Benchmarking

Mathematical Connections, Communication

Benchmarking

Why do we benchmark?

What are the benefits? Challenges?

Other points/concerns?

Professional Critical Analysis

Why do Students Need to Write to Learn Mathematics?

“One of the best and most practical ways to improve accuracy is the collaborative scoring of student work. This is also a superb professional learning experience”

Douglas Reeves:

Kelsey Shields and Candice Gale

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Connections Recognize and use connections

among mathematical ideas Understand how mathematical ideas

interconnect and build on one another to produce a coherent whole

Recognize and apply mathematics in contexts outside of mathematics

NCTM

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Piaget

knowledge is constructed as the learner strives to organize his or her experiences in terms of pre-existing mental structures or schemes

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Connections Math to real life Math to self Math to Math

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Communication works together with reflection to produce new relationships and connections. Students who reflect on what they do and communicate with others about it are in the best position to build useful connections in mathematics. (Hiebert et al., 1997, p. 6)

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Communication Organize and consolidate their

mathematical thinking through communication 

Communicate their mathematical thinking coherently and clearly to peers, teachers, and others

Analyze and evaluate the mathematical thinking and strategies of others;

Use the language of mathematics to express mathematical ideas precisely.   

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Through communication, ideas become objects of reflection, refinement, discussion, and amendment. The communication process also helps build meaning and permanence for ideas and makes them public (NCTM, 2000). When students are challenged to think and reason about mathematics and to communicate the results of their thinking to others orally or in writing, they learn to be clear and convincing. Listening to others’ thoughts and explanation about their reasoning gives students the opportunity to develop their own understandings.

-Huang

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Tell Me!

DescribeExplain JustifyDebateConvinceProove

"...It becomes evident to the students and teacher that mathematical communication is not about “answering the question using words, numbers, pictures, and symbols.” Instead, they realize that these forms of communication are selected and applied in order to create a precise mathematical argument, where labelled diagrams and/or numeric expressions and equations are viewed as being more precise, concise and persuasive forms than descriptive narratives. These discussion processes provoke students to use higher-order thinking skills, such as analysis, evaluation and synthesis, in order to improve their conceptual understanding, use of mathematical strategies and mathematical communication."

We look for connections throughout the paper

Some problems lend themselves to showcasing different processes and student abilities

We decide on what a correct answer looks like

We fit the rubric to the problem. Its tricky! We “tighten up” the rubric…but be

careful!

Benchmarking

Math to self and real life◦ “This reminds me of when my Mom made

cupcakes for my birthday”, or “Newspapers talk about statistics for sports and money”

Math to other subject: “This is like the graphs we did in science” or “we talked about statistics in health class” Math to Math: “ This is a linear relation”. “I could use an equation”. Or “I can make a rule”

Types of Connections we will see

Extending the solution or problem posing are a type of connection

When a student chooses a strategy, creates a representation, discusses the situation in mathematical terms, illustrates, or makes decisions about algorithms, they are really connecting mathematical ideas.

Introduce writing early, doesn’t have to be an exemplar

Emphasize vocabulary (communication) Discuss strategies, which are efficient Emphasize representations Have students highlight connections they

have recognized Emphasize good solid explanations that the

reader can follow (logic, reasoning and proof)

Assessing Exemplars

Templates

Problem Solving with Your Class

To close….From Carol Anne Tomlinson: