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What kinds of experiences should high school mathematics offer students? The key is to provide important mathematical opportunities centered on reasoning and sense-making. How can you and other high school mathematics teachers like you give students these kinds of experiences?
1.
A Teacher’s Guide to
Reasoning and Sense Making
hat kinds of experiences should high For instance, in high school literature
school mathematics offer students? courses, students often must analyze, interpret,
The key is to provide important or think critically about books that they are
mathematical opportunities centered on reading. Reasoning is important in all fields—
reasoning and sense making. How can you and particularly mathematics. Mathematical
other high school mathematics teachers like you reasoning involves drawing logical conclusions
give students these kinds of experiences? on the basis of assumptions and definitions.
Focus in High School Mathematics: Reasoning Sense making involves developing an
and Sense Making, a new publication from the understanding of a situation, context, or
National Council of Teachers of Mathematics, concept by connecting it with other knowledge.
offers guidelines to improve high school Reasoning and sense making are closely
mathematics by refocusing it in this way. interrelated.
Teachers will play a crucial role in realizing the Reasoning and sense making should occur
vision of this innovative publication from the in every mathematics classroom every day. In
nation’s leading advocate for more and better classrooms that encourage these activities,
mathematics. teachers and students ask and answer such
questions as “What’s going on here?” and “Why
do you think that?” Addressing reasoning and
What do reasoning and sense making
sense making does not need to be an extra
burden if you are working with students who
Reasoning and sense making refer to students’ are having a difficult time just in learning
abilities to think about and use mathematics procedures. On the contrary, the structure
in meaningful ways. In any subject, simply that a focus on reasoning brings can provide
exposing students to topics is not enough. Nor vital support for understanding and continued
is it enough for students to know only how to learning.
perform procedures. For example, in your own Often students struggle because they find
experiences with mathematics in high school, mathematics meaningless. Instruction that
you may have solved page after page of equations fails to help them find connections through
or factored page after page of polynomials. reasoning and sense making may lead to a
Students today must understand more about seemingly endless cycle of reteaching. However,
algebra than how to apply procedures. They with purposeful attention and planning,
need to develop critical thinking skills to succeed teachers can hold all students in every high
in the subject—and in other areas of life and school mathematics classroom accountable for
learning. personally engaging in reasoning and sense
2.
making, thus leading students to reason for Student 1: Oh, yeah, I remember—there’s a
themselves instead of merely observing and great big square root sign, but I
applying the reasoning of others. don’t remember what goes under
it.
What can you do in your classroom Student 3: I know! It’s x1 plus x2, all over 2,
to ensure that reasoning and sense isn’t it?
making are paramount? Student 4: No, that’s the midpoint formula.
You can make reasoning and sense making a
focus in any mathematics class. A crucial step is to The discussion continued in the same
determine how reasoning and sense making serve way until the teacher reminded the class of
as integral components of the material that you the formula. The next year, the same teacher
teach. decided to try a different approach—one with
Even with topics traditionally presented the potential to engage the students in reasoning
through procedural approaches, you can teach about the distance formula as they solved a
the concepts in ways that allow students to reason problem. The following scenario shows students
about what they are doing. Although procedural reasoning about mathematics, connecting what
fluency is important in high school mathematics, they are learning with the knowledge that they
it should not be sought in the absence—or at the already have and making sense of the distance
expense—of reasoning and sense making. formula:
What exactly do reasoning and sense
making “look like” in the mathematics classroom? Teacher: Let’s take a look at a situation in
The following example illustrates the need which we need to find the distance
to infuse reasoning and sense making into a between two locations on a map.
classroom experience. The scenario illustrates Suppose that this map [shown at
what frequently happens when students are the top of the next page] shows your
asked to recall a procedure taught without school; your house, which is locat-
understanding—in this case, the distance ed two blocks west and five blocks
formula. north of school; and your best
friend’s house, which is located
Teacher: Today’s lesson requires that we eight blocks east and one block
calculate the distance between south of school. Also suppose that
the center of a circle and a point the city has a system of evenly
on the circle to determine the cir- spaced perpendicular and paral-
cle’s radius. Who remembers how lel streets. How many blocks would
to find the distance between two you have to drive to get from your
points? house to your friend’s house?
