5. Session Outcome
• Understand the Standards for Mathematical
Practice
• Explore strategies for implementing the
Standards effectively
6. What does it mean to be
Mathematically Proficient?
7. Turn and Talk
• Are students who can remember formulas
or memorize algorithms truly mathematically
proficient, or are there other skills that are
necessary?
• Is the correct answer the ultimate goal of
mathematics, or do we expect a greater
level of competence?
8. For the first time, mathematical processes are
elevated to essential expectations, changing
our view of math to encompass more than just
content.
The goal now is to apply, communicate, make
connections, and reason about math content
rather than simply compute.
15. Understanding the Standard
• What do we do each day in our classroom
to build mathematical thinkers?
• What do we do to keep our students actively
engaged in solving problems?
• How do we help our students develop
positive attitudes and demonstrate
perseverance during problem solving?
16. How Do We Get There?
Brainstorming Strategies…
1. Choose an Operation
2. Draw a Picture
3. Find a Pattern
4. Make a Table
5. Guess and Check
6. Make an Organized List
7. Use Logical Reasoning
8. Work Backward Page 11
17. The Holiday Tree
The Partin family counted the different types of
ornaments on the town’s holiday tree. Here is
the list of what they saw.
stars – 24
gingerbread men – 14
snowflakes – 12
reindeer – 18
candy canes – 6
Six of the reindeer had red noses. What fraction
of the reindeer had red noses? Tell how you
would get the answer.
Page 14
18. The Holiday Tree
The Partin family counted the different types of ornaments
on the town’s holiday tree. Here is the list of what they
saw.
stars – 24
gingerbread men – 14
snowflakes – 12
reindeer – 18
candy canes – 6
What fraction of the ornaments were snowflakes?
What fraction of the ornaments were edible?
If 6 of the stars were silver, what fractions of the stars were
not silver?
Page 14
20. What questions could you ask?
Shipley Aquarium
Admission Cost
Adults - $8.00
Children (ages 3 and over) - $6.50
Children (ages 2 and under) – Free
Look on page 19 for some additional suggestions.
21. Traditional Problems vs.
Rich Problems
• We can ask questions that stifle learning by
prompting a quick number response.
– What is the answer to number 3 on your
worksheet?
– What is 5 x 4?
• We can ask questions that promote
discussion, thinking, and perseverance.
22. Sort the math questions.
Check your arrangement on page 22
24. Reflecting on strengthening student
problem solving experience…
1. Do I routinely provide opportunity for my students
to share their solutions and processes with
partners, groups, and the whole class?
2. Do I show my students that I value process (how
they did it) rather than simply the correct answer?
3. Do I pose problems that require perseverance?
Do I use thoughtful questions to guide and
encourage students as they struggle with
problems?
27. Understanding the Standard
• What can we do in our classrooms each day
to help students build a strong
understanding of numbers (quantities)?
• How do we help students convert problems
to abstract representations?
• What can we do to help students
understand what numbers stand for in a
given situation?
28. How Do We Get There?
• Number Webs
• Headline Stories
– Post It
– Book It
– Reverse It
– Match It
– Question It
• Pinch Cards
31. Headline Stories
• Headlines sum up a story.
• Equations are like newspaper headlines—short
and to the point.
• Equations are connected to word problems the
same way a headline is connected to a news
story.
• Using headlines can help you see students
understandings and misunderstandings.
34. Headline Stories
Headline: 52÷4 =
Headline stories can be as easy or as difficult
as you make them!
Students might be asked to write problems
about equations that include fractions,
decimals, percents, or variables.
Let’s look at some variations found on page 39.
35. Pinch Cards
Pinch cards are an all-pupil response technique.
There were 6 soccer teams in the league and 12 players on each
team. How many players were in the league?
The 4 members of the High Rollers Bowling Team scored 120, 136,
128, and 162. What was the team’s mean score?
page 41
36. Avoiding Key Words
• Key words are misleading.
• Many problems have no key words.
• The key word strategy sends a terribly
wrong message about doing mathematics.
A sense making strategy will always work.
Van de Walle & Lovin (2006)
37. Read and Discuss page 33
• What is contextualization and
decontextualization?
• Why is it important?
• Discuss at your table.
38. Contextualize and
Decontextualize
120 students and 5 chaperones went on the
field trip. Each bus held 35 people. How many
buses were needed?
Decontextualize: consider the data, the action of the
problem, and create an equation to represent the problem in
an abstract way
Contextualize: refer back to the context of the problem to
determine if the answer makes sense
39. • Understanding the units and quantities
within a problem is an important factor in
making sense of the numbers within the
problem.
