Focus
in High School
Mathematics
Putting Reasoning
and Sense Making
at the Center
Two Classrooms
In Ms. Stree’s class, the following question
was asked:
• The driving distance Boston to Chicago is
990 miles. Rico drives from Boston to
Chicago at an average speed of 50 mph
and returns at an average speed of 60
mph. For how many hours is Rico on the
road?
Graham, K., Cuoco, A., & Zimmermann, G. (2010). Focus in High School
Mathematics: Reasoning and Sense Making in Algebra. Reston, VA: NCTM.
Two Classrooms
In Ms. Taque’s class, this question was
asked:
• Rico drives from Boston to Chicago at an
average rate of 50 mph and returns by the
same route at an average speed of 60
mph. If he is on the road for 36 hours, how
far is it from Boston to Chicago?
Graham, K., Cuoco, A., & Zimmermann, G. (2010). Focus in High School
Mathematics: Reasoning and Sense Making in Algebra. Reston, VA: NCTM.
Two Classrooms
The two questions show the divide
between what happens in typical
classrooms. Are instructors asking
students to reason with mathematics and
reason through problems? Or are
teachers giving prompts that require little
engagement and reasoning?
Graham, K., Cuoco, A., & Zimmermann, G. (2010). Focus in High School
Mathematics: Reasoning and Sense Making in Algebra. Reston, VA: NCTM.
History of NCTM Standards
2000 - Principles and Standards for School
Mathematics
– Updated the 1989 standards, incorporating
• Professional Standards for Teaching Mathematics
(1991)
• Curriculum and Evaluation Standards for School
Mathematics (1995)
History of NCTM Standards
2006 - Curriculum Focal Points for Prekindergarten
through Grade 8 Mathematics: A Quest for
Coherence
– The most important mathematical topics for
each grade level, based on Principles and
Standards
But what about high school mathematics?
Reasoning and Sense Making
A focus on reasoning and sense making, when
developed in the context of important content, will
ensure that students can accurately carry out
mathematical procedures, understand why those
procedures work, and know how they might be
used and their results interpreted.
National Council of Teachers of Mathematics (2009). Focus in High
School Mathematics: Reasoning and Sense Making. Reston, VA, p.3
Reasoning and Sense Making
• Reasoning is not a set of topics but a stance
toward learning mathematics
• Reasoning and sense making throughout the
curriculum lends coherence across domains:
number, algebra, geometry, and statistics
• Emphasizes underlying connections
• Connections, in turn, promote reasoning and
sense making that strengthens coherence,
allowing streamlining of the curriculum
Reasoning and Sense Making
Reasoning and sense making is
a way to approach instruction no
matter what content you are
teaching.
What Is Reasoning and Sense
Making?
• Reasoning: The process of drawing conclusions
on the basis of evidence or stated assumptions.
• Sense making: Developing understanding of a
situation, context, or concept by connecting it
with existing knowledge.
Reasoning and Sense Making
Reasoning and sense making is an
evolution of NCTM’s longstanding
position that problem solving should
be the emphasis of mathematics
teaching and learning.
Reasoning and Sense Making
“The processes of mathematics—
Problem Solving, Reasoning and Proof,
Connections, Communication, and
Representation—are all manifestations
of the act of making sense and of
reasoning …”
National Council of Teachers of Mathematics (2009). Focus in High
School Mathematics: Reasoning and Sense Making. Reston, VA, p.5
Reasoning and Sense Making
Levels of reasoning
•
•
•
•
Informal explanation
Inductive observations
Justification
Formal deduction
Reasoning and Sense Making
Reasoning is the foundation of
mathematical competence
•
•
•
•
•
Conceptual understanding
Procedural fluency
Strategic competence
Adaptive reasoning
Productive disposition
National Council of Teachers of Mathematics (2009). Focus in High
School Mathematics: Reasoning and Sense Making. Reston, VA, p.12
Reasoning and
Sense Making
should occur every day in
the high school
mathematics classroom
The Examples in
Focus in High School Mathematics
• Demonstrate how reasoning and sense making
can be incorporated into the high school
curriculum.
• Paint an idealized picture of what reasoning and
sense making should look like in practice.
• Take a variety of formats.
Reasoning and Sense Making
Focus in High School Mathematics
highlights reasoning opportunities
in five content areas:
•
•
•
•
•
Number and Measurement
Algebraic Symbols
Functions
Geometry
Statistics and Probability
Using Tasks
Fuel for Thought
A teacher gives her students the following
prompt taken from an article in the New
York Times (Chang 2008) and asks them
to explain their reasoning.
