Mathematical (and not mathematical) Diagrams

A diagram of a problem situation in mathematics shows where the numbers come from. Numbers can be known, unknown, or variables (sets of numbers). For example:

We are pouring water into a water tank. 5/6 liter of water is being poured every 2/3 minute. How many liters of water will have been poured after one minute?

Where are the numbers going to come from? Not from “water tanks.” You change to gas tanks, swimming pools, or catfish ponds without changing the meaning of the word problem. All of the numbers referred to (given, implied or asked about) come from:

• The number of liters poured
• The number of minutes spent pouring
• The rate of pouring (which relates liters to minutes)

A diagram should show where each of these numbers comes from. We need a way to show liters and a way to show minutes. Examples of diagrams for this situation shown.

The examples range in abstractness. The least abstract strategy (Level1) is not a good reasoning tool because it fails to show where the numbers come from. The more abstract examples are easier to reason with, if the student can make sense of them. Our goal is to teach students to make sense of, produce and reason with abstract diagrams that show all the numbers and their relationships. A good practice is to first make a more concrete diagram in early sense-making, then revise it to a more abstract diagram for reasoning purposes.

SERP has been supported to conduct this work by the S.D. Bechtel, Jr. Foundation.

Strategic Education Research Partnership  •  1100 Connecticut Avenue NW, Suite 1310  •  Washington, DC  20036

serpinstitute.org  •  info@serpinstitute.org   •  (202) 223-8555

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Mathematical (and not mathematical) Diagrams

A diagram of a problem situation in mathematics shows where the numbers come from. Numbers can be known, unknown, or variables (sets of numbers). For example:

We are pouring water into a water tank. 5/6 liter of water is being poured every 2/3 minute. How many liters of water will have been poured after one minute?

Where are the numbers going to come from? Not from “water tanks.” You change to gas tanks, swimming pools, or catfish ponds without changing the meaning of the word problem. All of the numbers referred to (given, implied or asked about) come from:

• The number of liters poured
• The number of minutes spent pouring
• The rate of pouring (which relates liters to minutes)

A diagram should show where each of these numbers comes from. We need a way to show liters and a way to show minutes. Examples of diagrams for this situation shown.

The examples range in abstractness. The least abstract strategy (Level1) is not a good reasoning tool because it fails to show where the numbers come from. The more abstract examples are easier to reason with, if the student can make sense of them. Our goal is to teach students to make sense of, produce and reason with abstract diagrams that show all the numbers and their relationships. A good practice is to first make a more concrete diagram in early sense-making, then revise it to a more abstract diagram for reasoning purposes.

SERP has been supported to conduct this work by the S.D. Bechtel, Jr. Foundation.

Strategic Education Research Partnership

1100 Connecticut Avenue NW, Suite 1310

Washington, DC  20036

serpinstitute.org

info@serpinstitute.org

(202) 223-8555