Standard 3: Mathematics as Reasoning

In grades K-4, the study of mathematics should emphasize reasoning so that students can:

• Draw logical conclusions about mathematics

Benchmarks 1B (The Nature of Science: Scientific Inquiry)
Grades 3-5, page 11
Scientists do not pay much attention to claims about how something they know about works unless the claims are backed up with evidence that can be confirmed and with a logical argument.

Benchmarks 12E (Habits of Mind: Critical-Response Skills)
Grades 3-5, page 299
Recognize when comparisons might not be fair because some conditions are not kept the same.

• Use models, known facts, properties, and relationships to explain their thinking

Benchmarks 11B (Common Themes: Models)
Grades K-2, page 268
A model of something is different from the real thing but can be used to learn something about the real thing.

Benchmarks 1B (The Nature of Science: Scientific Inquiry)
Grades K-2, page 10
When people give different descriptions of the same thing, it is usually a good idea to make some fresh observations instead of just arguing about who is right.

Benchmarks 9E (The Mathematical World: Reasoning)
Grades 3-5, page 232
One way to make sense of something is to think how it is like something more familiar.

Benchmarks 11B (Common Themes: Models)
Grades 3-5, page 268
Seeing how a model works after changes are made to it may suggest how the real thing would work if the same thing were done to it.

Benchmarks 12B (Habits of Mind: Computation and Estimation)
Grades 3-5, page 290
Judge whether measurements and computations of quantities such as length, area, volume, weight, or time are reasonable in a familiar context by comparing them to typical values.

Benchmarks 12E (Habits of Mind: Critical-Response Skills)
Grades 3-5, page 299
Buttress their statements with facts found in books, articles, and databases, and identify the sources used and expect others to do the same.

Benchmarks 11B (Common Themes: Models)
Grades 3-5, page 268
Geometric figures, number sequences, graphs, diagrams, sketches, number lines, maps, and stories can be used to represent objects, events, and processes in the real world, although such representations can never be exact in every detail.

• Justify their answers and solution processes

Benchmarks 9E (The Mathematical World: Reasoning)
Grades K-2, page 232
People are more likely to believe your ideas if you can give good reasons for them.

Benchmarks 9E (The Mathematical World: Reasoning)
Grades 3-5, page 232
Reasoning can be distorted by strong feelings.

Benchmarks 12A (Habits of Mind: Values and Attitudes)
Grades 3-5, page 286
Offer reasons for their findings and consider reasons suggested by others.

• Use patterns and relationships to analyze mathematical situations

Benchmarks 11C (Common Themes: Constancy and Change)
Grades K-2, page 272
Things can change in different ways, such as size, weight, color, and movement. Some small changes can be detected by taking measurements.

Benchmarks 11C (Common Themes: Constancy and Change)
Grades 3-5, page 273
Some features of things may stay the same even when other features change. Some patterns look the same when they are shifted over, or turned, or reflected or seen form different directions.

Benchmarks 11C (Common Themes: Constancy and Change)
Grades 3-5, page 273
Things change in steady, repetitive, or irregular ways or sometimes in more than one way at the same time. Often the best way to tell which kinds of change are happening is to make a table or graph of measurements.
 

• Believe that mathematics makes sense

Benchmarks 2A (The Nature of Mathematics: Patterns and Relationships)
Grades 3-5, page 27
Mathematics is the study of many kinds of patterns, including numbers and shapes and operations on them. Sometimes patterns are studied because they help to explain how the world works or how to solve practical problems, sometimes because they are interesting in themselves.