what does concept-based curriculum look like in the math...
TRANSCRIPT
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
MODULE 1
What Does Concept-Based Curriculum Look Like in the Math Classroom?
Main Text: Chapter 1. Why Is It Important for My Students to Learn Conceptually? What are the facts, processes, and concepts in mathematics? Mathematics is an inherently conceptual language, so both the Structure of Knowledge and the Structure of Process apply to mathematics. What are concept-based curricula?
Concept-based is not a curricular program, but rather a design model that can overlay any curricula to enhance conceptual depth and understanding. Concept-based encourages the explicit delineation of what students must Know (factually), Understand (conceptually), and be able to Do (skill) within the particular school’s curricula. The traditional objectives-based model of curriculum design failed to clearly differentiate these facets (KUDs). (Erickson & Lanning, 2014)
In this module you will find the following resources:
• Examples of the Structure of Knowledge and the Structure of Process using linear functions
• A summary of the Structure of Knowledge and the Structure of Process
• A blank template of the Structure of Knowledge for you to use when you think about your own example
• An example of a traditional activity versus an activity that promotes synergistic thinking
• The synergistic thinking task template
• An outline of the five tenets of concept-based math curriculum and instruction
• Additional sample conceptual lenses relevant for mathematics
• Discussion questions for Module 1
• An opportunity for reflection
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M1.1: THE STRUCTURE OF KNOWLEDGE FOR LINEAR FUNCTIONS
Structure of Knowledge
Erickson, © 1995
Topic
Memorized
vocabulary,
definitions,
formulae,
mathematical
symbols
ConceptsConcepts
F
PRINCIPLE
GENERALIZATION
THEORY
A
C
T
S
F
A
C
T
S
F
A
C
T
S
F
A
C
T
S
F
A
C
T
S
F
A
C
T
S
y = mx + c
Gradient = riserun
Linear Functions
• Transformation
• Translation
• Constant rate of change
• Linear function
• Initial value
Linear functions showrelationships that exhibit
a constant rate of change.
Transformation: translation of a linear function changes
the initial value.
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M1.2: THE STRUCTURE OF PROCESS FOR LINEAR FUNCTIONS
Processes
–––––––––
Strategies
Skills
PRINCIPLEGENERALIZATION
THEORYMathematicians create
different representations;table of values, algebraic,geometrical to compareand analyze equivalent
linear functions.
Linear Functions
Table of valuesAlgebraicGeometricalEquivalentLinear functions
ConceptsConcepts
Structure of ProcessLanning, © 2012
Creating Representations
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M1.3: THE STRUCTURE OF KNOWLEDGE
Theory: A supposition or set of conceptual
ideas used to explain a phenomenon or
practice.
Principle: A generalization that rises to the
level of a law or theorem. Does not use
qualifiers.
Generalization: Two or more concepts stated
in a sentence of relationship that transfers
through time across cultures and across
situations. Supported by the facts.
Concepts: Mental constructs for a group of
objects or ideas with common attributes.
One or two words or a short phrase;
timeless, universal, abstract to different
degrees. Concepts transfer.
Topics: Specific; Locked in time, place, or
situation. Do not transfer.*
Facts: Provide support for principles and
generalizations. Locked in time, place, or
situation. Do not transfer.
*Because mathematics is a conceptual
language, the topics are actually broader
concepts that break down into
micro-concepts at the next level.
Topic
ConceptsConcepts
F
PRINCIPLE
GENERALIZATION
THEORY
A
C
T
S
F
A
C
T
S
F
A
C
T
S
F
A
C
T
S
F
A
C
T
S
F
A
C
T
S
H. Lynn Erickson © 1995
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M1.4: THE STRUCTURE OF PROCESS
Theory: A supposition or set of conceptualideas used to explain a phenomenon or practice.
Principle: A generalization that rises to the levelof a law or theorem. Does not use qualifiers.
Generalization: Two or more concepts statedin a sentence of relationship that transfers throughtime, across cultures, and across situations.
Concepts: Mental constructs for a group ofobjects or ideas with common attributes. Oneor two words or a short phrase; timeless,universal, abstract to different degrees. Conceptstransfer.
Processes: Actions that produce results. Aprocess is continuous and moves through stagesduring which input may transform the way aprocess flows.
Strategies: A systematic plan to help learnersadapt and monitor to improve learning. Strategiesare complex. Many skills are situated within astrategy.
Skills: Smaller operations or actions embedded instrategies that enable strategies to work. Skillsunderpin more complex strategies.
Processes
–––––––––
Strategies
Skills
PRINCIPLEGENERALIZATION
THEORY
ConceptsConcepts
Structure of ProcessLanning, © 2012
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M1.5: TEMPLATE FOR THE STRUCTURE OF KNOWLEDGE
The student understands that….
*Because mathematics is a conceptuallanguage, the topics are actually broaderconcepts that break down intomicro-concepts at the next level.
Topic*
ConceptsConcepts
F
PRINCIPLEGENERALIZATION
THEORY
A
C
T
S
F
A
C
T
S
F
A
C
T
S
F
A
C
T
S
F
A
C
T
S
F
A
C
T
S
H. Lynn Erickson © 1995
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M1.6: EXAMPLE OF SYNERGISTIC THINKING WHEN PLANNING TASKS
Synergistic thinking means that . . .
Traditional Activity:
Adapted to facilitate synergistic thinking:
Use a table of values to plot the following: y = 3x + 4; y = 3x + 5; y = x − 2; y = x + 1
Coordinates Game (see Chapter 4, Figure 4.8 for full details)
Ask student in position bottom left to be the coordinate (0,0). Now ask all students to work out their position and write their coordinates. Do not tell students what their coordinate is.
Ask students to stand up if they are x = 0, x = 1, then x = 2, then x = 3 and x = 4.
Next, go through y = 0, 1, 2, 3, 4.
Now show the card y = x and ask students to stand up if they follow this rule.
Next show the cards y = 2x, 3x, and 4x.
Now show y = x + 1 and ask students to stand up if they follow this rule. Then y = x + 2; y = x + 3.
Option: Ask students to change positions, and ask them to work their coordinates and repeat the same linear equations.
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M1.7: TEMPLATE FOR SYNERGISTIC THINKING WHEN PLANNING TASKS
Synergistic thinking means that . . .
Traditional Activity:
Adapted to facilitate synergistic thinking:
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M1.8: THE TENETS OF CONCEPT-BASED CURRICULUM AND INSTRUCTION
Explain each of the tenets of concept-based curriculum and instruction and give examples to illustrate your explanations.
1. Synergistic thinking
2. The conceptual lens
3. Inductive teaching
4. Guiding questions
5. Three-dimensional curriculum and design model
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M1.9: THE CONCEPTUAL LENS IN MATHEMATICS
A conceptual lens is a broad, integrating concept that focuses a unit of work to allow students to process the factual information, for example, the study of patterns when looking at different types of sequences, such as square and triangular numbers. Sample Conceptual Lenses for Mathematics
identity form errors randomness
change logic parts and wholes balance
systems relationships variation representation
transformations uncertainty precision order
patterns space structure/function quantity
accumulation development abstraction intuition
Adapted from Lynn Erickson (2007).
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Discussion Questions for Module 1
1. Create your own math examples for the Structure of Knowledge and the Structure of Process.
2. What are some examples of inductive teaching?
3. How do you use the conceptual lens to focus a unit of work? Choose one from the list in Figure M1.9 and provide an example of how you would use it.
4. Can you think of more conceptual lenses for math?
5. How do you promote synergistic thinking in your classroom?
6. How do you apply the philosophy and design model of concept-based curriculum to your own curriculum?
An Opportunity for Reflection
I used to think . . .
Now I think . . .
I understand that . . .
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
MODULE 2
How Do Teachers Write Quality Generalizations? What Do We Want Our Students to Understand From Their Program of Study?
Main Text: Chapter 3. What Are Generalizations in Mathematics?Generalizations give us explanations of why and what we want our students to comprehend in terms of the relationship between two or more concepts. Also known in education circles as enduring understandings, essential understandings, or big ideas of the unit, they summarize what we would like our students to take away from their unit of study. It is important when crafting generalizations to not merely write definitions or objectives. Effective math-ematics learning arises out of guiding students to particular principles or generalizations through inductive inquiry. Each unit of work contains six to eight generalizations.
This module contains the following resources:
• A table of verbs to avoid when crafting generalizations
• An exemplar of the scaffolding process when crafting generalizations
• A template for scaffolding generalizations
• A table of sample verbs for scaffolding level 2 and 3 generalizations
• A checklist for crafting generalizations
• Discussion questions for Module 2
• An opportunity for reflection
When crafting generalizations, remember these guidelines:
• Don’t just write definitions of terms
• Use your suggested sample verbs list
• The goal is to help students deeply understand as a result of studying the unit
• No pronouns, past tense verbs, proper nouns, or “no no” verbs such as is, are, affects, impacts, influences
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
• Ask yourself the following questions:
{{ Do the ideas grow in sophistication?
{{ Do the ideas become clearer at Level 2 because they are more specific (use more specific concepts)?
{{ Did I answer the question at each level? (Level 2 How or why? Level 3 So what is the significance?)
{{ Are the verbs active and present tense?
{{ Are the ideas based in fact? (Are they true?)
{{ Are the ideas important?
{{ Are the ideas developmentally appropriate?
• Start off with this stem: Students will understand that . . .
The generalization equation:
Generalization = concept 1 + strong verb + concept 2
(Generally you could have more than two concepts, but the concepts need to be compatible when scaffolding.)
Lynn Erickson (2007) states that avoiding certain “no no verbs” will strengthen the expression of a generalization (see Chapter 3). Take a look at the following Level 1 generalizations:
The gradients of two parallel lines are equal.
The area of a circle is found by multiplying the radius squared by π.
Speed is affected by distance and time.
