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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

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

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

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

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

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.

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.