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Mathematics PCK Workshop #1 Day 2. Ramallah, Palestine march 22 & 24, 2013. Dr. Elizabeth (Betsy) McEneaney [Mac-Uh-Ninny] Dept. of Teacher Education and Curriculum Studies UMass – Amherst A QUICK FORMATIVE ASSESSMENT! . Take a moment, please, and write….

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mathematics pck workshop 1 day 2

Mathematics PCK Workshop #1Day 2

Ramallah, Palestine march 22 & 24, 2013

Dr. Elizabeth (Betsy) McEneaney[Mac-Uh-Ninny]Dept. of Teacher Education and Curriculum StudiesUMass –
take a moment please and write


Take a moment, please, and write…
  • What “stood out” for you from Friday’s session?

What was something different and important that you remember?

after today i hope you will be able to
After today, I hope you will be able to:
  • Develop methods to support teachers’ focus on problem solving in mathematics;
  • Recognize strategic uses of technology to support effective teaching and learning in mathematics;
  • Select authentic learning assessments that engage students in active learning that connects mathematics content to real-world, interdisciplinary contexts.

“mistakes” aren’t usually random.

teachers learning from student mistakes
Teachers learning from student mistakes

Common misconceptions usually “make sense” to students in some way.

Our examples (“homework” from Friday)

A common misconception, and how we might guide students toward a better understanding.

  • THINK: Are there patterns across our examples in:
  • 1) The type of misconceptions and why it “makes sense”
  • 2) The kinds of approaches that help improve understanding?
my example a story problem
My example: 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?
think for a minute then pair up with a partner
Think for a minute, then pair up with a partner
  • What makes a problem a “great” math problem?
  • Do Palestinian teachers give students enough “great” math problems? Why or why not?
what is a rich math task worth solving
What is a “rich” math task worth solving?


  • Includes substantial math
  • Multiple “points of entry” – all can get started
  • Can be scaffolded
  • Interesting and motivating for students
  • Can be solved in different ways
  • Promotes discussion
  • Involves students in:
    • Interpreting
    • Testing/making conjectures
    • Proving/justifying
    • Explaining
    • Reflecting
  • Helps develop conceptual understanding of math

Source: Gojak, L. 2011. What’s Your Math Problem!?! Huntington Beach, CA, USA: Shell Education.

a teacher talks about making good math problems
A teacher talks about making “good math problems”

Mr. Dan Meyer

7th grade teacher

TED Talk

the king of problem solving polya
The King of Problem-Solving: Polya

But what do students often do??

  • Principles of Problem-Solving (How to Solve It)
  • 1) Understand the problem
  • 2) Devise a plan
  • 3) Carry out the plan
  • 4) Look back
problem solving strategies for students

Understanding the problem!

Problem-solving strategies for students

Developing a plan!

  • 1) Restate the problem in your own words
  • 2) Identify wanted, given and needed info
  • 3) Identify a subgoal
  • 4) Draw a picture
  • 5) Make a model
  • 6) Act it out
  • 7) Choose appropriate notation
  • 8) Look for a pattern
  • 9) Create a table
  • 10) Organize a list
  • 11) Guess and check
  • 12) Create and use a graph
  • 13) Solve a simpler problem
  • 14) Think of all possibilities
  • 15) Work backwards
  • 16) Change your point of view

Metacognition: Which are some favorite strategies FOR YOU? Which strategies do you use less often?

teacher role to improve problem solving
Teacher role to improve problem-solving
  • Strategy 1: GIVE GOOD PROBLEMS!!
  • Strategy 2: Ask good questions. [See handout.]
  • Strategy 3: Be less helpful. (Encourage patience!)
awareness of self as learner
Awareness of self as learner

Let’s solve some problems!!

while you work
While you work…
  • How did you make sense of the problem?
  • What helped you get started?
  • What problem-solving strategies did you use?
  • Did you need to adapt, add or change strategies?
  • What math concepts did you use?
  • How did you know if your solution made sense?
summarizing the math pck workshop 1
Summarizing the math PCK workshop #1
  • 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 considered what makes a math problem “worthy” of student engagement and persistence.
  • We reviewed Polya’s principles of problem solving.
  • We used specific problem solving strategies doing math, while thinking metacognitively.
  • We considered the role of teachers in supporting student problem solving.