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![Page 1: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/1.jpg)
A Framework for A Framework for Mathematical Knowledge for Mathematical Knowledge for
Teaching at the Secondary LevelTeaching at the Secondary Level
Conference on Knowledge of Mathematics for Teaching at the Secondary Level
Pat Wilson [email protected]
Kathy Heid [email protected]
Tucson, AZ --- March 2011
Mid Atlantic Center for Mathematics Teaching and LearningCenter for Proficiency in Teaching Mathematics
![Page 2: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/2.jpg)
Situations ProjectSituations Project
• Classroom-based situations
• A framework for mathematical knowledge at the secondary level
![Page 3: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/3.jpg)
Mathematical Knowledge for Teaching Mathematical Knowledge for Teaching at the Secondary Levelat the Secondary Level
Mathematics Classroom Created
Situations
Mathematics Knowledge for Teaching
![Page 4: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/4.jpg)
Mathematical Knowledge for Teaching Mathematical Knowledge for Teaching Mathematics at the Secondary LevelMathematics at the Secondary Level
What is MKTS?• MKTS is specialized mathematical knowledge
• Knowing mathematics
• Being able to and having a tendency to use the mathematics in appropriate circumstances
• MKTS is not pedagogical knowledge.
How is MKTS different from MKT elementary?
![Page 5: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/5.jpg)
Examples of SituationsExamples of Situations
• PromptPrompt
• Mathematical FociMathematical Foci
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Circumscribing Polygons
PromptIn a geometry class, after a discussion about circumscribing circles about triangles, a student asked, “Can you circumscribe a circle about any polygon?”
A few mathematical foci
• Every triangle is cyclic. This fact is core to establishing a condition for other polygons to be cyclic.
• A convex quadrilateral in a plane is cyclic if and only if its opposite angles are supplementary.
• Every planar regular polygon is cyclic. However, not every cyclic polygon is regular.
![Page 7: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/7.jpg)
Summing Natural Numbers
PromptStudents found a need to sum the numbers 1 to n. One student offered a formula, but was not sure if he remembered it correctly. n(n +1) / 2 Students wondered if this would always be a natural number.
A few mathematical foci• When n is a natural number,
n (n+1) / 2 is also a natural number.
• Strategic choices for pair-wise grouping of numbers is critical to the development of the general formula.
• The first n natural numbers form an arithmetic sequence.
• Geometric arrays can lead to the development of the formula.
![Page 8: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/8.jpg)
FrameworkFramework
Mathematical Knowledge for Teaching • Mathematical Proficiency• Mathematical Activity• Mathematical Work of Teaching
![Page 9: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/9.jpg)
Mathematical ProficiencyMathematical Proficiency
• Conceptual Understanding
• Procedural Fluency
• Strategic Competence
• Adaptive Reasoning
• Productive Disposition
• Historical and Cultural Knowledge
![Page 10: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/10.jpg)
At secondary level
Procedural Fluency: Knowing when and how to apply a procedure in typical settings
Algorithms are more complicated, settings are somewhat varied.
Productive Disposition: Tendency to notice and apply mathematics in the world around us
Mathematical applications are verging on modeling rather than solely recognizing or associating.
![Page 11: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/11.jpg)
At secondary level
Adaptive Reasoning: Adjust to changes in assumptions and conventions
More attention is paid to assumptions and their consequences. There is more overt consideration of conventions.
Strategic competence: Generating, evaluating, and implementing problem-solving strategies
Strategies are evaluated for their mathematical fidelity, rather than solely for the viability of the answers they generate.
![Page 12: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/12.jpg)
Conceptual Understanding: Knowing “why”
Every triangle is cyclic.
Every regular polygon is
cyclic.
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Historical and Cultural Knowledge
Awareness of how people from various times and culture conceptualize and express mathematical ideas
E.g., The story of Gauss and the sum of the natural numbers from 1 to 100.
1 + 2 + 3 + … + 98 + 99 + 100
100 + 99 + 98 + … + 3 + 2 + 1
Sum = (100)(101)/2
![Page 14: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/14.jpg)
Historical and Cultural Knowledge
E.g., The Japanese theorem for cyclic polygons states that no matter how we triangulate a cyclic polygon, the sum of the inradii is constant.
