Empowering Pre-Service & New Math Teachers to Use the Common Core Practice Standards

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Empowering Pre-Service & New Math Teachers to Use the Common Core Practice Standards

Tuesday, August 26, 2014

Dr. Tim Hudson,

Senior Director of Curriculum Design,

DreamBox Learning

Benjamin Braun,

Associate Professor of Mathematics,

the University of Kentucky

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Empowering Pre-Service & New Math Teachers to Use the

Common Core Practice Standards

August 26, 2014

Benjamin Braun, PhDAssociate Professor of Mathematics, U of Kentucky

Editor-in-Chief, American Mathematical Society blog “On Teaching and Learning Mathematics”

Twitter: @BraunMath

Tim Hudson, PhDSenior Director of Curriculum Design, DreamBox Learning

Former K-12 Mathematics Curriculum Coordinator, Parkway School District

Twitter: @DocHudsonMath

1970-1990

• “Back to basics” movements in 1970s led to influential reports arguing in favor of balance between conceptual and procedural understanding:o A Nation at Risk (1983, NCEE)o An Agenda for Action (1980, NCTM)

• NCTM Standards released in 1989.

2000-2001

• Role of standards-based assessment increased in early 2000’s with No Child Left Behind.

• At the same time, updated NCTM Standards released in 2000, and National Research Council report Adding It Up released in 2001.

2004

• Must read: “The Math Wars,” Alan H. Schoenfeld, Educational Policy, Vol. 18 No. 1, January and March 2004, pp. 253-286. (PDF versions available online.)

NCTM divided proficiency into two categories inPrinciples and Standards for School Mathematics (2000)

Content• Numbers and operations• Algebra• Geometry• Measurement• Data analysis and

probability

Process• Problem solving• Reasoning and proof• Making connections• Oral and written

communication• Uses of mathematical

representation

NRC emphasized five “strands” of proficiency in Adding It Up (2001)

• Conceptual understanding: comprehension of mathematical concepts, operations, and relations

• Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

• Strategic competence: ability to formulate, represent, and solve mathematical problems

• Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification

• Productive disposition: habitual inclination to see mathematics as sensible, useful and worthwhile, coupled with a belief in diligence and one’s own efficacy

Common Core Mathematical Practice Standards1. 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.

Implications for Pre-service Teachers?

Many (if not most) pre-service teachers at the elementary, middle, and secondary levels do not have a robust set of mathematical practices, as this has not been part of their own educational experience.


This creates a disconnect between content and methods courses, and also between pre-service coursework and in-service curriculum and assessment expectations.


Teacher educators, including faculty teaching both methods and content courses, need to ensure that pre-service teachers enter the beginning of their careers with an understanding that mathematical proficiency extends beyond content.


Challenges to incorporating practices in pre-service teacher courses include balancing practices and content, effectively training college faculty (including adjuncts and TAs), and building quality connections between content and methods instructors so these courses articulate well.


Even in non-CCSS states, the NRC and NCTM reports of the past 30+ years have had a major impact on curriculum and assessment, so this matters even in non-CCSS states.

Implications for Year 1-3 Teachers?

In situations where there has been inadequate Pre-Service training for mathematics teachers, new teachers need even more content-specific support.


Many schools, districts, and states do not have adequate mathematics curriculum leadership to support new math teachers in content-specific ways.


New teacher induction and PD often emphasizes other aspects of teaching instead of curriculum (i.e., classroom management, parent communication, or building culture).


What are math teachers hired to accomplish?

What is mathematics?

How do people learn mathematics?

Grant Wiggins

What’s the job of a teacher?The crying need for a genuine job

description.

grantwiggins.wordpress.com 7-25-14

Grant Wiggins

“A real job description would be written around key learning goals and Mission-related outcomes.

• What am I expected to cause in students?• What am I supposed to accomplish?

Whatever the answer, that’s my job.”


Must Cause 4 Things in Learners

1. greater interest in the subject and in learning than was there before, as determined by observations, surveys, and client feedback



2. successful learning related to key course goals, as reflected in mutually agreed-upon evidence



3. greater confidence and feelings of efficacy as revealed by student behavior and reports (and as eventually reflected in improved results)



4. a passion and intellectual direction in each learner as determined by student-initiated pursuits, observations, surveys, and behavior.


Does Your Mission Obligate Teachers to Achieve these Goals?

1. greater interest than was there before2. successful learning related to key goals3. greater confidence and feelings of efficacy4. a passion and intellectual direction

Do Teachers Hired at Your School Know these are their Goals?

Do Teachers Receive Feedback about how well they’ve Achieved these Goals?


