Transcript Standards for Mathematical Practice

Standards for Mathematical Practice
December 1, 2011
NCTM Principles
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Representation
Connections
Reasoning
Problem Solving
Communication
Make Sense of Problems and Persevere in
Solving Them
• A problem is a situation where you do not know
what to do.
• Teachers seldom are able to model problem solving
and when they do, it is an accident.
• Problems for students are usually exercises for
teachers.
Make Sense of Problems and Persevere in
Solving Them
• Teachers identify rich mathematical tasks that will lead
to the understanding of the content they are teaching
and which they believe will be “problems” for their
students.
• Teachers present these tasks to the students.
• Teachers use on-going formative assessment to modify
tasks so that all students are engaged and challenged.
Make Sense of Problems and Persevere in
Solving Them
• Students should be given rich mathematical tasks
presented as a “problem” for them to engage in and
make sense of (related to the task).
• Students should make and implement plans to try to
accomplish the goal of the task.
• Students should monitor progress and revise and refine
their plans based on the intermediate findings.
Make Sense of Problems and Persevere in
Solving Them
• Teachers should encourage students to compare
and evaluate results and processes.
• Teachers should provide students with information
about mathematical vocabulary, notations, and
conventions to enhance their ability to
communicate effectively with others.
Make Sense of Problems and Persevere in
Solving Them
• Problem-solving often results in the creation of
factual or procedural knowledge that can be used to
accomplish future tasks, which are no longer
problems but merely exercises.
Reason Abstractly and Quantitatively
• Use less than, greater than, and equal to as you
compare the following:
6
8
12 4
7
7
Reason Abstractly and Quantitatively
• Use less than, greater than, and equal to as you
compare the following:
6 pounds
8 ounces
12 nickels 4 dollars
7 meters
7 centimeters
Reason Abstractly and Quantitatively
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What is the sum of 4 and 3?
What is the sum of 4 nickels and 3 dimes?
What is the sum of 4 third-cups and 3 half-cups?
What is 4x + 3y?
Reason Abstractly and Quantitatively
• Compare the following rates:
2/3 meters per second
1.5 seconds per meter
Reason Abstractly and Quantitatively
• Why is the slope of a line change in y over change in
x?
Reason Abstractly and Quantitatively
• The school has 782 students that need to be
transported by bus. Each bus can transport 48
students. How many buses are needed?
Reason Abstractly and Quantitatively
• Karen has 12 yellow pencils and 8 red pencils. Does
she have more yellow pencils or red pencils? How
many more?
Construct a Viable Argument and Critique
the Reasoning of Others
• Often more is learned by being wrong than by being
right.
• Competition for grades and awards often make it
unwise to risk being wrong at school.
Construct a Viable Argument and Critique
the Reasoning of Others
• What is 7 times 8?
I don’t know, but I do know 5 times 8 is 40. So 6
times 8 would be 40 plus 8, 48. To get 7 times 8, I
have to add 8 more to 48. I know 48 is 2 away from
50, if you take 2 from 8 that leaves 6, so 7 times 8
must be 56.
Construct a Viable Argument and Critique
the Reasoning of Others
• I asked a class of third graders to find 203-78?
The class got 35, 125, 135, and 205. The most
popular answer was 275.
• Which answers are unreasonable? Why?
Construct a Viable Argument and Critique
the Reasoning of Others.
• Students measured the perimeter of a table and got the
data below (all measures were in cm):
168, 209, 241.5, 271, 400, 432, 436, 438, 440, 446,
450, 450, 450, 458, 460, 460, 460, 462, 464, 468,
470, 480, 494, 530
• Why is the range of this data so large?
Construct a Viable Argument and Critique
the Reasoning of Others
• If two line segments intersect (not at their end
points) and are perpendicular, what kind of
quadrilateral will be formed by connecting the end
points of the two intersecting segments?
Model with Mathematics
• Everyday Life
• Society
• Workplace
Model with Mathematics
• What time will I get home today?
• How much money do I need to budget to buy
Christmas gifts?
• Can I get a table top that is 7.5 feet wide through a
door that is 3 feet by 7 feet?
• What long distance calling plan should I buy?
Model with Mathematics
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Should I cut a cedar tree?
How do I calibrate a crop sprayer?
How much should I charge for a candy bar?
How do surveyors calculate some distances?
Where is the center of a room that is an isosceles
trapezoid?
Model with Mathematics
• If two candidates have very different plans for
collecting taxes, for which one should I cast my
vote?
• Should children have to be immunized to attend
school?
th
• What does it mean if a child or a school is at the 48
percentile?
Model with Mathematics
• Write a word problem that can be solved by
simplifying the following expression.
2
1
¸
3 2
Attend to Precision
• Ten refers to exactly ten and no more or less.
• -teen means one ten and some toward a second
group of ten.
• -ty means more than one group of ten.
