facilitators’ sessions

JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

FACILITATORS’ SESSIONSK-1, 2-3, and 4-6 Number and Computation


Before We Get Started…. What are your questions/concerns/joys

about acting as facilitators for these courses?

Write your questions/concerns/joys on an index cards.


….that’s one of the purposes of the courses.

So, what is number sense?

Building Number Sense


“…good intuition about numbers and their relationships.” It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989).

Flexibility in thinking about numbers and their relationships.

“Two hallmarks of number sense are flexible strategy use and the ability to look at a computation problem and play with the numbers to solve with an efficient strategy” (Cameron, Hersch, Fosnot, 2004, p. 5).

Number Sense

The History of the NCTM and Standards

Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989)

Professional Standards for Teaching Mathematics (NCTM, 1991)

Assessment Standards for School Mathematics (NCTM, 1995)

Principles and Standards for School Mathematics (NCTM, 2000)

Principles for School Mathematics

Equity Curriculum Teaching Learning Assessment Technology

Content Standards for School Mathematics

Number and Operations Algebra Geometry Measurement Data Analysis and Probability

Process Standards for School Mathematics

Problem Solving Reasoning and Proof Communication Connections Representation

Standards for the Professional Development of

Teachers of Mathematics

Experiencing Good Mathematics Teaching Knowing Mathematics and School

Mathematics Knowing Students as Learners of

Mathematics Knowing Mathematical Pedagogy Developing as a Teacher of Mathematics Teachers’ Role in Professional Development


Mathematician at Work Think of a mathematician at work. What

is this person doing? Where is this person? What tools is this person using?

Draw what you “see”.


If mathematics were an animal…. What would it be, and why?


Core Beliefs About mathematics, mathematics

teaching and learning. List 3-5 core beliefs about these ideas.


Kathy Statz, Third-Grade Teacher I got good grades in high school algebra. I

learned the procedures that the teacher demonstrated. I thought that was what mathematics was about; if you could memorize a procedure then you could do math. I didn’t even know that I didn’t understand math because I didn’t know that understanding was part of math….I have become a confident problem solver by working to understand my kids’ strategies (Thinking Mathematically, pp. 74-75).


Jim Brickweddle, First/Second-Grade Teacher

There are things [in math] that I don’t understand. I am a learner along with my students. Most of us teachers don’t use the invented algorithms that the kids do. Teachers who don’t have a broad understanding of math might end up restricting kids who are thinking outside the box. I saw a teacher ask a child to solve 20 x 64. The child said, “It will be easier if 20 is 4 times 5, then I can find what 5 64s are then add that 4 times.” The teacher wasn’t sure if this would work. I tell teachers, you need to be comfortable with feeling uncomfortable (Thinking Mathematically, pp. 111-112).


How would you respond to this student who answered the following task as shown?

Which of the following helps you with 12 – 7 = ? a. 12 + 7 = 19

b. 2 + 5 = 7c. 5 + 7 = 12d. 4 + 5 = 9

Share your ideas with a neighbor.


Possible Responses…depends on how you are listening

No. You should have chosen (c) because it is part of the fact family.

No. Choose another answer. What is 12 – 7? How did you get your answer? Explain how 2 + 5 = 7 helps you find the

answer. Can you show me your reasoning using

materials?


How you respond gives a glimpse into how you are listening, what you are listening for, what you ignore, and your beliefs about mathematics, mathematics teaching and learning.

It also sends a message to students about what is

important in your classroom.


Child’s explanation: 12 - 7 = ?

L: What is 12-7? C: 5 L: How did you get that? C: Well, I know 12 is 2 away from 10, so I broke 7

into a 2 and a 5. Then I took away 2 from both 12 and 7 so that I had a 10 and a 5. I know what 10 - 5 is. It’s 5.

L: Why did you choose (b)? C: Because it’s 2 + 5 = 7 and I used those

numbers to find the answer.


