algebra for college and career readiness: the role …...snakewood a very special snake has a...

Effective Teaching with Principles to Actions: Implementing College- and Career-Readiness

Standards

Gail Burrill Michigan State University

[email protected]

Algebra for College and Career Readiness: The Role of Discussions

Snakewood

A very special snake has a pattern on its skin. It is born with one red ring on its skin. Each year it sheds its skin. In year 2, a black ring develops in the middle of the red ring. In year 3, the same thing happens with each red ring. The black ring stays the same. (Roodhardt et al, 1997)

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Snakewood

A very special snake has a pattern on its skin. It is born with one red ring on its skin. Each year it sheds its skin. In year 2, a black ring develops in the middle of the red ring. In year 3, the same thing happens with each red ring. The black ring stays the same. (Roodhardt et al, 1997)

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Take a few minutes to model – either with cubes or by drawing - the snake over several generations.

overview

• Effective teaching and learning • Thinking about discussions • Conceptual and procedural knowledge

– systems of equations

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Standards for Mathematical Practice: Students

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the

reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated

reasoning CCSS, 2010

Teaching and Learning Practices

• Learning: Mathematical Practices for Students – habits of mind which students should employ

as they are engaged in doing mathematics

• Teaching: Mathematical Teaching Practices – teacher instructional behaviors that support

student learning

Principles to Action: Effective Teachers

• Establish mathematics goals to focus learning. • Implement tasks that promote reasoning and

problem solving. • Use and connect mathematical representations. • Facilitate meaningful mathematical discourse. • Pose purposeful questions. • Build procedural fluency from conceptual

understanding. • Support productive struggle in learning math. • Elicit and use evidence of student thinking.

(NCTM, 2014)

Think about the discussions related to Snakewood you had with your partner, at your tables and then whole group. Did any of these contribute to your learning? If so, how?

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Water Tower

Calculus class discussing the rate of change (slope) of the water flowing into a water tower using a graphical representation, in particular find the rate of change of the slope at 2.5. National Board for Professional Teaching Standards ATLAS library.

As you watch the clip, consider the question: What are students talking about?

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Functions and Sliders T3 Professional Development Workshop, 2012)

High school algebra class examining the effect of changing the parameters on the graph of the equation of a line. In particular, if the y –intercept is held constant, what happens as the coefficient of the variable is changed? As you watch the clip, think about the question: What are students talking about?

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How might you orchestrate a discussion to better support the calculus students? What questions could you pose that would engage them in thinking about the math not just how to do the math?

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• What is a mathematically productive discussion?

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• What is a mathematically productive discussion?

• What does a mathematically productive

discussion look like when you see/hear it?

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The “stuff” of algebra

• Variables • Expressions • Equations- relationships between expressions And what we do with the stuff: Manipulate, solve and graph, and Make connections among relationships and representations to solve problems

Systems of Equations

x + 2y = 17 x - 2y = 3 The graphs of the two equations shown above intersect at the point (x, y). Which is the value of x at the point of intersection? a) 3 ½ b) 5 c) 7 d) 10 e) 20

NAEP 2005

Systems of Equations

x + 2y = 17 x - 2y = 3 The graphs of the two equations shown above intersect at the point (x, y). Which is the value of x at the point of intersection? a) 3 ½ b) 5 c) 7 d) 10 e) 20 Results: a) 12% b) 18% c) 21% d) 31%* e) 12%

NAEP 2005

What is a solution?

Which of these ordered pairs is on the graph of 3x - 2y = 12?

A) (0, 4) B) (3, 4) C) (4, 0) D) (-6, 0)

Wisconsin, grade 10, 2006

What is a solution?

Which of these ordered pairs is on the graph of 3x - 2y = 12?

13% A) (-6, 0) 8% B) (0, 4) 25% C) (3, 4) 52%* D) (4, 0)

Wisconsin, grade 10, 2006





(NCTM, 2014)

• Conceptual Knowledge: – Makes connections visible, – enables reasoning about the mathematics, – less susceptible to common errors, – less prone to forgetting.

• Procedural Knowledge: – Strengthens and develops understanding – allows students to concentrate on relationships

rather than just on working out results NRC, 1999; 2001

6.PtA: Building procedural knowledge from conceptual knowledge?

