MEC Vision

We envision mathematics classrooms where students and teachers:

· Think, reason and use mathematics to make sense of the world around them

· Have a sound conceptual understanding of mathematics

· Use the tools of mathematics efficiently · Work persistently to solve real,

challenging and relevant problems · Engage actively in learning mathematics

Community Context

• Series of Community Math Nights• Leadership Development for Math Support

Team• Mathematics Content Courses for K-20

Teachers• Ongoing support for Administrators• IHE Partnerships• Work with Business and Community Leaders

MEC Course Series

• Patterns, Functions and Algebraic Reasoning• Numerical Reasoning• Geometric and Proportional Reasoning• Probability and Data Analysis• Extending Algebraic Reasoning Using A

Functional Point of View• Integrating Mathematical Ideas: Algebra,

Geometry, Probability and Statistics• Connecting Mathematics Content to Science

and Technology

Mathematical Goals

The workplace of today demands unprecedented mathematical understandings. Students must leave our schools…

• Quantitatively literate• Able to make informed judgments when dealing with data• Able to reason probabilistically, algebraically and geometrically,

and see connections within and among mathematical ideas• Persistent in solving complex and relevant problems, with a

disposition to search for patterns and relationships• Able to communicate mathematically and make convincing

arguments in defense of mathematical ideas

Mathematics Content Coursesfor K-20 teachers of mathematics

• K-20 teachers learn mathematics together in contexts that fully model what we hope to see in classrooms with regard to mathematics content, mathematical pedagogy and assessment practices.

Patterns, Functions & Algebraic Reasoning

Mathematics Content

• Teaching to the big ideas of linear, quadratic and exponential growth patterns

• Skills are learned and practiced in the context of their use

n+1

n = stage number T = number of tiles needed I saw these as half rectangles, so I made a duplicate and put them together to make a rectangle that is 2 by 3. Since in this case n=1, the rectangle is (n+1) by (n+2). Since the original shape was just half of this rectangle, the equation is T = (n+1)(n+2).

1 2 3 4 5 6 7 7 6 5 4 3 2 1 8 8 8 8 8 8 8 For n = 6 I thought of Gauss and used what he might have done. I paired the numbers 1 through 7, with 7 through 1, added the pairs and got 7 eights, or (n+1)(n+2). Since I double counted, I had to divide my result by 2. The expression for any step is (n+1)(n+2.)

I rearranged the tiles and saw this as part of a square. I completed the square and got a 2x2 square when n =1. I then divided the square in half and got (n+1)^2. I had to add back two half squares or (n+1) halves. The expression

for any stage is (n+1)^2 +

(n+1)1/2

Stage 1

Stage 2

n = Stage number I finished the square which was (n+1)2 and I had to take away the yellow which I recognized as triangular numbers, so I used GaussÕ theorem to subtract n(n+1) to get back to the original, so my espression was (n+1)2 Ð [1/2 n (n+1)].

For stage 6, I wrote the numbers 1 through 7, or 1 through (n+1) then thought about what Gauss might do. I paired the first number with the last, the second with the second to last, etc. and I had 3 pairs of 8. I used Gauss theorem for adding integers n x (n+1). Since the pattern started with 3 tiles instead of one, I had to change the expression to (n+1)(n+2)

1 2 3 4 5 6 7

8 8 8

½ (n+1) (n+2) ½ (n+1)2 + (n+1) ½ (n+1)2 – [½ n (n+1)]

Are these expressions equivalent? Use algebra to determine if these are three different ways of seeing the same thing.

Patterns, Functions & Algebraic Reasoning

Mathematics Content

• Teaching to the big ideas– linear, quadratic and exponential growth

• Skills are learned and practiced in the context of their use

• Analyzing situations from geometric, numerical/tabular, algebraic, verbal and graphical perspectives

Patterns, Functions & Algebraic Reasoning

Mathematics Content

• Teaching to the big ideas of linear, quadratic and exponential growth patterns

• Analyzing situations from geometric, tabular, algebraic,

• Skills are learned and practiced in the context of their use

• Numerical reasoning as the foundation for algebraic reasoning

48 x 27

48 x 2750 X 27 = 13502 X 27 = 54 1350 - 54 =1296

48 x 27

25 X 48 = 12002 X 48 = 96

1200+96 = 1296

48 x 27 24 X 54

50 X 24 = 12004 X 24 = 96

1200 + 96 = 1296

48 X 27

48 X (20 + 5 + 2)960 + 240 + 961200 + 96 = 1296

Mathematical Pedagogy

• Safe and productive learning environment• Inquiry-based learning• Focused mathematics• Talking mathematic is the norm

Mathematical Pedagogy

• Safe and productive learning environment• Inquiry-based learning• Focused mathematics• Talking mathematic is the norm• Instruction is differentiated to meet a range of needs

- expandable tasks - challenges - learning environment

Cubic Octahedra:

Mathematical Pedagogy

• Safe and productive learning environment• Inquiry-based learning• Focused mathematics• Talking mathematic is the norm• Instruction is differentiated to meet a range of needs• Small group tasks around significant problems• Individual and small group investigations are followed by whole group

processing of the mathematics and reflecting on learning• Assessment practices enhance learning while revealing levels of

understanding– Pre and post performance tasks– On demand tasks– Scored tasks– Rubrics– Portfolios

