Elementary Mathematics
Learning Community
Dialogue 2
November 2012
Office of Academics and Transformation
Division of Mathematics
NOTE:
One of the goals of this presentation is to discuss data driven decision making, and the
instructional implementation of data analysis. For this reason, at the actual presentation,
some slides with Interim Assessments items were shown. However, such slides have been
deleted from this power point presentation version due to test security.
Slides with answers to activities conducted with participants were also deleted due to
availability of this power point prior to conducting the PD session.
AGENDA
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Puzzles, Patterns, Learning in Context
What is Problem Solving?
Metacognitive Behaviors
What Do Good Problem Solvers Do?
Effective Instructional Procedures for Teaching Math
Problem Solving
• QMBA Item Analysis (K-2)
• Will CCSSM Matter in Ten Years?
• Instructional Coaching Cycle
Puzzles, Patterns, Learning in Context
• Maria saved $24. She saved 3 times
as much as Wayne.
Two interpretations of Division
Thomas H. Parker and Scott Baldridge
When we know the original amount and the number of parts,
we use partitive (sharing) division to find the size of each part.
We use “division
to find the
number
in each group.”
Thomas H. Parker and Scott Baldridge
When we know the original amount and the size or measure of one
part, we use measurement division to find the number of parts.
Sarah made 210 cupcakes. She put them into boxes of 10 each. How many boxes of cupcakes
were there?
We use “division to
find the number of
groups.”
Thomas H. Parker and Scott Baldridge
Partitive or Measurement Divison?
Measurement Division
Partitive Division
 Millie has 15 cookies. She puts 3 cookies in
each bag. How many bags can she fill?
 Millie has 5 bags of cookies. Altogether she has15
cookies. There are the same number of cookies in
each bag. How many cookies are in each bag?
Adapted from Cognitively Guided Instruction, University of Wisconsin-Madison, 1992
Partitive or Measurement Divison?
Measurement Division
Partitive Division
 Manuel has 24 pencils. They are packed 6 pencils
to a box. How many boxes of pencils
does he have?
 Manuel has 6 boxes of pencils with the same
number of pencils in each box. Altogether, he has
24 pencils. How many pencils are in each box?
Adapted from Cognitively Guided Instruction, University of Wisconsin-Madison, 1992
Einstein is quoted to have said :
“if he had one hour to save the world he would spend fifty-five
minutes defining the problem and only five minutes finding the
solution”.
What is Problem Solving?
Problem solving is a process and skill that
you develop over time to be used when
needing to solve immediate problems in
order to achieve a goal.
University of South Australia
New Research
• You're at a big group dinner and it's time to pay up, to
divide the total and multiply a certain percentage for the
tip. How many people tense up and say something like,
"Oh, I'm so bad at math"?
• Fear of math is everywhere - in the adult world where
there aren't official pop quizzes, and in schools where
the next generation of scientific problem-solvers are
struggling with homework.
• Researchers report in a new study in the journal PLOS.
One that this anxiety about mathematics triggers the
same brain activity that's linked with the physical
sensation of pain.
Elizabeth Landau - CNN.com Health Writer/Producer
Metacognition
• Several research studies have concluded that
metacognitive processes improve problem solving
performance.
(Artzt & Armour-Thomas, 1992; Goos & Galbraith, 1996;
Kramarski & Mevarech, 1997)
• Metacognition is also believed to help students develop
confidence to attempt authentic tasks (Kramarski,
Mevarech, & Arami, 2002), and to help students
overcome obstacles that arise during the problemsolving process (Goos, 1997; Pugalee, 2001; Stillman &
Galbraith, 1998).
Cognitive and Metacognitive Aspects of Mathematical Problem Solving: An Emerging Model
by Asmamaw Yimer and Nerida F. Ellerton
What is metacognition?
Metacognition is defined as "cognition about cognition",
or "knowing about knowing." It can take many forms; it
includes knowledge about when and how to use particular
strategies for learning or for problem solving.
Wikipedia
Categories of Cognitive and Metacognitive
Behaviors:
1. Engagement: Initial confrontation and making sense of
the problem.
2. Transformation-Formulation: Transformation of initial
engagements to exploratory and formal plans.
