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NCTM’s Focus in High School Mathematics: Reasoning and Sense Making

NCTM’s Focus in High School Mathematics: Reasoning and Sense Making. History of NCTM’s Standards. 2000 -- Principles and Standards for School Mathematics

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NCTM’s Focus in High School Mathematics: Reasoning and Sense Making

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  1. NCTM’s Focus in High School Mathematics: Reasoning and Sense Making

  2. History of NCTM’s Standards • 2000 -- Principles and Standards for School Mathematics • Updated the 1989 standards, incorporating the Professional Standards for Teaching Mathematics (1991) and the Evaluation Standards for School Mathematics (1995) • 2006 -- Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics • Set forth the most important mathematical topics for each grade level, based on Principles and Standards But what about high school mathematics?

  3. Focus in High School Mathematics: Reasoning and Sense Making • A high school mathematics program based on reasoning and sense making will prepare students for citizenship, for the workplace, and for further study. • Goal: • “To create a document to use as the conceptual framework to guide the development of future publications and tools related to 9-12 mathematics curriculum and instruction.” • Audience -- Everyone involved in decisions regarding high school mathematics programs, including.

  4. Definitions • Reasoning • Most generally, the process of drawing conclusions on the basis of evidence or stated assumptions. • Mathematical reasoning ranges from informal explanation and justification to formal deduction, as well as inductive observations. • Sense making • Development of understanding of a situation, context, or concept by connecting it with existing knowledge.

  5. Relationship of Reasoning and Sense Making

  6. Reasoning Habits • Reasoning and sense making should be a part of the mathematics classroom every day. • Reasoning habit: • “a productive way of thinking that becomes common in the processes of mathematical inquiry and sense making” • Should not be approached as a new list of topics to be added to the curriculum.

  7. Reasoning Habits • Analyzing a problem, for example… • Implementing a strategy, for example… • Seeking and using connections… • Reflecting on a solution to a problem, for example…

  8. Analyzing a problem • identifying relevant mathematical concepts, procedures, or representations… • defining relevant variables and conditions carefully… • seeking patterns and relationships.. • looking for hidden structure… • considering special cases or simpler analogs; • applying previously learned concepts… • making preliminary deductions and conjectures… • deciding whether a statistical approach is appropriate.

  9. Developing Reasoning and Sense Making • Reasoning levels • Empirical • Preformal • Formal • Tips for the classroom, for example: • Provide tasks that require students to figure things out for themselves. • Ask students questions that will press their thinking – for example, “Why does this work?” or “How do you know?”

  10. Reasoning in the Curriculum • Reasoning and sense making are integral to the experiences of all students across all areas of the high school mathematics curriculum. • A stance toward learning mathematics, not which will, of course, take time. • However, it also promises compensating efficiencies. • Less reteaching since they may better retain what they have learned. • Focus on underlying connections, less time on lists of skills.

  11. Content Strands • Specific content areas in which reasoning and sense making should be developed: • Reasoning with Numbers and Measurements • Reasoning with Algebraic Symbols • Reasoning with Functions • Reasoning with Geometry • Reasoning with Statistics and Probability

  12. Key Elements • Provide structure for how each content strand can be focused on reasoning and sense making. • Not intended to be an exhaustive list. • Instead, a lens through which to view the potential of high school programs for promoting and developing mathematical reasoning and sense making.

  13. Key Elements for Algebraic Symbols • Meaningful use of symbols. • Mindful manipulation. • Reasoned solving. • Connecting algebra with geometry. • Linking expressions and functions.

  14. Example 8 Task (intermediate algebra class) • Find a way of solving the equationx2 + 10x = 144 using an area model.

  15. Teacher: Can anybody see how to think ofx2 + 10x as an area?

  16. Student 2: Maybe if we knew what the area of the square was, we could just take the square root to find x. • Teacher: Is there a way of rearranging the figure into a square?

  17. Student 2: But it’s not a complete square. It’s missing a corner. • Teacher: What’s the area of the corner?

  18. Student 2: …Since the gray area is 144, the entire area of the big square is 144 + 25=169. • Student 1: And that means the side length of the square is 13, so x + 5 = 13, which means x = 8. • Student 2: Shouldn’t there be another solution since the (x + 5) is squared?

  19. Equity • Mathematical reasoning and sense making must be evident in the mathematical experiences of all students. • Courses that students take have an impact on the opportunities that they have for reasoning and sense making. • Students’ demographics too often predict the opportunities students have for reasoning and sense making • Expectations, beliefs, and biases have an impact on the mathematical learning opportunities provided for student.

  20. Coherence • Curriculum, instruction, and assessment form a coherent whole in order to support reasoning and sense making. • Alignment of Curriculum and Instruction implies well-designed curriculum and challenging tasks. • Assessment • High stakes tests need to include focus on reasoning and sense making. • Importance of formative assessment.

  21. Poses Questions for: • Students • Parents • Teachers • Administrators • Policymakers • Higher Education • Curriculum Designers

  22. Teachers • What can teachers do in their classrooms to be sure that reasoning and sense making are paramount? • What can teachers do to help students see the importance of mathematics for their lives as well as future career plans? • What can teachers do to make students’ high school mathematical experience more meaningful overall?

  23. Conclusion • The need has been evident for years, the time to act is now. • All stakeholders need to work together to make this happen.

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