Philipp am slides

The Nature of Knowledge for Teaching and Implications for

Research and Practice

Yopp Distinguished Speaker Series University of North Carolina, Greensboro

September 8, 2014

Dr. Randy Philipp

San Diego State University [email protected]

Thank You!

UNCG Mathematics Education Group in the School of Education The Graduate Students James D. and Johanna F. Yopp

Presentation Plan

•  Knowledge for Teaching and Knowledge for Teaching a Particular Subject

•  Assessing Specialized Content Knowledge

•  Rethinking Specialized Content Knowledge in a Particular Domain

•  Discussion

Presentation Plan




•  Discussion

What Knowledge of Mathematics Do Teachers Need?

Example, Javier, Grade 5 At the time of this interview, Javier had been in the United States about one year, and he did not speak English before coming to this country. (Javier, VC #6, 0:00 - 1:10)

6 × 12

= (5 × 12) + (1 × 12)

= [(12 × 10) × 12] + 12

= [12 × (10 × 12)] + 12

= [12 × (120)] + 12

= 60 + 12

= 72

One Representation of Javier’s Thinking

6 × 12

= (5 × 12) + (1 × 12) (Distributive prop. of x over +)

= [(12 × 10) × 12] + 12 (Substitution property)

= [12 × (10 × 12)] + 12 (Associative property of x)

= [12 × (120)] + 12

= 60 + 12

= 72


6 × 12

= (5 × 12) + (1 × 12) (Distributive prop. of x over +)

= [(12 × 10) × 12] + 12 (Substitution property)

= [12 × (10 × 12)] + 12 (Associative property of x)

= [12 × (120)] + 12

= 60 + 12

= 72


Place value

Unpacking The Knowledge Demands

What is the nature/category/classification of the knowledge required to… …solve 12 x 6 procedurally? …solve 12 x 6 using number sense? …understand Javier’s (and other students’) reasoning? …think to ask “How did you know that 12 x 5 is 60?” …think of a productive follow-up question to pose to

Javier after he solved this task? …situate the mathematical issues embedded in Javier’s

thinking in terms of the mathematics that has come before and how these ideas might unfold in future mathematics courses?

Knowledge for Teaching Ball, Hill, & Bass, 2005; Hill, Sleep, Lewis, & Ball, 2007

Common Content Knowledge—the knowledge teachers are responsible for developing in students

Specialized Content Knowledge—knowledge that is used in teaching, but not directly taught to students

Pedagogical Content Knowledge (Shulman, 1986)—the ways of represenCng and formulaCng the subject that make it comprehensible to others including knowledge of how students think, know, and learn.

Evaluate and understand the meaning of 12 ÷ 3.

Write a real-‐life story problem that could be represented by the expression 12 ÷ 3.

How might children reason about this task?

Unpacking The Knowledge Demands What is the nature/category/classification of the knowledge required to… …solve 12 x 6 procedurally? …solve 12 x 6 using number sense? …understand Javier’s (and other

students’) reasoning? …think to ask “How did you know

that 12 x 5 is 60?”

CCK

CCK, SCK

CCK, SCK, PCK

CCK, SCK, PCK

Consider an example of SCK in a field other than mathematics

Common Content Knowledge—the knowledge teachers are responsible for developing in students

Specialized Content Knowledge—knowledge that is used in teaching, but not directly taught to students

Pedagogical Content Knowledge (Shulman, 1986)—the ways of represenCng and formulaCng the subject that make it comprehensible to others including knowledge of how students think, know, and learn.

Is this knowledge taught? If so, where? If not, why not? Is this knowledge assessed by researchers? How?

Presentation Plan




•  Discussion

Principal Investigators Randy Philipp, PI Vicki Jacobs, co-PI

Faculty Associates Lisa Lamb, Jessica Pierson Research Associate Bonnie Schappelle Project Coordinators Candace Cabral Graduate Students John (Zig) Siegfried Others Chris Macias-Papierniak,

Courtney White Funded by the National Science Foundation, ESI-0455785

Participant Groups (N=129 with 30+ per group) PSTs, Prospective Teachers Undergraduates enrolled in a

first mathematics-for-teachers content course ________________________________________________________________________________________________________________________

IPs, Initial Participants 0 years of sustained professional development

APs, Advancing Participants 2 years of sustained professional development

ETLs, Emerging Teacher Leaders At least 4 years of sustained professional development and some leadership activities

_______________________________________________ SMSs, Strong Mathematics Students - Graduate or advanced undergrads taking advanced math courses

K–3

Tea

cher

s

Ones Task Andrew Task

PST IP AP/ETL SMS

Andrew ? Ones ?

Group Means by Task (0–4 scale)

PST IP AP/ETL SMS

Andrew 1.48 ? Ones 1.58 ?


PST IP AP/ETL SMS

Andrew ? 1.48 2.36 Ones ? 1.58 2.49


PST IP AP/ETL SMS

Andrew 1.67 1.48 2.36 ? Ones 0.31 1.58 2.49


PST IP AP/ETL SMS

Andrew 1.67 1.48 2.36 2.55 Ones 0.31 1.58 2.49


Two ETL’s solution to Andrew Task 1) Because he made 63 into 65 so that he could solve the problem. He got 40 and he subtracted the 2 in which he had added to simplify the problem.

2) 5 = 3 + 2 and if you only have 3 and you're subtracting 5, you can take away the 3 but you still have two more to take away, hence the -2.

