mathematical thinking: what every middle and high school teacher should know ben sinwell, pendleton...
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Mathematical Thinking: What Every Middle and High School Teacher Should Know
Ben Sinwell, Pendleton High SchoolEd Dickey, University of South
Carolina
2015 SCCTM ConferenceGreenville SC
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Mathematical Thinking
• What is it?• South Carolina Portrait of the College
and Career Ready student.• How do we develop mathematical
thinking?
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What is NOT Mathematical Thinking?
• Tricks and Memorizationhttp://www.pedagonet.com/maths/TrickBusters.pdf
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Example
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Intuitive vs. Mathematical Approaches
• Intuitive reasoning:27 equilateral triangle sized pieces, so $675 / 27 =
$25. The triangle is $25, the rhombus is $50 (twice as big), and trapezoid is $75 (3 times bigger)
• System of equations: r = 2e, t = 3e, and 2e + 5r + 5t = $675
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Student Work
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Student Work
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Student Work
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Portrait of College and Career Ready
p. 9 of http://ed.sc.gov/agency/ccr/Standards-Learning/Mathematics.cfm
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Portrait of College and Career Ready
• Academic Success and Employability• Interdependent Thinking and Collaborative Spirit• Intellectual Integrity and Curiosity• Logical Reasoning• Self-Reliance and Autonomy• Effective Communication
p.10 of http://ed.sc.gov/agency/ccr/Standards-Learning/Mathematics.cfm
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21st Century Knowledge and Skills
• Problem Solving• Critical Thinking• Communication• Collaboration• Self-management
From the National Research Council
FREE: http://tinyurl.com/q25wodk
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NCTM’s Principles to Actions
Available from NCTM.orgAs book ($28.95 or $23.16 for members)OrAs PDF ($4.99 or $3.99 for members)
http://www.nctm.org/store/Products/%28eBook%29-Principles-to-Actions-%28PDF-Downloads%29/
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Effective Mathematics Teaching Practices
1.Establish mathematics goals to focus learning.
2.Implement tasks that promote reasoning and problem solving.
3.Use and connect mathematical representations.
4.Facilitate meaningful mathematical discourse.
.
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Effective MathematicsTeaching Practices
5. Pose purposeful questions.
6. Build procedural fluency from conceptual understanding.
7. Support productive struggle in learning mathematics.
8. Elicit and use evidence of student thinking
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Processes & Practices
• SCCCR Mathematics Process Standards• Common Core Standards for Mathematical
Practices• Next Generation Science Practices
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SC Process StandardsA mathematically literate student can:• Make sense of problems and persevere in solving them.• Reason both contextually and abstractly.• Use critical thinking skills to justify mathematical reasoning and
critique the reasoning of others.• Connect mathematical ideas and real-world situations through
modeling.• Use a variety of mathematical tools effectively and strategically.• Communicate mathematically and approach mathematical
situations with precision.• Identify and utilize structure and patterns.
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SC Math Processes
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Next Gen Science Practices
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Mathematical Thinking• What it REALLY Means to Learn (Schwartz)• Habits of Mind (EDC, Cuoco et. al.)• National Board of Professional Teaching Standards• Anna Sfard and Paul Cobb• Mathematical Worlds (David Tall)• What it means to be SMART in math (Horn)
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What ‘Learning How to Think’ Really MeansBarry Schwartz, Swarthmore CollegeChronicle of Higher Education, June 2015http://chronicle.com/article/What-Learning-How-to-Think/230965/?cid=at&utm_source=at&utm_medium=en
• Is there a right way to think?• If so, what is it?
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What ‘Learning How to Think’ Really Means
Intellectual virtues (not skills):• Love of Truth• Honesty• Fair-mindedness• Humility
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What ‘Learning How to Think’ Really Means
• Perseverance• Courage• Good listening• Empathy and Perspective-taking• Wisdom
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Love of truth
• Current relativism based on perspectives• Different perspectives provide piece of truth
previously invisible• When people have respect for truth, they seek
it out and speak it in dialogue
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Honesty
• Allowing students to face the limits of what they themselves know
• Encourage students to own up to their mistakes
• Accept unpleasant truth and see what you can do about it instead of denying it
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Fair mindedness
• Psychologically we emphasize evidence that is consistent with our beliefs
• Evaluate the arguments of others in a manner that is fair
• Use reason less like a lawyer making a case than as a judge deciding one
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Humility
• Student should be allowed to face up to their own limitations and mistakes
• Seek help from others
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Perseverance
• “… little that is worth knowing or doing comes easily.”