Multiplying two negative numbers equals a positive.
Because they are constructed using “no no verbs,” the above statements do not reflect an understanding of why or how these concepts are significant.
Below is a table of “no no verbs” adapted from the work of Lynn Erickson.
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M2.1: NO NO VERBS WHEN CRAFTING GENERALIZATIONS
is are have influences
affects impacts implies equals
Adapted from Lynn Erickson (2007)
During the process of forming your generalizations (enduring understandings) for any unit, start off with the sen-tence “After this program of study, students will understand that . . .” Often initial ideas result in a Level 1 general-ization that includes “no no” verbs from Figure M2.1. To develop the quality of your conceptual statement to the next level, ask the questions How? or Why? Often this will provide a Level 2 generalization that can be used in the unit of work. Sometimes generalizations can scaffold to Level 3 by asking the question So what?
Generally a unit of work will have six to eight generalizations, which are a combination of Levels 2 and 3. The teaching target is the Level 2 generalization; use Level 3 for extension or clarification.
Here are a couple of examples of the scaffolding process:
FIGURE M2.2: SCAFFOLDING THE PYTHAGOREAN THEOREM
Level 1 Students will understand that . . .
Two side lengths of any right triangle affect the third side.
Level 2 How or why?
For right-angled triangles, the area of the square drawn from the hypotenuse represents the sum of the areas of the squares drawn from the other sides.
Level 3 So what? What is the significance or effect?
For right-angled triangles, knowing the area of the square drawn from the hypotenuse or an area drawn from one of the other sides allows calculation of an unknown side.
The teaching target in Level 2 generalization
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M2.3: SCAFFOLDING GENERALIZATION FOR SEQUENCES AND SERIES
Level 1 Students will understand that . . .
Arithmetic and geometric sequences and series are structured by a set of formulae and definitions.
This generalization contains a “passive voice” verb, which weakens the sentence. Flipping the sentence emphasizes the strong verb for a Level 2 generalization. This also contains a no no verb.
Level 2 How or why?
A set of formulae and definitions structure arithmetic and geometric sequences.
Whether arithmetic and geometric sequences and series share a common difference or a common ratio distinguish one from another.
Level 3 So what? What is the significance or effect?
Arithmetic and geometric sequences and series describe patterns in numbers supply algebraic tools that help to solve real-life situations.
The teaching target in Level 2 generalization
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
The following is a scaffolding template to support the crafting of generalizations developed by Lynn Erickson (2007, p. 93).
FIGURE M2.4: SCAFFOLDING GENERALIZATIONS TEMPLATE
Meso Concept:
Micro Concepts:
(see Chapter 3)
Level 1 Generalization: After this program of study, students will understand that . . .
Level 2 Generalization: Why or how?
Level 3 Generalization: So what? What is the significance? What is the effect?
Level 1 Generalization: After this program of study, students will understand that . . .
Level 2 Generalization: Why or how?
Level 3 Generalization: So what? What is the significance? What is the effect?
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Here are some Level 2 and 3 verbs to help support the process of writing generalizations.
FIGURE M2.5: SAMPLE VERBS TO USE WHEN CRAFTING GENERALIZATIONS
accelerate compile draw up gather
accomplish compose drive generate
achieve compute earn give
acquire conceive edit govern
activate condense elaborate guide
adapt conduct eliminate handle
address conserve empathize help
adjust consolidate employ hypothesize
administer construct enact identify
advance contribute encourage illustrate
allocate convert enforce imagine
analyze convey engineer implement
anticipate cooperate enhance improve on
approve coordinate enlist improvise
arrange correlate ensure increase
ascertain correspond equip inform
assemble create establish initiate
assess cultivate estimate innovate
assign deal evaluate inspect
assimilate decide examine inspire
assist define execute install
assure delegate expand instill
attain deliver expedite institute
attend demonstrate experiment instruct
balance design explain insure (ensure)
bring detect express integrate
bring about determine facilitate interpret
build develop fashion introduce
calculate devise finance invent
challenge direct fix investigate
chart discover follow judge
check display forecast justify
clarify distribute forge reconcile
classify document form record
collect draft formulate recruit
command dramatize function as rectify
communicate draw gain keep
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
kindle pilot render studylaunch pioneer reorganize substitutelead place repair succeedlearn plan report summarizelift play represent supersedelisten to predict research superviselocate prepare resolve supply maintain prescribe respond surveymake present restore symbolizemanage prevent retrieve synergizemanipulate problem solve revamp talkmarket process review teachmaster procure revise tellmediate produce revitalize tendmeet project revive testmemorize promote rout tracementor propose save trackminimize protect schedule translatemodel prove screen travel
monitor provide secure treatmotivate publicize select trimmove purchase sell trainnavigate question sense transcribenegotiate raise separate transfernominate read serve transformobserve realize service uncoverobtain reason show undertakeoffer receive simplify unifyoptimize recognize sketch uniteorchestrate recommend solidify updateorder set up solve upgradeorganize shape sort useoriginate share spark utilizeovercome shift speak validateoversee redesign spearhead verifypaint reduce staff widenparticipate re evaluate stimulate winperceive refer streamline withdrawperfect refine strengthen workperform regulate stress writepersuade relate stretchphotograph remember structure
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M2.6: CHECKLIST FOR CRAFTING GENERALIZATIONS
Are there no no verbs?
Do some of the generalizations in the unit answer the questions How? or Why?
Do some of the generalizations in the unit answer the So what? question?
Have you included approximately six to eight generalizations in your unit?
Have you thought about how you are going to draw the generalizations from your students?
Are your generalizations the most important big ideas, the enduring understandings, and what you would like students to understand from their program of study?
Are your generalizations statements that include two or more concepts in a related context?
Would you be happy if your students expressed this generalization to you?
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Discussion Questions for Module 2
1. Explain why the “no no” verbs result in weak generalizations.
2. What is the purpose of scaffolding generalizations?
3. What is the difference between a Level 1, 2, and 3 generalization?
4. How will you develop quality generalizations?
5. Provide five Level 2 or 3 verbs not on the list in Figure M2.5.
6. How will you draw generalizations from your students?
7. Which of these generalizations best describes the purpose of the quadratic formula? Explain the reason for your choice. Will your students choose the same one as you? Why or why not?
• The quadratic formula is xb b ac
a=− + −2 4
2.
• Solving a quadratic equation by using the quadratic formula graphically displays the x intercepts of a quadratic function.
• The quadratic formula tells us what x is equal to and allows us to solve quadratic equations.
8. Can you think of other ways that you can guide your students toward the generalizations in a unit of work?
An Opportunity for Reflection
3 things I learned:
2 things I found interesting:
1 thing I need to develop:
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
MODULE 3
Guiding Questions and Critical ContentHow Can I Draw Essential Understandings From Students?
Main Text: Chapter 4. How Do I Plan Units of Work for a Concept-Based Curriculum?Guiding and essential questions guide students’ thinking from the concrete to the conceptual level. When planning your unit of work, one of the critical content components is the guiding or essential questions. Guiding questions bridge the factual to the conceptual, which supports the generalizations of the unit. The questions can be catego-rized into three types: factual, conceptual, and debatable (or provocative). For each unit of work there may be six to eight generalizations, and each generalization may have three to five factual and conceptual questions. For the entire unit there maybe two to three debatable or provocative questions. Concept-based curriculum design is also focused on what students will know (factually), understand (conceptually), and be able to do (in terms of skills). These are known as the KUDs of the unit. What we want our students to understand is summarized by the gen-eralizations of the unit.
In this module you will find the following resources:
• An example of the three types of guiding questions
• A template for designing the three types of guiding questions—factual, conceptual, and provocative/debatable
• An example of how to use KUDs
• A template to plan your KUDs for a unit of work
• Discussion questions for Module 3
• An opportunity for reflection
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M3.1: EXAMPLE OF GUIDING QUESTIONS FOR GEOMETRY: ORDERED PAIRS
Generalization:
Ordered pairs define the relationship between values on a coordinate plane that describe real situations.
Factual Questions
What?
How do you plot (x, y) on a coordinate plane?
How do you draw an x and a y axis?
What coordinates are given by the origin?
Conceptual Questions
Why or How?
How are coordinates useful in real life?
Why is the order important when naming coordinates?
How do coordinates such as (1, 2), (2, 4), and (3, 6) express an x, y relationship?
Why can x and y be used to represent real-life situations?
Why is the coordinate plane known as a plane?
Debatable/Provocative Question Are points on the coordinate plane one- or two-dimensional?
Unit of Study:
GeometryGuiding Questions
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M3.2: EXAMPLE OF GUIDING QUESTIONS FOR GEOMETRY: TWO-DIMENSIONAL SHAPES
Generalization:
Properties of two-dimensional plane shapes allow mathematicians to classify and compare common attributes.
Factual Questions
What?
What are the properties of a quadrilateral?
What are the properties of a square?
What are the properties of a rectangle?
What are the properties of a parallelogram?
What are the properties of a rhombus?
What are the properties of a trapezoid (U.S.) or trapezium (UK)?
Conceptual Questions
Why or How?
Why can a square also be a rectangle?
How is a rhombus related to a parallelogram?
Debatable/Provocative Question Do planes (two-dimensional shapes, not the flying type!) exist in the real world?
Unit of Study:
GeometryGuiding Questions
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M3.3: TEMPLATE FOR WRITING GUIDING QUESTIONS
Generalization:
Factual Questions
What?
Conceptual Questions
Why or How?
Debatable/Provocative Question
Unit of Study:
Guiding Questions
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Generalization:
Factual Questions
What?
Conceptual Questions
Why or How?
Debatable/Provocative Question
Unit of Study:
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M3.4: EXAMPLE OF KUDS FOR GEOMETRY GRADE 8/YEAR 7
KnowUnderstand(Generalizations) Do
A pair of perpendicular number lines, called axes, define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates.