Systems: A learning trajectory What is central to thinking about solving systems of equations? • Conceptual understanding: variable, equation;

solution (1 and 2 variables) • Building Block: Linear • Building to

– Quadratic, Rational, Irrational, Exponential, Logarithmic, ..

• Misconceptions

• Building expressions • What is an equation? • What is a solution for equations in two

variables? • Linear combination method

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Organizing procedural knowledge: Sorting

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Create at least two grouping based on characteristics of the systems. Find a new system and see where it fits.

Engaging Activities

• Sorting – Which systems were problematic, if any, and why? – What did you learn about people’s thinking about

solving systems of equations? – What questions would you ask students during

the activity to help them make progress? – What follow up task could you have students do

after completing this task?

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Exit Ticket

• What are you wondering about related to discourse?

• Related to any of the other Principles for Effective Teaching?

• Other comments:

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References • Algebra Nspired. (2010). From Expressions to Equations. Texas Instruments

Education Technology. www.ti-mathnspired.com/ • Common Core Standards. College and Career Standards for Mathematics

2010). Council of Chief State School Officers (CCSSO) and (National Governor’s Association (NGA).

• Functions & Sliders. (2012). Video clip from T-Cubed Professional Development CCSS workshop by Brennan, B., Olson J. & the Janus Group. Curriculum Research & Development Group. University of Hawaii at Manoa, Honolulu HI

• National Assessment for Educational Progress (2005). Released Item. National Center for Educational Statistics.

• National Research Council. (1999). How People Learn: Brain, mind, experience, and school. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). Washington, DC: National Academy Press.

• National Research Council (2001). Adding It Up. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) Washington DC: National Academy Press. Also available on the web at www.nap.edu.

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http://www.ti-mathnspired.com/

http://www.nap.edu

References • National Council of Teachers of Mathematics, (2014). Principles to action:

Ensuring Mathematical success for all. (2014). Reston VA: Author • Roodhardt, A., Kindt, M., Burrill, G., & Spence, M. (1997). Patterns and

Symbols. From Mathematics in Context. Directed by Romberg, T. & de Lange, J. Chicago IL: Encyclopedia Britannica

• Wisconsin Department of Education. (2006). Released items mathematics grade 10 fall.

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Disclaimer The National Council of Teachers of Mathematics is a public voice

of 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 offering 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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Research suggests learning may be hindered by

• isolated sets of facts that are not organized

and connected or organizing principles without sufficient knowledge to make them meaningful (NRC, 1999)

Worthwhile Tasks

• Focused on important mathematics; clear mathematical goal

• Provide opportunities for discussion • Provoke thinking and reasoning about the

mathematics; high level of cognitive demand • Engage students in the mathematical practices

and process standards • Create a space in which students “wonder,

notice, are curious” SSTP, 2013

Tasks we give and questions we ask should ensure students

are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback.

encounter contrasting cases- notice new features and identify important ones.

struggle with a concept before they are given a lecture

develop both conceptual understandings and procedural skills

National Research Council, 1999; 2001

Goals

• Clearly articulated and well-defined lesson foci are key elements of successful instruction (Black et al., 2004; Smith & Stein, 2011; Hiebert et al, 2002; Chappuis & Stiggins, 2002)

• Ill defined and not well understood (learning goals, instructional goals, objectives, outcomes,…)

• Students will – explore the Pythagorean theorem, – work together in a group to graph linear equations, – have a discussion about lines

• Understanding is not a learning goal??

Learning Goals www.itma.vt.edu/modules/spring03/instrdes/introduction.htm)

• specify the essential knowledge, skills and dispositions students should take from the lesson;

• shape the structure of the lesson; • guide the selection of instructional materials and

resources; • suggest interventions to support learners when well

specified; • provide guidelines for student assessment; • suggest the types of evidence needed to determine if

the goals have been achieved.

http://www.itma.vt.edu/modules/spring03/instrdes/introduction.htm

Three Types of Goals

• performance goals focus on what students’ will do (e.g., compare and contrast linear and exponential functions);

• learning goals focus on learning at a general level (e.g., learn about linear equations);

• learning goals articulate the key ideas or concepts students are to learn (e.g., linear functions have a constant rate of change that is additive and exponential functions have a constant rate of change that is multiplicative). (Smith et al, 2014)

Goals depend

• On place in the learning trajectory, focus teacher wants to emphasize, prior knowledge,…

• Perhaps: Recognize and use the formula for area of a triangle in solving a problem

The “stuff” of algebra

• Variables • Expressions • Equations - relationships between expressions And what we do with the stuff: Manipulate, solve and graph, and Make connections among the relationships and representations to solve problems


Standards


[email protected]

Connecting Algebraic Concepts through Well Chosen Tasks

How can you plan for a mathematically productive discussion?

