Northwest Georgia RESA Summer Mathematics Institute

1

Northwest Georgia RESA Summer Mathematics
Institute
2

• Northwest Georgia RESA
• Summer Mathematics Institute
• Coosa Valley Technical College
• Rome, Georgia
• June 25, 2009
• Dexter Mills, Executive Director
• Karen Faircloth, Director of School
• Improvement Professional
• Learning

3
Contact Information
Danny Lowrance, Math Specialist W.L. Swain
Elementary 2505 Rome Rd SW Plainville, GA
30733 706-629-0141 dlowrance_at_gcbe.org
Northwest Georgia RESA Summer Mathematics
Institute
4
Facilitators for each Curriculum Band Claire
Pierce, Math I and II Independent Consultant
former DOE Math Program Manager Linda Segars,
Math I and II School Improvement Specialist for
Metro RESA Terry Haney, Grades 6-8 Math
Coordinator for Northwest Georgia RESA Danny
Lowrance, Grades 3-5 Math Specialist at W.L.
Swain Elementary School in Gordon County
Northwest Georgia RESA Summer Mathematics
Institute
5
Purpose The purpose of the Northwest Georgia
RESA Summer Mathematics Institute is to provide
ongoing professional learning experiences for
district teams in mathematics.  Each team should
consist of at least one representative from each
of the following curriculum bands  3-5, 6-8, and
Math I II.  Members of the teams may be
teachers and/or academic coaches, along with a
building-level and system-level
administrator.  Each representative will then
attend a session based on his or her appropriate
curriculum band.  During this extended session,
instructors for all curriculum bands will address
one specific content strand (algebra, geometry,
numbers and operations, data analysis) by
facilitating work on performance tasks and
pedagogy.    Other topics may include data-driven
teaching and learning, characteristics of the
standards-based classroom,  and ACTION planning
for mathematics. 
Northwest Georgia RESA Summer Mathematics
Institute
6
Content Topic Algebra Pedagogy Topic
Writing and Using Commentary Effectively
Northwest Georgia RESA Summer Mathematics
Institute
7
Essential Questions How do I effectively
integrate the Algebra standards into the
mathematics curriculum? How can teachers and
students write and use commentary effectively?
Why should I post the standards?
Northwest Georgia RESA Summer Mathematics
Institute
8
The Algebra Curriculum Ladder

• K Students will identify, create, extend, and
  transfer patterns from one representation to
  another.
•  
• 1 Students will build number patterns using
  various concrete representations.
• 2 Students will represent and interpret
  quantities and relationships using mathematical
  expressions and symbols (, lt, gt).
•  
•  

Northwest Georgia RESA Summer Mathematics
Institute
9
The Algebra Curriculum Ladder

• 3 Students will use mathematical expressions to
  represent relationships between quantities and
  interpret given expressions. This will include
  describing and extending patterns, describing and
  explaining a relationship represented by a
  formula, and using a symbol to represent an
  unknown and finding the value of the unknown.
• 4 Students will represent and interpret
  mathematical relationships in quantitative
  expressions. They will understand and apply
  patterns and rules, represent unknowns using
  symbols, and write and evaluate mathematical
  expressions.
•  
• 5 Students will represent and interpret the
  relationship between quantities algebraically.
  They will use variables for unknown quantities in
  algebraic expressions and investigate simple
  algebraic expressions. They will also determine
  that a formula will be reliable regardless of the
  type of number substituted for the variable.
•  

Northwest Georgia RESA Summer Mathematics
Institute
10
The Algebra Curriculum Ladder