• Labeling answers forces students to refer
back to the context of the problem.
42. Understanding the Standard
• What do we do in the classroom to get students to
justify their answer and defend their process for
finding the answer?
• How do we help students understand math skills
and concepts so they can construct viable
arguments?
• How do we help students consider and judge the
reasonableness of other answers and strategies?
43. How Do We Get There?
• Eliminate It
• Agree or Disagree?
• My 2 Cents
44. Eliminate It!
• As a group, decide on the concept that
should be eliminated with reasoning or math
data to back up your decision.
• There may be more than one way to
eliminate an item!
• Create your own.
49. Agree or Disagree?
• Jim has 12 pencils and Annie has 8. Jim has more
than Annie.
• 7 + 3 and 4 + 6 are the only ways to make 10.
• 9 is an even number.
• 6 tens and 3 ones is the same as 5 tens and 13
ones.
• 3 jars of peanut butter for $7.50 is a better deal
than 4 jars of peanut butter for $10.20.
page 53
51. Constructing Arguments
• Read page 44-46
• What is the difference between an assertion
and an argument?
• Be prepared to share your thinking.
52. Assertion vs. Argument
• Assertion: a statement of what students want us to
believe without support or reasoning.
– The answer is correct “because it is,” “because I know
it,” or “because I followed the steps.”
• Argument: a statement that is backed up with
facts, data, or mathematical reasons
• Constructing viable arguments is not possible for
students who lack an understanding of math skills
and concepts.
55. Understanding the Standard
• As teachers, we model with mathematics
routinely in our classrooms. Should students
be able to model? Why?
• How do students modeling mathematics
look?
• How does student modeling of mathematics
affect instruction?
56. How Do We Get There?
• Model It
• Part-Part-Whole mats (addition & subtraction)
• Bar Diagrams (multiplication & division)
• Bar Diagrams (solving equations)
57. How would you model…
• 123 + 57
• 1 – 1
3
• 3.4 + 5.07
66. Assessment Tips
• Tell me what your model represents.
• Why did you choose this model?
• Did creating a model help you any way? If so,
how?
• Did you get any insights by looking at your
model?
• Is there another way you might model this
problem or idea? How?
Page 75
68. Understanding the Standard
• What are tools used by our students?
• Why is it important to use tools?
• Tools enhance our students’ mathematical
power by assisting them as they perform
tasks.
• The ability to select appropriate tools is an
important reasoning skill.
69. Which tool is more efficient?
There is often more than one tool that
will work for a task, but some tools are
more efficient than others.
Paper & Pencil
Mental Math
Calculator
70. Solve using your assigned tool!
1. 5 x 6
2. 23 x 15
3. Estimate the cost of
2 pies @ $3.75 each Cereal @ $3.20 each
Milk @ $1.79 gal Bananas @ 59 cents/lb
1. 236 x 0 x 341
2. What comes next 3, 7, 15, 31, ___
3. A local TV store had a sale on TV’s. They sold 7 for
$1,699.95 each. They made a profit of $169.00 on
each TV. What did the store pay for the 7 TVs?
A. $1,183.00 C. $13,082.65
B. $10,716.65 D. $11,899.65
71. How Do We Get There?
• Use tools appropriately
• Number Lines (It’s Close to…)
• Rulers (broken ruler, magnified inch)
• Mental Math
– Number Partners
– In My Head?
72. • Students benefit from opportunity to select a
tool that makes sense for the math task and
to evaluate which tool is most efficient for
that task.
• Not only do our students need to be able to
select appropriate tools, they must be able
to effectively use those tools. (page 80-81)
75. Folding Paper
• Fold the strip in half. Open it, and mark ½ at
the center fold.
• Refold the strip in half and fold it in half
again. Label 1/4, 2/4, 3/4 on the three folds.
76. Questions
• Are the sections equal in size?
• Do the fraction labels make sense? Why?
• Where is 0? Why?
• Where is 1? Why?
• Why are 1/2 and 2/4 on the same fold?
77. Folding Paper
• Refold the paper and then fold it in half one
more time.
• Open the paper, place a mark on each fold
and indicate what each of the new marks
represent.
78. Folding Paper
• Why is there more than one fraction on
some folds?
• Does it make sense that those fractions are
on the same fold? Why?
• Which of those fractions is easiest to
understand? Would you say 1/2 or 2/4 or
4/8? Why?
79. Number Partners (Mental Math)
• Find a Number Partner that Makes 10
5 4 9 3 6 1 7 2 5 8
Page 86
What are some modifications for this task?
