Standards for Mathematical Practice
1. Make sense of complex problems and persevere in solving them.2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.
Grouping the SMP’s
Common Core State Standards for Mathematics:
Shifts and Implications for Mathematics Instruction
Morning Session
Focus, Coherence, and Rigor; Realizing the Common Core in
Elementary Mathematics Understand how the Common Core influences
our Instruction Identify the 3 major shifts in math education
Last week, I was at a restaurant with a group of friends. The bill came and this is what was
said…
“I’m not good at math, you figure
it out.”
Out to lunch…
Next time we go…“I’m not good reading, can you read this
to me?”
Same Reaction?
Illiteracy Innumeracy
Why is it socially acceptable to be
innumerate?
Innumeracy21% of Americans possess numeracy skills at the lowest level . . . [which] means that
people cannot . . . work out the change from $2 when buying
goods worth $1.58.
(Murray, 2000. p. 2)www.lausd.net/District_8/math/secmath/0607/math_anx_ncsm.ppt
Procedures = Understanding
17- 6
Explain to your partner how you would use the traditional algorithm to solve this
Procedures = Understanding
Now explain to your partner how you would use the traditional algorithm to solve this 10
- 6
Procedures = Understanding
17- 6
10- 6
How did changing the numbers influence the “difficulty” of the task?
Procedures = Understanding
10- 6
https://www.teachingchannel.org/videos/subtraction-as-a-confidence-builder?fd=1
Do they understand?
SMP IntegrationTraditional U.S. ProblemWhich fraction is closer to 1: 4/5 or 5/4 ?
Same problem with SMP integration4/5 is closer to 1 than is 5/4. Using a number line, explain why this is
so…
Discuss the shift with this question…
Procedures & Algorithms
At your table, fill your large paper with examples of algorithms/procedures you learned as a student. Examples:Foil method in algebraButterfly to compare fractionsStandard Algorithm for multiplication or division
AlgorithmsChoose one procedure or algorithm
that you learned as a student.E.g., Finding common denominator,
FOIL, Butterfly to compare fractions, multiplication or division algorithm, dividing fractions
Do a sample problem using it… Then explain it to your neighbor –
explain the process and why it works
AlgorithmsWhat algorithms did you explain?
How does understanding the math relate to following the steps of the algorithm?
BreakWhen you come back, please sit in
grade level groups.
Welcome BackAccording to the
Common Core, What is MOST IMPORTANT at your grade level?
Answer by placing one sticker on the chart.
The Three Shifts in Mathematics
Focus Coherence Rigor
Authors of the Common Core Math
Standards have emphasized 3 shifts
from old state standards
(like the NCSCOS)
Shift One: Focus strongly where the Standards focus
Significantly narrow the scope of content and deepen how time and energy is spent in the math classroom
Focus deeply only on what is emphasized in the standards, so that students gain strong foundations
Why Focus?
The U.S. curriculum before the Common Core was known as ‘a mile wide and an inch deep.’What did this look like in the NCSCOS?
Focus is necessary in order to achieve the rigor set forth in the standards
Mathematics topics intended at each grade
1 Schmidt, Houang, & Cogan, “A Coherent Curriculum: The Case of Mathematics.” (2002).
By at least 2/3 of US States By at least 2/3 of A+ Countries
The Shape of
Math Instruction
Teach Less, Learn More!
https://www.teachingchannel.org/videos/common-core-state-standards-for-math?fd=1
How will teaching fewer “topics” in each grade change your planning?
Content Emphasis
Designed to help teachers focus on what is most important at each grade level.
Focus
Content Emphases by Cluster
Not all of the content in a given grade is emphasized equally in the standards.
Some clusters require greater emphasis based on the depth of the ideas, the time that they take to master, and/or their importance to future mathematics or the demands of college and career readiness.
An intense focus on the most critical material at each grade allows depth in learning, which is carried out through the Standards for Mathematical Practice.
Focus does not mean neglecting certain standards. Neglecting material will leave gaps in student skill and understanding and may leave students unprepared for the challenges of a later grade.
Sample Content Emphases
What do you notice?What’s missing?
Focus – your grade levelWith your grade level team
Look at the Content Emphasis
In YOUR copy of the Common Core –highlight areas of focus at your grade level.
Focus
With your grade level team, discuss…
K-2 Teachers Look at the supporting
standards at your grade level.
Generate specific examples of how each supporting standard relates to major standards
3-5 Teachers Identify major changes
within your grade level. How has the focus
changed from the 2003 NCSCOS?
Focus
FocusWould you
change where you placed your sticker?
Focus
Break!~
Coherence Within GradeUse the handout of your Grade Level
ClustersAnalyze how these clusters are classified
How do these clusters overlap or address similar “larger concepts”?
