Mathematics PCK Workshop #1Day 1RAMALLAH, PALESTINE MARCH 22 & 24, 2013

Dr. Elizabeth (Betsy) McEneaney[Mac-Uh-Ninny]Dept. of Teacher Education and Curriculum StudiesUMass – Amherstemcenean@educ.umass.edu

IntroductionsYOUR NAMEWHERE YOU WORK AND WHAT YOU DO THERE WHAT YOU HOPE TO LEARN IN THESE TWO DAYSONE INTERESTING THING ABOUT YOU THAT NOT MANY PEOPLE KNOW

Goal for this workshop: Enhance your capacity to prepare in-service teachers to use effective strategies for interpreting and transforming mathematics content knowledge into pedagogically appropriate teaching strategies and learning environments in mathematics, grades 5 to 10.

Honoring your experience and intuition…

Honoring your experience and intuition…

Much of this you already know…

We will review, synthesize, brainstorm, adapt.

By the end of the day today, I hope you will be able to:

Identify major math concepts and skills that students need;

Anticipate and diagnose some typical conceptual difficulties;

Identify and incorporate students' prior knowledge effectively;

’

You hope to:

Our Mission in Math•Consolidate good “procedural knowledge”•Cultivate good “conceptual knowledge”

through active learning

•For both “concrete operational” and “formal operational” learners•And for those who don’t know that they should care•All while making it FUN!

Awareness of self as learner One aspect of METACOGNITIVE SKILLS We want students to develop this awareness So… teachers must have this awareness So… teacher educators must have this awareness

Your Math Autobiography Please take 5 MINUTES to write down some thoughts about your experiences with math so far. We will share later.

Such as:

Was there a time that you really ENJOYED doing math? Describe the context and the feelings you had.

Was there a time that you really DISLIKED doing math? What was that like?

How do you study math? What works for you?

How do you use math in your life?

What are some variations we see?WHAT VARIATIONS WILL WE SEE AMONG TEACHERS AND STUDENTS?DOES THIS AFFECT HOW WE DESIGN INSTRUCTION?

The learner-centered classroomFIRST, A “SILENT CONVERSATION”

SILENT CONVERSATION ACTIVITY

No talking, please. In your group, write answers, comments, reflections on the chart paper. Later, you may write comments to other people’s reflections. We will summarize later.

The “look” of a great math class Teacher: Setting and sharing goals Giving “wait time” Checking for understanding Encourage mathematical thinking

◦ Represent in multiple ways◦ Integrate technology wisely◦ Gain from “mistakes”

/misconceptions

Students: Actively engaging Working together Using what they know Persevering Taking Risks

From: Moynihan, C. 2012. Math Sense: The Look, Sound and Feel of Effective Instruction. Stenhouse Publishers.

The “sound” of a great math class

Students: Justifying and clarifying their thinking Analyzing the thinking of others

Teacher: Supporting math discourse/”math talk”

Engaging ALL students

BOTH: Active listening, Using math vocabulary, encouraging risk taking

The “feel” of a great math classPurpose

Building Confidence

PrideCollaboration

Creativity

Myths, counterproductive beliefs and misconceptions about math

Free Write (2 minutes) If we ask people on the street what they think about:

◦ MATH in general◦ SCHOOL MATH

What to you think they would say?

Which of those ideas are: Category A: UNTRUE, and we need to change that misconception!! Category B: UNTRUE, but it’s OK. Category C: TRUE! Category D: TRUE, unfortunately. We need to change the conditions

that make it true.

Thinking about student misconceptions in specific math topics

An insight from science education: Misconception = “Everyday” conception A mistake makes “sense” in a particular context

Example: ½ + ¼ = ?