80. In My Head? (Mental Math)
Do I use paper & pencil or do it in my head?
–734 x 82
–63 x 4
–1/4 + 2/8
–930 ÷ 3
Page 86-87
Students need to identify tools that increase their efficiency with math tasks.
83. Understanding the Standard
• Why is precision important in mathematics?
• What does it mean to be precise?
• What can we do in the classroom to
promote precise communication in
mathematics?
84. How Do We Get There?
• Estimate and Exact
• Vocabulary
– Word Webs
– Word Walls
– Sort and Label
– Mystery Words
– Translate the Symbol
– Word Boxes
85. Estimate and Exact
• Buying bags of candy to put in party treat
bags
• Measuring the dimensions of the doorway to
install a screen door
• Buying pizzas for a class party
• Buying carpeting for a living room floor
86. Estimation Skills
• Will the sum of 8 + 7 be greater than or less than
20? Why?
• Is the difference of 81 and 29 closer to 40, 50, or
60? Why?
• Is the sum of 1/3 + 4/8 greater than or less than 2?
Why?
• How would you estimate the product of 2.4 and
63? Will the product be between 2 x 60 and 3 x
60? Why or why not?
91. Translate the Symbol
• 4 dollars and 10 cents is greater than 4
dollars and 5 cents
• One-fourth of 16 is 4
• Doubling a number then adding six more
93. Which Is More Challenging?
• 4 dollars and 10 cents
is greater than 4
dollars and 5 cents
• One-fourth of 16 is 4
• Doubling a number
then adding six more
• 12 = 7 + 5
• 4 + x = 6
• 3 x 4 > 2 x 5
99. Understanding the Standard
• How do we help students discover patterns
in the number system?
• What can we do to help students make
sense of mathematics through the use of
structure?
101. Properties, Patterns, and Functions
There were 10 children at the party. How
many were boys and how many were girls?
Boys Girls
0 10
1 9
2 8
3 7
4 6
5 5
6 4
7 3
8 2
9 1
10 0
102. Perimeter Patterns
If there was a row of 50 connected
equilateral triangles, what would the
perimeter measure?
103. Number Flexibility
There was 1 ½ cupcakes left on the plate and
Liam and Molly decided they would eat them.
How much might each person have eaten?
Be ready to justify your answers.
104. How Do We Get There?
• Pattern Cover Up
• Pattern in the Hundreds Chart or
Multiplication Chart
• Ratio Tables
• Number Lines
106. Hundreds Chart
On a hundred chart, can students explain the vertical and
horizontal pattern? Do they see diagonal patterns?
107. Multiplication Chart
Can students explain vertical and horizontal patterns? Do
they see diagonal patterns? Can they find patterns that
explore equivalent fractions or proportions?
108.
109.
110. Ratio Table
Margo was painting flowers on the classroom
mural. Every flower had 3 leaves. How many
leaves were on the mural after 7 flowers had
been painted.
Number of flowers 1 2 3 4 5 6 7
Number of leaves 3 6 9 12 15 18 21
111. Ratio Table
• Each chicken has two legs. How many legs
are on 4 chickens?
• 2 jars of peanut butter cost $4.50. How
much do 8 jars cost?
• To make one dozen ice-cream sandwiches,
Katie used ¾ gallon of ice cream. How
much ice cream did she need for 60 ice
cream sandwiches?
115. Understanding the Standard
• Why is it important for students to recognize
repetition and reason why it is happening?
• When you think about repeated reasoning,
what do you think about?
116. How Do We Get There?
• Exploring Repetition
• Finding Shortcuts
• Organizing and Displaying Data to Discover
Rules
• Classroom Investigations
117. Exploring Repetition
2 + 1 = 3
5 + 1 = 6
3 + 1 = 4
6 + 1 = 7
4 + 1 = 5
What do you notice?
What conclusion can you draw?
119. Investigations to Find Shortcuts
• Read first paragraph under heading
Investigations to Find Shortcuts - page 124
• Turn and Talk – about your experience with
“Memorization”
120. Investigations to Find Shortcuts
• What do you notice?
• Do you see any pattern?
• Did you notice anything
interesting about the
solution?
121. Organizing and Displaying Data
to Discover Rules
Alice jumps rope faster then anyone in her class.
She can jump 8 times in 4 seconds. How long will it
take her to jump 40 times? Justify your answer.
Page 127
122. Orchestrating Classroom Investigations
to Discover Shortcuts
1. Opportunities for all students to gather data
with partners or teams.