The properties of two-dimensional figures
Ordered pairs define the relationship between values on a coordinate plane that describe real-life situations.
Properties of two-dimensional plane shapes allow mathematicians to classify and compare common attributes.
Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
Unit of Study:
Geometry
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M3.5: TEMPLATE FOR WRITING KUDS
KnowUnderstand
(Generalizations) Do
.
Unit of Study:
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Discussion Questions for Module 3
1. Why should you include KUDs in your unit planning?
2. Where do generalizations fit in your KUDs?
3. What are the advantages of using guiding questions when planning a unit of work?
4. How will you use the three types of guiding questions in your practice?
5. How will you use the provocative/debatable questions?
An Opportunity for Reflection
Write down three do’s and three don’ts based on what you have learned about guiding questions in this module.
Three Do’s Three Don’ts
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
MODULE 4
Math Unit PlanningWhat Does Concept-Based Unit Planning Look Like?
Main Text: Chapter 4. How Do I Plan Units of Work for a Concept-Based Curriculum?Time should be invested in planning a unit of work to ensure that students are given opportunities to draw the essential understandings from it. A unit web is an essential tool for planning a concept-based unit of work. A unit web contains critical content topics and concepts and gives an overview of the depth and breadth of the unit of instruction. Erickson has said the more complete the planning web, the stronger the conceptual understanding. When planning a unit of work for mathematics, it is important to include a strand titled “Mathematical Practices,” which represents the process generalizations. The other strands around the unit web represent the knowledge generalizations.
In this module you will find the following resources:
• A generic math unit web template
• A template for a concept-based unit planner
• A template for a concept-based weekly planner
• Discussion questions for Module 4
• An opportunity for reflection
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
FIGUR
E M4.
1: GE
NERI
C MAT
HEM
ATIC
S UNI
T WEB
TEM
PLAT
E
Uni
t Ti
tle Uni
t St
rand Mic
ro C
onc
epts
Gen
eral
izat
ion
Co
ncep
tual
Len
s:
Uni
t St
rand
1
Mic
ro C
onc
epts
Uni
t St
rand
2
Mic
ro C
onc
epts
Uni
t Ti
tle
Gen
eral
izat
ion
Gen
eral
izat
ion
Gen
eral
izat
ion
Gen
eral
izat
ion
Uni
t St
rand
Mat
hem
atic
sP
ract
ices
Uni
t St
rand
3
Mic
ro C
onc
epts
Mic
ro C
onc
epts
Pro
cess
Gen
eral
izat
ion
Pro
cess
Gen
eral
izat
ion
Kno
wle
dg
e St
rand
s
Big
Idea
s, E
ssen
tial
Und
erst
and
ing
s
The
Co
mp
one
nts
of
a U
nit
Web
fo
r M
ath
Gen
eral
izat
ion
Gen
eral
izat
ion
Gen
eral
izat
ion
Gen
eral
izat
ion
Key
Co
ncep
tso
f th
e un
it
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M4.2 TEMPLATE FOR CONCEPT-BASED UNIT PLANNER
(Continued)
Unit Title: Conceptual Lens: Time:
Allocation:
Grade/Year:
Unit Overview: Concepts in the Unit:
We would like students to know . . .
We would like students to understand that . . .
We would like students to be able to do . . .
Generalization Guiding Questions
Factual:
Conceptual:
Factual:
Conceptual:
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Generalization Guiding Questions
Factual:
Conceptual:
Factual:
Conceptual:
Factual:
Conceptual:
Debatable Unit Questions
FIGURE M4.2 (CONTINUED)
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
FIGUR
E M4.
3 TE
MPL
ATE F
OR CO
NCEP
T-BA
SED
WEE
KLY
PLAN
NER
Wee
kly P
lann
er
Wee
k C
onc
epts
G
ener
aliz
atio
nLe
arni
ng E
xper
ienc
e D
iffer
enti
atio
n St
rate
gie
s
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Discussion Questions for Module 4
1. What is the purpose of a unit web?
2. What is the purpose of a unit planner?
3. Why do we need to include KUDs (know, understand, and do) when planning units of work?
4. What is the purpose of a weekly planner?
5. How will you develop unit planners in the future?
6. How will you resource your units with concept-based learning experiences?
An Opportunity for Reflection
I was surprised . . .
I learned . . .
I feel . . .
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
MODULE 5
How Do I Design Inductive, Inquiry-Based Math Tasks?
Main Text: Chapter 5. How Do I Captivate Students? Eight Strategies for Engaging the Hearts and Minds of StudentsOne of the tenets of concept-based curriculum and instruction is the use of inductive teaching approaches. This means students are given specific numerical examples to work out and are guided to the generalizations. It may be appropriate to use different levels of inquiry—structured, guided, or open—depending on teacher experience and student readiness. The levels of inquiry also provide a differentiation strategy.
In this module you will find the following resources:
• Examples of different levels of inquiry using inductive approaches
• An example of an inductive inquiry task and the use of a hint jar
• A template for planning inductive inquiry tasks
• Discussion questions for Module 5
• An opportunity for reflection
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M5.1: EXAMPLE OF STRUCTURED INQUIRY TASK: LINEAR EQUATIONS
Investigating Straight LinesPart 1
The cost of a taxi is a flat rate of $1 and then $3 for every mile travelled. Complete this table in which y represents the total cost of the taxi ride and x represents miles travelled.
x (miles) 0 1 2 3 4 5 6 7 8
y (total fare)
Draw the y and x axes below and plot the points from the table.
Can you think of a function that represents the total fare for a taxi ride travelling x miles?
y =
What is the cost per mile? How is this represented on the graph?
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Part 2
The cost of a taxi is a flat rate of $2 and then $4 for every mile travelled. Complete this table in which y represents the total cost of the taxi ride and x represents miles travelled.
x (miles) 0 1 2 3 4 5 6 7 8
y (total fare)
Draw the y and x axes below and plot the points from the table.
Can you think of a function that represents the total fare for a taxi ride travelling x miles?
y =
What is the cost per mile? How is this represented on the graph?
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Part 3
The cost of a taxi is a flat rate of $3 and then $9 for every mile travelled. Complete this table in which y represents the total cost of the taxi ride and x represents miles travelled.
x (miles) 0 1 2 3 4 5 6 7 8
y (total fare)
Draw the y and x axes below and plot the points from the table.
Can you think of a function that represents the total fare for a taxi ride travelling x miles?
y =
What is the cost per mile? How is this represented on the graph?
Can you think of a function that represents the total fare for a taxi ride travelling x miles if the flat rate was m and the cost per mile was b?
y =
What is the cost per mile? How is this represented on the graph?
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M5.2: EXAMPLE OF GUIDED INQUIRY TASK: LINEAR EQUATIONS
Investigating Straight LinesPart 1
The cost of a taxi is a flat rate of $1 and then $3 for every mile travelled.
Draw the y and x axes below and plot the points that represent the total cost (y) of a taxi ride for different miles (x) travelled.
Can you think of a function that represents the total fare for a taxi ride travelling x miles?
y =
What is the cost per mile? How is this represented on the graph?
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Part 2
The cost of a taxi is a flat rate of $2 and then $4 for every mile travelled.
Draw the y and x axes below and plot the points that represent the total cost (y) of a taxi ride for different miles (x) travelled.
Can you think of a function that represents the total fare for a taxi ride travelling x miles?
y =
What is the cost per mile? How is this represented on the graph?
Can you think of a function that represents the total fare for a taxi ride travelling x miles if the flat rate was m and the cost per mile was b?
y =
What is the cost per mile? How is this represented on the graph?
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M5.3: EXAMPLE OF OPEN INQUIRY TASK: LINEAR EQUATIONS
Investigating Straight LinesInvestigate the effects of the parameters m and b on the linear function y = mx + b and explain what happens when m and b take different values. Use real-life examples to illustrate your explanations.
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M5.4: EXAMPLE OF INDUCTIVE INQUIRY TASK: ADDING AND SUBTRACTING NEGATIVE NUMBERS
Temperature ScalesThe lowest temperature recorded in Washington is -26°C.
Mark this on the temperature scale on the right with a W.
A heat wave sweeps Washington and takes away this cold temperature, resulting in a temperature of 0°C.
This could be represented as
-26 take away –26
Or in mathematical symbols
-26 - (-26) =
Another way to express this heat wave is to add 26
-26 + 26 =
What conclusion can you make about subtracting a negative number? Provide an example of your own that shows how you take away something that is already negative.
0°C
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M5.5: EXAMPLE OF INDUCTIVE INQUIRY TASK: USING THE TRI-MIND ACTIVITY
Hint Jar
You owe your mum $10. This means you have -$10. How do you take this away? Explain and show the sum.
You tell someone either to eat, to not eat, or to NOT not eat. What does NOT not eat mean?
Good things happen to good people = GOOD (positive)
Good things happen to bad people = BAD (negative)
Bad things happen to good people =
Bad things happen to bad people =
Think of your own analogy and write it here:
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M5.6: EXAMPLE OF INDUCTIVE INQUIRY TASK: ADDING AND SUBTRACTING POSITIVE AND NEGATIVE NUMBERS
Hot Air BalloonsThe hot air balloon basket is floating in the sky with the balloons and weights. Fill in the following table:
Take Away or Add Balloons Basket Moves up (+) or Down (-)
Add 3 balloons
Take away 3 balloons
Take Away or Add Weights Basket Moves up (+) or Down (-)
Add –3 weights
Take away –3 weights
Write your own sums that represent the four operations above:
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
What generalizations can you make about adding and subtracting positive and negative numbers? Provide several examples to illustrate your explanations.