Two ways to incorporate mathematical practices in teacher training and

professional development

Using Writing Assignments

Pre-service teachers need to write and discuss mathematical practices explicitly in content courses. This can’t be “left to the methods courses” to handle, there must be a dialogue.

Using Writing Assignments

One of the best tools we have for this task is writing assignments.

• Personal writing, e.g reflective essays• Expository writing, e.g. report on good

contact/practices contact points• Critical writing, e.g. analyzing practices of

peers

Example of a Personal Essay Prompt

There are eight Standards for Mathematical Practice in the Common Core State Standards for Mathematics. Select at least three of these standards to consider. For each of the standards that you select, discuss a situation where you have observed one of your classmates demonstrating that practice in their work. This situation might have arisen from in-class group work, from working with a study group on homework, from hallway discussions of a problem, etc, but you should discuss a moment when your classmate was using one of these practices when working on a mathematical problem. You should explicitly connect each situation with the written description of the related practice standard given in the Common Core.

Low-threshold high-ceiling problems

• Nothing is better than doing mathematics while receiving quality feedback for developing good mathematical practices, which teachers must have if they are to help others develop them.

• Students need to engage with low-threshold-high-ceiling (LTHC) problems.

• Open (unsolved) problems in math are a great source of LTHC problems!

K-5 level open problem #1

Fibonacci Primes• The Fibonacci numbers are

1,1,2,3,5,8,13,21,34,55,... obtained by adding the two previous numbers to get the next in the sequence.• OPEN QUESTION: Are there infinitely

many prime Fibonacci numbers?


Fermat Primes• The Fermat numbers are 2^(2^n)+1 for

all non-negative integers n, e.g. 3, 5, 17, 257, 65537,...• OPEN QUESTION: Are there infinitely

many prime Fermat numbers?


Collatz Conjecture• Given a positive integer n, if it is odd then

calculate 3n+1. If it is even, calculate n/2. Repeat this process with your new number.• Example: 1,4,2,1,4,2,1,4,2,1,...• Example: 5,16,8,4,2,1,...• OPEN QUESTION: If you start with any

positive integer, does this process always end by cycling through 1,4,2,1,4,2,1,...?


Erdos-Strauss Conjecture• OPEN QUESTION: For every positive

integer n larger than 1, does there exist a solution to4/n = 1/x + 1/y + 1/zusing positive integers x, y, and z?

• Example: 4/5 = 1/2 + 1/5 + 1/10

Higher-level open problems

• Parity of the partition function

• Irrationality of Euler-Mascheroni constant

• Lagarias’s reformulation of the Riemann Hypothesis

Many more are available at: http://en.wikipedia.org/wiki/List_of_conjectures

http://en.wikipedia.org/wiki/List_of_conjectures

K-5 Level Closed ProblemOn day three of the bicycle race, Donald’s

time was:3 hours, 4 minutes, and 11 seconds.

Keina’s time was:2 hours, 58 minutes, and 39 seconds.

How long was Keina finished before Donald crossed the finish line?

Donald & Keina

Hours Minutes Seconds

3 4 112 58 39

\ 3 X71\

61

3

\2 \

6 3\5 1

50 2304 – 298 = ?

44 academic pure mathematicians were asked to estimate (make reasonable guesses) for 20 multiplication and division problems (Ex. 482 x 51.2 and 546 ÷ 33.5)

Strategy Frequency UsedUse of fractions 40%

Using “nicer” numbers 17%

Rounding two numbers 16%

Rounding one number 8%

Factorization 8%

Standard algorithms 4%

Distributive Property 3%

Computational Estimation Strategies of Professional Mathematicians, Dowker, Journal for Research in Mathematics Education, Vol. 23 ©1992

Oxford University 1992

44 academic pure mathematicians were asked to estimate (make reasonable guesses) for 20 multiplication and division problems (Ex. 482 x 51.2 and 546 ÷ 33.5)

Strategy Frequency UsedUse of fractions 40%

Using “nicer” numbers 17%

Rounding two numbers 16%

Rounding one number 8%

Factorization 8%

Standard algorithms 4%

Distributive Property 3%



“To the person without number sense, arithmetic is a bewildering territory in which any deviation from the known path may rapidly lead to being totally lost. The person with number sense…

has, metaphorically, an effective ‘cognitive map’ of that same territory.”



Summary• It is well-established that mathematical proficiency involves

both practices and content.

• All teachers need support in developing skillful approaches to teaching both mathematical content and practices.

• An excellent way for pre-service and new in-service teachers to develop their understanding of the practices is to work on LTHC problems themselves, then reflect on the mathematical practices they used in their own work.

• Everyone - teachers and students - benefit the most from receiving quality feedback when developing their content knowledge and mathematical practices.

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