• 6 Students will understand the concept of ratio
  and use it to represent quantitative
  relationships. They will consider relationships
  between varying quantities which includes
  analyzing and describing patterns from rules,
  tables, and graphs using manipulatives and
  drawings to solve problems involving proportional
  relationships describing and graphing
  proportional relationships using y kx using
  proportional reasoning to solve problems.
• Students will also evaluate algebraic
  expressions, including those with exponents, and
  solve simple one-step equations using each of the
  four basic operations.
• 7 Students will represent and evaluate
  quantities using algebraic expressions. This
  includes translating verbal phrases to algebraic
  expressions simplifying and evaluating
  expressions using commutative, associative, and
  distributive properties adding and subtracting
  linear expressions. Students will also
  understand and apply linear equations in one
  variabledefine a variable, write an equation,
  solve the equation, and interpret the solution.
  Students will be able to use addition and
  multiplication properties of equality to solve
  one- and two-step linear equations.
• Students will understand relationships between
  two variables. They will plot points on the
  coordinate plane represent, describe, and
  analyze relations from tables, graphs, and
  formulas describe how change in one variable
  affects the other variable describe patterns in
  the graphs of proportional relationships, both
  direct (y kx) and
• inverse (y k/x).
•  
•  
•  

Northwest Georgia RESA Summer Mathematics
Institute
11
The Algebra Curriculum Ladder

• 8 Students will use algebra to represent,
  analyze, and solve problems. This includes
  representing a situation using expressions or
  equations in one variable simplifying algebraic
  expressions solving algebraic equations in one
  variable (including absolute value) solving
  equations involving several variables for one
  variable in terms of the others interpreting
  solutions in problem contexts. Students will
  also understand and graph inequalities in one
  variable.
• Students will understand relations and linear
  functions. They will recognize a relation as a
  correspondence between varying quantities
  recognize a function as a correspondence between
  inputs and outputs where the output for each
  input must be unique use tables to describe
  sequences recursively. Students will graph and
  analyze graphs of linear equations and
  inequalities (interpreting slope as rate of
  change determining the meaning of slope in a
  given situation graphing equations of the form y
  mx b and ax by c solving problems
  involving linear relationships).
• Students will understand systems of linear
  equations and inequalities and use them
• to solve problems. This includes writing
  systems for a context, solving the system,
• and interpreting the solution in context.
•  
•  

Northwest Georgia RESA Summer Mathematics
Institute
12
The Algebra Curriculum Ladder

• Math I Students will explore and interpret
  characteristics of functions, using graphs,
  tables, and simple algebraic techniques. They
  will also graph transformations (including
  vertical shifts, stretches and shrinks, and
  reflections across the x and y axes) investigate
  and explain characteristics of functions (domain,
  range, zeros, intercepts, intervals of increase
  and decrease, maximum and minimum values, and end
  behavior) determine graphically and
  algebraically whether a function has symmetry and
  whether it is even, odd, or neither.
• Students will factor expressions by GCF,
  grouping, trial and error, and special products.
  They will simplify and operate with radical
  expressions, polynomials, and rational
  expressions. They will also solve simple
  quadratic and radical equations.
•  

Northwest Georgia RESA Summer Mathematics
Institute
13
A task
Northwest Georgia RESA Summer Mathematics
Institute
14
Balance Scale Algebra

• Opening Tell me what you know about balance
  scales .

• M4A1. Students will represent and interpret
  mathematical relationships in quantitative
  expressions.
• Understand and apply patterns and rules to
  describe relationships and solve problems.
• Represent unknowns using symbols, such as ? and
  ?.
• Write and evaluate mathematical expressions using
  symbols and different values.

Northwest Georgia RESA Summer Mathematics
Institute
15
Process Standards

• M4P1. Students will solve problems (using
  appropriate technology).
• Build new mathematical knowledge through problem
  solving.
• Solve problems that arise in mathematics and in
  other contexts.
• Apply and adapt a variety of appropriate
  strategies to solve problems.
• Monitor and reflect on the process of
  mathematical problem solving.
• M4P2. Students will reason and evaluate
  mathematical arguments.
• Recognize reasoning and proof as fundamental
  aspects of mathematics.
• Make and investigate mathematical conjectures.
• Develop and evaluate mathematical arguments and
  proofs.
• Select and use various types of reasoning and
  methods of proof.
• M4P3. Students will communicate mathematically.
• Organize and consolidate their mathematical
  thinking through communication.
• Communicate their mathematical thinking
  coherently and clearly to peers, teachers, and
  others.
• Analyze and evaluate the mathematical thinking
  and strategies of others.
• Use the language of mathematics to express
  mathematical ideas precisely.