Positive
Negative
- - -
+
+
+
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M5.7: TEMPLATE FOR DESIGNING INDUCTIVE INQUIRY TASKS
Designing Inductive Inquiry Tasks
Generalization:
Guiding questions
Factual Questions
Conceptual Questions
Debatable Questions
Inductive Student Task Prompts
Specific numerical example
Second specific numerical example
Third specific numerical example, if appropriate
Students form generalizations
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Discussion Questions for Module 51. What is the difference between inductive and deductive teaching approaches?
2. What are the different levels of inquiry?
3. When and how would you use the different levels of inquiry?
4. How do you use the hint jar for the activity on negative numbers?
An Opportunity for Reflection
Write a headline about inductive inquiry tasks
This thinking routine helps you to summarize the essence or the most important ideas to you regarding inductive inquiry tasks.
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
MODULE 6
Designing Differentiated TasksHow Do I Plan Tasks to Ensure Every Student Learns?
Main Text: Chapter 5. How Do I Captivate Students? Eight Strategies for Engaging the Hearts and Minds of Students, and Chapter 6. How Do I Know My Students Understand the Concepts? Assessment StrategiesGenerally, ongoing formative assessment is employed throughout a unit of work. This is often not graded as it pro-vides feedback on how your students are progressing in their learning and informs future instruction. Diagnostic assessment allows you to identify the starting point for individual students, to facilitate differentiation of instruction. Summative assessment allows you to evaluate learning at the end of a unit, which may be used for future teaching and learning plans. The suggested assessments may be used for diagnostic, formative, or summative assessment. For example, I use Frayer’s model as a diagnostic at the beginning of a unit to assess students’ preexisting knowledge; I also use it during a unit to check for understanding and as a summative assessment to give students opportunities to show me what they know. Included in this chapter are differentiation strategies such as the tri-Mind thinking styles and hint cards.
In this module you will find the following resources:
• Examples of pre-assessment strategies and formative and summative assessments
• Examples of tri-mind tasks for differentiating thinking styles
• A template for designing tri-mind tasks
• Discussion questions for Module 6
• An opportunity for reflection
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M6.1: EXAMPLE OF A PRE-ASSESSMENT MATCHING ACTIVITY USING VOCABULARY WORDS
MatchingMatch the word with the correct definition.
rhombus/diamond four-sided 2D shape
rectangle four-sided 2D shape, opposite sides parallel, opposite sides equal
square four-sided 2D shape, pair of opposite sides parallel
parallelogram four-sided 2D shape, equal sides, 4 right angles, opposite sides parallel
trapezoid (U.S.) or trapezium (UK)
four-sided 2D shape, all sides equal length, opposite sides parallel, opposite angles equal
quadrilateral four-sided 2D shape, opposite sides parallel, opposite sides equal in length, 4 right angles
kite adjacent sides equal in length
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M6.2: EXAMPLE OF A PRE-ASSESSMENT MATCHING ACTIVITY USING VISUAL CUES
MatchingMatch the vocabulary words with the pictures.
rhombus/diamond
rectangle
square
parallelogram
trapezoid (U.S.)/trapezium (UK)
quadrilateral
kite
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M6.3: EXAMPLES OF QUICK QUESTIONS FOR PRE-ASSESSMENT
Quick Questions on Mini Whiteboards
Starting off with quick questions informs you which concepts students already know and which they do not. Students hold up their answers on mini whiteboards so you can quickly and easily identify which students already know these concepts. Different stations could be set up in the classroom after this pre-assessment to address these ideas. Students who already know these concepts well can move on to line symmetry.
What is an acute angle?
What is an obtuse angle?
What is a reflex angle?
What is a right angle?
Draw a set of parallel lines and include an explanation.
Draw a set of perpendicular lines and include an explanation.
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M6.4: EXAMPLE OF USING A CONCEPT MAP FOR PRE-ASSESSMENT
Concept MapsIn pairs, write down what you know about angles.
Angles
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M6.5: EXAMPLE USING FRAYER’S MODEL FOR ANGLES
Frayer’s Model
Definition Facts, Attributes, Characteristics
Examples Non-Examples Angles
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M6.6: EXAMPLE OF USING WINDOW PANES FOR ASSESSMENT (DIAGNOSTIC, FORMATIVE, OR SUMMATIVE) OF TRIGONOMETRY
Window PanesWrite down what you know about trigonometry.
Trigonometry
Right-angled trigonometry Three-dimensional trigonometry
Non-right-angled trigonometry Explain how right-angled and non-right-angled trigonometry are connected to three-dimensional trigonometry.
You can also use window panes by including your KUDs (what we want our students to Know, Understand, and Do) and asking some questions here for pre-assessment.
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
FIGUR
E M6.7
: EXA
MPL
E OF A
TRI-M
IND
ACTIV
ITY FO
R M
EASU
REM
ENT
Mea
sure
men
t1.
Cho
ose
one
of t
he t
hree
op
tions
bel
ow.
2. F
ind
tw
o ot
her
peo
ple
who
hav
e al
so m
ade
the
sam
e ch
oice
as
you.
3. W
ork
on y
our
task
tog
ethe
r an
d b
e re
ady
to s
hare
.
Mak
e a su
mm
ary a
nd ex
plai
n all
the k
ey co
ncep
ts in
the t
opic
of
mea
sure
men
t. Ex
plai
n how
you w
ould
use a
ll th
e con
cept
s of
mea
sure
men
t in r
eal l
ife.
Crea
te a
grap
hic (
visu
al) o
r ski
t of h
ow al
l the
conc
epts
of
mea
sure
men
t are
rela
ted.
Prov
ide e
xam
ples
to il
lust
rate
yo
ur po
ints
.
My
cho
ice
is:
I cho
se t
his
bec
ause
:
The
mem
ber
s o
f m
y g
roup
are
:O
utlin
e o
f in
div
idua
l ro
les
and
act
ion
pla
n:
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
FIGUR
E M6.8
: EXP
LANA
TION
OF TR
I-MIN
D AC
TIVITY
Top
ic f
or
Tri-M
ind
:1.
Cho
ose
one
of t
he o
ptio
ns b
elow
.
2. F
ind
tw
o ot
her
peo
ple
who
hav
e al
so m
ade
the
sam
e ch
oice
as
you.
3. W
ork
on y
our
task
tog
ethe
r an
d b
e re
ady
to s
hare
.
Anal
ytic
al Th
inke
rsEx
ample
: Writ
e a de
finitio
n of th
e top
ic.
Prac
tical
Thin
kers
Exam
ple: S
how
the a
pplic
ation
of th
e top
ic. Ho
w w
ould
you u
se th
is?Cr
eativ
e Thi
nker
sEx
ample
: Cre
ate a
meta
phor
, visu
al, or
skit.
My
cho
ice
is:
I cho
se t
his
bec
ause
:
Stud
ents
may
rec
og
nize
the
ir p
refe
rred
sty
le; h
ow
ever
, it
is a
lso
imp
ort
ant
to e
nco
urag
e st
uden
t to
try
diff
eren
t ap
pro
ache
s an
d d
iffer
ent
thin
king
sty
les.
The
mem
ber
s o
f m
y g
roup
ar
e:O
utlin
e o
f in
div
idua
l ro
les
and
act
ion
pla
n:
In a
ny g
roup
wo
rk, i
t is
imp
ort
ant
that
eac
h m
emb
er h
as a
cl
earl
y d
efin
ed r
ole
so
all
stud
ents
are
act
ivit
y in
volv
ed.
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Discussion Questions for Module 61. What is the purpose of pre-assessment?
2. What are other differentiation strategies for the math classroom?
3. How would you use formative assessment strategies in your own classroom?
4. What are some other ways you can use tri-mind activities in your classroom?
An Opportunity for Reflection
Think, Puzzle, Explore
1. What do you think you know about differentiation?
2. What questions or puzzles do you have about differentiation?
3. How can you explore differentiation further to support your own classroom practice?
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
MODULE 7
Designing Math Performance TasksHow Do I Engage Students With Authentic Performance Tasks?
Main Text: Chapter 6. How Do I Know My Students Understand the Concepts? Assessment Strategies, and Chapter 7. How Do I Integrate Technology to Foster Conceptual Understanding?Performance tasks allow students to demonstrate what they need to know, understand, and do from a unit of work. Lynn Erickson has developed the “What, Why, and How” guide to support the process of designing a performance task. To develop a performance of deep understanding, Erickson (2007) encourages teachers to “pick up exact language from the generalization (the Why) and build that language into the task to ensure the tie between the task and the conceptual understanding.”
RAFTS (Holston & Santa, 1985) is another tool for writing performance tasks, which helps to identify role, audience, format, topic, and the use of strong verbs and adjectives, all put together in an engaging scenario. Many lesson plans have students use skills to learn factual content and do not extend student thinking to the conceptual level. These are referred to as “activities.” Look at the following examples and determine which is a performance task of deep understanding and which is a lower level activity.
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M7.1: PERFORMANCE TASK VS. ACTIVITY
Performance Task or Avtivity?
Use the ratios of sine, cosine, and tangent to find the unknown angles and unknown sides of the following right-angled triangles.
Island School was built in in its current location in 1973 and the buildings, while well maintained, need to be rebuilt. The buildings are literally falling apart! The architects would like to preserve the playgrounds that all the buildings are centered around and need to know the heights of all seven blocks for the new plan. Find out the heights of all the blocks at Island School and make a scale drawing of the current school. This scale drawing should be a bird’s-eye view and include all dimensions including the height. Design your own school ensuring that you accommodate all faculties and facilities listed below. You may use any format to present your findings.
In this module you will find the following resources:
• Templates for designing a math performance task
• Examples of task formats and roles
• Discussion questions for Module 7
• An opportunity for reflection
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M7.2: TEMPLATE FOR DESIGNING A MATH PERFORMANCE TASK USING WHAT, WHY, AND HOW
Unit Title/Key Topic:
What Investigate . . . (unit title or key topic)
Why In order to understand that . . . (generalization)
How Performance task/engaging scenario for students
Adapted from Lynn Erickson (2007).