• Developing the “muscle” versus excavating a natural resource
• Carol Dweck’s growth mindset
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Courage
• Standing up for what you believe is true• Taking risks to pursue intellectual paths that
may not pan out
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Good Listening
• Students cannot learn from others, or their teachers, without listening.
• Good listening takes courage because your views and plans may be challenged
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Empathy
• Society has moved from “authority” to “shared decision” making
• Especially in medicine, law, and education• Good teachers avoid one-size-fits-all lesson
plans• Reach every student where she or he is BUT• Must gain insight into the thoughts and
aspirations of each student
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Wisdom
• Finding balance between the “mean” vs “extreme”
• Wisdom manages the other intellectual virtues
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Wisdom• Timid vs. reckless• Careless vs. obsessive• Flighty vs. stubborn• Speaking up vs. listening up• Trust vs. skepticism• Empathy vs. detachment
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How teachers encourage these virtues…
• Model them yourselves in your everyday behavior
• The questions we ask teach students how to ask questions
• How we pursue dialogue models reflectiveness• Students watch who we call on or don’t and
learn about fairness
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How teachers encourage these virtues…
• We are always modeling and students are always watching
• Teach when and how to interrupt and• Teach how to listen by doing this ourselves• Admit we don’t’ know something to
encourage both intellectual honesty and humility
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Habits of MindStudents should be:• Pattern Sniffers• Experimenters• Describers• Tinkerers
• Inventors• Visualizers• Conjecturers• Guessers
From: Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle
for mathematics curricula. Journal of Mathematical Behavior, 15, 375-407.
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Describers• Give precise descriptions of the steps in a process. • Invent notation.• Argue. • Write. Students should develop the habit of writing
down their thoughts, results, conjectures, arguments, proofs, questions, and opinions about the mathematics they do, and they should be accustomed to polishing up these notes every now and then for presentation to others. From: Cuoco et al., 1996
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What are the next 2 terms?
• 5, 8, 11, 14, …
• 4, 6, 8, 10, …
• 10, 20, 30, 40 …
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What is the pattern?
• 5, 8, 11, 14, …
• 4, 6, 8, 10, …
• 10, 20, 30, 40 …
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Results of Pattern SniffingTo get the next term, double the previous term and
subtract the term before that.An= 2An-1 – An-2
• 5, 8, 11, 14, … • • 4, 6, 8, 10, … • • 10, 20, 30, 40, …
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Conjecturers
• Will An= 2An-1 – An-2 work for all arithmetic sequences?
• What about geometric sequences?
• How about 1, 4, 9, 16, 25, …?
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National Board of Professional Teaching Standards
• Standard VIWays of Thinking Mathematically
• Investigate and explore patterns• Discover structures• Explore relationships• Formulate and solve problems• Justify and communicate• Question and extend
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AYA Mathematics• Reasoning correctly using processes such as classification,
representation, deduction and induction;• Using heuristics as a key strategy to guide solutions to
mathematical problems, such as testing extreme cases, conducting an organized search of specific examples, and using different problem representations;
• Modeling mathematical relations in problem situations—describing important relationships through symbolic expressions and other representations;
• Connecting ideas, concepts, and representations across the strands of mathematics.
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AYA Mathematics• Students’ mathematical achievement … dependent on their
ability to conceptualize and analyze mathematics, • Discover structures and establish relationships, to explore
justification and proof, and to formulate and solve problems. • Teachers know that they must develop students’ mental
acuity as well as pencil-and-paper skills. • Technology tools to help develop students’ reasoning,
mathematical thinking, and discourse.• Accomplished teachers are able to use applications such as
graphing technology, interactive geometry software, and computer algebra systems to support student inquiry, conjecture, and proof.
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Seeing Structure in Expressions
Resistors in Parallel
+ +
As increases what happens to the overall resistance if others are constant?
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Computer Algebra System
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Computer Algebra System
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How Do We Learn Mathematics?
• Anna Sfard, Professor of Mathematics Education at the University of Haifa
• Paul Cobb, Professor in the Dept. of Teaching & Learning at Vanderbilt University
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How Do We Learn Mathematics?• Acquisitionism: “Portrays mathematics as pre-
given structures and procedures” that are acquired “through passive ‘transmission’ or actively through the learner’s own constructive efforts.”
• Existing knowledge “acquired or reconstructed by the learner”
Sfard, A. and Cobb, P. (2014). Reasoning in mathematics education: What it can teach us about human learning? In Sawyer, K. R. (Ed.), The Cambridge handbook of the learning sciences (2nd Ed., pp. 545-564). Cambridge: Cambridge University Press.
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How Do We Learn Mathematics?• Participationism: “Portrays mathematics as a
form of human activity rather than as something to be ‘acquired’ and thus view learning as process of becoming a participant in this distinct type of activity”
• “One of the many human ways of doing things…”Sfard, A. and Cobb, P. (2014). Reasoning in mathematics education: What
it can teach us about human learning? In Sawyer, K. R. (Ed.), The Cambridge handbook of the learning sciences (2nd Ed.,
pp. 545-564). Cambridge: Cambridge University Press.
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In your own words, how do we find a solution to the system of equations below?
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In your own words, how do we find a solution…
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In your own words, how do we find a solution…
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In your own words, how do we find a solution…
You write the y-intercept then You graph it and see where the[y] intercept
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Three Worlds of Mathematics
• David Tall• Transition in thinking from school mathematics
to formal mathematics• Recognition of patterns• Repetition of sequences of actions (automatic)• Language to describe and refine
Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5-24.
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Three Worlds of MathematicsWorlds
1. (conceptual) Embodied2. (proceptual) Symbolic3. (axiomatic) Formal
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Three Worlds of Mathematics
• “Failure to compress counting procedures into thinkable concepts can lead to learning facts by rote.”
• Each traveller’s journey through the Worlds is different
• “met-befores”– Numbers to fractions– Obstacles help us learn and progress in Worlds
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Commutative Vector Addition
• Embodied World: truth of u + v = v + u follows from parallelogram (trace finger)
• Symbolic World: vectors are matrices were addition is commutative (component part)
• Formal World: commutativity holds as part of definition of a vector space (axiom)
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Being Mathematically Smart: Is it calculating quickly and accurately?1. We tend to rank people “on [this] one dimension
of mathematical competence. This rank order usually relates to students’ academic status, and students tend to be aware of it.”
2. We need to create a “multidimensional competence space.”From: Horn, I. (2012). Strength in numbers:
Collaborative learning in secondary
mathematics. Reston, VA: NCTM, pp. 30-31.
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Student Work
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Student Work
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What Does It Mean to be Smart in Mathematics?
Ilana Horn, Vanderbilt Universityhttps://teachingmathculture.wordpress.com/2014/03/17/what-does-it-mean-to-be-smart-in-mathematics/
• posing interesting questions (Fermat);• making astute connections (Wiles);• representing ideas clearly (Poincaré);• developing logical explanations (Klein);• working systematically (Appel and Haken); and• extending ideas (irrational/complex number systems).
Horn calls these “vital mathematical competencies”
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Lesson Plan Template
Source: Escondido Unified High School District https://docs.google.com/document/d/1ekwZBu4N1Xmy4B1Y0kpCyr7BihhtKQlSzmzbvL1WmN4/mobilebasic?pli=1
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Lesson Plan Template
Source: Escondido Unified High School District https://docs.google.com/document/d/1ekwZBu4N1Xmy4B1Y0kpCyr7BihhtKQlSzmzbvL1WmN4/mobilebasic?pli=1
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Lesson Plan Template
![Page 65: Mathematical Thinking: What Every Middle and High School Teacher Should Know Ben Sinwell, Pendleton High School Ed Dickey, University of South Carolina](https://reader035.vdocuments.site/reader035/viewer/2022062315/5697bf991a28abf838c91ebe/html5/thumbnails/65.jpg)
Lesson Plan Template
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Closing• Nova: The Great Math Mystery• Is Math a discovered from nature• Or invented by man?• http://video.pbs.org/video/2365464997/
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The Great Math Mystery