Northwest Georgia RESA Summer Mathematics
Institute
16
Process Standards

• M4P4. Students will make connections among
  mathematical ideas and to other disciplines.
• a. Recognize and use connections among
  mathematical ideas.
• Understand how mathematical ideas interconnect
  and build on one
• another to produce a coherent whole.
• c. Recognize and apply mathematics in contexts
  outside of mathematics.
• M4P5. Students will represent mathematics in
  multiple ways.
• Create and use representations to organize,
  record, and communicate
• mathematical ideas.
• Select, apply, and translate among
  mathematical representations to solve
• problems.
• Use representations to model and interpret
  physical, social, and
• mathematical phenomena.

Northwest Georgia RESA Summer Mathematics
Institute
17
Work Period

• Use symbols for the unknown weights to write a
  number sentence that represents the balanced
  scale. Then, find the missing weights.

Note Objects that have the same size and shape
also have the same weight.
Northwest Georgia RESA Summer Mathematics
Institute
18
Closing

• Be prepared to justify your answer.
• How does your work meet the standard?
• Did you apply a pattern or a rule to describe a
  relationship between two unknowns?

Northwest Georgia RESA Summer Mathematics
Institute
19
How can questions be categorized? Questions can
be coded by DOK (Depth of Knowledge) Levels.
Northwest Georgia RESA Summer Mathematics
Institute
20
Depth of Knowledge

• Powerful tool to determine the relationship
  between Instruction and Assessment in terms of
  cognitive demand
• Initially developed in collaboration with CCSSO
  (Council of Chief State School Officers)
• By late 2005, used in over 17 states and other
  countries More have joined since then.
• Aligns state tests, standards, curriculum

Northwest Georgia RESA Summer Mathematics
Institute
21
Table Talk

• Read the descriptors for each level of
  mathematics. Annotate/highlight key terms and
  ideas.
• Discuss the progressions of cognitive complexity
  evidenced in the levels.
• How might this tool be used to impact classroom
  assessment and instruction?
• Be prepared to briefly share with the whole group.

22
More About Level 4

• DOK 4 requires high cognitive demand and is very
  complex. Students are expected to make
  connections relate ideas within the content or
  among content areasand have to select or devise
  one approach among many alternatives on how the
  situation can be solved.
• Due to the complexity of cognitive demand, DOK 4
  often requires an extended period of time.

23
Depth of Knowledge Levels
Given the equation below, solve for the ?.

• 3 ? 9

Level 1 This question is basically a
computation problem. This is a missing addend
question the requires a one-step process.
Northwest Georgia RESA Summer Mathematics
Institute
24
Depth of Knowledge Levels
Extend the following pattern.
21, 25, 29, 33, __
Level 2 This problem is asking the student to
find the numeric pattern and extend the pattern.
This item requires students to make some
decisions as to how to approach the problem. This
pattern is increasing and by what numeric amount.
Northwest Georgia RESA Summer Mathematics
Institute
25
Depth of Knowledge Levels

• A pattern was used to determine the number of
  black tiles and the number of white tiles in each
  figure below.

If the pattern continues, how many black tiles
will be in figure 5? How do you know?
Northwest Georgia RESA Summer Mathematics
Institute
26
Depth of Knowledge Levels

• Level 3 For this item, students identify a
  geometric pattern and the numerical pattern
  associated with it. Solution is 5 justifications
  will vary but may include an explanation and a
  sketch. Requires reasoning, generalize from given
  facts, applies geometric pattern and number
  pattern, and requires to draw conclusions.