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M7.3: TEMPLATE FOR DESIGNING A MATH PERFORMANCE TASK USING RAFTS (ROLE, AUDIENCE, FORMAT, TOPIC, AND STRONG VERBS AND ADJECTIVES)
Unit Title/Key Topic:
R = Role
A = Audience
F = Format
T = Topic
S = Strong Verbs and Adjectives
Adapted from Holston and Santa (1985).
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M7.4: EXAMPLES OF TASK FORMATS AND TASK ROLES
Task Formats
Lab report Movie/Documentary Panel discussion
Newsletter Movie review Experiment
Cartoon Debate Diorama & explanation
Advertisement Action plan Applet using software
Jingle/Song Invention Museum exhibit
Play/Skit Journal entry Sculpture or drawing
Letter Critique Musical story
Top tips Story board Police report
Animation 3D model Board game
Short story Proposal Map
Website Travel brochure Biography
Task Roles
Scientist, doctor Movie maker Panel discussion
Secretary Movie reviewer Scientist
Cartoonist Debater Set designer
Advertiser Leader, hero Software designer
Musical ad maker Inventor Museum curator
Playwright Journal writer Artist
Writer Reviewer Musician
Friend, citizen News presenter Policeperson
Expert in a field/Teacher 3D model maker Game maker
Storyteller Proposal writer Cartographer
Web page designer Travel agent, tour guide Biographer
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
Discussion Questions for Module 71. What is the purpose of a task planner?
2. Why is it important to design a task with an engaging scenario?
3. How can you plan for differentiation when designing a performance task?
4. What is the difference between a performance task and an activity?
5. Why is it critical to reflect the exact language of the generalization in the actual task?
6. Can you think of other formats and roles not listed in Figure M7.4?
An Opportunity for Reflection
PPositive:
MMinus:
IInteresting:
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
MODULE 8
Developing the Concept-Based TeacherHow Do I Develop as a Teacher to Integrate Concept-Based Curriculum in My Planning and Instruction?
Main Text: Chapter 8. What Do Ideal Concept-Based Math Classrooms Look Like?The journey to developing as a concept-based teacher will require a commitment to fostering and provoking deep conceptual understanding in our students. Lois Lanning (Erickson & Lanning, 2014) has developed rubrics to help the developing concept-based teacher identify progress in terms of lesson planning, instruction, and the under-standing of the concept-based model overall (Figures M8.1–M8.3).
Identify which level you are at now for each of the rubric components in concept-based curriculum and instruction and use the target-setting templates (Figures M8.4–M8.6) to reflect on how you are going to make personal progress as a concept-based teacher. Use the template in Figure M8.7 to draft your personal action plan.
Once you have created your personal action plan, set targets as a faculty using the SMART goals faculty develop-ment plan in Figure M8.8. The SMART goals template can also be used schoolwide. The use of SMART goals will ensure your targets are specific, measurable, achievable/assignable, realistic, and time bound.
In this module you will find the following resources:
• A rubric for concept-based instruction
• A rubric for concept-based lesson planning
• A rubric for understanding concept-based curriculum and instruction
• Target-setting templates for each of the rubrics: concept-based instruction, concept-based lesson planning, and understanding concept-based curriculum and instruction
• A personal action plan template
• A SMART faculty development plan template
• Discussion questions for Module 8
• An opportunity for reflection
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
Novi
ce
Emer
ging
M
aste
r
Less
on
Op
enin
g
•Le
sson
op
ens
by
laun
chin
g d
irect
ly
into
an
activ
ity w
ithou
t p
rovi
din
g a
n ov
ervi
ew o
r cl
ear
dire
ctio
ns•
The
less
on’s
targ
et g
ener
aliz
atio
n is
p
oste
d o
r st
ated
at
the
beg
inni
ng
of t
he le
sson
rat
her
than
dra
wn
from
st
uden
ts t
hrou
gho
ut•
The
less
on o
pen
ing
is a
ccur
ate
but
bla
nd a
nd le
aves
stu
den
ts
dis
inte
rest
ed a
s it
is m
ore
of a
te
ache
r’s m
onol
ogue
•Th
e le
sson
op
enin
g s
ets
the
stag
e fo
r in
duc
tive
teac
hing
tha
t w
ill d
raw
out
co
ncep
tual
und
erst
and
ing
s fr
om t
he
stud
ents
(e.g
., ex
amp
les
are
pos
ted
, in
trig
uing
que
stio
ns p
rese
nted
, an
inte
rest
ing
sce
nario
sha
red
, rel
evan
t co
ncep
ts p
osed
) but
the
op
enin
g is
ov
erly
det
aile
d a
nd t
oo lo
ng
•Th
e op
enin
g c
lear
ly c
omm
unic
ates
an
eng
agin
g, c
aptiv
atin
g o
verv
iew
of
the
less
on t
hat
conn
ects
and
ext
end
s p
revi
ous
lear
ning
Dur
ing
Inst
ruct
ion
refle
cts
mod
elin
g,
faci
litat
ing
, an
d m
edia
ting
co
ncep
tual
un
der
stan
din
g
•Th
e le
sson
follo
ws
the
writ
ten
less
on p
lan
but
sin
ce t
he p
lan
doe
s no
t ad
dre
ss a
ll th
e el
emen
ts o
f an
effe
ctiv
e co
ncep
t-b
ased
less
on,
inst
ruct
ion
falls
sho
rt•
Con
cep
ts a
re s
omew
hat
app
aren
t in
th
e le
sson
with
litt
le a
tten
tion
to h
ow
to u
se c
once
pts
to
crea
te in
telle
ctua
l en
gag
emen
t an
d d
eep
en s
tud
ents
’ un
der
stan
din
g•
Focu
s is
less
on
the
tran
sfer
of
lear
ning
and
mor
e on
tas
k co
mp
letio
n
•Fo
llow
s th
e co
ncep
t-b
ased
less
on
pla
n p
resc
riptiv
ely
•M
aint
ain
the
emot
iona
l eng
agem
ent
of s
tud
ents
by
usin
g e
xam
ple
s an
d
reso
urce
s•
Ther
e is
som
e g
rad
ual r
elea
ses
of re
spon
sib
ility
of l
earn
ing
from
te
ache
r to
stu
den
t, b
ut t
he t
each
er is
as
sum
ing
mos
t of
the
cog
nitiv
e w
ork
•Th
e ex
ecut
ion
of t
he le
sson
pla
n is
wel
l pac
ed, w
hile
bei
ng fl
exib
le
and
resp
onsi
ve t
o an
ticip
ated
st
uden
t ne
eds
(and
nee
ds
that
aris
e th
roug
hout
the
less
on)
•Th
ere
is a
con
scio
us a
nd c
onsi
sten
t d
evel
opm
ent
of s
tud
ents
’ syn
erg
istic
th
inki
ng t
hrou
gh
inst
ruct
iona
l te
chni
que
s an
d t
houg
ht p
rovo
king
ex
amp
les
and
thr
oug
h le
arni
ng
exp
erie
nces
and
reso
urce
s th
at
brid
ge
to a
dee
per
, con
cep
tual
idea
(u
nder
stan
din
g)
FIGUR
E M8.1
: RUB
RIC:
CONC
EPT-
BASE
D IN
STRU
CTIO
N BY
DR.
LOIS
A. L
ANNI
NG
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
Novi
ce
Emer
ging
M
aste
r
•In
stru
ctio
n em
plo
ys d
iffer
ent
kind
s of
que
stio
ns a
s th
e m
ain
tool
for
enco
urag
ing
the
tra
nsfe
r of
lear
ning
b
ut s
till o
ver-
relie
s on
fact
ual
que
stio
ns•
Inst
ruct
ion
rem
ains
pre
dom
inan
tly
teac
her
cent
ered
•St
uden
t p
artic
ipat
ion
is
pre
dom
inan
tly in
resp
onse
to
teac
her
que
stio
ning
and
eva
luat
ion
•M
ost
stud
ent’s
are
eng
aged
in t
he
lear
ning
whi
le c
lust
ers
of s
tud
ents
m
ay re
mai
n of
f tas
k or
dis
inte
rest
ed
due
to
inef
fect
ive
leve
l of c
halle
nge
or la
ck o
f rel
evan
ce
•Te
achi
ng u
ses
a va
riety
of t
echn
ique
s to
sup
por
t th
e tr
ansf
er o
f lea
rnin
g
and
dee
pen
ing
of u
nder
stan
din
g
(e.g
. que
stio
ns, a
skin
g fo
r ot
her
exam
ple
s/no
n-ex
amp
les
of t
he s
ame
conc
ept
or g
ener
aliz
atio
n, fe
edb
ack,
an
d b
y as
king
stu
den
ts fo
r an
ana
lysi
s of
the
ir re
ason
ing
with
sup
por
ting
ev
iden
ce)
•G
rad
ual r
elea
se o
f res
pon
sib
ility
and
ow
ners
hip
for
lear
ning
from
tea
cher
to
stu
den
t is
cle
ar•
Con
tinuo
us m
onito
ring
of s
tud
ents
’ in
dep
end
ent
and
col
lab
orat
ive
gro
up
wor
k w
ith t
imel
y re
leva
nt fe
edb
ack
and
que
stio
ns t
hat
faci
litat
e an
d
med
iate
the
lear
ning
pro
cess
Less
on
Clo
sing
•Th
e te
ache
r re
cap
s th
e le
arni
ng
exp
erie
nces
in t
he le
sson
•Th
ere
is c
losi
ng a
sses
smen
t (fo
rmat
ive
or s
umm
ativ
e) o
f the
kn
owle
dg
e an
d s
kills
stu
den
ts
lear
ned
and
an
atte
mp
t to
det
erm
ine
stud
ents
’ lev
el o
f con
cep
tual
un
der
stan
din
g•
Rele
vant
pra
ctic
e b
eyon
d t
he le
sson
is
ass
igne
d
•Ev
iden
ce o
f lea
rnin
g (f
orm
ativ
e or
su
mm
ativ
e) t
he le
sson
’s ta
rget
ed
know
led
ge,
ski
lls, a
nd u
nder
stan
din
g
is c
olle
cted
•C
olla
bor
ativ
ely
the
teac
her
and
st
uden
ts re
flect
on
and
ana
lyze
the
su
cces
s of
the
lear
ning
(pro
cess
and
p
rod
uct)
•St
uden
ts le
arn
how
the
lear
ning
will
b
uild
futu
re le
arni
ng t
arg
ets
SOU
RCE:
Tra
nsiti
onin
g t
o C
once
pt-
Bas
ed C
urric
ulum
and
Inst
ruct
ion,
Cor
win
Pre
ss P
ublis
hers
, Tho
usan
d O
aks,
CA
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
Com
pone
nts o
f Les
son P
lan
Novi
ce
Emer
ging
M
aste
r
Less
on
op
enin
g: A
n ex
plic
it an
d
eng
agin
g s
umm
ary
of t
he w
ork
to b
e ac
com
plis
hed
tha
t tr
igg
ers
syne
rgis
tic t
hink
ing
Lear
ning
tar
get
s: W
hat
stud
ents
ar
e ex
pec
ted
to
know
(fac
tual
kn
owle
dg
e), u
nder
stan
d
(gen
eral
izat
ion)
, and
be
able
to
do
(ski
lls)
Gui
din
g q
uest
ions
: The
thr
ee
diff
eren
t ty
pes
of q
uest
ions
(fa
ctua
l, co
ncep
tual
, pro
voca
tive)
se
rve
as a
brid
gin
g t
ool f
or
conc
eptu
al t
hink
ing
and
pro
ble
m
solv
ing
.