• Students learn to judge the quality of their own work

Mathematical Pedagogy

• Safe and productive learning environment• Inquiry-based learning• Focused mathematics• Talking mathematic is the norm• Instruction is differentiated to meet a range of needs• Small group tasks around significant problems• Individual and small group investigations are followed by whole

group processing of the mathematics and reflecting on learning• Assessment practices enhance learning while revealing levels of

understanding• Students learn to judge the quality of their own work• Professional articles read and discussed

MEC Content Courses…

• Deepen teachers’ mathematical understandings

• Change teachers’ mathematical dispositions and their relationship with mathematics

• Teach teachers how to teach mathematics• Promote transfer to classroom practice• Create learning communities

Teacher Pre and Post Scores On Quadratic Growth Performance Task Scored On Six-Point Rubric Performance Assessment Pre-Post Median Scores (Patterns Course) (N=70)

Conceptual Understanding (CU)

Processes and Strategies (PS)

Communication (COM)

Accuracy (ACC)

PRE POST PRE POST

PRE POST

PRE POST

Median 2.000 4.000 2.000

4.000

2.000 4.000 4.000 5.000

0

5

1 0

1 5

2 0

2 5

3 0

P S _ P R E

P S _ P O S T

P S _ P R E2 0 2 4 2 1 4 1 0

P S _ P O S T 0 4 1 9 2 8 1 9 0

1 2 3 4 5 6

0

5

1 0

1 5

2 0

2 5

3 0

C U _ P R E

C U _ P O S T

C U _ P R E1 8 2 0 2 3 8 1 0

C U _ P O S T 0 1 2 2 2 5 2 0 2

1 2 3 4 5 6

0

5

1 0

1 5

2 0

2 5

3 0

C O M _ P R E

C O M _ P O S T

C O M _ P R E2 2 2 8 1 4 3 3 0

C O M _ P O S T 0 3 1 9 2 8 1 6 4

1 2 3 4 5 6

0

1 0

2 0

3 0

4 0

5 0

A C C _ P R E

A C C _ P O S T

A C C _ P R E2 1 3 9 1 0

A C C _ P O S T 5 2 6 3 9

1 2 3

Participant 1 Participant 2 Participant 3 Day 1 Day 4 Day 8 Day 1 Day 4 Day 8 Day 1 Day 4 Day 8

Understanding of Mathematical Ideas

Uses variables to describe unknowns

X X X X X X

Explains why equations make sense geometrically

X X X X

Represents linear and quadratic equations in variety of ways

X X X X

Productive Disposition Persists when answer is not known

X X X X X

Asks for guidance but not answers

X X X X

Tries variety of strategies for approaching problem

X X X

Inquiry and Reflection Makes extensions and connections beyond immediate problem

X X X

Explores why it works and whether it will always work

X

Confusion and mistakes lead to further exploration

X X X

Communication Explains reasoning fluently

X X X

Asks probing questions X X Shares ideas with class X X X X X

Year 1 is baseline data Year 2 is after MECÕs 9-day course, Patterns, Functions and Algebraic Reasoning Year 3 teachers have each had one or two MEC courses Random Sample of Tea chers, each observed by two trained evaluators

Year 1 (n=11) Year 2 (n=12) Year 3 (n=40) G5 1

G6 1

G7 4

G8 5

G5 1

G6 1

G7 7

G8 3

G5 10

G6 9

G7 9

G8 12

Bessemer Fairfield Hoover Homewood Jefferson Count y Shelby Count y

Fairfield Hoover Homewood Jefferson Count y Shelby Count y Trussville

Bessemer Fairfield Hoover Homewood Jefferson Count y Mountain Brook Shelby Count y Trussville

Each of the categories below consists of five items. Items are rated on a five-point scale ranging from 0 to 4. Observers select a Ò0Ó if the characteristic never occurred in the lesson to a Ò4Ó if the item was very descriptive of the lesson. Median Ratings by Category

RTOP Categories Year Median 1 4 2 11.75 Lesson Design/Implementation 3 14.5 1 5.5 2 10.75 Propositional Knowledge 3 13.25 1 4 2 13 Procedural Knowledge 3 13.5 1 4 2 12.75 Communicative Interaction 3 13.5 1 5.5 2 14 Student/Teacher Re lationships 3 15.5

What kinds of change do researchers and evaluators

see in classrooms?• Number talks happen frequently.• Questioning permeates the environment

(student to student, student to teacher, teacher to student).

• Diverse ways of seeing are being encouraged and examined.

• Having students justifying their ideas orally and in writing is the norm.

Advancing the Teaching of Mathematics:What will it take?• Mathematics courses for K-20 classroom teachers• Support assessments that are aligned with the goal of helping

all students think, reason, and put mathematics to work in powerful ways to solve real, complex and relevant problems

• Invest in pre-K-20 mathematics partnerships• Engage with parents and the public in pursuit of quality

mathematics in schools• Invest in leadership development• Partner with business and community leaders• Close collaboration between internal (district) leaders and

external leaders