Grade 6 Math Cohort

Summer Institute – Meeting 1

HISTORY

This cohort met over the past year (or two?) with a focus on Social Studies

Cohort met at the end of last year and asked to change their focus to math as the comfort level of SS had increased

Reasons for Change of Focus “Sucking Slough Water in Math”

The need to meet more often It was felt that students had no number

sense and were unable to compose and decompose numbers

No transference The need to develop further understanding

and use of math processes Time for collaboration, building, and

producing Would like to invite other teachers to join

Agenda (unless you change it)9:20 – 12:00

Meet and greet one another Big Ideas - Focus on what we believe makes a

successful math student Discussion and work around Math Processes Begin looking at building a thinking classroom

1:30 – 3:30 Time for you to plan, build and collaborate

How are you arranging your year? Is here where you need to be?

Introductions

Name

School

Thoughts on Math

You’ll want to

know me b/c

Have you Heard/Felt?

Why do we have to learn/teach this?

Don’t tell me why, just tell me how?

Parent: I was bad at math so that why ___ doesn’t like it.

You’ll need this in high school

What are Big Ideas?

These are ideas that underpin a great number of problems, concepts, or ideas that we want students to learn.

A big idea is NOT a topic like fractions, but might be an idea like a fraction only makes sense if you know the whole of which it is a fraction.

Some people use language like “key concepts” or “enduring understandings”.

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Making Big Ideas Explicit

We cannot assume that students will see the big ideas if we do not bring them to the student’s attention.

Many we do not know what the big ideas in a lesson are, even if they know the lesson goal. This is the WHY ARE WE TEACHING THIS?

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Provide an Internal Map of the subjectInterconnections, importance

Helping Interpret CurriculumAssess the attention or emphasis required

to achieve outcome

Avoids looking at outcomes as a checklist of concepts or skills to accomplish

What Teaching to Big Ideas Means

Basis to look critically at lessons and activities

Big Ideas are not meant to be implicit, rather EXPLICIT

Instruction & Assessment are planned around big ideas

What Teaching to Big Ideas Means

i.e. Estimate & measure length, height and distance using non-standard units

What does this outcome mean to you? What are the big ideas?Working with nonstandard units helps

describe the same object with diff measurements

Any measurement can be determined in a variety of ways

The choice of units affects the numerical value of the measurement

So What do I Mean?

Table Talk: As a group you will have approximately 10 minutes to answer the following:

What skills do students need to

have in order to be successful in math?

Big Ideas of Math vs Big Ideas of Math Learner

Debrief: Key Words

How do we grow, assess and evaluate _____?

What does _____ look like?

What does _____ sound like?

Do we have rubrics? Checklists or assessment tools we can share? Collaboratively create?

So if we believe what we say…

Break?

Placemat Activity

Create a placemat with as many sides as you have members

Each participant will record three different numbers that describe them in some way

GO!

Placemat Activity

As a group, choose one number from each section so that the chosen numbers have something in common (mathematical or contextual)

Choose a speaker who will share what your middle number are and how you came up with your middle category

Mathematical processes

It’s the HOW we teach

1. 2. 3. 4. 5. 6. 7.

What are the Mathematical Processes?

GRADE SIX NumberPatterns & Relations

Shape & SpaceStatistics & Probability

Total Per Strand

Communication [C] 6 5 9 4 24

Connections [CN] 8 5 6 3 22

Mental Math & Estimation [ME]

4 1 1 1 7

Problem Solving [PS]

6 5 5 4 20

Reasoning [R] 7 5 4 3 19

Visualization [V] 6 3 8 1 18

Technology [T] 3 0 3 3 9

           

Total Per Process 40 24 36 19 119

Mathematical Processes

[C] Communication

[CN]Connections

[ME]Mental Mathematics and Estimation

[PS]Problem Solving

[R]Reasoning

[T]Technology

[V]Visualization

Philosophy is Constructivism

Mathematical Processes & Curriculum

As a table group….Choose an outcome from an agreed

strand

On the chart paper come up with an activity or activities that would support each process of your outcome.

Be prepared to share with the large group.

The Seven Math Processes

Communication Connections Mental Math and Estimation Problem Solving Reasoning Technology Visualization

The Seven Math Processes

Communication Connections Mental Math and Estimation Problem Solving Reasoning Technology Visualization

Mental Math and Estimation

35 – 16 = 92 – 56 =1001 – 692

=

Use mental math to find these differences.

As you solve each question, keep track of the processes you are using.