3. Implementation: A monitored acting on plans and
explorations.
4. Evaluation: Passing judgments on the appropriateness
of plans, actions, and solutions to the problem.
5. Internalization: Reflecting on the degree of intimacy
and other qualities of the solution process.
Cognitive and Metacognitive Aspects of Mathematical Problem Solving: An Emerging Model
by Asmamaw Yimer and Nerida F. Ellerton
Problem solving and Metacognition
“Without metacognitive monitoring, students are
less likely to take one of the many paths available
to them, and almost certainly are less likely to
arrive at an elegant mathematical solution.”
Cognitive and Metacognitive Aspects of Mathematical Problem Solving: An Emerging
Model
by Asmamaw Yimer and Nerida F. Ellerton
Problem Solving
Mathematical problem solving is a complex cognitive
activity involving a number of processes and strategies.
Problem solving has two stages:
1. problem representation
2. problem execution
Successful problem solving is not possible without first
representing the problem appropriately.
Appropriate problem representation indicates that the
problem solver has understood the problem and serves
to guide the student toward the solution plan. Students
who have difficulty representing math problems will have
difficulty solving them.
Math Problem Solving for Upper Elementary Students with Disabilities
by Marjorie Montague, PhD
Visualization
A powerful
problem-solving
strategy…
Math Problem Solving for Upper Elementary Students with Disabilities
by Marjorie Montague, PhD
Visualization
Problem Solving
1. READ the problem for
understanding.
2. PARAPHRASE the problem by
putting it into their own words.
3. VISUALIZE or draw a picture or
diagram.
4. HYPOTHESIZE by thinking about
logical solutions.
5. ESTIMATE or predict the answer.
6. COMPUTE.
7. CHECK.
Math Problem Solving for Upper Elementary Students with Disabilities
by Marjorie Montague, PhD
Instructional Procedures
The content of math problem solving
instruction are the cognitive processes
and metacognitive strategies that good
problem solvers use to solve
mathematical problems.
~Marjorie Montague
Math Problem Solving for Upper Elementary Students with Disabilities
by Marjorie Montague, PhD
Problem Solving
Effective instructional
procedures for
teaching math
problem solving!
Math Problem Solving for Upper Elementary Students with Disabilities
by Marjorie Montague, PhD
Instructional Procedures
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Explicit Instruction
Sequencing and Segmenting
Drill-repetition and Practice-review
Directed Questioning and Responses
Control Difficulty or Processing Demands of the Task
Technology
Group Instruction
Peer Involvement
Strategy Cues
Verbal Rehearsal
Process Modeling
Visualization
Role Reversal
Performance Feedback
Distributed Practice
Mastery Learning
Math Problem Solving for Upper Elementary Students with Disabilities
by Marjorie Montague, PhD
Part A
A restaurant makes a special seasoning for all its grilled
vegetables. Here is how the ingredients are mixed:
1/2 of the mixture is salt
1/4 of the mixture is pepper
1/8 of the mixture is garlic powder
1/8 of the mixture is onion powder
When the ingredients are mixed in the same ratio as shown above,
every batch of seasoning tastes the same.
Study the measurements for each batch in the table. Fill in the blanks
so that every batch will taste the same.
The Charles A. Dana Center
at the University of Texas at
Austin and Agile Mind, Inc.
Answers
The Charles A. Dana Center
at the University of Texas at
Austin and Agile Mind, Inc.
Part B
A restaurant makes a special seasoning for all its grilled vegetables.
Here is how the ingredients are mixed:
1/2 of the mixture is salt
1/4 of the mixture is pepper
1/8 of the mixture is garlic powder
1/8 of the mixture is onion powder
The restaurant mixes a 12-cup batch of the mixture every week. How
many cups of each ingredient do they use in the mixture each week?
The Charles A. Dana Center
at the University of Texas at
Austin and Agile Mind, Inc.
Answers
The Charles A. Dana Center
at the University of Texas at
Austin and Agile Mind, Inc.
