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

History of NCTM’s Standards 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) 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?

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.

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.

Relationship of Reasoning and Sense Making

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.

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…

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.

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?”

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.

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

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.

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

Example 8 Task (intermediate algebra class) Find a way of solving the equation x x = 144 using an area model.

Teacher: Can anybody see how to think of x x as an area?

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?

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

Student 2: …Since the gray area is 144, the entire area of the big square is =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?

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.

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.

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

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?

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