Lear
ning
exp
erie
nces
: Int
elle
ctua
lly
eng
agin
g s
tud
ent
wor
k th
at
pro
vid
es o
pp
ortu
nitie
s fo
r st
uden
ts
to p
ract
ice
thei
r le
arni
ng a
nd t
o ar
rive
at t
he t
arg
et g
ener
aliz
atio
n (c
once
ptu
al u
nder
stan
din
g)
Less
on o
pen
s b
y st
atin
g t
he
activ
ities
stu
den
ts w
ill
exp
erie
nce
in t
he le
sson
an
d m
ay in
clud
e st
atin
g
the
gen
eral
izat
ion
the
less
on w
ill b
e te
achi
ng
tow
ard
Wha
t st
uden
ts m
ust
Kno
w
and
/or
be
able
to
Do)
is
liste
d in
the
less
on p
lan
Que
stio
ns in
the
less
on p
lan
focu
s he
avily
on
fact
ual
know
led
ge
and
rout
ine
skill
s
Less
on o
pen
ing
con
tain
s a
conc
eptu
al
lens
but
the
wea
k tie
to
cont
ent
fails
to
eng
age
syne
rgis
tic t
hink
ing
Targ
ets
iden
tify
wha
t st
uden
ts m
ust
Kno
w, U
nder
stan
d (g
ener
aliz
atio
n),
and
Do,
but
the
re m
ay b
e m
ore
lear
ning
tar
get
s th
an c
an b
e ac
com
plis
hed
in-d
epth
with
in t
he
less
on t
imef
ram
e
Less
on q
uest
ions
refle
ct d
iffer
ent
typ
es
(fact
ual,
conc
eptu
al, a
nd p
ossi
bly
p
rovo
cativ
e q
uest
ions
) an
d a
ntic
ipat
e st
uden
t m
isco
ncep
tions
The
stud
ent
wor
k at
tem
pts
to
pur
sue
conc
eptu
al u
nder
stan
din
gs
but
m
ay n
ot p
rovi
de
a cl
ear
pat
hway
w
ith e
noug
h ex
amp
les
or
scaf
fold
s fo
r st
uden
ts t
o re
aliz
e th
e co
ncep
tual
und
erst
and
ing
(g
ener
aliz
atio
n)
Less
on p
lan
show
s ef
fort
s to
des
ign
stud
ent
wor
k to
eng
age
stud
ents
’ in
tere
st a
nd o
ffers
som
e st
uden
t ch
oice
The
less
on o
pen
ing
eng
ages
syn
erg
istic
th
inki
ng b
y as
king
stu
den
ts t
o co
nsid
er t
he
know
led
ge
and
/or
skill
s th
ey w
ill b
e le
arni
ng
thro
ugh
a co
ncep
tual
que
stio
n(s)
or
lens
Lear
ning
tar
get
s re
pre
sent
wha
t st
uden
ts a
re
exp
ecte
d t
o K
now
, Und
erst
and
, and
Do;
the
lim
ited
num
ber
of l
earn
ing
tar
get
s al
low
s fo
r in
-dep
th, f
ocus
ed in
stru
ctio
n an
d le
arni
ng
Pote
ntia
l que
stio
ns a
re o
f diff
eren
t ty
pes
(fa
ctua
l, co
ncep
tual
, pro
voca
tive)
and
are
lis
ted
thr
oug
hout
the
pla
n
The
less
on p
lan
show
s a
del
iber
ate
effo
rt
to u
se q
uest
ions
to
help
stu
den
ts b
ridg
e fr
om t
he fa
ctua
l to
the
conc
eptu
al le
vel o
f un
der
stan
din
g
The
stud
ent
wor
k re
qui
res
stud
ents
to
cog
nitiv
ely
wre
stle
with
and
syn
thes
ize
the
know
led
ge,
ski
lls, a
nd c
once
pts
und
er
stud
y in
rele
vant
con
text
s th
at le
ad t
o th
e re
aliz
atio
n of
the
gen
eral
izat
ion
FIGUR
E M8.2
: THE
DEV
ELOP
ING C
ONCE
PT-B
ASED
TEAC
HER:
CONC
EPT-
BASE
D LE
SSON
PLAN
NING
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
Com
pone
nts o
f Les
son P
lan
Novi
ce
Emer
ging
M
aste
r
Ass
essm
ent
met
hod
s: A
sses
smen
t ty
pes
are
sel
ecte
d a
ccor
din
g
to t
he le
sson
’s le
arni
ng t
arg
ets
(kno
w, u
nder
stan
d, d
o) a
nd t
o th
e as
sess
men
t p
urp
oses
(for
mat
ive
& s
umm
ativ
e) in
ord
er t
o ca
ptu
re
evid
ence
of s
tud
ents
’ lea
rnin
g
(pro
cess
& p
rod
uct)
whi
ch w
ill t
hen
info
rm in
stru
ctio
n.
Diff
eren
tiat
ion:
Les
son
adju
stm
ents
ar
e p
lann
ed, a
s ne
eded
, for
the
co
nten
t st
uden
ts a
re e
xpec
ted
to
mas
ter,
the
pro
cess
stu
den
ts
will
use
to
acce
ss t
he c
onte
nt,
and
the
pro
duc
t st
uden
ts w
ill
pro
duc
e to
sho
w t
heir
lear
ning
. Th
e co
ncep
tual
und
erst
and
ing
(g
ener
aliz
atio
n) a
ll st
uden
ts a
re
exp
ecte
d t
o re
aliz
e re
mai
ns
cons
iste
nt fo
r al
l stu
den
ts.
The
targ
eted
kno
wle
dg
e an
d s
kills
are
ind
icat
ed in
th
e p
lan,
but
the
lear
ning
ex
per
ienc
es d
o no
t re
qui
re s
tud
ents
to
app
ly
thei
r le
arni
ng in
rele
vant
co
ntex
ts, w
hich
wou
ld
clea
rly le
ad t
o co
ncep
tual
un
der
stan
din
g a
nd t
rans
fer
acro
ss le
arni
ng s
ituat
ions
The
stud
ent
wor
k re
qui
red
in
the
less
on p
rimar
ily
relie
s on
wor
kshe
ets,
d
isco
nnec
ted
ski
lls,
and
fact
s th
at a
re n
ot
auth
entic
or
inte
llect
ually
/em
otio
nally
eng
agin
g fo
r st
uden
ts
Ass
ignm
ents
are
inte
llect
ually
and
em
otio
nally
eng
agin
g b
ut a
re n
ot a
t th
e ap
pro
pria
te le
vel o
f cha
lleng
e fo
r al
l stu
den
ts
Ass
essm
ent
typ
es a
re v
arie
d a
nd
help
mon
itor
stud
ents
dev
elop
ing
kn
owle
dg
e an
d s
kills
Ass
essm
ents
of u
nder
stan
din
g, a
ligne
d
to t
he t
arg
et g
ener
aliz
atio
n, a
re n
ot
clea
r
Stud
ent
wor
k is
at
the
app
rop
riate
leve
l of
chal
leng
e, is
inte
llect
ually
and
em
otio
nally
en
gag
ing
, mea
ning
ful,
and
rele
vant
to
the
dis
cip
line,
and
pro
vid
es a
pp
rop
riate
stu
den
t ch
oice
The
lear
ning
exp
erie
nces
are
del
iber
atel
y de
signe
d
to e
nhan
ce th
e tra
nsfe
r of l
earn
ing
acro
ss o
ther
di
scip
lines
ans
situ
atio
ns
Ass
essm
ent
typ
es a
re v
arie
d s
o th
ey a
sses
s st
uden
ts’ d
evel
opin
g k
now
led
ge,
ski
lls, a
nd
und
erst
and
ing
s (g
ener
aliz
atio
n) a
nd a
llow
fo
r tim
ely
feed
bac
k
Ass
essm
ents
pro
vid
e re
leva
nt in
form
atio
n ab
out
stud
ents
’ pro
cess
of l
earn
ing
as
wel
l as
the
ir le
arni
ng p
rod
ucts
Stud
ent
self-
asse
ssm
ent
is v
alue
d
Plan
s fo
r d
iffer
entia
tion
to m
eet
the
need
s of
al
l lea
rner
s ar
e in
clud
ed a
nd s
upp
ort
all
stud
ents
mee
ting
a c
omm
on c
once
ptu
al
und
erst
and
ing
(gen
eral
izat
ion)
Less
on
des
ign:
In a
ded
uctiv
e le
sson
des
ign,
the
tea
cher
sta
tes
the
lear
ning
tar
get
s (in
clud
ing
g
ener
aliz
atio
n) t
o th
e le
arne
rs a
t th
e b
egin
ning
of t
he in
stru
ctio
n.