Interim Assessments
Edusoft Reports
 Performance Band Report (Question Groups)
 Item Analysis per Period
2012-2013 Benchmark Analysis
Performance Band Report
Item Analysis
Edusoft Guides
• Guide to Creating Interim Assessment Reports by SubGroups (All, Ethnicity, Ed Program, and ELL)
• Guide to Performance Band, Item Analysis, and Item
Response Reports
• Edusoft Sub-Group (Demographic) Selection
Item Specifications
Benchmark Clarifications
 Explain how the achievement of the
benchmark will be demonstrated by
students for each specific item type. In
other words, the clarification statements
explain what the student will do when
responding to questions of each type.
Item Specifications
Content Limits
 Define the range of content knowledge and
degree of difficulty that should be assessed in
the items for the benchmark.
 Benchmark content limits are to be used in
conjunction with the General Content Limits
identified for each grade level in the
Specifications. The content limits defined in the
Individual Benchmark Specifications section
may be an expansion or further restriction of
the General Content Limits by Grade Level
specified earlier in the Specifications.
Item Specifications
Item Specifications
• MA.4.A.6.6
QMBA: Grades K – 2 (CCSSM)
 Performance Band Reports
 Item Analysis
Only three questions above 50% correct!
What are the
Instructional
Implications?
Re-teaching MA.4.A.6.6
• Resources
• Instructional Strategies
• Differentiated Instruction
Re-teaching MA.4.A.6.6
GO Math!
Resources
About Mini Bats and How to Use Them
Interim assessments are a formative assessment that informs whether the
student attained an understanding of the standards taught in the quarter.
Fall Interim assesses Quarter 1 concepts.
Winter Interim assesses Quarter 2 concepts.
Mini BATs are mini benchmark assessments that assess whether, after
remediation, a student has gained an understanding of the concepts of a
benchmark he/she showed weakness on the Interim.
Quarter 1 Mini BATs- are used during Quarter 2 based on remediation
needed as per Fall Interim Assessment results.
Quarter 2 Mini BATs- are used during Quarter 3 based on remediation
needed as per Winter Interim Assessment results.
Quarter 3 Mini BATs- can be used either in Quarter 3 or 4.
Quarter 4 Mini BATs- can be used in Quarter 4.
Other Assessment Resources
SAT-10 Dailies
The SAT-10 Dailies is an instructional tool developed to help teachers reinforce
instruction on basic mathematics concepts taught to students, using items that
also reflect the content and process clusters measured by the Stanford
Achievement Test-10.
The daily practice provides opportunities for both multiple-choice and student
constructed responses during the school year leading up to the week of testing.
Suggested Usage:
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Dailies should be used each day of the week, only one practice sheet per
day.
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Dailies are recommended to be used during the transition time into the
mathematics instructional block; preferably during the first five minutes of
class. During such time, students answer the questions in 3 minutes or less
and the teacher briefly reviews responses.
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Effort grades may be assigned. Teachers should not assign a grade to
Dailies since they do not represent the students’ understanding of the
concept(s) of the day’s lesson.
Will CCSSM Matter in
Ten Years?
Will CCSSM Matter in Ten Years?
 During which part of the day could we work as a grade-level team
to discuss the Standards for Mathematical Practice and develop
our understanding?
 How can we work collaboratively to implement the Standards for
Mathematical Practice into our lesson designs? Which initial
tasks, activities, or formative assessments can we use?
 What student behaviors might define the Standards for
Mathematical Practice?
 Given a specific CCSSM Content Standard. Examine the meaning
of the Standard, and analyze and interpret new or unfamiliar
content. How can we deepen our understanding of the math?
NCTM
Vol. 19, No. 2 | teaching children mathematics • September 2012
MACC.1.OA.4.7:
Understand the meaning of the equal
sign, and determine if equations
involving addition and subtraction are
true or false. For example, which of the
following equations are true and which
are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5,
4 + 1 = 5 + 2.
MACC.1.OA.4.7:
Mathematically proficient students communicate precisely by
engaging in discussion about their reasoning using appropriate
mathematical language. The terms students should learn to use
with increasing precision with this cluster are: equations, equal,
the same amount/quantity as, true, false
Unpacking
What do these standards mean a child will
know and be able to do?
