Using Math to Teach Math: Mathematicians and educators investigate the mathematics needed for teaching
Mark Hoover Thames, Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA 94720-5070, www.msri.org

Criteria for Mathematical Explanation

Generated by Prospective Teachers

Makes clear at the outset what is being explained, and why to start there, and carefully connects the explanation to the question or idea being explained

Starts from the beginning, and traces the logical flow of the reasoning

Is logical and complete, makes conclusion clear and links back to original question or claim or problem

Might number the steps if appropriate, or label parts of a diagram

Strives to be as simple and clear as possible

Defines terms as needed, uses available definitions as needed

Uses representations accurately (algebraic, geometric, etc.), and combining representations

Links the language and diagrams clearly to the steps of the argument

Shows what something means or why it is true, and is convincing to the person to whom you are explaining

Is calibrated to the context (considers the person to whom you are explaining, and what is already established as true and does not need more explanation)

Adapted from the 2004 Summer Institute sponsored by the Center for Proficiency in Teaching Mathematics, University of Michigan.

"While mathematical explanation does not lend itself to simple formulation, if teachers had tools for discussing explanation, they could be more explicit with students. For instance, identifying the steps in mathematical explanations and talking about when steps can be left out and when they must be included would help students who are not as able to pick up on subtle messages about acceptable and unacceptable steps. Short of this, indirect messages will continue to be more likely understood by students who share the teacher’s background than by those who don’t."

Order these subtraction problems from easiest to hardest for students learning the standard subtraction algorithm, and explain the reasons for your ordering:

(a) 322 - 115

(b) 302 - 115

(c) 329 - 115

Writing Prompts for potential writing assignments: The first prompt in each group is from "A Writing Workshop in Mathematics: Community Practice of Content Discourse" by Linda Fernstein from the November, 2007 issue of Mathematics Teacher Magazine, the rest in each group are just creations of similar types of prompts to start thinking and capture thinking, please feel free to join in.

A scientific calculator has a pi key. Explain when it should be used and how it can be used effectively

Type 2, *, enter into your calculator. Press enter again, what mathematical operation is the calculator doing? How else can you get the calculator to create the same sequence of numbers?

What is the largest area of a rectangle that has a perimeter of 48 yards? Write an explanation detailing how you determined your answer.
What is the relationship between the length and width of a soccer field? Write an explanation detailing the process you used to determine your answer.
How does the area of the bri

What concepts underlie the formula for the volume of a cone? For example, what physical assumptions must be made? Explain how the formula was derived and give some real-world situations for its use.

Using Math to Teach Math: Mathematicians and educators investigate the mathematics needed for teachingMark Hoover Thames, Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA 94720-5070, www.msri.org

- Criteria for Mathematical Explanation

Generated by Prospective Teachers- Makes clear at the outset what is being explained, and why to start there, and carefully connects the explanation to the question or idea being explained
- Starts from the beginning, and traces the logical flow of the reasoning
- Is logical and complete, makes conclusion clear and links back to original question or claim or problem
- Might number the steps if appropriate, or label parts of a diagram
- Strives to be as simple and clear as possible
- Defines terms as needed, uses available definitions as needed
- Uses representations accurately (algebraic, geometric, etc.), and combining representations
- Links the language and diagrams clearly to the steps of the argument
- Shows what something means or why it is true, and is convincing to the person to whom you are explaining
- Is calibrated to the context (considers the person to whom you are explaining, and what is already established as true and does not need more explanation)

Adapted from the 2004 Summer Institute sponsored by the Center for Proficiency in Teaching Mathematics, University of Michigan.Writing Prompts for potential writing assignments: The first prompt in each group is from "A Writing Workshop in Mathematics: Community Practice of Content Discourse" by Linda Fernstein from the November, 2007 issue of

Mathematics Teacher Magazine, the rest in each group are just creations of similar types of prompts to start thinking and capture thinking, please feel free to join in.What is the largest area of a rectangle that has a perimeter of 48 yards? Write an explanation detailing how you determined your answer.

What is the relationship between the length and width of a soccer field? Write an explanation detailing the process you used to determine your answer.

How does the area of the bri

What concepts underlie the formula for the volume of a cone? For example, what physical assumptions must be made? Explain how the formula was derived and give some real-world situations for its use.