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

Mathematical Thinking

• What is it?• South Carolina Portrait of the College

and Career Ready student.• How do we develop mathematical

thinking?

What is NOT Mathematical Thinking?

• Tricks and Memorizationhttp://www.pedagonet.com/maths/TrickBusters.pdf

Example

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

Student Work

Student Work

Student Work

Portrait of College and Career Ready

p. 9 of http://ed.sc.gov/agency/ccr/Standards-Learning/Mathematics.cfm

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

21st Century Knowledge and Skills

• Problem Solving• Critical Thinking• Communication• Collaboration• Self-management

From the National Research Council

FREE: http://tinyurl.com/q25wodk

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/

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.

.

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

Processes & Practices

• SCCCR Mathematics Process Standards• Common Core Standards for Mathematical

Practices• Next Generation Science Practices

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.

SC Math Processes

Next Gen Science Practices

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)

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?

What ‘Learning How to Think’ Really Means

Intellectual virtues (not skills):• Love of Truth• Honesty• Fair-mindedness• Humility

What ‘Learning How to Think’ Really Means

• Perseverance• Courage• Good listening• Empathy and Perspective-taking• Wisdom

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

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

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

Humility

• Student should be allowed to face up to their own limitations and mistakes

• Seek help from others

Perseverance

• “… little that is worth knowing or doing comes easily.”

• Developing the “muscle” versus excavating a natural resource

• Carol Dweck’s growth mindset

Courage

• Standing up for what you believe is true• Taking risks to pursue intellectual paths that

may not pan out

Good Listening

• Students cannot learn from others, or their teachers, without listening.

• Good listening takes courage because your views and plans may be challenged

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

Wisdom

• Finding balance between the “mean” vs “extreme”

• Wisdom manages the other intellectual virtues

Wisdom• Timid vs. reckless• Careless vs. obsessive• Flighty vs. stubborn• Speaking up vs. listening up• Trust vs. skepticism• Empathy vs. detachment

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

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

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.

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

What are the next 2 terms?

• 5, 8, 11, 14, …

• 4, 6, 8, 10, …

• 10, 20, 30, 40 …

What is the pattern?

• 5, 8, 11, 14, …

• 4, 6, 8, 10, …

• 10, 20, 30, 40 …

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, …

Conjecturers

• Will An= 2An-1 – An-2 work for all arithmetic sequences?

• What about geometric sequences?

• How about 1, 4, 9, 16, 25, …?

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

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.

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.

Seeing Structure in Expressions

Resistors in Parallel

+ +

As increases what happens to the overall resistance if others are constant?

Computer Algebra System

Computer Algebra System

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

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.

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.

In your own words, how do we find a solution to the system of equations below?

In your own words, how do we find a solution…

In your own words, how do we find a solution…

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

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.

Three Worlds of MathematicsWorlds

1. (conceptual) Embodied2. (proceptual) Symbolic3. (axiomatic) Formal

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

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)

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.

Student Work

Student Work

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”

Lesson Plan Template

Source: Escondido Unified High School District https://docs.google.com/document/d/1ekwZBu4N1Xmy4B1Y0kpCyr7BihhtKQlSzmzbvL1WmN4/mobilebasic?pli=1

Lesson Plan Template

Source: Escondido Unified High School District https://docs.google.com/document/d/1ekwZBu4N1Xmy4B1Y0kpCyr7BihhtKQlSzmzbvL1WmN4/mobilebasic?pli=1

Lesson Plan Template

Lesson Plan Template

Closing• Nova: The Great Math Mystery• Is Math a discovered from nature• Or invented by man?• http://video.pbs.org/video/2365464997/

The Great Math Mystery