ interest in math.
Should the engagement
project be retained in
future years?
Evidence of Course Effectiveness
First Attempts on Texas 8-12 Mathematics Certification Exam
93%
83%
72%
Number of students attempting exam
P < 0.002
Perceptions of Capstone Course
I hope future TNT
students take Math 4050
in the same format.
After taking this class, I
feel much better prepared
to be a math teacher.
Summary
• Core components of our capstone course
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•
•
•
•
Monday/Wednesday lectures
Certification reviews
Friday Q&A
Course project
Illustrates importance of theorem-proving skills
Low-key way to address student misconceptions
Low-key way to deepen knowledge of Precal/Calculus
Prepare students for certification exam and classroom
Complements the UTeach teacher-education sequence.
MeanGreenMath.com
Feel free to subscribe,
promote on Twitter,
etc.
4050 Webpage
Friday Q&A, projects,
certification reviews,
and class outline
Conference Survey
Friday Question-and-Answer Sessions
A student presents the following work:
𝟑 𝒙+𝟐
=
𝟖
𝟏𝟔
𝟑−𝟐 𝒙+𝟐−𝟐
=
𝟖−𝟐
𝟏𝟔 − 𝟐
𝟏
𝒙
=
𝟔 𝟏𝟒
𝟕
=𝒙
𝟑
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Friday Question-and-Answer Sessions
A student asks, “What is 𝟎−𝟕 ?”
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Friday Question-and-Answer Sessions
A junior-high student is excited: “Miss Jones, I’ve
found a new way to get the average of two
numbers. Take 12 and 18, for example. Subtract
12 from 18; that’s 6. Divide by 2; that’s 3. Add 3
to 12, and you get the average, 15.”
If you were Miss Jones, what would you tell the
student?
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Friday Question-and-Answer Sessions
A student presents the following work:
𝒙𝟐 − 𝟏𝟒𝒙 + 𝟐𝟒 = 𝟑
𝒙 − 𝟏𝟐 𝒙 − 𝟐 = 𝟑 ∙ 𝟏
𝒙 − 𝟏𝟐 = 𝟑 𝒐𝒓 𝒙 − 𝟐 = 𝟏
𝒙 = 𝟏𝟓 𝒐𝒓 𝒙 = 𝟑
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Friday Question-and-Answer Sessions
A student presents the following work:
𝟒𝒙𝟐 − 𝟐𝟑𝒙 + 𝟏𝟓 = 𝟎
𝒕𝟐 − 𝟐𝟑𝒕 + (𝟏𝟓)(𝟒) = 𝟎
𝒕𝟐 − 𝟐𝟑𝒕 + 𝟔𝟎 = 𝟎
𝒕 − 𝟐𝟎 𝒕 − 𝟑 = 𝟎
𝒕 = 𝟐𝟎 𝒐𝒓 𝒕 = 𝟑
𝟐𝟎
𝟑
𝒙=
= 𝟓 𝒐𝒓 𝒙 =
𝟒
𝟒
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions
Friday Question-and-Answer Sessions
A student asks: “I noticed the following pattern:
𝟐𝟓𝟔 × 𝟐𝟓𝟔 = 𝟔𝟓, 𝟓𝟑𝟔
𝟐𝟓𝟕 × 𝟐𝟓𝟕 = 𝟔𝟔, 𝟎𝟒𝟗
𝟐𝟓𝟖 × 𝟐𝟓𝟖 = 𝟔𝟔, 𝟓𝟔𝟒
Is there a reason why the last two digits are
perfect squares? I know it usually doesn’t work
out that way.”
History
Mon/Wed lectures
Certification review
Friday Q&A
Projects
Student perceptions