Sustaining High CogniCve Demand in Intermediate Algebra Jason Slowbe - - PDF document

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Sustaining High CogniCve Demand in Intermediate Algebra Jason Slowbe - - PDF document

Mar 27, 2023 155 likes •356 views

7/15/16 Sustaining High CogniCve Demand in Intermediate Algebra Jason Slowbe Great Oak High School San Diego, CA @TheSlowbe jason.slowbe@gmail.com #NCTMinst Slides and handout will be on Ins1tute website: h3p://nctm.org/hs16

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Jason Slowbe

Great Oak High School San Diego, CA @TheSlowbe jason.slowbe@gmail.com #NCTMinst

Sustaining High CogniCve Demand in Intermediate Algebra

  • Slides and handout will be on Ins1tute website:

h3p://nctm.org/hs16

  • Everything will be on my own website:

h3p://theslowbe.weebly.com

  • Please tweet throughout today’s session

2

@theslowbe #NCTMinst

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Goals for this workshop

  • Avoiding focus on “answer-geFng”
  • What is Cogni1ve Demand?
  • Modifying exis1ng problems and tasks, not

“reinven1ng the wheel”

  • Work 1me crea1ng a shared resource to take

back to our classrooms and colleagues

4 @theslowbe

#NCTMinst

Toward BeJer PD…

Ar1cle on my website, h3p://theslowbe.weebly.com

Johnson, Karen. “5 Things Teachers Want from PD, and How Coaching and Collabora1on Can Deliver Them – If Implementa1on Improves”. EdSurge.com, June 28, 2016 h3ps://www.edsurge.com/news/2016-06-28-5-things-teachers-want-from- pd-and-how-coaching-and-collabora1on-can-deliver-them-if-implementa1on- improves

5 @theslowbe

#NCTMinst

6 @theslowbe

#NCTMinst

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7 @theslowbe

#NCTMinst

8 @theslowbe

#NCTMinst

Our Google Doc…

  • Go to h3p://theslowbe.weebly.com, go to this

workshop’s page, open the Google Doc

  • Populate the document with some “ordinary”

Algebra 2 problems from each unit

9 @theslowbe

#NCTMinst

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If there was one “big” thing you could change about Algebra 2, what would it be?

Phil Daro on “answer-geSng”

  • h3ps://www.youtube.com/watch?

v=DgTnmRyV9bc

  • Call to ac1on:

Delay answer-ge2ng

11 @theslowbe

#NCTMinst

Swooping in

  • h3p://www.1mssvideo.com/97

12 @theslowbe

#NCTMinst

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Goals for this workshop

  • Avoiding focus on “answer-geFng”
  • What is Cogni1ve Demand?
  • Modifying exis1ng problems and tasks, not

“reinven1ng the wheel”

  • Work 1me crea1ng shared resource to take back

to our classrooms

13 @theslowbe

#NCTMinst

What is CogniCve Demand?

Peg Smith, Juli Dixon, others:

  • Task selec1on and implementa1on are

important…

  • …but what about Algebra 2 content

specifically?

14 @theslowbe

#NCTMinst

What is CogniCve Demand?

This is a bridge of length three. Determine the number of beams in a bridge of any length.

15 @theslowbe

#NCTMinst

Dixon Nolan Adams Mathema1cs 2016

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High CogniCve Demand Task

This is a bridge of length three. Describe how the following generaliza1ons were visualized:

16 @theslowbe

#NCTMinst

Dixon Nolan Adams Mathema1cs 2016

3+4 n−1

( )

n+2n+ n−1

( )

3n+ n−1

( )

Levels of CogniCve Demand

  • Memoriza1on
  • Procedures without connec1ons
  • Procedures with connec1ons
  • Doing mathema1cs

17 @theslowbe

#NCTMinst Lower-Level Demands Higher-Level Demands

Smith & Stein, “Selec1ng and Crea1ng Mathema1cal Tasks: From Research to Prac1ce”, 2012

What “High CogniCve Demand” Looks Like

  • Messy!
  • Students “grapple with complexity”
  • Students respond directly to other students
  • Students do the sense-making
  • Leverage Standards for Mathema9cal Prac9ce