In a
n in
duc
tive
des
ign,
stu
den
ts
cons
truc
t th
eir
und
erst
and
ing
s th
roug
h an
inq
uiry
pro
cess
.
Clo
sing
: Pla
n fo
r a
way
tha
t ev
iden
ce
of le
arni
ng c
an b
e re
view
ed
colle
ctiv
ely
Ass
essm
ent
typ
es a
re li
mite
d
so it
is d
iffic
ult
to k
now
th
e d
egre
e of
stu
den
ts’
lear
ning
and
pro
gre
ss
tow
ard
con
cep
tual
un
der
stan
din
g
Plan
s fo
r d
iffer
entia
tion
may
be
stat
ed b
ut la
ck
rele
vanc
e to
ind
ivid
ual
stud
ent
lear
ning
nee
ds
The
less
on d
esig
n is
d
educ
tive
(e.g
., ob
ject
ive
to e
xam
ple
vs.
exa
mp
le t
o g
ener
aliz
atio
n)
Plan
s fo
r d
iffer
entia
tion
are
incl
uded
fo
r st
uden
ts w
ho n
eed
sup
por
t (e
.g.,
spec
ial e
duc
atio
n, E
LL) i
n th
e ar
eas
of c
onte
nt, p
roce
ss, a
nd
pro
duc
t
Mis
conc
eptio
ns a
re g
ener
ally
ad
dre
ssed
with
the
cla
ss a
s a
who
le
The
less
on d
esig
n at
tem
pts
to
use
ind
uctiv
e te
achi
ng b
ut t
he e
xam
ple
s p
rese
nted
onl
y va
gue
ly il
lust
rate
the
ta
rget
ed c
once
ptu
al u
nder
stan
din
gs
A d
educ
tive
des
ign
may
als
o b
e in
clud
ed t
o su
pp
ort
the
lear
ning
of
foun
dat
iona
l fac
ts a
nd s
kills
Diff
eren
tiatio
n is
bas
ed o
n an
ana
lysi
s of
m
ultip
le d
ata
poi
nts
that
reve
al in
div
idua
l st
uden
t le
arni
ng n
eed
s
Spec
ific
acco
mm
odat
ions
are
read
ily
avai
lab
le b
ased
on
antic
ipat
ed s
tud
ent
mis
conc
eptio
ns a
nd n
eed
s
The
less
on d
esig
n is
prim
arily
ind
uctiv
e,
req
uirin
g s
tud
ents
to
eng
age
in a
m
ultif
acet
ed in
qui
ry p
roce
ss a
nd t
o re
flect
on
the
con
nect
ions
acr
oss
the
exam
ple
s p
rese
nted
so
that
stu
den
ts c
an fo
rmul
ate
and
def
end
the
ir g
ener
aliz
atio
ns
A d
educ
tive
des
ign
may
als
o b
e in
clud
ed t
o su
pp
ort
the
lear
ning
of f
ound
atio
nal f
acts
an
d s
kills
SOU
RCE:
Eric
kson
and
Lan
ning
(201
4, p
. 62)
.
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
FIGUR
E M8.3
: THE
DEV
ELOP
ING C
ONCE
PT-B
ASED
TEAC
HER:
UND
ERST
ANDI
NG CO
NCEP
T-BA
SED
CURR
ICUL
UM A
ND IN
STRU
CTIO
N
Novi
ceEm
ergi
ngM
aste
r
Sup
po
rt f
or
Co
ncep
t-B
ased
Tea
chin
g a
nd
Lear
ning
•N
ames
one
or
two
reas
ons
for
conc
ept-
bas
ed t
each
ing
and
le
arni
ng
•A
rtic
ulat
es t
he m
ajor
reas
ons
for
conc
ept-
bas
ed t
each
ing
and
le
arni
ng
•A
rtic
ulat
es t
he re
ason
s fo
r co
ncep
t-b
ased
cu
rric
ulum
and
inst
ruct
ion,
citi
ng re
leva
nt
sup
por
ting
rese
arch
Co
mp
one
nts
of
Co
ncep
t-B
ased
Cur
ricu
lum
and
In
stru
ctio
n
•C
once
pts
(mac
ro-
mic
ro)
•C
once
ptu
al le
nses
•Sy
nerg
istic
thi
nkin
g•
Gen
eral
izat
ions
•G
uid
ing
que
stio
ns
leve
led
and
alig
ned
w
ith g
ener
aliz
atio
ns•
Crit
ical
kno
wle
dg
e an
d k
ey s
kills
•Pe
rfor
man
ce
asse
ssm
ents
ver
sus
activ
ities
•C
an d
efin
e so
me
of t
he
elem
ents
of a
thr
ee-
dim
ensi
onal
con
cep
t-b
ased
cur
ricul
um
mod
el v
ersu
s a
two-
dim
ensi
onal
mod
el•
From
a li
st o
f con
cep
ts,
dis
ting
uish
es m
acro
- fr
om m
icro
-con
cep
ts•
Can
def
ine
syne
rgis
tic
thin
king
but
can
not
yet
exp
lain
met
hod
s to
cre
ate
syne
rgis
tic
thin
king
by
stud
ents
•U
ses
corr
ect
conc
ept-
bas
ed
term
inol
ogy
but
may
not
be
clea
r ab
out
the
ratio
nale
for
each
co
mp
onen
t•
Whe
n re
view
ing
con
cep
t-b
ased
cu
rric
ulum
and
/or
inst
ruct
ion,
re
cog
nize
s th
e th
ree-
dim
ensi
onal
co
mp
onen
ts•
Exp
lain
s th
e d
iffer
ence
bet
wee
n m
acro
- an
d m
icro
-con
cep
ts a
nd
stat
es w
hy it
is im
por
tant
to
know
th
e d
iffer
ence
•D
efin
es s
yner
gis
tic t
hink
ing
and
g
ives
at
leas
t on
e ex
amp
le o
f an
inst
ruct
iona
l tec
hniq
ue/s
trat
egy
to
eng
age
stud
ents
’ syn
erg
istic
thin
king
•Ex
pla
ins
the
vario
us c
omp
onen
ts o
f con
cep
t-b
ased
cur
ricul
um u
sing
cor
rect
ter
min
olog
y an
d t
he r
atio
nale
for
each
•W
hen
revi
ewin
g a
tw
o-d
imen
sion
al c
urric
ulum
an
d/o
r le
sson
, sug
ges
ts c
hang
es t
hat
will
m
ove
the
curr
icul
um o
r le
sson
to
a th
ree-
dim
ensi
onal
mod
el•
Dem
onst
rate
s a
solid
und
erst
and
ing
of
mac
ro-
and
mic
ro-c
once
pts
by
exp
lain
ing
how
to
em
plo
y th
em a
ccur
atel
y an
d e
ffect
ivel
y in
co
ncep
t-b
ased
cur
ricul
um a
nd in
stru
ctio
n•
Can
exp
lain
the
val
ue o
f syn
erg
istic
thi
nkin
g
and
offe
rs m
ore
than
one
per
sona
lly c
reat
ed
exam
ple
of i
nstr
uctio
nal t
echn
ique
s/st
rate
gie
s th
at c
lear
ly e
ngag
e st
uden
ts’ s
yner
gis
tic
thin
king
Co
mm
itm
ent
to
Co
ntin
ued
Lea
rnin
g•
Part
icip
ates
in c
once
pt-
bas
ed p
rese
ntat
ions
; at
tem
pts
ste
ps
to
adap
t tw
o-d
imen
sion
al
less
ons
to t
hree
-d
imen
sion
al le
sson
s
•Pa
rtic
ipat
es in
pro
fess
iona
l d
evel
opm
ent
pre
sent
atio
ns o
n co
ncep
t-b
ased
cur
ricul
um a
nd
inst
ruct
ion
and
follo
ws
up b
y at
tem
ptin
g t
o p
ut n
ew le
arni
ng in
to
pra
ctic
e•
Acc
epts
form
al o
r in
form
al c
oach
ing
an
d m
ento
ring
sup
por
t•
Ind
epen
den
tly re
ads
text
s on
co
ncep
t-b
ased
cur
ricul
um a
nd
inst
ruct
ion
•Pa
ssio
nate
ly s
uppo
rts
conc
ept-
base
d
curr
icul
um a
nd in
stru
ctio
n as
evi
denc
ed b
y ac
tive
part
icip
atio
n in
pre
sent
atio
ns o
r boo
k st
udy
grou
ps w
ith fo
llow
-up
impl
emen
tatio
n th
at in
clud
es o
ngoi
ng c
omm
unic
atio
n an
d
colla
bora
tive
wor
k w
ith o
ther
s (e
.g.,
coac
hing
an
d m
ento
ring)
•D
emon
stra
tes
cont
inue
d e
ffort
to
dee
pen
un
der
stan
din
g t
hrou
gh
shar
ing
lear
ning
and
le
adin
g p
rofe
ssio
nal d
evel
opm
ent
•C
ontin
ually
refle
cts
upon
and
refin
es p
ract
ice
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
FIGUR
E M8.4
: CON
CEPT
-BAS
ED IN
STRU
CTIO
N: TA
RGET
SETT
ING
Com
pone
nt of
Conc
ept-
Base
d Ins
truc
tion
Leve
l (No
vice
, Em
ergi
ng,
or M
aste
r)W
hy ar
e you
at th
is lev
el?