Different kinds of listening Evaluative

Response seeking Listening for particular responses Set learning trajectory

Interpretive Information seeking Making sense of students’ sense-making Listening for particular responses Set learning trajectory

Hermeneutic Moving with the students Mathematical ideas are locations for exploration Student contributions essentially direct the learning

trajectory of the classDavis, B. (1997). Listening for differences: An evolving conception of mathematics teaching. Journal for Research in Mathematics Education 28(3), 355-376. Reston, VA: NCTM.


Beliefs and Listening Evaluative

Math is about getting answers Strategies to get answers are decided by the teacher

Interpretative Math is about making sense Reasoning is big part of mathematics learning and

assessment Strategies to get answers could be decided by the student

Hermeneutic Math is about exploring ideas and making sense Teaching math is about capitalizing on students’ ideas Strategies to get answers are decided by the student


Challenging Beliefs Making beliefs explicit Supportive environment Where in the sessions do you remember

either your beliefs or a colleague’s beliefs were challenged?

Engaging in class activities, reading Young Mathematics at Work, responding to blogs, observing videos, interviewing children, etc. – all in attempts to develop different perspectives about mathematics, mathematics teaching and learning.


Mathematical Proficiency Conceptual understanding Procedural fluency Strategic competence Adaptive reasoning Productive disposition

National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.


Think about the following problem: 40,005 – 39,996 = ___.

A student with weak procedural skills may launch into the standard algorithm, regrouping across zeros (this usually doesn’t go well), rather than notice that the number 39,996 is just 4 away from 40,000 and 5 more mean the difference is 9.


Constructivism…

What does this mean to you?


Continuum of Understanding

Relational InstrumentalUnderstanding Understanding


Mathematical Example of Instrumental Understanding

7 x 8 = ? Knows the number 5,6 and 7,8 go in that

order. So, remember that those numbers “go together.”

7 x 8 = 56

InstrumentalUnderstanding


Relational Understanding: 7 x 8 = ?

7 7777777

14 x 2

28 x 2

56Seven 7’s are 49, so all I need is one more 7.

I know five 8’s is 40 and two 8’s is 16. 40 and 16 are 56.

4 times 7 is 28. Double that would be 56.

10 x 8 is 80. Take away 3 8’s or 24 is…60, then 56.

RelationalUnderstanding

3 times 8 is 24, double that to get 48. I need one more 8 to get 56.


Continuum of Understanding

Relational InstrumentalUnderstanding Understanding

Perturbation = Disequilibrium = Learning


How do you deal with resistant teachers?

What is the origin of the resistance and fear? Overwhelming to make (any) changes. Uncomfortable with not knowing. Possible approaches Make it less overwhelming.

Choose a part of your practice to focus on one semester. Change is a process, not an event. The process is a marathon not a sprint.

Share what we know from research and international studies Research shows…. International comparisons….

Keep the focus on the students and what is best for them in terms of learning for understanding.


What if teachers have questions I am unable to answer?

Let’s brainstorm ideas!


Standards for Teaching Mathematics

Worthwhile Mathematical Tasks Teacher’s and Students’ role in Discourse Learning Environment Analysis of Teaching and Learning


Hiking Club ProblemTwo-thirds of the students in the school’s

hiking club have climbed Massanutten Mountain, one-half have climbed Afton Mountain, and one-fourth have climbed both of these mountains. Only two students in the club have not climbed either mountain. How many students are in the club?


Questions to consider: What is the purpose of the problem? What prior knowledge and experiences can

students draw on to solve the problem? What mathematics do the students need to

know to solve the problem? How will I present this problem? What questions will I ask struggling

students? How might students solve the problem?


Procedural Steps

141

32

21

x

1123

128

126

x

11211

x121

x


2 students represent of the hiking club.

2 x 12 = 24 students in the hiking club.

121


Your Joys and Concerns/QuestionsWhat concerns/questions do you need

more support with?

facilitators’ sessions

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