supporting rigorous mathematics teaching and learning

© 2013 UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Tennessee Department of EducationHigh School MathematicsAlgebra 2

Engaging In and Analyzing Teaching and Learning

RationaleAsking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true….…Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

By engaging in a task, teachers will have the opportunity to consider the potential of the task and engagement in the task for helping learners develop the facility for expressing a relationship between quantities in different representational forms, and for making connections between those forms.

Common Core State Standards for Mathematics, 2010


Session Goals

Participants will:

• develop a shared understanding of teaching and learning; and

• deepen content and pedagogical knowledge of mathematics as it relates to the Common Core State Standards (CCSS) for Mathematics.


Overview of Activities

Participants will:

• engage in a lesson; and

• reflect on learning in relationship to the CCSS.


Looking Over the Standards• Look over the focus cluster standards.

• Briefly Turn and Talk with a partner about the meaning of the standards.

• We will return to the standards at the end of the lesson and consider:

What focus cluster standards were addressed in the lesson?

What gets “counted” as learning?


Missing Function Task

If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning.

x f(x)-2 0-1 10 21 32 4

The Structures and Routines of a Lesson

The Explore Phase/Private Work Time

Generate Solutions

The Explore Phase/Small Group Problem Solving

1. Generate and Compare Solutions

2. Assess and Advance Student Learning

MONITOR: Teacher selects examples for the Share,

Discuss, and Analyze Phase based on:

• Different solution paths to the

same task

• Different representations

• Errors

• Misconceptions

SHARE: Students explain their methods, repeat others’

ideas, put ideas into their own words, add on to ideas

and ask for clarification.

REPEAT THE CYCLE FOR EACH

SOLUTION PATH

COMPARE: Students discuss similarities and

difference between solution paths.

FOCUS: Discuss the meaning of mathematical ideas in

each representation

REFLECT: By engaging students in a quick write or a

discussion of the process.

Set Up of the Task

Share, Discuss, and Analyze Phase of the Lesson

1. Share and Model

2. Compare Solutions

3. Focus the Discussion on

Key Mathematical Ideas

4. Engage in a Quick Write


Solve the Task(Private Think Time and Small Group Time)

• Work privately on the Missing Function Task.

• Work with others at your table. Compare your solution paths. If everyone used the same method to solve the task, see if you can come up with a different way.

• Consider what each person determined about g(x).


Expectations for Group Discussion

• Solution paths will be shared.

• Listen with the goals of:– putting the ideas into your own words;– adding on to the ideas of others;– making connections between solution paths;

and– asking questions about the ideas shared.

• The goal is to understand the mathematics and to make connections among the various solution paths.


Missing Function Task

If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning.

x f(x)-2 0-1 10 21 32 4


Discuss the Task(Whole Group Discussion)

• What do we know about g(x)? How did you use the information in the table and graph and the knowledge that h(x) = f(x) · g(x) to determine the equation of g(x)?

• How can you use what you know about the graphs of f(x) and g(x) to help you think about the graph of h(x)?

• Predict the shape of the graph of a function that is the product of two linear functions. Explain from the graphs of the two functions why you have made your prediction.


Reflecting on Our Learning

• What supported your learning?

• Which of the supports listed will EL students benefit from during instruction?

• Which CCSS for Mathematical Content did we discuss?

• Which CCSS for Mathematical Practice did you use when solving the task?

Linking to Research/LiteratureConnections between Representations

Pictures

WrittenSymbols

ManipulativeModels

Real-worldSituations

Oral Language

Adapted from Lesh, Post, & Behr, 1987

Five Different Representations of a Function Language

TableContext

Graph Equation

Van De Walle, 2004, p. 440




• Which of the supports listed will EL students benefit from during instruction?



The CCSS for Mathematical ContentCCSS Conceptual Category – Number and Quantity

The Real Number System (N-RN)

Extend the properties of exponents to rational exponents.

N-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Common Core State Standards, 2010, p. 60, NGA Center/CCSSO

The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra

Seeing Structure in Expressions (A–SSE)

Write expressions in equivalent forms to solve problems.

A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★

A-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ͌ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

A-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.★

★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.


The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra

Arithmetic with Polynomials and Rational Expressions (A–APR)

Understand the relationship between zeros and factors of polynomials.

A-APR.B.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

A-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.


The CCSS for Mathematical ContentCCSS Conceptual Category – FunctionsBuilding Functions (F–BF)

Build a function that models a relationship between two quantities.

F-BF.A.1 Write a function that describes a relationship between two quantities.★

F-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.

F-BF.A.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★

★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific

modeling standards appear throughout the high school standards indicated with a star ( )★ . Where an entire domain is marked with a star, each standard in that domain is a modeling standard.





• Which of the supports listed would EL students benefit from during instruction?



What math practices made it possible for us to learn?

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Common Core State Standards for Mathematics, 2010

Research Connection: Findings by Tharp and Gallimore

• For teaching to have occurred - Teachers must “be aware of the students’ ever-changing relationships to the subject matter.”

• They [teachers] can assist because, while the learning process is alive and unfolding, they see and feel the student's progression through the zone, as well as the stumbles and errors that call for support.

• For the development of thinking skills—the [students’] ability to form, express, and exchange ideas in speech and writing—the critical form of assisting learners is dialogue -- the questioning and sharing of ideas and knowledge that happen in conversation.

Tharp & Gallimore, 1991


Underlying Mathematical Ideas Related to the Lesson (Essential Understandings)• The product of two or more linear functions is a polynomial

function. The resulting function will have the same x-intercepts as the original functions because the original functions are factors of the polynomial.

• Two or more polynomial functions can be multiplied using the algebraic representations by applying the distributive property and combining like terms.

• Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x1, the point (x1, f(x1)+g(x1)) will be on the graph of the sum f(x)+g(x). (This is true for subtraction and multiplication as well.)

Essential Understandings

24

EU #1a: Functions are single-valued mappings from one set—the domain of the function—to another—its range.

EU #1b: Functions apply to a wide range of situations. They do not have to be described by any specific expressions or follow a regular pattern. They apply to cases other than those of “continuous variation.” For example, sequences are functions.

EU #1c: The domain and range of functions do not have to be numbers. For example, 2-by-2 matrices can be viewed as representing functions whose domain and range are a two-dimensional vector space.

EU #2a: For functions that map real numbers to real numbers, certain patterns of covariation, or patterns in how two variables change together, indicate membership in a particular family of functions and determine the type of formula that the function has.

EU #2b: A rate of change describes how one variable quantity changes with respect to another—in other words, a rate of change describes the covariation between variables.

EU #2c: A function’s rate of change is one of the main characteristics that determine what kinds of real-world phenomena the function can model.

Essential Understandings

25

EU #3a: Members of a family of functions share the same type of rate of change. This characteristic rate of change determines the kinds of real-world phenomena that the function can model.

EU #3c: Quadratic functions are characterized by a linear rate of change, so the rate of change of the rate of change (the second derivative) of a quadratic function is constant. Reasoning about the vertex form of a quadratic allows deducing that the quadratic has a maximum or minimum value and that if the zeroes of the quadratic are real, they are symmetric about the x-coordinate of the maximum or minimum point.

EU #5a: Functions can be represented in various ways, including through algebraic means (e.g., equations), graphs, word descriptions, and tables.

EU #5b: Changing the way that a function is represented (e.g., algebraically, with a graph, in words or with a table) does not change the function, although different representations highlight different characteristics, and some may only show part of the function.

EU #5c: Some representations of a function may be more useful than others, depending on the context.

EU #5d: Links between algebraic and graphical representations of functions are especially important in studying relationships and change.

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