Student 1: Isn’t there a formula for that? Student 1: Well, we would have to drive ten
Student 2: I think it’s x1 plus x2 squared, or blocks to the east and six blocks
something like that. to the south, so I guess it would be
sixteen blocks, right?
2 The National Council of Teachers of Mathematics
3.
10 Student 3: Wait a minute—you just drew a
Your house right triangle, because the streets
8
are perpendicular.
6
4
Student 4: So that means we could use the
School
Friend’s house Pythagorean theorem:
02 + 62 = c2, so c = 136
5 10 15 20
Teacher: But what if you could use a he- Student 2: But how many blocks would that
licopter to fly straight to your be?
friend’s house? How could you
Student 3: Shouldn’t the distance be be-
find the distance “as the crow
tween eleven and twelve blocks,
flies”? Work with partners to es-
since 121 < 136 < 144? Actual-
tablish a coordinate-axis system
ly, it’s probably closer to twelve
and show the path that you’d have
blocks, since 136 is much closer to
to drive to get to your friends
144 than to 121.
house. Then work on calculating
the direct distance between the The teacher then extended the discussion
houses if you could fly. to consider other examples and finally to
develop the general formula. By having the
Student 1: [working with students 2, 3, and 4]:
students approach the distance formula from the
What if we use the school as the
perspective of reasoning and sense making, she
origin? Then wouldn’t my house
increased their understanding of the formula
be at (–2, 5) and my friend’s
and why it is true, making it more likely that they
house be at (8, −1)?
would be able to retrieve, or quickly recreate, the
Student 2: Yeah, that sounds right. Here, formula later.
let’s draw the path on the streets The focus of every mathematics class
connecting the two houses, and should be on helping students make sense of the
then draw a line segment con- mathematics for themselves. Bringing this focus
necting the two houses. to instruction depends on—
Student 1: Maybe we could measure the • selecting worthwhile tasks that engage
length of a block and find the dis- and develop students’ mathematical
tance with a ruler. understanding, skills, and reasoning;
8
12 • creating a classroom environment in which
6
serious engagement in mathematical
10 Your
house
4 thinking is the norm;
8
•
2
effectively orchestrating purposeful
6
School
discourse aimed at encouraging students to
4 Friend’s house
–2 reason and make sense of what they are
5 10 15 20
A Teacher’s Guide to Reasoning and Sense Making 3
4.
• using a range of assessments to monitor (for example, choosing a model for
and promote reasoning and sense making, simulating a random experiment);
both in identifying student progress and in
— defining relevant variables and conditions
making instructional decisions;
carefully, including units if appropriate;
• constantly reflecting on teaching practice
— seeking patterns and relationships (for
to be sure that the focus of the class in on
example, systematically examining cases
reasoning and sense making (based on
or creating displays for data);
recommendations in Mathematics Teaching
Today [NCTM 2007]). — looking for hidden structures (for example,
drawing auxiliary lines in geometric
The teacher in the preceding example
figures, finding equivalent forms of
performed each of these actions with the
expressions that reveal different aspects
apparent goal of helping students move beyond
of the problem);
simply knowing how to find the distance by using
a formula, to understanding and making sense — considering special cases or simpler
of the formula itself. analogs;
— applying previously learned concepts to
What should you expect students to the problem, adapting and extending as
be able to do? necessary;
Focus in High School: Reasoning and Sense Making — making preliminary deductions and
describes reasoning habits that should become conjectures, including predicting what a
routine and fully expected in all mathematics solution to a problem might look like or
classes at all levels of high school. Approaching putting constraints on solutions; and
these reasoning habits as new topics to be taught — deciding whether a statistical approach is
is not likely to have the desired effect. The appropriate.