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How can you make a mathematically productive discussion happen? Good discussion does not happen by accident - it takes

Planning, planning, planning….planning

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What is Changing?

• As the step changes in the figures below, the _____________ also changes.

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Step 1 Step 2 Step 3

Peterson, 2006

With your partner, pick one of the attributes in our list and investigate how it changes. Make a conjecture and try to prove it. How would a graph, a table, and/or an equation support your conclusion? (If time, explore a 2nd or 3rd property)

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• In what order would you choose to discuss the solutions? Why?

• What connections would you want students to discuss? How would you help them see those connections?

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Possible learning goals

• Distinguish between recursive and closed form rules for sequences

• Distinguish between linear and quadratic relationships relationships

• Describe a geometric pattern by an arithmetic expression

• Recognize and be able to describe the components of an arithmetic sequence

• Explain what rate of change means in different situations

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• Which changes led to linear equations and which lead to quadratic? Is there an explanation?

• What were the advantages and disadvantages of different approaches (symbolic vs. tables vs. graphs vs diagrams)?

• What is the distinction between patterns and proof? Is this important? Why or why not?

• Identify where it was important to “attend to precision”.

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Processing the activity • What answers do you expect to see?

(Anticipating) • What are students doing? (Monitoring) • What responses are worth discussing?

(Selecting) • How will you sequence the responses?

(Sequencing) • What is the mathematical punchline?

(Connecting)

9 Smith & Stein, 2011

The 5 Practices

• Anticipate • Monitor • Select • Sequence • Connect Smith & Stein, 2011

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Think of the three opportunities for discussion: with your partner as you investigated the change, explaining and questioning each other about your approaches and conclusions on the posters around the room, and the group discussion at the end about the mathematics. What was useful about each of those discussions?

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Bungee Barbie

Illuminations, 2015

What is important about having students talk?

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Switching from structure in geometric structures to structure from an algebraic perspective:

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“look before you leap”

In your groups, choose three or four of the problems in 1-5 and decide what “look before you leap” means.

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Compare the nature of the discussions for tasks involving “Look before you Leap” and those in the worksheet. http://www.math-

aids.com/Algebra/Algebra_1/Equations/Multiple_Step_Integers.html http://www.math-

aids.com/Algebra/Algebra_1/Equations/Multiple_Step_Integers.html

Another look at structure

Select two whole numbers a and b, not too large. Let a be the larger of the two; find

a2+b2 a2-b2 2ab.

17 Pythagorean Triples, 2010

Write down at least two observations about the numbers in the table.

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Write down at least two observations about the numbers. Share your observations with others in your group. Choose one observation and prove your claim.

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What level of math talk was going on during these tasks?

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(NCTM, 2014)

Think about all of the tasks we have done so far. • Pythagorean Triples • What is Changing? • Look before you Leap • Snakewood • Solving Systems of Equations • Sorting

What are some characteristics of tasks that promote discussion?

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• Illuminations, Barbie Bungee. Accessed 2/5/15. http://illuminations.nctm.org/uploadedFiles/Content/Lessons/Resources/6-8/Barbie-AS-Project.pdf

• Hufferd-Ackles, K., Fuson, K., & Sherin, M. (2004). Describing Levels and Components of a Math-Talk Learning Community. Journal for Research in Mathematics Education. 35(2), 81-116.

• Smith, M., & Stein, M.(2011). Five practices for orchestrating productive mathematics discussions. Reston VA: National Council of Teachers of Mathematics.

• Peterson, B. (2006). Linear and quadratic change: A problem from Japan. The Mathematics Teacher, 100(3). Reston VA: National Council of Teachers of Mathematics.

