Common Core State Standards for Mathematics:
Shifts and Implications for Mathematics Instruction
Schedule 11:00-11:30 Auditorium- ALL 11:30-2:00 Break-out rooms
Group A- Amy Room 108 A (Science comes 1:00-1:30) Group B- Marilyn Room 108 B (Science comes 1:30-2:00) Group C- Susan Room 209 (Science comes 12:30-1:00) Group D- Drew Room 112 (Drew’s group stay in auditorium 11:30-12:00
for Science, then move to room 112
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, have Drew
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
Making Banners
You have 9 yards of fabric.You need 2 of a yard to make a banner.How many banners could you make?
We have number lines, graph
paper, and blank paper available.
Approaches? Now let’s change the numbers…
Making Banners
You have 9/4 yards of fabric.You need 1/2 of a yard to make a banner.How many banners could you make?Solve 2 ways– number line/picture AND then computation
We have number lines, graph
paper, and blank paper available.
Under every other chair there is a solution to this problem. With a partner, show how how
a student might arrive at this solution.
Making Banners
Find your group4 ½ 4 2 9/8
How are your solutions the same?How are they different?
Making Banners
Listen as groups share what they noticed4 ½ 4 2 9/8
A Lesson in the “old” DaysWhat did math class look like when you were a student?
First, invert the 1/2 so you now have 9/4 x 2/1
Next, multiply 9x2 to get 18, then 4x1 to get 4.Now we need to reduce so you take 18 divided by 4 to get 4 r 2 or 4 2/4 that needs to then be reduced to 4 1/2
4 ½ 4 2 9/8
Which answer does method this lead to?
“Don’t Ask Why, Just Invert and Multiply!”
During and after reading a selected text, we ask…
“What do you notice in the text?”
“Do you agree with the characters decision
to…/why?”
After solving a math problem, we ask…
“Did you get the right answer?”
Numbers….9 divided by 2???
How did changing the numbers influence the “difficulty” of the task?
9/4 divided by 1/2
Procedures = Understanding
17- 6
10- 6
How did changing the numbers influence the “difficulty” of the task?
Procedures = Understanding
17- 6
By place value…start at the right…Ones column: 7- 6 = 1. Tens column: Bring down the 1.I have a 1 in both places. Answer is 11.
Procedures = Understanding
10- 6
By place value…start at the right…Ones column: I can’t do 0 minus 6.
Go next door… change the 1 to a 0 in the tens
Change the 0 to a 10 in the ones.Ones: 10 – 6 = 4Tens: 0 – 0 = 0.I have 0 tens and 4 ones. My answer is 4.
Do they understand?
Turn and TalkThink of a mathematical procedure
you learned as a student.
How does that make it difficult for you to explain your mathematics behind your solution?
FoilButterflyStandard Algorithm for …Change mixed numbers to improper fractions
Break
Welcome BackWhat is MOST
IMPORTANT at your grade level?
Answer by placing one sticker on the chart.
The Three Shifts in Mathematics
Focus specificity and concentration on fewer “big ideas”
Coherence: Think across grades and link to major topics within grades
Rigor: Require conceptual understanding, fluency, and application
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
It starts with Focus
The current U.S. curriculum is ‘a mile wide and an inch deep.’
Focus is necessary in order to achieve the rigor set forth in the standards
The following graphs refer to mathematics instructionWith a partner, predict what these represent
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
Comparing Charts
Compare our school chart to this one
What do you notice?
K 12Number and Operations
Measurement and Geometry
Algebra and Functions
Statistics and Probability
Traditional U.S. Approach
Complete the Diagram Below
Algebra
K 1 2 3 4 5 6 7 8 High School
-Number and Operations—Fractions
-Expressions and Equations
-Operations and Algebraic Thinking
-Number and Operations—Base Ten
-The Number System
Focusing attention within Number and Operations
Operations and Algebraic Thinking Expressions and Equations
Algebra
Number and Operations—Base Ten
The Number System
Number and Operations—Fractions
K 1 2 3 4 5 6 7 8 High School
How do they compare?
Content Emphasis-Released by Achieve.org-Non-profit group of educational leaders who have committed to provide resources at no cost.-Designed to help teachers focus on what is most important at each grade level.
Focus
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
What is vital at my grade level?
Complete this sentence with only one mathematical concept;
“By the end of ___ grade, if my students knew _____ they would be
considered mathematically proficient.”
Focus
Walk and Trade
Shift Two: Coherence 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.
Coherence Within GradeLook within your grade and find where
important concepts are connected across domains.
Create a product that communicates your findings
Coherence Within GradeGrade Levels Share
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
Hundreds, Tens, and Ones
Hundreds, Tens, and Ones
Compare the two Hundreds, Tens, and Ones activities thinking about how each provides…Solid conceptual understandingProcedural skill and fluencyApplication of skills in problem solving
situations
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 72B 620C 720D 7,200
Common Core
Rigor
Compare the two items, which provides for…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!
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!
Afternoon SessionEnsuring the Standards for
Mathematical Practice at the student level.
Illustrating the Mathematical Practices
Afternoon SessionEnsuring the Standards for
Mathematical Practice at the student level.
A pen for 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 2 ½ feet from the perimeter of the Dane’s pen shown below,
how much fencing will she need?
7 ¼
3 5
2
Solutions
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 )
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).