“Out of the Starting Gate:Preparing for a Year of Hands-On,

Brains-On Mathematics”

Primary Break Out Session

Primary “Break Out” Session Agenda

9:45 – 10:25 Memorization or Automaticity?10:25 – 10:45 Games Break10:45 – 11:30 Mental Mathematics Strategies11:30 – 12:15 Lunch12:15 – 1:05 Algorithms vs. Number Sense1:05 – 1:20 Games Break1:20 – 2:15 Children’s Mathematical

Models

Learning Intentions

Participants will begin to become familiar with the “Young Mathematicians at Work” book they received.

Participants will explore and discuss ideas about manipulatives and models, mental mathematics, personal strategies and number sense.

Participants will begin to work towards a sequence of lessons they can use in the classroom.

Participants will experience games they can use in the classroom.

What is your “relationship” to the new pedagogy?

It’s a “blind date”. We’re “just friends”. “Starry-eyed” and on our way. We’re trying again. Married with children (and pets)!

A Bit of Graphing Fun

http://nlvm.usu.edu

Introductions: Your name, the grade level(s) of your assignment “Where are you at?”

Constructing Number Sense, Addition and Subtraction

Chapter One: the new pedagogy Chapter Two: choosing tasks and structuring lessons Chapter Three: counting Chapter Four: place value Chapter Five: children’s mathematical models Chapter Six: basic facts Chapter Seven: algorithms vs. number sense Chapter Eight: mental mathematics Chapter Nine: assessment strategies Chapter Ten: professional development

Addition and Subtraction Facts

“Can you do addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one?”

“I don’t know,” said Alice. “I lost count.”

Lewis Carroll Through the Looking Glass

A/B partner talk. What do I think of when I think about

teaching “Basic Facts”? Share. Recorder (Volunteer) record ideas on a chart.

Teaching Basic Facts

To Memorize or not to memorize? Understanding what it means to add and

subtract is necessary before facts can become automatic BUT understanding does not necessarily transfer to automaticity.

Understanding is necessary but not sufficient.

A Fact’s a Fact

What are the strategies that children commonly invent and use?

Work with a partner on the sheet “A Fact’s a Fact”.

Share strategies with the group.

Common Strategies from YMAW

Doubles plus or minus one 6 + 7 = 6 + 6 + 1 = 13

Working with the structure of “5” 6 + 7 = 5 + 1 + 5 + 2 = 10 + 3 = 13

Making tens 9 + 7 = 10 + 6 = 16 Compensating 6 + 8 = 7 + 7 = 14 Using known facts

6 + 8 = 14 so. . . 7 + 8 = 14 + 1 = 15

Memorization or Automaticity?

Memorization of basic facts refers to committing the results of unrelated operations to memory so that thinking is unnecessary. (Fosnot)

Teaching facts for automaticity relies on thinking and focusing on relationships.

The issue is not whether facts should eventually be memorized but how the memorization is achieved.

Isn’t memorization faster? “NO” according to a study by Kamii (1994) Two grade one classes were compared. One

teacher focused on relationships and worked towards automaticity. The other class memorized facts with drill sheets and flashcards.

The automaticity group outperformed the memorization group in being able to produce correct answers within 3 seconds. 76% compared to 55% for memorization.

When relationships are the focus there are far fewer facts to remember.

Also, if a child forgets an answer, they have a quick way to come up with it by going back to a fact they already know.

Focusing on Relationships

Doubles, doubles, doubles! Children need to know doubles and “combinations

that make ten” because these form the basis for other strategies.

Fortunately, most children learn these facts relatively easily. They construct understanding of these facts in a number of ways using visuals, hand shapes, stories and experience.

“Summing” Up.

As children use “strategies” they develop a better sense of the relationships among numbers. It is the automaticity of facts and the development of number relationships that form the basis for efficient calculating and the further development of number theory. (Fosnot/Dolk)

What’s on Your Mind?

Grade 2 example: It is expected that students will: apply mental math strategies, such as - using doubles - making 10 - one more, one less - two more, two less - building on a known fact - addition for subtraction to determine basic addition facts to 18 and related

subtraction facts.

Assessment Strategies

Have students explain their thinking when doing mental math. Observe and record their use of strategies. Consider whether they

know a range of strategieschoose efficient and appropriate strategies

depending on the contextbecome progressively more fluent in their

computations (quicker and more accurate)

Assessment Strategies (cont.)

Students may develop personal strategies that make sense to them for mentally determining answers for basic facts.

Encourage students to develop and share their personal strategies with others.

Encourage students to compare strategies and discuss their efficiency.

Using Manipulatives

Counters and cubes are great for counting, classifying patterning and probability activities.

These manipulatives can reinforce low level counting skills as children learn basic facts.

Manipulatives such as Cuisenaire rods and base ten blocks are designed from an adult perspective. Most children cannot use these manipulatives to construct that conceptual understanding – they just don’t see it that way.

The Rekenrek

Instead of making materials based on adults’ mathematical ideas, we need to look to children’s cognitive development for design ideas.

The “rekenrek” supports children as they develop fact strategies.

There are a number of ways to use this idea – you don’t need to purchase a “rekenrek”.