Source Adapted from Massachusetts
Comprehensive Assessment System, Mathematics,
Grade 4 (released items 2002, item 26)
27
Based on the rigor of our Georgia Performance
Standards Mathematics Curriculum, 55 of the
questions on the CRCT/EOCT must be at DOK 2 or
above.
28
Task 2 Creating a CRCT Poster
Northwest Georgia RESA Summer Mathematics
Institute
29
Opening

• Answer the following questions in complete
  sentences. You have 5 minutes.
• What are some strategies that you use when taking
  a test like the CRCT?
• How do you approach a problem when you are
  uncertain of the correct answer.
• Most tests like the CRCT have one correct answer
  and three incorrect answers for each item. How
  do you think test writers get their incorrect
  answer choices?

Northwest Georgia RESA Summer Mathematics
Institute
30
Opening Guidelines for creating your poster

• Your poster will include 4 sample problems based
  on the algebra standards.
• Choose problems in which you can accurately show
  how to deal with common mistakes and
  misconceptions.
• One problem should be DOK Level 1 one problem
  should be DOK Level 2 two problems should be
  DOK Level 3.
• Give a brief description of the standards that
  each problem addresses.
• Show your work and give and explanation for the
  correct answer.
• For each distracter, give a detailed description
  of why the
• answer is incorrect.

Northwest Georgia RESA Summer Mathematics
Institute
31
Opening Looking at a sample poster
Northwest Georgia RESA Summer Mathematics
Institute
32
Work Period

• Begin working with your group to create and solve
  problems for your poster.
• You should work through each problem on notebook
  paper and make certain that your group members
  agree on the answer before writing it on the
  poster.
• Discuss information about common mistakes and
  misconceptions with your group members.

Northwest Georgia RESA Summer Mathematics
Institute
33
Closing

• Gallery Walk to view Posters in Progress
• As you view each of the posters, give feedback on
  Post-It notes.
• Be prepared to discuss the process of creating
  the posters and what you noticed about the work
  of other groups.
• How can we link your work and the work of others
  to the standards?

Northwest Georgia RESA Summer Mathematics
Institute
34
(No Transcript)
35
(No Transcript)
36
Writing and Using Commentary

• Teacher Commentary should
• Use the language of the standards.
• Provide descriptive and specific comments related
  to the learning goals.
• Include honest and constructive guidance about
  steps to take or strategies to try next.
• Celebrate success and/or progress toward the
  learning goals.

Northwest Georgia RESA Summer Mathematics
Institute
37
The parts of a performance standard

• STANDARDS AND ELEMENTS ADDRESSED
• THE TASK
• STUDENT WORK
• COMMENTARY
• Teacher-written
• Student-written

Northwest Georgia RESA Summer Mathematics
Institute
38
A feedback system depends on
Vision
and
Reality.
Feedback exists between the two. Grant Wiggins
Northwest Georgia RESA Summer Mathematics
Institute
39
Good commentary provides evidence that shows
the level of proficiency of the work as
measured against the standard, cites
specific strategies used, and uses the language
of the standards.
Why are these things important?
Northwest Georgia RESA Summer Mathematics
Institute
40
Feedback is specific, and descriptive
information you can use.
Feedback is not advice, praise,
or blame. Grant Wiggins
Northwest Georgia RESA Summer Mathematics
Institute
41
Time to Write
Using the standards and the poster gallery
provided, write commentary you could use as a
teaching tool for your fellow students.
Northwest Georgia RESA Summer Mathematics
Institute
42
Lets go back and look at some commentary fellow
students wrote about your work. What did you
notice about the samples of commentary?
Northwest Georgia RESA Summer Mathematics
Institute
43
a sculptor chips away at a block of marble for
days and daysand a horse or a man emerges.
But an ordinary person could chip away at
the same block of marble for months and nothing
at all might emerge. The difference is in the
quality of attention. Its the intention
The difference between assessment that is
busywork and assessment that reflects the essence
of our teaching is what we and our students make
of what we collect (Calkins, p. 325).
Northwest Georgia RESA Summer Mathematics
Institute
44
Vertical Design Team Activity

• In your group align the questions with each
  corresponding grade, standard, and element.
• Discuss in your group why you made your decision.
  