How do Supporting Clusters align with Major Clusters.
CoherencePick a unit from Investigations that
incorporates multiple clustersWhat are the pre-requisite skills you
hope students have before this unit starts?
As students are doing the work in the unit, what is the Major Cluster?What other clusters are also
incorporated?
CoherenceMove and group with teachers from
multiple grade levels
Exchange ideas and discuss what your grade level came up with
Coherence Wrap-up When teaching the Common Core, teachers should: Carefully connect the learning within and across
grades so that students can build new understanding onto foundations built in previous years.
Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new set of events, but an extension of previous learning.
Shift Three: Rigor
The CCSSM require a balance of: Solid conceptual understanding Procedural skill and fluency Application of skills in problem solving situations
This requires balanced intensity in time, activities, and resources in pursuit of all three
Rigor
Place Value…
Why do we need rigor when we work with place value concepts?
Hundreds, Tens, and Ones
In groups with teachers from different grade levels, analyze the two Hundreds, Tens, and Ones activities. How does each provide…Solid conceptual understandingProcedural skill and fluencyApplication of skills in problem solving
situations
Hundreds, Tens and OnesDo these promote Conceptual
Understanding?Why or why not?
Hundreds, Tens and OnesDo these promote Procedural Skill and
Fluency?Why or why not?
Hundreds, Tens and OnesDo these promote skill application in
problem solving situations?Why or why not?
DPI 3-5 Released Item
Mrs. Gregory assigned a project that required each of her 20 students to use 36 toothpicks. How many toothpicks did the students use?
A)72 B)620 C)720 D)7,200
Common Core
Rigor
Compare the two previous multiplication items, and discuss how each does/does not provide the opportunity for students to demonstrate…Solid conceptual understandingProcedural skill and fluencyApplication of skills in problem solving
situations
Three Mathematical Shifts
Which shift will have the greatest impact on student learning?
Decide with your team and be prepared to share!
C.C. Item ExamplesInside Mathematics
http://insidemathematics.org/index.php/home
Tools for Educators Choose your grade level Mars Tasks
These should be similar to future Common Core assessment Items…
Realizing the Promise of the Common Core
It will take hard work on our part to move a society from this…
10 - 6
To This!
Choose 3 Ways!
4 Teachers $20.00
Burrito - $4.12 Soda - ?
February 21Ensuring the Standards for
Mathematical Practice at the student level.
Illustrating the Mathematical Practices
February 21Ensuring the Standards for
Mathematical Practice at the student level.
A pen for Aramis and Elise (Great Danes) is shown below. Amy wants to add a safe place for her
Chihuahua to run around the great Danes (and not get stepped on!). If she adds a fence that is 2½
feet from the current perimeter, how much fencing will she need for the Chihuahua’s run?
7 ¼
3 5
2
The Chihuahua run must be built 2½ feet from the edge of the Great Dane pen!
Standards for Mathematical Practice
4 and 5 In pictures and words, show how you
used either mathematical practice 4 or 5 as you solved this task
(you might have to pull these out and re-read )
Solutions
Linking Content and Practice
Numbers and Operations in Base Ten (NBT) K-2: 1.NBT.5 and 1.NBT.6
3-5: 4.NBT.5 and 4.NBT.6
Practice 4Model with Mathematics…
What fraction of this shape is red? https://www.teachingchannel.org/videos/what-fract
ion-of-this-shape-is-red?fd=1 Explain how the students engaged in one of the 3
shifts? (Focus, Coherence, Rigor) How could Mr. D extend this lesson to develop
students thinking around adding and subtracting fractions or decomposition of fractions?
Practice 5Use Appropriate Tools Strategically…
On the count of 3, you have 30 seconds to find as many colored tiles as possible. When the time is up, please bring them back to your group.
Complete the tally chart with your groups tiles. Then construct a bar graph with your data. Combine group data to make a class bar graph. Range, Median, Mode What conclusions can you draw?? How might the data relate to
animals in the wild? How does hiding the tiles around the room affect the lesson?
An Experience with Students
The following slides are from DPI.
The Task
What would that look like…
With our students?
Step 1: Introduce/Review Standards for Mathematical Practice # 4 and #5
Date:
Briefly describe how you will introduce/review Standards for Mathematical Practice # 4 and #5
Step 2: Apply Standards for Mathematical Practice # 4 and #5 to a math lesson.Select a lesson that provides students the opportunity to choose appropriate tools strategically and model with mathematics.
Briefly describe lesson chosen:
Toward the end of the lesson, ask students to illustrate how they used mathematical practice #4 OR #5 in their work today.
Step 3: Bring illustrations to math planning.As a team, choose 2 illustrations (one that shows a solid understanding and one that shows minimal understanding). Send illustrations in to Amy and Marilyn (use envelope provided).