Paradigm Shifts In Teaching
Higher Level High School
Mathematics
iMATHination Conference 2014
Q-Center, Saint Charles, IL
Saturday, January 25, 2014
Mr. Lamont Holifield, Presenter
Agenda
• Understanding the Urgency
• What is Math? and Why Do We All Need It?
• What’s Going Wrong In the Classroom?
• Key Strategies and Ways of Working
• Making a Difference on the Front Line
Disturbing Statistics!!
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In a recent international assessment of mathematics performance conducted in forty
countries across the world, the US ranked a lowly twenty eight. When the level of
spending was taken into account, the US sank to the bottom of the forty countries.
Interest in mathematics is declining. Dr. Boaler, who was a professor at Stanford
University, reported that the average number of math majors over the past ten years was
sixteen out of approximately 1,470 students. Furthermore it had dropped by 19 percent at
other four year institutions at the same time.
Nonresident aliens receive approximately 44 percent of all master’s degrees and 35
percent of all bachelor’s degrees in engineering, mathematics and information science in
the United States.
-Taken from What’s Math Got to Do With It, Boaler
Turn and Talk
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What’s most surprising about this information?
What are the ramifications?
What can we as mathematics educators do to reverse these disturbing statistics?
“
Lots of people think knowledge is what we want, and I
don’t believe that, because knowledge is astonishingly
transitory. We don’t employ people as knowledge bases,
we employ people to actually do things . . . Knowledge
bases comes out of books. So I want flexibility and
continuous learning . . . And I need team work.
Ray Peacock, Research Director at Phillips Laboratories
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What does this appear to be saying about the mathematics of work and life?
How does this compare with the mathematics of the classroom?
What are the similarities and what are the differences?
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What is Mathematics?
In pairs or alone, please prepare two definitions of mathematics. As you craft the definitions,
do it from the following lenses.
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If we asked your students to define mathematics, what would they say?
If we asked you for your definition, what would you say?
What is Mathematics?
Student’s Perspective
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Math is number.
Mathematics is a lot of rules.
Mathematician’s Perspective
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Mathematics is a study of patterns.
Mathematics is a set of connected ideas.
Mathematics is a human activity, a social
phenomena, a set of methods used to help
illuminate the world.
“
Computation involves going from a question to an
answer. Mathematics involves going from an
answer to a question. Such work requires creativity,
original thinking and ingenuity.
—Peter Hilton, an algebraic topologist
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Mathematics is a performance, a living act, a way of
interpreting the world. Imagine music lessons in which
students worked through hundreds of hours of sheet music,
adjusting the notes on the page , receiving checks and
crosses from the teachers, but never playing the music.
”
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How are mathematics and music related?
What does this comparison seem to say about the way we approach mathematics?
What’s Going Wrong in the Classroom?
Sample Scenarios
Topic: Variables
Presentation 1
Solve the following:
1. 9 + 18 − 2
2. 2 (𝑏 + 2)
Presentation 2
The Hickson household contains 3 people of different generations. The
total age of the three family members is 90.
Use variables and create an equation to:
(a)
(b)
Find reasonable ages for the 3 Hicksons.
Find another set of reasonable ages.
One student in solving this problem wrote:
𝐶 + 𝐶 + 20 + 𝐶 + 40 = 90
What do you think C means here?
How do you think the student got 20 and 40?
What set of ages do you think the student came up with?
Interview with a Student of Mathematics
• Interviewer: What does it take to be successful in a mathematics class?
• Student 1: Pay careful attention! Listen to the teacher!
• Interviewer: How is mathematics similar to other subjects.
• Student 1: It really isn’t. In math you have to remember, in other subjects
you can think about it.
It is ironic that math—a subject that should be all
“ about
inquiring, thinking, and reasoning—is one that
students have come to believe requires
NO THOUGHT!
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How does this quote strike you?
What can we do about it?
What implications does this quote have on our practice?
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Mathematics Problem to Consider
A woman is on a diet and goes into a shop to buy some turkey slices. She is
1
given three slices which together weigh of a pound, but her diet says that she
1
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is only allowed to eat of a pound? How much of the three slices can she eat
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while staying true to her diet?
Which of these represent the correct way to think?
Question: What is 18 𝑋 5?
Student 1 Response
Student 2 Response
18 + 2 = 20
10 𝑋 5 = 50
20 𝑋 5 = 100
8 𝑋 5 = 40
50 + 40 = 90
5 𝑋 2 = 10
100 − 10 = 90
Which of these represent the correct way to think?
Question: What is 18 𝑋 5?
Student 3 Response
Student 4 Response
15 𝑋 5 = 75
5 𝑋 18 = 10 𝑋 9
3 𝑋 5 = 15
75 + 15 = 90
10 𝑋 9 = 90
“
The beauty of mathematical problems is that they
can be seen or approached in different ways and,
although many have one answer, they can be
answered using different approaches.