• Write down specifically why your group aligned
  each specific question with which standard.

Northwest Georgia RESA Summer Mathematics
Institute
45
A task
Northwest Georgia RESA Summer Mathematics
Institute
46

• GEORGIA PERFORMANCE STANDARDS
• ALGEBRA
• Students will represent and investigate
  mathematical expressions algebraically by using
  variables.
• M5A1. Students will represent and investigate
  mathematical expressions algebraically by using
  variables.
• Use variables such as n or x, for unknown
  quantities algebraically.
• Investigate simple algebraic expressions by
  substituting numbers for the unknown.
• Determine that a formula will be reliable
  regardless of the type of number (whole number or
  decimals) substituted for the variable.

Northwest Georgia RESA Summer Mathematics
Institute
47
GEORGIA PERFORMANCE STANDARDS (PROCESS
STANDARDS) M5P3. Students will communicate
mathematically. a. Organize and consolidate
their mathematical thinking through
communication. b. Communicate their
mathematical thinking coherently and
clearly to peers, teachers, and others. c.
Analyze and evaluate the mathematical thinking
and strategies of others. d. Use the
language of mathematics to express mathematical
ideas precisely. M5P5. Students will
represent mathematics in multiple ways. a.
Create and use representations to organize,
record, and communicate mathematical
ideas.
Northwest Georgia RESA Summer Mathematics
Institute
48
Algebraic Expression Card Sort

• Lets investigate your skill to represent and
  interpret the relationships between quantities
  algebraically by attempting to put this puzzle
  back together again.
• Created by Michelle Parker, Gordon County Schools

Northwest Georgia RESA Summer Mathematics
Institute
49
Opening

• Take five minutes to write down as many phrases
  as you can that correlate to the four basic
  computation operations. Beware some verbal
  phrases are tricky.

Northwest Georgia RESA Summer Mathematics
Institute
50
Work Period

• Put the puzzle back together again, but remember
  each side must match any side of another card
  that touches that side.
• For example if a side of a card says the sum of
  some number and 5 the card that is next to or
  shares that side must match with x 5.
• If you are having difficulty remember to draw a
  picture or use a number line for help.

Northwest Georgia RESA Summer Mathematics
Institute
51
Closing

• Name the algebraic expressions that were hard to
  match. Why were they hard?
• Did drawing a picture or using a number line help
  you make your decision on which cards to place
  next to each other?
• How does your work meet the standard?

Northwest Georgia RESA Summer Mathematics
Institute
52

• GEORGIA PERFORMANCE STANDARDS
• ALGEBRA
• Students will represent and investigate
  mathematical expressions algebraically by using
  variables.
• M5A1. Students will represent and investigate
  mathematical expressions algebraically by using
  variables.
• Use variables such as n or x, for unknown
  quantities algebraically.
• Investigate simple algebraic expressions by
  substituting numbers for the unknown.
• Determine that a formula will be reliable
  regardless of the type of number (whole number or
  decimals) substituted for the variable.

Northwest Georgia RESA Summer Mathematics
Institute
53
GEORGIA PERFORMANCE STANDARDS (PROCESS
STANDARDS) M5P3. Students will communicate
mathematically. a. Organize and consolidate
their mathematical thinking through
communication. b. Communicate their
mathematical thinking coherently and
clearly to peers, teachers, and others. c.
Analyze and evaluate the mathematical thinking
and strategies of others. d. Use the
language of mathematics to express mathematical
ideas precisely. M5P5. Students will
represent mathematics in multiple ways. a.
Create and use representations to organize,
record, and communicate mathematical
ideas.
Northwest Georgia RESA Summer Mathematics
Institute
54
CRCT Data Interpretation Activity