Games Break!Brilliant BunniesAddition BingoI Spy 15Multiplication BingoOn Target

Resources for “Automaticity”

Mental Math Strategies classroom charts from Kelowna.

Manipulatives like ten frames, Rekenrek Storybooks

Find a “same grade” partner.

Kindies ducks Grade 1 horses Grade 2 cows Grade 3 pigs Grade 4 cats

Find your partners by their sound!

Partner Activity

Choose a strategy from your grade level. Come up with a lesson. Ideas from the book can be found on

Page 99 (Doubles - Madeline) Page 101 (Doubles - Shoes) Page 102 (Making Ten - Apples) Page 103 (Making Ten – Games) Page 109 (Structure of Fives, Compensation – Double

Decker Bus, Bunk Beds, Attendance Charts)

Lunch!!

See you back here at 12:45

Algorithms vs. Number Sense

We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others.

- Blaise Pascal

Mental Computation

76 + 29 =

(Do it in your head, keep it in your mouth, thumbs up)

How did you figure that out?

Algorithms versus Number Sense

A/B partners What do you know about algorithms and

personal strategies? Share what you learned or heard from your

partner.

Calculating with Number Sense

Calculate 3996 + 4246 What would someone with good number

sense do? What would a mathematician do? BOTH would look at the numbers first to

decide on a strategy.

3996 + 4246

3996 is close to the “friendly” number 4000. Now the problem could be

4000 + 4242 234 + 136 could become 235 + 135 289 + 79 could be 300 + 80 – 11 – 1

Try to find numbers that can’t be made friendly!

Choose Algorithms?

Possibly when you are trying to calculate long columns of numbers.

Usually today when we have long columns of numbers we use a calculator.

Fosnot/Dolk argue that using number sense “makes more sense”.

History of Algorithms

Algorithms – the borrowing and carrying procedures were invented by the Arab mathematician Muhammed ibn Musa al-Khwrizmi in the early part of the ninth century.

Algorithms replaced the abacus.

Children’s Struggles

Remembering all the steps in algorithms is difficult for children.

Even more difficult is understanding the steps and the place value.

With the algorithm the numbers are not treated as quantities; they are treated as digits.

Kamii’s Research (1989)

Kamii found with children’s invented strategies for addition and subtraction they went from left to right, from the largest units to the smallest.

Kamii felt that algorithms which work from right to left hinder understanding of place value.

7 + 52 + 186 with second graders.

Answers in the algorithm group ranged from 9308 to 29. 12% had the correct answer.

Answers in the group that had some algorithms ranged from 989 to 114. 26% had the correct answer

In the NO algorithm group the answers ranged from 617 to 138

45% had the correct answer

Children who are NOT taught algorithms get better and better at decomposing and tinkering with numbers. They become better and better mathematical thinkers. The process takes time; it doesn’t happen overnight. But it does happen.

(Fosnot & Dolk)

“Say Something” Strategy

Choose a partner Read “Summing Up” pages 123 to 125 in

“Young Mathematicians at Work” Read silently to the designated stopping point. When each partner is ready, stop and “say

something”. Continue to the end of the selection.

Video Lesson

Look at the structure of the lesson, how the problem is posed, instructions to the children, the teacher’s role during the workshop and the “congress”.

..\Hawkins.mov

Calculating with Number Sense.

Children’s strategies do become more efficient over time.

However many children get stuck splitting numbers into place value components.

Children need support to develop powerful strategies like keeping one number whole in addition (e.g. 368 + 208) or using constant difference in subtraction (e.g. 342 – 37).

This can be done using mental math mini-lessons for numbers greater than 18 (see Chapter 8).

Games Break!

Four Sums in a RowReach for the Stars!Pig Game!Closest to 100

Mental Math Mini-Lessons

10 minutes at the beginning of a lesson More guided than math workshop

investigations. Designed to highlight specific strategies. Teacher prepares a string of questions to

scaffold children as they construct the targeted strategy (see page 128)

Tens Strategies from YMAW

Leap ten and correct (grade 3, A6b, A7b)15 + 9 = 15 + 10 – 1

Next nearest ten (grade 3, A6b)15 + 9 = 15 + 5 + 4

Compensation makes ten15 + 9 = 14 + 10 (grade 3, A6b)

(Grades 1 – 3, A9)

Strategies for Addition from YMAW

Doubles and near doubles (grade 3, A6c) Place value splitting (grade 3, A6a) Jumping by tens (grade 3, A6a) Friendly numbers (grade 3, A6a, A6b) Swapping

(Grades 1 – 3, A9)

Strategies for Subtraction from YMAW

Adding on vs. Removing (Grade 3, A7b) Doubles and near doubles (Grade 3, A7c) Backward jumps of ten (Grade 3, A7c) Next friendly ten Constant difference Canceling out common amounts

(Grades 1 – 3, A9)

Planning a Mini-lesson

Choose a strategy to highlight Carefully create a string Present one equation at a time Be flexible – you may find the children aren’t

ready or take off in a different direction Use the rekenrek, ten frames or the open

number line as models of and for thinking

Planning Math Workshop

Find your partner.Continue working on a lesson plan or sequence

that you have already started, or look for new ideas:

How Many People Are on the Bus? Page 81 A Walk in the City Page 86 Ms. DiBrienza LOVES to read. Page 91