Some common misconceptions in Grades 5-10 mathematics (Betsy’s “Favorites”)

“Multiplication makes bigger, division makes smaller.” 4.442 is a decimal, it can’t be written as a fraction. ½ of 6, 0.5 of 6 and 50% of 6 are all different quantities. 1/8 = 0.8

1.25 > 1.4

Probability of flipping a coin and getting heads, then tails. ½+ ½ = 1

Adapted from: Misconceptions in Mathematics. www.counton.org

A story problem Hassan is on the wrestling team. To be in class one, he can weigh at most 48 kg, and to be in class two he can weigh more than 48 kg but no more than 55 kg. Hassan is hoping to wrestle in class two. How much is he allowed to weigh?

What common mistakes do students make?

Perhaps my all-time favorite:

Maybe it’s a favorite of yours, too! What mistake do students make, and what is the underlying misconception?

Adapted from: Collins, A. and Dacey, L. 2011. The Xs and Whys of Algebra. Stenhouse Publishers.

And the list goes on…THE TAKE AWAY? “MISTAKES” AREN’T USUALLY RANDOM. TEACHERS NEED TO PAY ATTENTION TO MISTAKES AND LEARN FROM THEM.

What should math teachers know?

Shulman’s view of teacher knowledge

Shulman’s Pedagogical Content Knowledge (PCK):• Represent subject so that it is

comprehensible to others• Understand student

conceptions/misconceptions• Curricular knowledge

Mathematical Knowledge for Teaching (MKT)

MKT: The deep knowledge of mathematics that allows for Shulman’s PCK

Where does MKT fit on

the diagram?

Thames and Ball article in your packet

Their approach: Look at real teaching, decide what kind of math knowledge is needed.

Teachers need a LOT of math, even in the lower grades

It isn’t the same kind of math that mathematicians need.

Example: Adding and subtracting with near-ten

45 – 19 =

100

90 91 92 93 94 95 96 97 98 99

80 81 82 83 84 85 86 87 88 89

70 71 72 73 74 75 76 77 78 79

60 61 62 63 64 65 66 67 68 69

50 51 52 53 54 55 56 57 58 59

40 41 42 43 44 45 46 47 48 49

30 31 32 33 34 35 36 37 38 39

20 21 22 23 24 25 26 27 28 29

10 11 12 13 14 15 16 17 18 19

0 1 2 3 4 5 6 7 8 9

45 – 19 =

USING THE HUNDREDS CHART

Math knowledge needed:

MKT is multi-facetedIncluding the ability to:

Some questions to think about:

In small groups:

What do you think mathematicians think about Ball’s idea of “Mathematical Knowledge for Teaching”? Does it matter what they think?

If Ball is correct, what does this mean for how we train and develop math teachers?

Extending Shulman’s theory: Technology

TPACK: Technological Pedagogical Content Knowledge

FREE WRITE:How do you see Technological knowledge influencing what math teachers need to know?

Instructional Models How are your teachers currently structure their math lessons?

Does that structure support deep and active learning?

Two alternative models5E AND “LAUNCH, EXPLORE, SUMMARIZE”TAKE A FEW MINUTES TO READ THE DESCRIPTIONS OF THESE TWO MODELS

Give feedback to a teacherVIDEO OF A 6 T H GRADE MATH CLASSWOULD YOU WANT THIS TEACHER TO CHANGE INSTRUCTIONAL MODEL AND APPROACH?

Designing Worthy Tasks in Math WORTHY: Having great merit, character, or value.

(Worthy for whom??)

How can we increase the worthy tasks we ask students to work on? Is everything that happens in a great math class worthy work?

Another Silent ConversationTHE PROBLEMS WITH MATH “PROBLEMS”

A teacher talks about making “good math problems”

Mr. Dan Meyer

7th grade teacher

TED Talk

Summarizing the day We clarified the goals we share for these PCK workshops.

We developed greater awareness of ourselves as math learners.

We reviewed common misconceptions about math and math learning.

We addressed a handful of common math mistakes and their associated “misconceptions.”

We reviewed Shulman’s work on teacher knowledge and extensions such as Ball’s MKT and TPACK.

We considered what makes a math problem “worthy” of student engagement and persistence.