The importance of coherent lessons in high school mathematics
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COHERENCE. The importance of Coherent Lessons in HIGH school Mathematics. College and Career Readiness Conference. Summer 2014. SHIFTS. COHERENCE. Coherence comes from the verb to cohere , which means: To stick together in a way that resists separation;

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The importance of Coherent Lessons in HIGH school Mathematics

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The importance of coherent lessons in high school mathematics

COHERENCE

The importance of Coherent Lessons in HIGH school Mathematics

College and Career Readiness Conference

Summer 2014


The importance of coherent lessons in high school mathematics

SHIFTS


The importance of coherent lessons in high school mathematics

COHERENCE

  • Coherence comes from the verb to cohere, which means:

  • To stick together in a way that resists separation;

  • To be logically linked and naturally related.

  • Think of Rice Krispies Treats…


The importance of coherent lessons in high school mathematics

We want our mathematics curriculum to have coherence… related concepts that stick together in a way that is obvious, understandable, connected, and reliable… for teachers and students alike!


Coherence

COHERENCE

A purposeful placement of standards to create logical sequences of content topics that bridge across the grades, as well as across standards within each grade.


The importance of coherent lessons in high school mathematics

TODAY’S OUTCOME

Participants will:

Explore in depth the “shift” of COHERENCE and its impact on mathematics content, instruction, andassessmentof the Maryland CCR Standards in mathematics.


Four corners

Four Corners

Teacher, Not HS Math

HS Math Teacher, taught Functions

HS Math Teacher, hasn’t taught Functions

Not a Teacher


Coherence1

COHERENCE

CONTENT

The writers of the CCSS:

  • Built COHERENCE into the content standards.

  • Clarified and shared their vision of grade-to-grade and within-grade COHERENCEin the Progressions documents.


Grade to grade coherence

GRADE-TO-GRADE COHERENCE

In the following task from Illustrative Mathematics, complete the following steps:

Compute the correct answer.

List all of the mathematics concepts/skills you used.

Write down the grade at which the concepts/skills are taught and the applicable domain, cluster, or standard.

List mathematics concepts/skills in later grades/courses that build upon/expand the information from #2.

Reflect on what you just have written. What do you notice about the task’s reliance on COHERENCE?

http://www.illustrativemathematics.org/illustrations/1564


The importance of coherent lessons in high school mathematics

Parking Lot!

A parking lot charges $0.50 for each half hour or fraction thereof, up to a daily maximum of $10. Let C(t) be the cost in dollars of parking for minutes.

a. Complete the table below.

http://www.illustrativemathematics.org/illustrations/588


The importance of coherent lessons in high school mathematics

Do calculations here:

DS

http://www.illustrativemathematics.org/illustrations/588


The importance of coherent lessons in high school mathematics

Sketch graph here

DS

http://www.illustrativemathematics.org/illustrations/588


The importance of coherent lessons in high school mathematics

Questions:

Answers

http://www.illustrativemathematics.org/illustrations/588


Grade to grade

GRADE-TO-GRADE

INSTRUCTION


Within grade coherence

WITHIN-GRADE COHERENCE

In the following task, complete the following steps:

Compute the correct answer.

List all of the mathematics concepts/skills you used.

Write down the grade at which the concepts/skills are taught and the applicable domain, cluster, or standard.

List mathematics concepts/skills in later grades/courses that build upon/expand the information from #2.

Reflect on what you just have written. What do you notice about the task’s reliance on COHERENCE?


The importance of coherent lessons in high school mathematics

INSTRUCTION

An equipment rental company has 2 different penalty options for returning a piece of equipment late:

Option 1: On day 1, the penalty is $10. On day 𝟐, the penalty is $2𝟎. On day 3, the penalty is $30. On day 4, the penalty is $4𝟎and so on, increasing by $10 each day the equipment is late.

Company 2: On day 1, the penalty is $𝟎.𝟎𝟏. On day 𝟐, the penalty is $𝟎.𝟎3. On day 3, the penalty is $𝟎.09. On day 4, the penalty is $𝟎.27 and so on, triplingin amount each additional day late.

Jim rented a backhoe using option because he thought it had the better late return policy. The job he was doing with the backhoe took longer than he expected, but it did not concern him because the late penalty seemed so reasonable. When he returned the backhoe 12 days late, he was shocked by the penalty fee. What did he pay, and what would he have paid if he had used Option 1 instead?


The importance of coherent lessons in high school mathematics

INSTRUCTION

Additional Questions for discussion:

Write the formula for the sequence in option 1.

Describe the sequence.

Write a formula for the sequence in option 2.

Describe the sequence.

5. Which of the 2 options grows more quickly? Why?


The importance of coherent lessons in high school mathematics

INSTRUCTION

Do calculations here:

DS


The importance of coherent lessons in high school mathematics

INSTRUCTION

Do calculations here:

DS


Within grade coherence1

WITHIN-GRADE COHERENCE

INSTRUCTION


Coherence2

COHERENCE

ASSESSMENT

Evidence of Coherencein assessment is not tremendously different from evidence of coherencein classroom instruction.


The importance of coherent lessons in high school mathematics

COHERENCE

ASSESSMENT

High School Content

Interpreting Functions (F-IF)

  • Major- Understand the concept of a function an due function notation

  • Interpret functions that arise in applications in terms of context

  • Supporting - Analyze functions using different representations

    Building Functions(F-BF)

  • Supporting - Build a function that models a relationship between two quantities

  • Additional- Build new function from existing functions


Coherence3

COHERENCE

https://www.parcconline.org/samples/mathematics/high-school-functions

ASSESSMENT


Coherence4

COHERENCE

https://www.parcconline.org/samples/mathematics/high-school-functions

ASSESSMENT

Do calculations here:

DS


Coherence5

COHERENCE

  • Which other Standards for Mathematical Practice are assessed in this task? MP6- Attend to Precision

  • Content standards F-­‐IF.9. Compare

  • properties of two functions each represented in a different way(algebraically, graphically,

  • Numerically in tables, or by verbal

  • descriptions).

  • (According to PARCC)

ASSESSMENT


Coherence6

COHERENCE

  • Type I-Balance of conceptual understanding,

  • fluency, and application

  • Type I- Machine scorable, innovative computer-based formats

  • Claim A- Major content with connections to

  • practices

  • Scoring- Full credit requires selecting correct

  • answers from drop down menu. Partial credit

  • can be given for each answer that is correct.

  • (According to PARCC)

ASSESSMENT


The importance of coherent lessons in high school mathematics

Shift of Coherence

In reality, the “key” to making the Shift of Coherencea success rests on the shoulders of curriculum writers and classroom teachers.

Instruction must take advantage of every opportunity to buildconnections between topics both within a grade and from grade to grade.


Grade to grade coherence activity

Grade-to-Grade Coherence Activity

INSTRUCTION

ASSESSMENT

COHERENCE


What are the muddiest points

What arethe muddiest points?

Record any question you still have after today’s presentation on your post-it note. Please provide your name and email address.

Stick your post-it on the front wall as you leave today. Thank you!


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