2. Creating compilations of class data.
3. Observing compiled data and discussing
insights.
Page 128
124. Our “To Do” List
• Introduce all teachers to the practice standards.
• Provide examples to illustrate the standards.
• Encourage ongoing reflection about the standards.
– Professional development
– Faculty meetings
– Book study or PLCs
– Grade level teams
– Math coaches
126. Practice the Mathematical Practices
Three different veterinarians each help a total
of 63 dogs and cats in a week, but each
veterinarian helps a different number of dogs
and cats. How many dogs and cats could
each veterinarian have helped?
Survey participants: first timers, math coaches, classroom teachers, central office, principals, etc… This will allow us to modify presentation accordingly.
Thumbs up if you agree with these norms. Are there other norms we need to add so that we have the best possible learning experience for all?
We will not use all slides but we have included additional slides for your convenience so that you may modify accordingly to your needs for a school or district. Example: If used in PLC’s – may want to take one Practice Standard at a time and develop it more thoroughly.
Share outcomes
Discuss
Discuss
Increase attention on problem solving, discussion, and justification of thinking and decrease attention on rote practice, rote memorization of rules, and teaching by telling
Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
These are reflective questions notes in the book. Participants turn and talk.
Have participants brainstorm strategies student use to solve problems. Then click and the list noted in book will appear.
In this question, discussion occurs on how to find an answer, not necessarily the answer. This is followed up over the next few days with different questions.
Using the same data, different questions can be asked that lead to strong classroom discussion.
Quick responses do not require mathematical thinking, do not create proficient mathematicians and do not promote the practices.
What would be the benefit of doing this activity with teachers?
Participant refer to book for additional ideas to develop the practice.
Turn and Talk
Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
Have participants create a number web. Carousel of webs: think about fractions, decimals, whole numbers…choose a number and then as a group at least two ways to represent it before the buzzer. Rotate to the next poster Have groups take post it notes with them. If they disagree with a representation, they can note it and tell why.
This is an example from the book on pg. 34.
What misconceptions does this student have? The calculation is correct, but the student does not have an understanding of multiplication.
Student created a problem that shows an understanding of multiplication.
Writing problems to contextualize math helps students with their ability to solve problems because they strengthen their understanding of the link between real situations and math equations.
Each student is given a pinch card, which is created with an index card that is printed with the operation signs in the same location on the front and back of the card. Math word problems are posed to students who then pinch the sign they would use to solve the problem. Word problems should match students’ grade levels and math computation skills.
Often the key word or phrase in a problem suggests an operation that is incorrect. For example: Maxine took the 28 stickers she no longer wanted and gave them to Zandra. Now, Maxine has 37 stickers left . How many stickers did Maxine have to begin with? Especially when you get away from overly simple problems found in primary textbooks and a child that has been taught to rely on key words is left with no strategy. The most important approach to solving any contextual problem is to analyze the structure and make sense of it. The key word approach encourages students to ignore the meaning of the problem and look for an easy way out.
To successfully interpret and solve math problems, students must be able to decontextualize and contextualize problems.
Labeling answers: Numeric part of the answer attests to student’s computational accuracy Label attests to whether the student knows what the label represents
Participant refer to book for additional ideas to develop the practice.
Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
Discuss
Three strategies for constructing and critiquing arguments.
Students are presented with four math concepts and asked to decide on the one that should be eliminated based on mathematical fact or reason. There may be more than one way to eliminate an item. The challenge s to construct a clear argument for eliminating the item selected. Do these, share, create their own.
Teacher poses a math statement and asks students to agree or disagree with the statement. Students must include math data or reasoning to support their decision. This student agrees that 75% is more than 2/3, converting 75% to a fraction and then drawing a diagram to compare fractions.
This student disagrees that 5 nickels are worth more than 3 dimes. She provides information about the value of the coins and clearly explains her computations to prove her thinking.
Cards for each table. Each participant takes a card and explains if they agree or disagree and why.
Students are presented with sample arguments with flawed or incomplete logic. Students work individually or with partners to improve the sample. This student critiques the argument and offers some ideas as to why it is faulty. This helps avoid the “teacher is always right” mentality.
The act of constructing arguments challenges students to think about the math they are doing and leads them to discover mistakes as well as insights while they struggle with the construction of their arguments.
Participant refer to book for additional ideas to develop the practice.
Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
Students need lots of experiences constructing math models. Students are challenged to think about the math and determine a way to represent it.
How would you model this situation?