![Page 15: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/15.jpg)
Mathematical ActivityMathematical Activity
• Mathematical Noticing
• Mathematical Reasoning
• Mathematical Creating
• Integrating Strands of Mathematical Activity
![Page 16: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/16.jpg)
Mathematical Noticing
Structure of mathematical systems
Symbolic form
Form of an argument
Connections within and outside of mathematics
![Page 17: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/17.jpg)
Summing Natural Numbers
PromptStudents found a need to sum the numbers 1 to n. One student offered a formula, but was not sure if he remembered it correctly. n(n +1) / 2 Students wondered if this would always be a natural number.
A few mathematical foci• When n is a natural number,
n (n+1) / 2 is also a natural number.
• Strategic choices for pair-wise grouping of numbers is critical to the development of the general formula.
• The first n natural numbers form an arithmetic sequence.
• Geometric arrays can lead to the development of the formula.
![Page 18: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/18.jpg)
Mathematical Noticing:Structure of mathematical systems
The set of natural numbers from 1 to n is an ordered sequence. Symmetry and the ordered nature of the sequence allow for rearrangements that facilitate finding a sum. Under any rearrangement, the cardinality and the sum of the elements remains the same.
![Page 19: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/19.jpg)
Mathematical Noticing:Structure of mathematical systems
. S 1 2 3 ... nS n n 1 n 2 ... 1
S S 1 n 2 n 1 3 n 2 ... n 1 2S n 1 n 1 n 1 ... n 1
2S n n 1
S n n 1
2
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Mathematical Noticing:Structure of mathematical systems
.
2 sum n(n 1) sum n(n 1)
2
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Mathematical Noticing:Structure of mathematical systems
.
sum n2
2
1
2n
n(n 1)
2
![Page 22: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/22.jpg)
Mathematical Noticing:Structure of mathematical systems
.
sum =
n 1
2gn
n(n 1)
2
![Page 23: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/23.jpg)
Mathematical Reasoning
• Justifying/Proving
• Reasoning when conjecturing and generalizing
• Constraining and extending
![Page 24: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/24.jpg)
Mathematical ReasoningConstraining and extending
We have looked at the sum of the first n natural numbers.
What is the sum of the squares of the first n natural numbers?
What is the sum of the kth powers of the first n natural numbers?
![Page 25: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/25.jpg)
Circumscribing Polygons
PromptIn a geometry class, after a discussion about circumscribing circles about triangles, a student asked, “Can you circumscribe a circle about any polygon?”
A few mathematical foci
• Every triangle is cyclic. This fact is core to establishing a condition for other polygons to be cyclic.
• A convex quadrilateral in a plane is cyclic if and only if its opposite angles are supplementary.
• Every planar regular polygon is cyclic. However, not every cyclic polygon is regular.
![Page 26: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/26.jpg)
Mathematical ReasoningConstraining and extending
Regular polygons are cyclic. What are the conditions under which non-regular polygons are cyclic?
![Page 27: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/27.jpg)
Mathematical Creating
• Representing
• Defining
• Modifying/transforming/manipulating
![Page 28: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/28.jpg)
Mathematical Creating :Representing
.
sum =
n 1
2gn
n(n 1)
2
![Page 29: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/29.jpg)
Mathematical Creating :Representing
.
sum n2
2
1
2n
n(n 1)
2
![Page 30: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/30.jpg)
Mathematical Work of Mathematical Work of TeachingTeaching
Teachers of secondary mathematics need the mathematical knowledge to be able to:
• know and do mathematics themselves
• facilitate their students’ development of mathematical knowledge
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Mathematical Work of TeachingMathematical Work of Teaching
• Analyze mathematical ideas
• Access and understand the mathematical thinking of students
• Know and use the curriculum
• Assess the mathematical proficiency of learners
• Reflect on the mathematics of practice
![Page 32: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/32.jpg)
Analyze mathematical ideas
Mathematics is dense, often succinct, elegant, but teachers need to see behind the elegant product.
Teachers need mathematical knowledge that helps them:• Probe; pull apart; unpack; dissect• Recognize conventions vs. core ideas• Understand the structure of mathematics
![Page 33: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/33.jpg)
Analyze mathematical ideasProbe; pull apart; unpack; dissect
What is an inverse?