Unpacking
In order to determine whether an equation is true or false, First Grade
students must first understand the meaning of the equal sign. This is
developed as students in Kindergarten and First Grade solve
numerous joining and separating situations with mathematical tools,
rather than symbols. Once the concepts of joining, separating, and
“the same amount/quantity as” are developed concretely, First
Graders are ready to connect these experiences to the corresponding
symbols (+, -, =). Thus, students learn that the equal sign does not
mean “the answer comes next”, but that the symbol signifies an
equivalent relationship that the left side ‘has the same value as’ the
right side of the equation.
Unpacking
When students understand that an equation needs to “balance”, with
equal quantities on both sides of the equal sign, they understand
various representations of equations, such as:
• an operation on the left side of the equal sign and the answer on the
right side (5 + 8 = 13)
• an operation on the right side of the equal sign and the answer on
the left side (13 = 5 + 8)
• numbers on both sides of the equal sign (6 = 6)
• operations on both sides of the equal sign (5 + 2 = 4 + 3).
Once students understand the meaning of the equal sign, they are
able to determine if an equation is true (9 = 9) or false (9 = 8).
MACC.3.NF.1.1:
Understand a fraction 1/b as the
quantity formed by 1 part when a
whole is partitioned into b equal parts;
understand a fraction a/b as the
quantity formed by a parts of size 1/b.
MACC.3.NF.1.1:
Mathematically proficient students communicate precisely by engaging
in discussion about their reasoning using appropriate mathematical
language. The terms students should learn to use with increasing
precision with this cluster are: partition(ed), equal parts, fraction,
equal distance ( intervals), equivalent, equivalence, reasonable,
denominator, numerator, comparison, compare, ‹, ›, = , justify
Unpacking
What do these standards mean a child will
know and be able to do?
Unpacking
This standard refers to the sharing of a whole being partitioned. Fraction
models in third grade include only area (parts of a whole) models (circles,
rectangles, squares) and number lines. Set models (parts of a group) are
not addressed in Third Grade.
In 3.NF.1.1 students start with unit fractions (fractions with numerator 1),
which are formed by partitioning a whole into equal parts and reasoning
about one part of the whole, e.g., if a whole is partitioned into 4 equal
parts then each part is ¼ of the whole, and 4 copies of that part make the
whole.
Next, students build fractions from unit fractions, seeing the numerator 3
of ¾ as saying that ¾ is the quantity you get by putting 3 of the ¼ ’s
together.
There is no need to introduce “improper fractions" initially.
Unpacking
(Progressions for the CCSSM; Number and Operation – Fractions, CCSS Writing Team, August 2011, page 2)
Unpacking
(Progressions for the CCSSM; Number and Operation – Fractions, CCSS Writing Team, August 2011, page 3)
MACC.5.NF.2.4a:
Interpret the product (a/b) × q as a parts
of a partition of q into b equal parts;
equivalently, as the result of a sequence
of operations a × q ÷ b. For example, use
a visual fraction model to show (2/3) × 4 =
8/3, and create a story context for this
equation. Do the same with (2/3) × (4/5) =
8/15. (In general, (a/b) × (c/d) = ac/bd.)
Coaching Cycle
Instructional Coaching
 Instructional coaches are onsite professional developers who teach
educators how to use proven teaching methods.
 They employ a variety of professional development procedures to
foster widespread, high-quality implementation of interventions,
providing “on-the-job learning.”
 Instructional coaches take a partnership approach, and thus they
respect teachers’ professionalism and focus their efforts on
conversations that lead to creative, practical application of
research-based practices.
 Instructional coaches see themselves as equal partners with
teachers in the complex and richly rewarding work of teaching
students.
 Instructional coaches work in partnerships to accelerate teachers’
professional learning through mutually enriching, healthy
relationships.
 Instructional coaches are colleagues, friends, and confidants who
listen with care and share valuable information with teachers at the
time when teachers most need it.
Kansas Coaching Project
Maria Teresa Diaz-Gonzalez, District Instructional Supervisor
mtdiaz@dadeschools.net
305-995-2763
Maria Campitelli, District Curriculum Support Specialist
mcampitelli@dadeschools.net
305-995-2927
Isis Casares, District Curriculum Support Specialist
isis17casares@dadeschools.net
305-995-7280