what does concept-based curriculum look like in the math...

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


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

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S

F

A

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F

A

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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.


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


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


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


THEORY

ConceptsConcepts

Structure of ProcessLanning, © 2012


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


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


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.


FIGURE M1.7: TEMPLATE FOR SYNERGISTIC THINKING WHEN PLANNING TASKS

Synergistic thinking means that . . .

Traditional Activity:

Adapted to facilitate synergistic thinking:


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


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).


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 . . .


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



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


• 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.


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


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


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?


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


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


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?



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?


3 things I learned:

2 things I found interesting:

1 thing I need to develop:


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.


• 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




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


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)?


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


FIGURE M3.3: TEMPLATE FOR WRITING GUIDING QUESTIONS

Generalization:

Factual Questions

What?


Why or How?

Debatable/Provocative Question

Unit of Study:

Guiding Questions


Generalization:

Factual Questions

What?


Why or How?

Debatable/Provocative Question

Unit of Study:


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


FIGURE M3.5: TEMPLATE FOR WRITING KUDS

KnowUnderstand

(Generalizations) Do

.

Unit of Study:



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?


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


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.


• A generic math unit web template

• A template for a concept-based unit planner

• A template for a concept-based weekly planner



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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:


Generalization Guiding Questions

Factual:

Conceptual:

Factual:

Conceptual:

Factual:

Conceptual:

Debatable Unit Questions

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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?


I was surprised . . .

I learned . . .

I feel . . .


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.


• 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




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?


Part 2


x (miles) 0 1 2 3 4 5 6 7 8

y (total fare)



y =



Part 3


x (miles) 0 1 2 3 4 5 6 7 8

y (total fare)



y =


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 =



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.


y =



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.


y =


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 =



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.


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


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:


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:


What generalizations can you make about adding and subtracting positive and negative numbers? Provide several examples to illustrate your explanations.

Positive

Negative

- - -

+

+

+


FIGURE M5.7: TEMPLATE FOR DESIGNING INDUCTIVE INQUIRY TASKS

Designing Inductive Inquiry Tasks

Generalization:

Guiding questions

Factual Questions


Debatable Questions

Inductive Student Task Prompts

Specific numerical example

Second specific numerical example

Third specific numerical example, if appropriate

Students form generalizations


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?


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.


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.


• 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




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


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


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.


FIGURE M6.4: EXAMPLE OF USING A CONCEPT MAP FOR PRE-ASSESSMENT

Concept MapsIn pairs, write down what you know about angles.

Angles


FIGURE M6.5: EXAMPLE USING FRAYER’S MODEL FOR ANGLES

Frayer’s Model

Definition Facts, Attributes, Characteristics

Examples Non-Examples Angles


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.

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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?


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?


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.


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.


• Templates for designing a math performance task

• Examples of task formats and roles




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).


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).


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


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?


PPositive:

MMinus:

IInteresting:


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.


• 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



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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


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


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?


Connect: How are the ideas connected to what you already knew?

Extend: What ideas extended your thinking?

Challenge: What is still challenging for you?

what does concept-based curriculum look like in the math...

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