Grade 3: NumberDescribe and apply mental mathematics strategies for subtracting two 2-digit numerals, such as:• taking the subtrahend to the nearest multiple of ten and then compensating• thinking of addition• using doubles.[C, ME, PS, R, V]

Grade 4: NumberDemonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3 and 4-digit numerals) by:• using personal strategies for adding and subtracting• estimating sums and differences• solving problems involving addition and subtraction.[C, CN, ME, PS, R]

Mental Math & Estimation

Mental Math and EstimationTypes of Calculations Used in Everyday Life

• 200 volunteers recorded all computation over a 24-hour period

• 84.6% involved some form of mental math

• Only 11.1% involved a written component

• Almost 60% of all calculations required only an estimate rather than an exact answer

What mathematics do adults really do in everyday life?- Northcote, M., & McIntosh, M. (1999)

Mental mathematics is the cornerstone of all estimation processes

0 1

Draw a number line like this one.

Place the fractions 3/8 and 4/7 on the number line.

Explain to a partner how you decided where to place each fraction.

Mental Math and Estimation

Benchmarks and Referents

Benchmark: something (for example a number) that serves as a reference to which something else (another number) may be compared.

Glossary: Alberta Online Guide

• Place given numerals on a number line with benchmarks 0, 5, 10 and 20• Order a given set of decimals by placing them on a number line that contains benchmarks, 0.0, 0.5, 1.0.

• Using 0, ½ , 1 to compare and order fractions• Estimate the quotient of two given positive fractions and compare the estimate to whole number benchmarks

• Estimate the square root of a given number that is not a perfect square using the roots of perfect squares as benchmarks.

Benchmarks and Referents

Referent: a personal item that is used to estimate.

• Known quantities: five-frame ten-frame• Using 10 and 100 as a referent for estimating quantities• Real-life referents for measurement units: cm, m, mm, g, kg, mL, L, cm2, m2, cm3, m3, minute, hour

1 mm is about the thickness of a dime1 L is like the small milk container50 g is the mass of a chocolate bar

To estimate the length of my eraser, I use my referent for a cm, the width of my baby finger, and mentally iterate it.

“ME” In Your Curriculum…

Kindergarten – 2 (Number)

Grade One – 8 (Number)

Grade Two – 8 (Number/S&S)

Grade Three – 12 (Number/S&S)

Grade Four – 6 (N/S&S)

Mental Math and Estimation

Mathematical Processes

Determine the surface area of:• right rectangular prisms• right triangular prisms• right cylindersto solve problems.[C, CN, PS, R, V]

COMMUNICATION CONNECTIONS PROBLEM_SOLVING REASONING VISUALIZATION

Grade 8: Shape and Space (Measurement) Outcome #3

What processes are embedded in this outcome?

Analyzing the Processes

Learning through problem solving should be the focus of mathematics at all grade levels.

- Alberta Program of Studies

Analyzing the ProcessesProblem Solving

Math Processes: Visualization

Visualization “involves thinking in pictures and images and the ability to perceive, transform and recreate different aspects of the visual-spatial world.”

- Thomas Armstrong

Are these two shapes congruent?

Visualization is fostered through the use of concrete materials, technology and a variety of visual representations.

- Alberta Program of Studies

Math Processes: Visualization

What are some activities you use to focus on the Math processes?

What direction?

The driving force behind learning

can be boiled down to ten common “needs”. These student needs are identified by Sfard (2003) and must be present if effective learning is to take place.

Need for Meaning

Learners look for order, logic, causal dependencies behind things, events, and experiences.

Because the need for meaning motivates us to learn, instruction that focuses on it is more effective than instruction that ignores this need.

Abbott and Ryan (1999) describe learning as an active process in which an individual assimilates new facts and experiences into a pre-existing web of knowledge and understanding.

Need for Structure The need for structure follows from the need for

meaning. Meaning involves relations among concepts, not just concepts as such. Understanding of these concepts requires the ability to see structure from the relations among concepts.

If understanding means seeing structure, then the connections between concepts already learned and new concepts being introduced must be an integral part of the curriculum and instruction.

Such connections must include not only real-world

applications and relevance, but also assistance in building mathematical abstractions, so students can see how the results can be transferred from one context to another (Wu, 1997).

Need for Repetitive Action A person who has created meaning and

structure for a mathematical concept is aware of a repetitive, constant structure of certain actions. He is able to think and speak about the process as an object whose inner structures do not have to be remembered each time one deals with it.

A reasonable level of mastery of basic skills is an important element in constructing mathematics knowledge (Fuson & Briars, 1990; Fuson & Kwon, 1992; Hiebert & Wearne, 1996; Siegler, 2003; Stevenson & Stigler, 1992).

Need for Difficulty

True learning implies coping with difficulties.

The goal of learning is to advance a student from abilities he now possesses to those he has not yet developed. The best way to accomplish this is to present the student with tasks beyond his present developmental level but within his zone of proximal development (ZPD). Tasks must be demanding, but still within reach of the student.

