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

Take a moment, please, and write… What “stood out” for you from Friday’s session? A QUICK FORMATIVE ASSESSMENT! What was something different and important that you remember?

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 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 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?

Focus on: Problem Solving

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? 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 What’s Your Math Problem!?! Huntington Beach, CA, USA: Shell Education. WHY SO FEW *GREAT* PROBLEMS IN MATH CLASSROOMS?

A teacher talks about making “good math problems” Mr. Dan Meyer 7 th grade teacher TED Talk

The King of Problem-Solving: Polya Principles of Problem-Solving (How to Solve It) 1) Understand the problem 2) Devise a plan 3) Carry out the plan 4) Look back AND MANY HEURISTICS… But what do students often do??

Problem-solving strategies for students 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 Understanding the problem! Developing a plan! Metacognition: Which are some favorite strategies FOR YOU? Which strategies do you use less often?

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 LET’S SOLVE SOME PROBLEMS!!

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 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.