NCTM’s Focus in High School Mathematics:

Reasoning and Sense Making

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?

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 equationx2 + 10x = 144 using an area model.

• Teacher: Can anybody see how to think ofx2 + 10x 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 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?

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.