Attend to Precision
• What is a circle?
• What is an angle?
• What do we mean when we say a segment is 15 cm
long?
Attend to Precision
• 3 red tiles + 4 blue tiles = 4 blue tiles + 3 red tiles
The above statement is an example of the
commutative property for addition.
• 3 red tiles + 4 blue tiles = 4 red tiles + 3 blue tiles
If the statement above is true, it is because of the
transitive property of equality not the commutative
property for addition.
Attend to Precision
What property do you see?
• 2 packages x 6 cookies/package =
6 packages x 2 cookies/package
• 2 packages x 6 cookies/package =
6 cookies/package x 2 packages
• 2 feet x 6 feet = 6 feet x 2 feet
Attend to Precision
What does “=“ mean?
•1/5 gallon = 2/10 gallon
•1 gallon/5 miles = 2 gallons/10 miles
•1 woman/ 5 people = 2 women/10 people
•2x + 3 = 15
•2(x+3) = 2x + 6
Attend to Precision
• Why is 2(x + 5) = 2x + 10?
• Why is 2x + 5x = 7x?
Attend to Precision
• What do we mean in mathematics class by
“cancel”? Is cancel a mathematical term?
• What understanding could we promote by avoiding
the use of pronouns without antecedents?
Attend to Precision
What are:
• Solutions?
• X-intercepts?
• Zeros?
• Roots?
Attend to Precision
• What connections do we need to make with ELA
that will promote vocabulary development in
mathematics?
• Many of our terms are compound words. When and
how do students best learn about compound words
in general? Are we using the same strategies in
mathematics vocabulary development that are used
in ELA and other disciplines?
Attend to Precision
Triangle indicates 3 angles.
Quadrilateral indicates 4 sides.
• Why not trilateral or quadrangle?
Pentagon is 5 sides.
Hexagon is 6 sides.
• Why not trigon and tetragon?
Attend to Precision
• Are there ways we can activate students’ prior
knowledge by using terminology and examples they
would recognize from earlier grades?
• What are the appropriate mathematics vocabulary
words for each grade?
• Are all teachers using the same mathematical
symbols and conventions?
Select Appropriate Tools and Use Them
Strategically.
• With and without manipulatives.
• With and without technology.
Select Appropriate Tools and Use Them
Strategically
• Technology is not evil and has not caused students
to lack automaticity with facts or fail to develop
procedural fluency.
Select Appropriate Tools and Use Them
Strategically
• Computer software – graphing utilities, spread sheets,
dynamic geometry packages, fluency development
programs, etc.
• Measuring tools – rulers, meter sticks, measuring tapes,
scales, balances, graduated cylinders, protractors,
clocks, thermometers, etc.
• Constructions tools- compass and straightedge
• Other – manipulatives, scissors, gridded paper, etc.
Select Appropriate Tools and Use Them
Strategically
• Teachers will not select the tool and direct the students
on the strategy for using the tool in each problem or
task.
• Students engaged in a problem or task will realize a
need for a tool, think about the purpose for using the
tool, find the tool or go to the teacher for help with
finding a tool that allows them to collect the
information they need or create the product they need
to accomplish their goal.
Look For and Make Use of Structure
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Properties of equality
Field Properties
Properties of inequality
Number naming conventions have structure
Classification of numbers, 2-D figures, and 3-D objects
imposes structure
• Logic has structure
• Algebraic notation imposes structure
Look For and Make Use of Structure
• Structure does not force everyone to think the same
way. If structure is understood, then students can be
flexible in their thoughts and make decisions based on
context.
• The structure of the base ten number system and the
distributive property allow us to generalize an
algorithm for multiplying multi-digit numbers. In some
cases, we can modify that algorithm to do the work
more efficiently.
Look For and Make Use of Structure
• Consider 8 x 24:
8(20 + 4) = 160 + 32 = 192
8(25 – 1) = 200 – 8 = 192
Look For and Make Use of Structure
• The Distance Formula, the Pythagorean Theorem, and
the one of the trigonometric identities are not three
different things they are the same thing.
• Think of the difference in the x-coordinates as “a”, the
difference in the y-coordinates as “b”, and the distance
itself as “c”.
• Think of sin(θ) as “a”, cos(θ) as “b”, and since the unit
circle is being used we know “c” is 1 unit.
Look For and Make Use of Regularity in
Repeated Reasoning
• Perhaps the writers of the CCSS for mathematics
intentionally avoided saying “look for patterns” because
educators have (over time) developed some very
limited ideas about what patterning is and why it might
be important.
• Maybe because patterning was being done with very
young children, most curriculum material on patterning
is somewhat elementary and isn’t transferring to more
rigorous mathematics situations.
Look For and Make Use of Regularity in
Repeated Reasoning
Activity: Fat Is
CCSS Standards for Mathematical Practice
Http://www.youtube.com/watch?v=m1rxkW8ucAI