Northwest Georgia RESA Summer Mathematics
Institute
55
Perception vs. Reality
Northwest Georgia RESA Summer Mathematics
Institute
56
Common Perceptions Openings, work periods, and
closings must meet exact time constraints.
While there are time suggestions for each
portion of the instructional framework, times
will vary depending on the type of lesson and
the content. Every concept must be completely
discovered by students. Discovery-based
lessons are highly encouraged as often as
possible however, time does not permit every
lesson to be completely based on discovery.
Northwest Georgia RESA Summer Mathematics
Institute
57
Common Perceptions Skills lessons are never
appropriate. Skills are a crucial part of our
mathematics instruction. Skills lessons should
be embedded within tasks as often as possible.
When they are taught in isolation, skills should
be brought back into a context as soon as
possible. Direct instruction is never
appropriate. Some information will need to be
presented in the form of direct instruction,
with lecture and note taking. Think of this
time as a DIALOGUE as opposed to a
MONOLOGUE.
Northwest Georgia RESA Summer Mathematics
Institute
58
Common Perceptions All work must be done in
pairs or in groups. The standards-based
classroom should incorporate a mix of group
work, partner work, and individual
accountability. Closings must always include
formal student presentations.
While student presentations are one of the most
effective methods of solidifying student
learning, not every lesson lends itself to this
type of closing. Sometimes a whole group
discussion with strategic questioning is just
as effective.
Northwest Georgia RESA Summer Mathematics
Institute
59
Common Perceptions Every student must play a
major role in the closing every day. Our goal
should be to involve as many students as possible
each day (in meaningful ways). Using the status
of the class sheet allows teachers to make note
of students who either make formal presentations
or who contribute to the class discussions
through meaningful questions and comments. For
example, a closing may involve 1-4 students
giving formal presentations, with the remainder
of the class giving feedback and asking
questions.
Northwest Georgia RESA Summer Mathematics
Institute
60
Common Perceptions Commentary should be
lengthy. Commentary can be of varying lengths,
depending on the purpose and the scope of the
work. The length also depends on the number of
standards being addressed. Commentary should
always be written for every student on a
particular task or assignment.
While our goal should be to have multiple pieces
of commentary for each student over the course
of the year to show growth, it is not
necessary to write commentary for each student
on every task!
Northwest Georgia RESA Summer Mathematics
Institute
61
Common Perceptions All commentary should be
written by the teacher. The ultimate goal with
commentary is to give specific ways that students
have met or exceeded the standard, or next
steps to use in order to make the work better.
It should also be our goal to teach students how
to evaluate their own work. Consequently, we
should begin to train our students how to write
commentary for their work on the work of
others. Commentary is mainly used for student
work displays. It is important
that student work and commentary be displayed but
only if it is being used as a teaching tool.
Commentary may be public or private. Some
commentary may only be used by the teacher and
an individual student. Some of this commentary
may be verbal. Ultimately, it is a tool to
improve student achievement by giving students
a true understanding of how their work stacks
up with respect to the standards.
Northwest Georgia RESA Summer Mathematics
Institute
62
Reflection
63
Reflection

• Take out a sheet of paper and write a reflection
  on the tasks and activities from today. Your
  ticket out the door is to turn in your individual
  reflection before you leave.
• Remember to make sure that you have read Inside
  the Black Box before tomorrow.

Northwest Georgia RESA Summer Mathematics
Institute
64
Essential Questions How do I effectively
integrate the Algebra standards into the
mathematics curriculum? How can teachers and
students write and use commentary effectively?
Why should I post the standards?
Northwest Georgia RESA Summer Mathematics
Institute
65
Questions, Comments, and Concerns

Northwest Georgia RESA Summer Mathematics
Institute
66
Contact Information
Danny Lowrance, Math Specialist W.L. Swain
Elementary 2505 Rome Rd SW Plainville, GA
30733 706-629-0141 dlowrance_at_gcbe.org
Northwest Georgia RESA Summer Mathematics
Institute