Need for Relevance and Significance Significance is the ability to understand

and appreciate the place and importance of what is to be learned

Significance means linking new knowledge to existing knowledge, so again stresses the importance of helping students build connections.

Recent research has shown that people tend to do significantly better in applying mathematics to real-life problems than in attempting to deal with the same mathematical content in the context of typical school problems (Nunes, Schlieman, & Carraher, 1993; Saxe, 1991; Schlieman & Carraher, 1996).

Need for Social Interaction There is an inherent social nature to learning

and making meaning. The most obvious forms are student-teacher

or student-student exchanges, but even interaction with a textbook is a form of social interaction (Sfard, 2003).

Cooperative learning is another form of learning interaction that does not have the teacher in the central role. It has become popular because of research findings suggesting positive effects on student achievement of collective effort (O'Connor, 1998; Siegler, 2003; Webb, 1991; Webb & Farivar, 1994)

Need for Symbolic-Verbal Interaction Interaction in learning means communication, and

communication means using both language (speech) and symbols (written language as well as special mathematics symbols) to convey thoughts.

If mathematics learning is to take place in an interactive setting, students must be encouraged to “talk” mathematics. Research shows that classroom discussion provides many learning opportunities (Ball, 1991;Cobb, Wood, & Yackel, 1991, 1993; Lampert, 1990; Schoenfeld, 1996).

The role of the teacher is particularly important. For the discourse to be effective, students must be taught to communicate mathematically.

Need for Well-Defined Discourse Refers to all communication

practices of the classroom - both written and verbal.

Need for Belonging

Learning by participation requires one to be a part of a learning community.

Implications for mathematics education: Students need to feel respected and free to speak their mind in the classroom.

However, the extent to which students value belonging to a mathematical community is influenced by the value given to mathematics by the wider community culture (Comiti & Ball, 1996).

Need for Balance

To meet learners’ many varied needs, the pedagogy must be varied and rich in possibilities.

It does not imply that old and new are mutually exclusive. For instance, the profound constructivist views of the learner building his or her own knowledge sometimes become trivialized into “teaching by telling”.

Balance individual and cooperative learning opportunities along with problem solving and basic skills acquisition.

The reality is that there must be a bit of everything in the classroom: problem solving as well as skills practice, teamwork as well as individual learning and teacher exposition, real-life problems as well as abstract problems, learning by talking as well a silent learning (Sfard, 2003).

So what are we really trying to achieve?

Our goal should be to have every student become mathematically literate!

These students are in grade 6

Already these students have learnedHow to avoidHow to copeHow to make excusesHow to ‘stop’ thinking

The average person (adult) spends 13 seconds on a problem before giving up

Before we can tackle number concept we need to create mathematical thinkers & ENGAGE OUR LEARNERS

A Mathematical Concept

Does ½ always equal ½?

Explain your answer

Student Created Inquiry

3.47 is the answer. What is the question?

EZ Rock is giving away 1242 tickets for the Billy Joel contest on November 3.

They will give away ½ of the tickets each day of the contest. How many days will it take before the station has only one ticket left to give away?

Explain your answer.

Real Life Situation

Based on problem from Children Are Mathematical Problem Solvers by NCTM

Scratch and Save Fall Sale

A department store in Edmonton is offering a scratch and save event in which you get to scratch 3 different amounts and use all three (imagine that!). You scratched 10%, 25% and 40%. You found the perfect shirt for work. Its original price is $40 Does it makes a difference which order the cashier applies the savings?

Provide evidence to support your answer.

Cell Phone Problem

You just received a cell phone for your birthday. You have been given the responsibility of choosing a cell phone plan.

Cell Phone Problem

1. Which is this the best plan for you? Why?

2. Support your choice with mathematics.

3. Create a visual that includes your answer to the question above and explain the process you went through to make your decision.

4. Are there any circumstances when you would choose the other plan?

Cell Phone Plans

Plan A: 250 minutes for $40 and each additional minute will cost $0.08

Plan B: 250 minutes for $30 and each additional minute will cost $0.13

Breakthrough Mathematics, PearsonEd 2006

Three-Way Sharing

Danny , Connie, and Jane have eight cookies to share among themselves. They decide that they each do not need to receive the same number of

cookies, but each person should receive at least one cookie.

If the children do not break any of the cookies in how many different ways

can they share the cookies?

Children are Mathematical Problem Solvers by NCTM

Creating Wonder

Draw a picture of Infinity

Idea from Marion Small (PRIME – Nelson)

Inquiry/Research QuestionHow much is our classroom worth?

Explain your reasoning.

How Many Sums?How many possible solutions can you find to this problem?

Use the numbers 1 through 9 only once

This Afternoon

Future Dates Topics What do you need in between

meetings?

Yours to work