Fuel for Thought
Which of the following would save more
fuel?
a) Replacing a compact car that gets 34 miles
per gallon (MPG) with a hybrid that gets 54
MPG
b) Replacing a sport utility vehicle (SUV) that
gets 18 MPG with a sedan that gets 28 MPG
c) Both changes save the same amount of fuel.
Fuel for Thought
Discuss with a small group
possible solutions for this
problem.
Fuel for Thought
Share your solutions
Fuel for Thought
Sample response:
I see that the change from 34 to 54 MPG
is an increase of 20 MPG, but the 18 to 28
MPG change is only a change of 10 MPG.
So, replacing the compact car saves more
fuel.
Fuel for Thought
Sample response:
The change from 34 MPG to 54 MPG is
an increase of about 59% while the
change from 18 to 28 MPG is an increase
of only 56%. So the compact car is a
better choice.
Fuel for Thought
Sample response:
I thought about how much gas it would take to
make a 100-mile trip.
Compact car:
100 miles/54MPG = 1.85 gallons used
100 miles/34MPG = 2.94 gallons used
SUV:
100 miles/28MPG = 3.57 gallons used
100 miles/18MPG = 5.56 gallons used
Fuel for Thought
The compact car saved 1.09 gallons while
the SUV saved 1.99 gallons for every 100
miles. That means you actually save more
gasoline by replacing the SUV.
Reasoning and Sense Making
Technology
• Technology can be used to advance the
goals of reasoning and sense making
• Technological tools can relieve students of
burdensome calculations and can facilitate
the search for patterns and relationships
and the formation of conjectures.
Fuel for Thought
Using Tasks
Fuel for Thought
Key Mathematical Elements
• Number and Measurement (reasonableness of answers and
measurements)
• Functions (multiple representations of functions)
Reasoning Habits
• Analyzing a problem (seeking patterns and relationships)
• Reflecting on a solution (interpreting a solution; reconciling
different approaches; refining arguments)
Strategy: Multiple Entry Points
Use problems that have multiple entry
points so that students at different levels
of mathematical experience and with
different interests can all engage
meaningfully in reasoning about a
problem.
How can you get started?
Think about any standard topic in your
curriculum.
• Recast the content as questions that
students can explore
• Resist the temptation to tell students the
content - Believe that students can
investigate and derive relationships and
mathematical concepts.
How can you get started?
Consider finding x- and yintercepts of a linear function
What ways do you currently use to teach this?
How can you get started?
Finding x- and y-intercepts
• Ask students to look at a graph and
identify the intercepts
• Ask students why these points may be
of interest to someone trying to interpret
the graph
• Create a context for a graph that
requires interpretation of one of the
intercepts
How can you get started?
Finding x- and y-intercepts
• Ask students to create their own rules
and to compare these to the textbook
rules. Why do the rules work?
• Ask students to graph a “messy”
equation that’s difficult to graph: 33x +
14y = 231
Review of Tips for the Classroom
• Provide tasks that require students to figure things
out for themselves
• Ask students to restate problems in their own words
• Give students time to analyze a problem intuitively,
explore the problem further using models, and then
proceed to a more formal approach
• Resist the urge to tell students how to solve
problems
• Ask questions that will press students thinking
National Council of Teachers of Mathematics (2009). Focus in High School
Mathematics: Reasoning and Sense Making. Reston, VA
Review of Tips for the Classroom
• Provide adequate wait time
• Encourage students to ask probing questions of
themselves and each other
• Expect students to communicate their reasoning to their
classmates and teacher verbally and in writing
• Highlight exemplary examples and have students reflect
on what makes them effective
• Establish a classroom climate in which students feel
comfortable sharing their mathematical arguments and
critiquing the arguments of others
National Council of Teachers of Mathematics (2009). Focus in High School
Mathematics: Reasoning and Sense Making. Reston, VA
Review of Tips for the Classroom
• Considering and evaluating alternative explanations
• Understanding the allowable scope of conclusions
• Determining whether a conclusion based on the context
is plausible
Shaughnessy, J. M., Chance, B., & Kranendonk, H. (2009). Focus in High School Mathematics:
Reasoning and Sense Making in Statistics and Probability. Reston, VA: NCTM.
Reasoning and Sense Making
Equity
• High expectations for all students
• Courses offer rich opportunities for
reasoning and sense making
• Allow all students to see mathematics
as important for their lives and future
careers
Published in 2009
This will be followed by a book that offer examples of
ways to incorporate technology to promote reasoning
and sense making.
Focus in High School Mathematics:
Reasoning and Sense Making
• For more information:
www.nctm.org/hsfocus
• Materials include an executive
summary and brochures for
o
o
o
o
o
Teachers
Students
Administrators
Policymakers
Families