18 @theslowbe

#NCTMinst

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Rethinking “I-We-You”

  • Not “I-we-you” = gradual release
  • Instead “You-we-I” = produc1ve struggle

19 @theslowbe

#NCTMinst

What “High CogniCve Demand” Looks Like – Teacher moves

  • Facilita1ng discourse among students

– “What did she say?” – “Is that what she said?” – “You seem to disagree. Why?” – Summarize different responses, then say “It sounds like you have more to talk about” and walk away – S: “I don’t get it” T: “Then you have a ques1on to ask (other student)” – Teacher rarely gives authorita1ve answer to ques1on

20 @theslowbe

#NCTMinst

What “High CogniCve Demand” Looks Like – Teacher moves

  • Ques1on students’ correct answers! Give no

indica1on of correctness

  • “I saw a student do ______. What do you think?”
  • Have [student 2] use [student 1]’s approach to

solve the problem

  • Wait 1me

21 @theslowbe

#NCTMinst

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CogniCve Demand in Algebra 2

  • Promp1ng students to understand each others’

thinking keeps students engaged longer and at a higher cogni1ve level

  • Leveraging the Mathema9cs Teaching Prac9ces

to focus on processes, not just answers

  • Give students more credit: give them the
  • pportunity then let them pursue it

22 @theslowbe

#NCTMinst

Goals for this workshop

  • Avoiding focus on “answer-geFng”
  • What is Cogni1ve Demand?
  • Modifying exis1ng problems and tasks, not

“reinven1ng the wheel”

  • Work 1me crea1ng shared resource to take back

to our classrooms

24 @theslowbe

#NCTMinst

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Pose Purposeful QuesCons

25 @theslowbe

#NCTMinst

From NCTM’s Principles to Ac9ons (pp35-36): “Purposeful ques9ons allow teachers to discern what students know and adapt lessons to meet varied levels of understanding, help students make important mathema9cal connec9ons, and support students in posing their own ques9ons.”

Modifying Problems to Increase CogniCve Demand

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#NCTMinst Apply a Context Multiple Representations Open-Middle Generalize Reversibility Uniqueness Flexibility Defend/Dispute a Claim

Generalize

Then: Write an equa1on for an ellipse that contains the point (6,0). Now: Write 2 more equa1ons of ellipses that all contain the point (6,0). Write a general rule for any ellipse containing (6,0) Write a general rule for any ellipse containing (a,0)

27 @theslowbe

#NCTMinst

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Generalize

Then: Determine the end behavior of y = 2x – 3 Now: Write a rule to determine the end behavior of any exponen1al func1on: y = a b (x – c) + d

28 @theslowbe

#NCTMinst

Reversibility

29 @theslowbe

#NCTMinst

Then: Find the first four iterates of f(n) = n2 + 2 where n0 = -1. Now: Given -1, 3, 11, 123, … Explain the pa3ern and determine a (recursive) formula

Reversibility

30 @theslowbe

#NCTMinst

Then: Solve this system of equa1ons: Now: Create a system of equa1ons whose solu1on is (3,0,1)

x +2y + z = 4 y − z = −1 −2x + z = −5

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Reversibility

31 @theslowbe

#NCTMinst

Then: Find csc(A): Now: 5/4 is the cosecant of which angle? Y E A 10 8 6

Uniqueness

“Is there more than one solu1on?” “How do you know?” Notes:

  • Ask this ques9on regularly, even when there is
  • nly 1 unique solu9on
  • Use interes9ng ques9ons regularly that have

non-unique solu9ons

32 @theslowbe

#NCTMinst

Flexibility

NCTM on “procedural fluency”:

  • Apply procedures accurately, efficiently, and

flexibly

  • Transfer procedures to different problems and

contexts

  • Build or modify procedures from other

procedures

  • Recognize when one strategy/procedure is

more appropriate to apply than another

33 @theslowbe

#NCTMinst

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Flexibility

Then: Factor completely: y = x3 – 5x2 + 3x – 15 Now: Factor completely: y = (x3 – 6x2 + 5x) + (x2 – 2x – 15) Now factor again using a different approach. Which do you prefer? Write an expression for which you prefer the other approach.

