NCTM`s High School Curriculum Project:

NCTM’s Focus in High School
Reasoning and Sense Making
History of NCTM’s Standards
• 2000 -- Principles and Standards for School
– 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.
• 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
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
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
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
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?
• 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
• Students’ demographics too often predict the
opportunities students have for reasoning and sense
• Expectations, beliefs, and biases have an impact on the
mathematical learning opportunities provided for student.
• Curriculum, instruction, and
assessment form a coherent whole in
order to support reasoning and sense
• Alignment of Curriculum and Instruction implies
well-designed curriculum and challenging tasks.
• Assessment
– High stakes tests need to include focus on reasoning and sense
– Importance of formative assessment.
Poses Questions for:
Higher Education
Curriculum Designers
• 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?
• 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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