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
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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
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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
x2 + 10x = 144 using an area model.
• Teacher: Can anybody see how to think of
x2 + 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:
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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.