34 @theslowbe

#NCTMinst

Flexibility

Then: Write an equa1on for: Now: Write two equa1ons using two different

  • func1ons. Which do you prefer? Why? Sketch

a graph that would make you prefer the other func1on

35 @theslowbe

#NCTMinst

Apply a Context

  • Recall:
  • Create your own context for:

B = 4n + (n-2)

36 @theslowbe

#NCTMinst

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Apply a Context

37 @theslowbe

#NCTMinst

http://www.nba.com/gametracker/#/20160508/CLEATL/lp/analysis

How is “root multiplicity” represented in this graph? What do they tell us about the game?

MulCple RepresentaCons

38 @theslowbe

#NCTMinst

www.nctm.org

MulCple RepresentaCons

39 @theslowbe

#NCTMinst

Then: The height of a hot air balloon can be modeled by h(t) = –0.08t4 + 4.7t3 – 84.3t2 + 539t . Find all local extrema. Now: Create a be3er representa1on for the company to display in their adver1sement that gives more clear details about the flight.

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Open-Middle

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#NCTMinst

Open-Middle

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#NCTMinst

Open-Middle

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Defend/Dispute a Claim

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y = a sin(b(x – c)) + d Esme: “There is only one value for b that makes this func1on have period 3π.” Do you agree or disagree with Esme? Write a paragraph to explain your reasoning.

Defend/Dispute a Claim

44 @theslowbe

#NCTMinst

Esme: “To find the complex roots of a quadra1c, I can reflect the parabola ver1cally over its vertex then use the real x-intercepts.” Do you agree or disagree with Esme? Support your answer mathema1cally.

Defend/Dispute a Claim

45 @theslowbe

#NCTMinst

Esme: “To find the complex roots of a quadra1c, I can reflect the parabola ver1cally over its vertex then use the real x-intercepts.”

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Defend/Dispute a Claim

46 @theslowbe

#NCTMinst

Esme: “Every polynomial has a unique factoriza1on.” Do you agree or disagree with Esme? Support your answer mathema1cally.

Phillips Exeter Academy

h3p://www.businessinsider.com/what-its-like-to-a3end-phillips-exeter-academy-2014-11?op=1

47 @theslowbe

#NCTMinst

48 @theslowbe

#NCTMinst

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Modifying Problems to Increase CogniCve Demand

49 @theslowbe

#NCTMinst Apply a Context Multiple Representations Open-Middle Generalize Reversibility Uniqueness Flexibility Defend/Dispute a Claim

ModificaCon and ReflecCon

Designate 1 person from your group to share: 1) one of your favorite modifica1ons 2) one reflec1on on the strategies

50 @theslowbe

#NCTMinst

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52 @theslowbe

#NCTMinst

https://www.edsurge.com/news/2016-06-28-5-things-teachers-want-from-pd-and-how-coaching-and-collaboration-can-deliver-them- if-implementation-improves

In the figure [below], arc BD is centered at A, and it has the same length as tangent segment

  • BC. Explain why sector ABD has the same area

as triangle ABC.

53 @theslowbe

#NCTMinst

Leave with a problem… @dynamic_math

h3ps://www.geogebra.org/m/npZNjN2K h3ps://www.geogebra.org/tbrzezinski

54 @theslowbe

#NCTMinst

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Jason Slowbe

Great Oak High School San Diego, CA @TheSlowbe jason.slowbe@gmail.com #NCTMinst

Sustaining High CogniCve Demand in Intermediate Algebra

Disclaimer

The National Council of Teachers of Mathematics is a public voice

  • f mathematics education, providing vision, leadership, and

professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all

  • students. NCTM’s Institutes, an official professional development
  • ffering of the National Council of Teachers of Mathematics,

supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of

  • viewpoints. The views expressed or implied in the Institutes,

unless otherwise noted, should not be interpreted as official positions of the Council.

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