crowded high school mathematics curriculum
• Implementing a strategy, for example—
affords little room for introducing them in
this way. Instead, you should give attention to — making purposeful use of procedures;
reasoning habits and integrate them into the — organizing the solution, including
existing curriculum to ensure that your students calculations, algebraic manipulations,
both understand and can use what you teach and data displays;
— making logical deductions based on
Reasoning habits involve— current progress, verifying conjectures,
and extending initial findings; and
• Analyzing a problem, for example—
— identifying relevant mathematical concepts, — monitoring progress toward a solution,
procedures, or representations that including reviewing a chosen strategy
reveal important information about the and other possible strategies generated
problem and contribute to its solution by oneself or others.
4 The National Council of Teachers of Mathematics
5.
• Seeking and using connections across different What can you do to help students
mathematical domains, different contexts, understand the importance of
and different representations. mathematics in their lives and future
• Reflecting on a solution to a problem, for career plans?
example— Knowing and using mathematics in meaningful
— interpreting a solution and how it answers ways are important for all students, regardless
the problem, including making decisions of their post–high school plans. Whether
under uncertain conditions; the students attend college and major in
mathematics or go straight into the workforce
— considering the reasonableness of a
after graduation, they will need to have
solution, including whether any numbers
confidence in their knowledge of and ability to
are reported to an unreasonable level of
use mathematics.
accuracy;
To help students realize the importance of
— revisiting initial assumptions about the mathematics in their lives, you should recognize
nature of the solution, including being and demonstrate the need for mathematics
mindful of special cases and extraneous reasoning habits and content knowledge as
solutions; essential life skills. You must show how these
skills can ensure your students’ success for many
— justifying or validating a solution,
years to come—not just in the next mathematics
including proof or inferential reasoning;
course that the students may take.
— recognizing the scope of inference for a In addition, you should demonstrate an
statistical solution; awareness of the wide range of careers that
— reconciling different approaches to solving involve mathematics, including finance, real
the problem, including those proposed estate, marketing, advertising, forensics, and even
by others; sports journalism. Exposing students to the ways
in which fields such as these use mathematics
— refining arguments so that they can be
will help them appreciate the importance of
effectively communicated; and
mathematics in their own lives.
— generalizing a solution to a broader class Beyond showing the relevance of
of problems and looking for connections mathematics in an array of careers, you should
to other problems. also emphasize its practical value in offering
approaches to real problems. Seek contexts in
which your students can see that mathematics
Many of these reasoning habits fit in more
can be a useful and important tool for making
than one category, and students should move
decisions. In doing so, you will help students
naturally and flexibly among them as they solve
recognize the benefit of mathematical reasoning
problems and think about mathematics. Focus
and its importance for their adult lives. Such
in High School: Reasoning and Sense Making offers
lessons can contribute to the development of a
examples of ways to promote these habits in the
productive disposition toward mathematics.
high school classroom.
A Teacher’s Guide to Reasoning and Sense Making 5
6.
What can you do to make your
students’ high school mathematical
experiences more meaningful overall?
You can be an important advocate beyond
your own classroom for more meaningful high
school mathematics. Compared with teachers
of mathematics in the middle and elementary
grades—or with school administrators at
any level—high school mathematics teachers
generally have stronger, more extensive
mathematics backgrounds and have taken higher-
level mathematics courses.
Because of these experiences, high school
mathematics teachers are the most likely to see
mathematics as a coherent subject in which the
reasons that results are true are as important as
the results themselves. You can play a vital role in
communicating that message to other decision
makers in your school.
For the experience of learning high school
mathematics to change and become something
that is meaningful to your students, you must
begin today to focus your content and instruction
on reasoning and sense making. In addition,
you are in a unique position to work with
administrators and policymakers to achieve the
goal of broadly restructuring the high school
mathematics program to reflect this focus.
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