(operation & domain required)
Types of inverses (additive, multiplicative, function)
Conventions:
notation (-1)
operation for inverse function
Strategies and rationale for finding inverse
What is a function? (Need to restrict the domain)
![Page 34: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/34.jpg)
Access and understand the mathematical thinking of students
Students often express ideas in vague ways with imprecise terms.
Teachers need mathematical knowledge that helps them:• Find key mathematical ideas within the
students’ thinking;• See important mathematics within
misconceptions; and • Understand the role of mathematical rigor.
![Page 35: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/35.jpg)
Access and understand the mathematical thinking of studentsFind key mathematical ideas within the
students’ thinking
Students found a need to sum the numbers 1 to n. n(n +1) / 2
Students wondered if this would always be a natural number.
Odd number/2 ≠ integer
Why is n(n+1) always even?•If n is even, n+1 is odd•If n is odd, n +1 is even•Even x odd = even
Why are even numbers 2n? Odd numbers 2n +1?
What about (n2 + n)/2?
![Page 36: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/36.jpg)
Know and use the curriculum
Curricula help to organize mathematical ideas.
Teachers need mathematical knowledge that helps them:• Understand the mathematical scope and
sequence in the curricula so it can be used or modified (know the flow);
• Know how concepts are related or build on each other; and
• Provide rationales for mathematical ideas.
![Page 37: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/37.jpg)
Assess the mathematical knowledgeof learners
Assessing mathematical knowledge is uncovering understanding as well as misconception.
Teachers need mathematical knowledge that helps them:• Assess more than procedural knowledge;• Identify the key concepts within problems or
exercises; and• Ask formative and summative questions.
![Page 38: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/38.jpg)
Assess the mathematical knowledge of learners
Assess more than procedural knowledge
There are 49 even numbers from 1 to 99.
The mean of 48 of these numbers is .
Which even number was not included in the calculating of this mean?
From Taiwan’s May 2009 Basic Competency Test, as reported by Lo & Tsai in MTMS, March 2011.
49 5
12
![Page 39: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/39.jpg)
Reflect on the mathematics of practice
Revisiting mathematical ideas after a lesson or unit can provide new mathematical insights.
Teachers need mathematical knowledge that helps them:
• Recognize assignments that hide, distort or illuminate the mathematics;
• Understand cultural factors that enhance or detract from the mathematics. (e.g., vocabulary, contexts or problems); and
• Recognize implicit mathematical ideas that need explanation? (e.g.,domain, convention, orientation of drawing)
![Page 40: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/40.jpg)
Reflect on the mathematics of practice Recognize implicit mathematical ideas that need
explanation? (e.g.,domain, convention, orientation)
Can a linear equation have neither an x-intercept nor a y-intercept?
A teacher asked her students to sketch the graph of
A student responded, “That’s impossible! You can’t take the square root of a negative number!”
y
x
f (x) x
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FrameworkFramework
Mathematical Knowledge for Teaching • Mathematical Proficiency• Mathematical Activity• Mathematical Work of Teaching
![Page 42: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/42.jpg)
This presentation is based upon work supported by the Center for Proficiency in Teaching Mathematics and the
National Science Foundation under Grant No. 0119790 and the Mid-Atlantic Center for Mathematics Teaching and Learning
under Grant Nos. 0083429 and 0426253 . Any opinions, findings, and conclusions or recommendations
expressed in this presentation are those of the presenter(s) and do not necessarily reflect the views of the National Science
Foundation.
![Page 43: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/43.jpg)
Question 1
What is the role of mathematical actions in mathematical knowledge for teaching at the secondary level and how can this be built into their formal mathematical background?
![Page 44: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/44.jpg)
Question 2
How could we develop and measure mathematical knowledge that enables the work of teaching?
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Questions
What is the role of mathematical actions in mathematical knowledge for teaching at the secondary level and how can this be built into their formal mathematical background?
How could we develop and measure mathematical knowledge that enables the work of teaching?
![Page 46: A Framework for Mathematical Knowledge for Teaching at the Secondary Level Conference on Knowledge of Mathematics for Teaching at the Secondary Level Pat.](https://reader035.fdocuments.in/reader035/viewer/2022081504/56649d9e5503460f94a88b20/html5/thumbnails/46.jpg)