• Pythagorean Triples. Geometry Nspired. (2009). Texas Instruments Education Technology. www.ti-mathnspired.com/




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Standards


[email protected]

Creating Opportunities for Mathematical Discussions

My Favorite No

• Solve the equation the way you think your students might:

• 2x - 3(x - 4) = 18+x

2 https://www.youtube.com/watch?v=Rulmok_9HVs

Equivalent Expressions

Albert wants to simplify the expression: 8(3–y) + 5(3–y) Which of the following is equivalent to the expression

above? 29% A. 39 – y

40%* B. 13(3 – y) 7% C. 40(3 – y)

24% D. 13(6 – 2y)

Florida 2006, Grade 9

Now what? 13(3-y) 39-13y 13(3-x) 39-y (24-8y)+(15-5y) 13(3+y) 24+15 39(y) 39-3y 5(3-y)+8(3-y) 7(3-y)+5(3-y) 8(3+y)+3(5+y)

Wilson, 1011

A Statistical Exploration: Fair share

The total number of goals scored by all the teams in the tournament is 54. If the teams were fairly matched, they would have scored 6 goals each. Use post-its and the number line on the chart to make a graph showing number of goals each team might have scored.

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When would the teams be most evenly matched? The least? Another way to ask this- when would there be the greatest variability among the teams? The least?

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Which tournament had the greatest variation in teams?

Building Concepts, 2015

Why MAD?

• The average temperatures in San Francisco and St. Louis are both 57º.

• Hummmmmmm?

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• How might the concept of MAD be transformed to a notion of standard deviation?

How would you find the standard deviation in the number of goals for teams with scores for the tournament of 2, 6, 8, 2, 8, 7, 3, 10?

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MAD to SD

standard deviation= where xi= a data value, x is the mean and n is the number of data values.

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To PROBE or uncover students’ thinking. • understand how students are thinking about the problem. • discover misconceptions. • use students’ understanding to guide instruction.

To PUSH or advance students’ thinking. • make connections • notice something significant. • justify or prove their thinking.

The only reasons to ask questions are: (Black et al., 2004)

Never Say Anything a Kid Can Say (Reinhart, 2000)

• Every time you are tempted to give a student an answer, ask them a question instead.

• Ask questions that surface student thinking – Process questions not product questions

• Consider wait time • Encourage students to ask questions

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Asking Questions

• Ask questions that probe and push student thinking such as: – “What else do you notice?” – “What happens when…?” – “How do you know?”

Black et al, 2004 pg 12

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Framework for asking questions

1. Mathematical concepts 2. Procedures 3. Connections 4. Structure 5. Comparisons & contrasts 6. Prior Knowledge

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1. Mathematical concepts (What is a deviation from the mean?)

2. Procedures (Why do you have to divide by the number of teams? Why do we take the absolute value? What would happen if you added the differences?) 3. Connections (How does the MAD related to the standard deviation?)

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4. Structure (Can a graphical representation help us understand MAD? ) 5. Comparisons & contrasts (How will the MAD change if you add 3 goals to each team’s scores? If you double each team’s scores?) 6. Prior knowledge (What do you need to know in order to understand and work with the MAD?)

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Strategies to create discourse

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What is an equation?

Lesson Lab

What is an equation?

Lesson Lab

Is this an equation? x/3 = 20 x = y 3x = 5 (x+5)(x+2) x = 2x 0 = 2

Examples vs Non-Examples

Which of the following could be solutions to a system of equations? Explain why or why not. • {1} • (1,-1) • ø • (1,-1), (1,1), (-1,1), (-1,-1) • (1,2,3) • {(x,y)/y=2x-3} • x=0 & y=2 • x=0 • y=x

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Engaging Activities

• Examples vs Non-Examples – What would your learning objective be that would

allow you to use this activity? – What previous knowledge do students need in

order to complete this task? – What questions would you ask students during

the activity to help them make progress? – What would you have students do after

completing the task?

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Systems of equations

Jeopardy • Write an equation whose solution is (2, 3) • Write two equations whose solution is (2, 3)

Participation quiz (PCMI, 2011)

• High school algebra class working on factoring

• You have expectations about the way discussions should happen in your classroom. Do your students know what they are?

• As you watch, what norms are being established to encourage discussion?

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Why are norms important?

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Norms for students

• Take turns • Listen to others ideas • Disagree with ideas not people • Be respectful • Helping is not the same as giving answers • Confusion is part of learning • Say your “becauses” • “I can’t do that yet?”