Result unknown 3 x 6 = ?
number in group unknown 18 divided by 3 = 6
Number of boxes unknown 18 divided by 6 = 3
Participant refer to book for additional ideas to develop the practice.
Formal and informal Spontaneous , informal interview can yield insights about the reasoning and skills. Formal interviews might be done at a specific time which students bring their model to our desk and we pose questions like the following:
Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection. If we attempt to place each of these standards in a separate compartment, we will surely become frustrated and confused.
While we may immediately think of tools as math devices like rulers, compasses, protractors etc…, the CCSS Standards for Mathematical Practices have a much broader definition of tools. Tools are what support students to perform the task. Concrete materials like base-ten blocks, connecting cubes, and 10 x 10 grids can be tools, so grid paper, number lines, and hundreds charts. Calculators, pencil and paper, and even mental math are also tools. Tools enhance our students’ mathematically power by assisting them as they perform tasks.
Students begin to decide which tools will best meet their needs.
Students benefit from opportunity to select a tool that makes sense for the math task and to evaluate which tool is most efficient for that task. Not only do our students need to be able to select appropriate tools, they must be able to effectively use those tools. – page 80-81 – using tools appropriately!
Number lines are powerful tools for computation and estimation tasks. The ability to view th quantities on a number line proves the comparison. It’s close too… helps students justify their decision, it’s visually exploring numbers.
Students need a clear understanding of the tools purpose. A common error occurs when our students do not accurately align the beginning of the ruler with the beginning of the object.
The understanding of equivalent fractions enhances our students abilities to use rulers accurately.
Increase or decrease the complexity. Simplify task, students could identify just one set of number partners or put fewer numbers in the box. To increase complexity, include numbers that don’t have partners or ask students to create their own number sets and target numbers.
Develops computational fluency and mental math skills. 734 x 82 – paper and pencil 63 x 4 - 1/4 + 2/8 930 ÷ 3
Participant refer to book for additional ideas to develop the practice.
Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
Math relies on precision in computation and communication. To communicate effectively about math content, students need to know the words that express that content. Students must understand the math and know that words that express that understanding. Understanding symbols is more than the ability to read them. Being able to name a symbol does not ensure students understand the meaning of the symbol.
When is precision important? Pose situations in which students must decide whether an estimate or an exact answer is needed. Students develop arguments for whether an exact or an estimated answer makes more sense in the situation
Partner talk
Quick and effective way to explore math ideas and expand math vocabulary. Questioning: Why did you think of that word? How does it relate to our word of the day? What is…? Can you give me an example of…? When have we talked about … before?
Students identify similarities and differences between math concepts. Sort and label cards
Give each pair of students a set of word cards and have partners take turns picking a card at random. Have students describe the word, without actually saying the word. The partner should figure out the mystery word from the clues. Continue to give clues until the word is identified. Talk about how to differentiate.
Place math expressions, equations, or inequalities on index cards Students select a card and explain it to their partner or explain in writing
Word boxes work well for assessment and instruction Table group activity
Participant refer to book for additional ideas to develop the practice.
Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
There is structure in math. People who see that structure find that math makes sense. Proficient students: See that numbers are flexible Understand properties Recognize patterns and functions
See solutions on pages111-112
Students deepen their understanding of fractions as they decompose 1 ½
Discussion on how this can be used in the classroom.
Ratio tables make sense when two pieces of data are connected. Supports students as they develop understanding and automaticity with multiplication and division.
A number line help students visualize properties.
Participant refer to book for additional ideas to develop the practice.
Think about what this standard means and looks like in a classroom. Here are some questions to help you with your reflection.
Students find ways to minimize their efforts in mathematics through shortcuts after discovering repetition.
What conclusion can your draw? Page 121 Check your prediction with your partner! When adding by one it is always the next number in the counting squence.
Share objectives
Memorize math rules without understanding. These are tricks to make math earsier but there are no “tricks” in math. It is the understanding of math that makes it easier. Page 124 - more
What do you notice? Do you see any pattern? Did you notice anything interesting about the solution? It is the same number just with a 1 in front of it. The one represents a ten.
Organize data into table so it is easier to see patterns – ultimately to discover insights. Pose problems in which students are challenges to extend patterns and look for a generalization, or algebraic relationship, to explain the pattern.
It is important for students to discover math rules, generalization, and shortcuts. The goal is to set up investigations that allow them ti have “ah-ah” moments. This slide contains some critical components of these investigations.
Participant refer to book for additional ideas to develop the practice.
Share objectives
Additional task to practice using Standards for Mathematical Practice – if time permits.
Additional task to practice using Standards for Mathematical Practice – if time permits.