63 – 23 40 – 2 38

63 - 25 -2 40 38

SMS’s solution to Andrew Task Explain why Andrew’s strategy makes mathematical sense. He makes the problem simpler by subtracting 20 from 60 and 5 from 3 and adding the results. Please solve 432 – 162 = ☐ by applying Andrew’s reasoning. 432 −162 0 − 30 300 270

63 - 25 -2 40 38

The Land of Specialized Content Knowledge

The$Land$of$SCK$

A$Mathema2cal$Path$to$the$Land$of$SCK$

A$Path$Through$Children’s$Mathema2cal$Thinking$to$the$

Land$of$SCK$

The$Land$of$Specialized$Content$Knowledge$

PST IP AP/ETL SMS

Andrew 1.67 1.48 2.36 2.55 Ones 0.31 1.58 2.49 ?


PST IP AP/ETL SMS

Andrew 1.67 1.48 2.36 2.55 Ones 0.31 1.58 2.49 0.94


Presentation Plan




•  Discussion

Project Z: Mapping Developmental Trajectories of Students’ Conceptions of Integers

•  Lisa Lamb, Jessica Bishop, & Randolph Philipp, Principal Investigators

•  Ian Whitacre, Faculty Researcher

•  Spencer Bagley, Casey Hawthorne, Graduate Students •  Bonnie Schappelle, Mindy Lewis, Candace Cabral, Project

researchers

•  Kelly Humphrey, Jenn Cumiskey, Danielle Kessler, Undergraduate Student Assistants Funded by the National Science Foundation, DRL-0918780

Solve each of the following and think about how you reasoned. If you have time, solve another way.

1)   3 – 5 = ___

2)   -6 – -2 = ___

3)   - 2 + ___= 4

4)   ___+ -2 = -10

So, Why Negative Numbers? Even secondary-school students who can successfully operate with negatives have trouble explaining.

KCC Montage, High School, 2:35

��So, Why Negative Numbers?��

Many middle-school students do not understand what they are doing with negatives.

Valentin, Grade 7, 1:25

��So, Why Negative Numbers?��

Many young children hold informal knowledge about negatives on which instruction might be based.

Rosie, 1st grade, ___ + 5 = 3, 1:23

One Last Reason… •  Negative numbers comprise (almost) half of the

reals!

Why Study Negative Numbers?

•  The literature that exists tends to either point out student difficulties, or offer purported instructional paths.

•  Too little literature documents students’ informal understandings.

•  Can we connect the goals of integer instruction to something other than procedures? And if so, what might that be?

Ways of Reasoning

Students who have negative numbers in their numeric domains typically approach integer tasks using one of the following five ways of reasoning:

Order-based reasoning Analogically based reasoning Formal mathematical reasoning Computational reasoning

Ways of Reasoning Order-based Analogically based Formal mathematical Computational

RandyLogNec6, Grade 1, 1:11-1:43 –2 + 5 = __

Ways of Reasoning Order-based Analogically (Analogy- based) Formal mathematical Computational

Roland, Grade 4, 0–0:49 –5 + –1

Ways of Reasoning Order-based Analogically based Formal mathematical Computational

Roland, Grade 4, 0–0:48, -5 - -3 -5 – -3

Teachers’ Knowledge About Integers

To begin to make sense of how teachers think about integers, we interviewed 10 seventh-grade teachers to determine their understanding of integers and their perspectives about teaching integers and about students’ thinking. We posed integer tasks, we asked them about their teaching, and we showed them video clips of children solving open number sentences.

Research Questions 1)  Do 7th-grade teachers answer integer tasks

correctly, and if so, how successfully?

2)  Do 7th-grade teachers invoke ways of reasoning, or rely instead upon procedures?

3) Do 7th-grade teachers recognize ways of reasoning in students?

4) How explicitly do 7th-grade teachers articulate the kinds of reasoning that they use or that students use?

5) What are the seventh-grade teachers’ goals for integer instruction?

Research Questions 1)   Do 7th-grade teachers answer integer tasks






Yes, very successfully.



2)   Do 7th-grade teachers invoke ways of reasoning, or rely instead upon procedures?




Yes, very successfully.

They invoke ways of reasoning.







Yes, very successfully. They invoke ways of reasoning.

This is mixed.








This is mixed.

Not at all.








This is mixed.

Not at all.

Almost entirely procedural/rule-based.

Brief Examples 1)  Do 7th-grade teachers answer integer tasks







This is mixed.

Not at all.

Almost entirely procedural/rule-based.


Example: -3 – ! = 2 Raymond: “So I am thinking about the number line…So I am starting somewhere… and what do I do to end up at positive two? I am moving one, two, three, four, five––five units to the right.” “I am moving the opposite direction, so I would write down negative five here.”

Order-based reasoning

Formal reasoning

Teachers invoked ways of reasoning. Even with the result unknown sentence, -3 + 6 = !, all but one invoked reasoning.

Yes, they do.

3)  Do 7th-grade teachers recognize ways of reasoning in students?

Roland, -5 – (-3) Jessica: “He seems to have a solid understanding

that adding negatives to negatives gets you further away. So in his mind, he is saying, so subtracting, that must bring me closer.”

Raymond: “I think he got confused. There’s no context involved…I don’t quite understand him when he used the opposite. Opposite of what? …He said, “minus minus. I don’t believe that he knows the meaning of “minus minus.”

This is mixed.

Three Implications for Teacher Integer Study

•  Revise the Specialized Content Knowledge About Integers


•  Revise the Specialized Common Content Knowledge About Integers



•  Consider the Challenge in Teachers’ Adopting New Goals for Integer Instruction



•  Consider the Challenge in Teachers’ Adopting New Goals for Integer Instruction

•  Stop Seeking the Holy Grail for Integer Instruction: There is No One Best Approach or Model for Teaching Integers

The relationship among three types of knowledge for mathematics teaching.

Discussion

Thank you.