Targ
et Se
tting
How
will
you p
rogr
ess t
o the
ne
xt lev
el?
Less
on
op
enin
g
Dur
ing
a le
sso
n
Inst
ruct
ion
refle
cts
mod
elin
g,
faci
litat
ing
, an
d m
edia
ting
co
ncep
tual
un
der
stan
din
g
Less
on
clo
sing
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
FIGUR
E M8.5
: CON
CEPT
-BAS
ED LE
SSON
PLAN
NING
: TAR
GET S
ETTIN
G
Com
pone
nt of
Conc
ept-B
ased
Le
sson
Plan
ning
Leve
l (No
vice
, Em
ergi
ng, o
r Mas
ter)
Why
are y
ou at
this
level?
Targ
et Se
tting
How
will
you p
rogr
ess t
o the
next
level?
Less
on
op
enin
g: A
n ex
plic
it an
d e
ngag
ing
su
mm
ary
of t
he w
ork
to b
e ac
com
plis
hed
th
at t
rigg
ers
syne
rgis
tic
thin
king
Lear
ning
tar
get
s: W
hat
stud
ents
are
exp
ecte
d
to k
now
(fac
tual
kn
owle
dg
e), u
nder
stan
d
(gen
eral
izat
ion)
, and
be
able
to
do
(ski
lls)
Gui
din
g q
uest
ions
: The
th
ree
diff
eren
t ty
pes
of
que
stio
ns (f
actu
al,
conc
eptu
al, p
rovo
cativ
e)
serv
e as
a b
ridg
ing
too
l fo
r co
ncep
tual
thi
nkin
g
and
pro
ble
m s
olvi
ng.
Lear
ning
exp
erie
nces
: In
telle
ctua
lly e
ngag
ing
st
uden
t w
ork
that
p
rovi
de
opp
ortu
nitie
s fo
r st
uden
ts t
o p
ract
ice
thei
r le
arni
ng
and
to
arriv
e at
the
ta
rget
gen
eral
izat
ion
(con
cep
tual
un
der
stan
din
g)
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
Com
pone
nt of
Conc
ept-B
ased
Le
sson
Plan
ning
Leve
l (No
vice
, Em
ergi
ng, o
r Mas
ter)
Why
are y
ou at
this
level?
Targ
et Se
tting
How
will
you p
rogr
ess t
o the
next
level?
Ass
essm
ent
met
hod
s:
Ass
essm
ent
typ
es a
re
sele
cted
acc
ord
ing
to
the
less
on’s
lear
ning
ta
rget
s (k
now
, un
der
stan
d, d
o) a
nd t
he
asse
ssm
ent
pur
pos
es
(form
ativ
e &
sum
mat
ive)
in
ord
er t
o ca
ptu
re
evid
ence
of s
tud
ents
’ le
arni
ng (p
roce
ss &
p
rod
uct),
whi
ch w
ill t
hen
info
rm in
stru
ctio
n.
Diff
eren
tiat
ion:
Les
son
adju
stm
ents
are
p
lann
ed, a
s ne
eded
, for
th
e co
nten
t st
uden
ts
are
exp
ecte
d t
o m
aste
r, th
e p
roce
ss
stud
ents
will
use
to
acce
ss t
he c
onte
nt,
and
the
pro
duc
t st
uden
ts w
ill p
rod
uce
to s
how
the
ir le
arni
ng.
The
conc
eptu
al
und
erst
and
ing
(g
ener
aliz
atio
n) a
ll st
uden
ts a
re e
xpec
ted
to
real
ize
rem
ains
co
nsis
tent
for
all
stud
ents
.
(Con
tinue
d)
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
Com
pone
nt of
Conc
ept-B
ased
Le
sson
Plan
ning
Leve
l (No
vice
, Em
ergi
ng, o
r Mas
ter)
Why
are y
ou at
this
level?
Targ
et Se
tting
How
will
you p
rogr
ess t
o the
next
level?
Less
on
des
ign:
In a
d
educ
tive
less
on
des
ign,
the
tea
cher
st
ates
the
lear
ning
ta
rget
s (in
clud
ing
g
ener
aliz
atio
n) t
o th
e le
arne
rs a
t th
e b
egin
ning
of t
he
inst
ruct
ion.
In a
n in
duc
tive
des
ign,
st
uden
ts c
onst
ruct
the
ir un
der
stan
din
gs
thro
ugh
an in
qui
ry p
roce
ss.
Clo
sing
: Pla
n fo
r a
way
tha
t ev
iden
ce
of le
arni
ng c
an b
e re
view
ed c
olle
ctiv
ely.
FIGUR
E 8.3:
(CON
TINUE
D)
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
FIGUR
E M8.6
: UND
ERST
ANDI
NG CO
NCEP
T-BA
SED
CURR
ICUL
UM A
ND IN
STRU
CTIO
N: TA
RGET
SETT
ING
Com
pone
nt of
Und
erst
andi
ng Co
ncep
t-Bas
ed Cu
rric
ulum
and
Inst
ruct
ion
Leve
l (No
vice
, Em
ergi
ng,
or M
aste
r)W
hy ar
e you
at th
is lev
el?
Targ
et Se
tting
How
will
you p
rogr
ess t
o the
ne
xt lev
el?
Sup
po
rt f
or
conc
ept-
bas
ed t
each
ing
and
le
arni
ng
Co
mp
one
nts
of
conc
ept-
bas
ed:
C
once
pts
(mac
ro-m
icro
)C
once
ptu
al le
nses
S
yner
gis
tic t
hink
ing
G
ener
aliz
atio
nsG
uid
ing
que
stio
ns le
vele
d &
alig
ned
with
g
ener
aliz
atio
nsC
ritic
al k
now
led
ge
& k
ey s
kills
P
erfo
rman
ce a
sses
smen
ts v
ersu
s ac
tiviti
es
Co
mm
itm
ent
to c
ont
inue
d le
arni
ng
Retrieved from the companion website for Concept-Based Mathematics by Jennifer T. H. Wathall. Thousand Oaks, CA: Corwin. Reproduction authorized only for the local school site or nonprofit organization that has purchased this book.
FIGURE M8.7: TEMPLATE FOR PERSONAL ACTION PLAN
Personal Action PlanList your top five skills (things that you do well):
1.
2.
3.
4.
5.
List your top five strengths (could be a knowledge area):
1.
2.
3.
4.
5.
List three things you want to develop in terms of your teaching (list in order of priority):
1.
2.
3.
When and how will you achieve these three goals?
Retr
ieve
d f
rom
the
com
pan
ion
web
site
for
Con
cep
t-B
ased
Mat
hem
atic
s b
y Je
nnife
r T.
H.
Wat
hall.
Tho
usan
d O
aks,
CA
: C
orw
in.
Rep
rod
uctio
n au
thor
ized
onl
y fo
r th
e lo
cal
scho
ol s
ite o
r no
npro
fit o
rgan
izat
ion
that
has
pur
chas
ed t
his
boo
k.
Ad
apte
d fr
om D
oran
, G. T
. (19
81).
Ther
e’s
a S.
M.A
.R.T
. way
to
writ
e m
anag
emen
t’s g
oals
and
ob
ject
ives
. Man
agem
ent
Revi
ew, 7
0(11
), 35
–36.
FIGUR
E M8.8
: TEM
PLAT
E FOR
SCHO
OL O
R FA
CULT
Y DE
VELO
PMEN
T PLA
N US
ING S
MAR
T GOA
LS
SCHO
OL/F
ACUL
TY D
EVEL
OPM
ENT P
LAN
FOR
DEVE
LOPI
NG CO
NCEP
T-BA
SED
TEAC
HERS
Spec
ific
Tar
get
WW
WW
W
Wha
t, W
hy, W
ho, W
here
, W
hich
Mea
sura
ble
How
will
I kn
ow w
hen
this
is a
ccom
plis
hed
?
Ach
ieva
ble
/A
ssig
nab
le
Who
will
do
this
?
Rel
evan
t/R
ealis
tic
Is t
his
rele
vant
to
our
scho
ol d
evel
opm
ent
pla
n?
Tim
e Fr
ame
Wha
t tim
e fr
ame
shou
ld w
e se
t?
Ach
ieve
d?
Y/N
Wha
t: u
nit
web
s an
d u
nit
pla
nner
sW
hy: t
o w
rite
conc
ept-
bas
ed
units
of w
ork
and
cra
ft
gen
eral
izat
ions
for
a un
itW
ho: t
he w
hole
facu
ltyW
here
: mid
dle
sch
ool
Whi
ch: G
rad
e 8
Whe
n w
e ha
ve w
ritte
n un
it p
lann
ers
for
each
of t
he s
ix u
nits
in
Gra
de
8
The
Gra
de
8 te
achi
ng t
eam
Th
e sc
hool
dev
elop
men
t p
lan
has
iden
tifie
d
conc
ept-
bas
ed
curr
icul
um a
nd
inst
ruct
ion
as a
ta
rget
, and
the
sen
ior
lead
ersh
ip t
eam
has
al
loca
ted
col
lab
orat
ive
sess
ions
to
sup
por
t th
is p
roce
ss.
– C
olla
bor
atio
n on
ce p
er w
eek
for
90 m
inut
es–
Thre
e m
onth
s
Discussion Questions for Module 8
1. How will you develop certain aspects of concept-based lesson planning and instruction?
2. How will you encourage your faculty to start this journey and lead the change toward concept-based cur-riculum?
3. How will you use your current resources, planners, and standards to design a concept-based curriculum?
4. Which rubrics will you use from this module and why?
An Opportunity for Reflection
Connect: How are the ideas connected to what you already knew?
Extend: What ideas extended your thinking?
Challenge: What is still challenging for you?