”
Ponder the Following:
Suppose you were asked to present a lesson on expanding binomials.
In other words, students must be able to expand the following binomial:
(2𝑥 + 4)5
• What would be the layout of the lesson?
• How would you introduce the lesson?
• What was the rationale for the choices you made?
Common Core State Standards (CCSS)
“These standards are not intended to be
new names for old ways of doing
business. They are a call to take the next
step. It is time for states to work
together to build lessons from two
decades of standards-based reforms. It
is time to recognize that standards are
not just promises to our children, but
promises we intend to keep”
-National Governor’s Association for Best Practices and Council of Chief State School
Officers
Eight Mathematical Practices and Their Intent
Mathematical Practice 1: Make Sense of Problems and Persevere In Solving Them
Intent of the Practice
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Students make conjectures about the meaning of a solution and plan a solution
pathway.
Students try special cases or simpler form to gain insight. (They hypothesize and
test conjectures.)
Students monitor and evaluate their progress and discuss with others.
Students under multiple approaches and ask the question, “Does this solution
make sense?”
Eight Mathematical Practices and Their Intent
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Mathematical Practice 2: Reason Abstractly and Quantitatively
Intent of the Practice
Students can decontextualize a problem by representing the problem symbolically for a
solution.
Students can contextualize a problem by attending to the meaning of the quantities
involved in the problem.
Students can create a coherent representation of the task or problem presented.
Student can attend to the meaning of the quantities involved in a problem as well as the
units appropriate to problem.
Eight Mathematical Practices and Their Intent
Mathematical Practice 3: Construct Viable Arguments and Critique the Reasoning of Others
Intent of the Practice
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Students make conjectures and can explore the truths of those conjectures.
Students justify their conclusions and communicate them to others.
Students compare the effectiveness of two plausible arguments.
Students listen, read and respond to the arguments of others for sense making and clarity.
Eight Mathematical Practices and Their Intent
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Mathematical Practice 4: Model with Mathematics
Intent of the Practice
Students represent mathematical concepts by using such tools as diagrams, tables, charts,
graphs, calculators, graphing calculators, and various forms of area or volume models.
Students use symbols and tools to represent real-world situations.
Students routinely interpret their mathematical results in the context of the problem
situation.
Students are comfortable making assumptions and approximations to simplify a
complicated real-life situation and are comfortable testing those assumptions.
Eight Mathematical Practices and Their Intent
Mathematical Practice 5: Use Appropriate Tools Strategically
Intent of the Practice
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Students choose an appropriate tool for the problems and tasks presented in class and
homework. Is the tool necessary to the mathematical task or problem?
Students know the limits of each tool for providing accurate solutions to the problems.
Students detect tool-generated errors by estimating reasonable solutions without the tool.
Students use tools to explore and deepen their discovery and understanding of concepts.
Eight Mathematical Practices and Their Intent
Mathematical Practice 6: Attend to Precision
Intent of the Practice
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Student communicate precisely to others.
Students use clear definitions of terms in discussing their reasoning.
Students express numerical answers with a degree of precision appropriate for the
problem context.
Students calculate accurately and efficiently.
Students are careful about specifying units of measures and using proper labels.
Eight Mathematical Practices and Their Intent
Mathematical Practice 7: Look for and Make Use of Structures
Intent of the Practice
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Students’ attention is consistently drawn to the structure of mathematics as it occurs. For
example, the teacher consistently asks, “What do you notice is happening here? How does
this compare to our previous experiences?”
Students are engaged in exploring numerical and visual patterns that reveal structures.
Students can use strategies that shift the perspective of a problem—like auxiliary line in
geometry.
Eight Mathematical Practices and Their Intent
Mathematical Practice 8: Look For and Express Regularity in Repeated Reasoning
Intent of the Practice
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Students notice and discuss if their results seem reasonable while solving the problem and
when they solved the problem.
Students notice and can articulate patterns in calculations that can become generalized
properties or formulas.
Valuables Ways of Working
Lectures
Student-Led
Discussions
Individual Work
Types of Tasks
Long Applied Project
Short Questions
Contextual and Abstract Investigations
Five Practices for Orchestrating Productive
Mathematics Discussions
Anticipating
Connecting
Sequencing
Monitoring
Selecting
Credits and Resources Used
Common Core Mathematics in a PLC at Work, Timothy D. Kanold and
Matthew Larson
Five Practices for Orchestrating Productive Mathematics Discussion, Margaret
S. Smith and Mary K. Stein
What’s Math Got To Do With It?, Jo Boaler
Contact Informati0n
Mr. Lamont Holifield, M.A.
Urban Prep High School – Bronzeville Campus
2710 S. Dearborn Street, Chicago, IL 60616
Mobile: 773-307-0734
Email: lholifield@urbanprep.org