Horn, 2012

Norms have a purpose and need to be clear about this purpose. The enable students to achieve the math goals of learning content and how to think mathematically. Help students grow as listeners and as questioners Enable students to can take charge of their own learning and that of their peers.

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Norms for teachers

• Listen for what can be learned about students' thinking rather than for correct answers

• Identify & check a “hinge point” in the lesson where student understanding is critical for moving on

• “no hands up, except to ask a question” Leahy et al, 2005

• Be relentless in asking what does it mean/why it works • Maintain neutral stance with respect to answers • Record responses so everyone can think about them • Wait time before responses/after response • Deflect questions to students • Offer examples/counterexamples to test understanding • Plan questions/discussion in advance

Burrill, 2013

Things to get “math talk” going in your classroom

• Sentence starters • Gallery walk • Questioning framework • Favorite no • Dyads • Pair-share • ……

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Mathematical Discourse

• Where did we see it today?

• Where did we see it yesterday?

• How does it look in your classroom?

• What can you do to get it there more often?

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Exit ticket

• What was your biggest take away? • What ideas will you take from the institute to

do in your classroom?

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References • Black, P. Harrison, C., Lee, C., Marshall, E., & Wiliam, D. (2004). “Working

Inside the Black Box: Assessment for Learning in the Classroom,” Phi Delta Kappan, 86 (1), 9-21.

• Building Concepts: Mean as Fair Share (2015). Texas Instruments Education Technology. education.ti.com

• Burrill, G. (2014). Reflecting on Practice, Summer School Teacher Program, Park City Mathematics Institute.

• Burrill, G. (2013). Talk at National Council of Supervisors of Mathematics annual meeting.

• Florida Department of Education (2006). FCAT Mathematics Released Items, Grade 9.

• Horn, I. (2012). Strength in numbers: Collaborative learning in secondary mathematics. Reston VA: National Council of Teachers of Mathematics

• Kader, G., & Memer, J. (2008). Contemporary curricular Issues: Statistics in the middle school: Understanding center and spread, pp. 38-43

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References • Lesson Lab (2004). Equations. Video of Nick Branca teacher workshop • National Assessment for Educational Progress (2005, 2011). Released

Item. National Center for Educational Statistics. • Park City Mathematics Teacher Program. (2010). Reflecting on Practice. • Reinhart, Steven C., (2000). Never say anything a kid can say! Mathematics

Teaching in the Middle School. Apr. 478–83. • Wilson, J. (2011). Personal correspondence

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References

• Geometry Nspired. (2009). Texas Instruments Education Technology. www.ti-mathnspired.com/

• Black, P. & Wiliam, D. (1998). “Inside the Black Box: Raising Standards Through Classroom Assessment”. Phi Delta Kappan. Oct. pp. 139-148.

• Black, P. Harrison, C., Lee, C., Marshall, E., & Wiliam, D. (2004). “Working Inside the Black Box: Assessment for Learning in the Classroom,” Phi Delta Kappan, 86 (1), 9-21.

• Common Core Standards. College and Career Standards for Mathematics 2010). Council of Chief State School Officers (CCSSO) and (National Governor’s Association (NGA).

• Functions & Sliders. (2012). Video clip from T-Cubed Professional Development CCSS workshop by Brennan, B., Olson J. & the Janus Group. Curriculum Research & Development Group. University of Hawaii at Manoa, Honolulu HI



• Hufferd-Ackles, K., Fuson, K., & Sherin, M. (2004). Describing Levels and Components of a Math-Talk Learning Community. Journal for Research in Mathematics Education. 35(2), 81-116.

• Lesson Lab (2004). Equations & Buying CDs. Classroom video • National Assessment for Educational Progress (2005). Released Item.

National Center for Educational Statistics. U.S. Department of Education. • National Research Council (2001). Adding It Up. Kilpatrick, J., Swafford,

J., & Findell, B. (Eds.) Washington DC: National Academy Press. Also available on the web at www.nap.edu.

• Smith, M., & Stein, M.(2011). Five practices for orchestrating productive mathematics discussions. Reston VA: National Council of Teachers of Mathematics

• James Zull, ( 2002). The Art of Changing the Brain: Enriching the Practice of Teaching by Exploring the Biology of Learning. Association for Supervision and Curriculum Development, Alexandria, Virginia

http://www.nap.edu



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algebra for college and career readiness: the role …...snakewood a very special snake has a...

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