Making Sense of Numbers ISK Parent

K 1 2 3 4 5 6 7 8 HS Counting

& Cardinality

Algebra

Number and

Quantity

Mod

elin

g

Operations & Algebraic Thinking

Expressions and

Equations

Number & Operations in Base Ten

The Number System

Number & Operations Fractions

Ratios & Proportional

Relationships

Functions

Measurement & Data

Statistics & Probability

Geometry

Use appropriate tools strategically

Attend to precision

Look for and make sense of structure

Look for and express regularity in repeated reasoning

STANDARDS FOR MATHEMATICAL PRACTICE

Make sense of problems and persevere in solving them

Reason abstractly and quantitatively

Construct viable arguments and critique the reasoning of others

Model with mathematics

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Handout #2

STANDARDS FOR MATHEMATICAL PRACTICES

Graphic

4 5/29/12

WHAT DOES IT LOOK LIKE IN THE TASKS WE ASK CHILDREN TO DO?

¢ We do not “teach” the practices, rather we design our instruction around tasks that give students the opportunity to use and develop the practices in learning and doing mathematics everyday, not only in mathematics class but in all aspects of their education and lives that include mathematical thinking.

FROG PROBLEM

¢ Work with a small group (2-3 people). ¢ Record your solution. ¢ Try to solve the problem in at least two different

ways. Pay attention to the mathematical processes you

are using as you solve this problem.

FROG PROBLEM ¢ Grade 1 - Frogs in the Pond ¢ There were 12 frogs at the pond. Some were

swimming and some were sunning themselves on a log. There were more frogs swimming than sunning.

¢ How many frogs were swimming and how many were sunning?

¢ Use pictures, words and numbers to prove that your answer makes sense.

¢ Can you find more than one way to do this?

“The level and kind of thinking in which students engage determines what they will learn.”

Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human,

1997

9

NUMBER SENSE? ¢ Students with number sense develop multiple

meanings of numbers, know how operations work and how to apply them, and use numbers fluently (accurately, efficiently, flexibly)

CPR C–Conceptual Understanding � What students will be expected to KNOW. These

are the ideas that students should be developing �

P–Procedural Understanding �What students will be expected to DO. These are the procedures and skills that enable students to access the ideas of mathematics. �

R – Representational Understanding � How students will SHOW what they know and

can do. These are the drawings, models, and ways that students make their thinking visible. �

BUILDING CONCEPTUAL UNDERSTANDING

Concrete Manipulatives

DOING

Pictorial Representation

SEEING

I I I I

I I I I

Abstract Symbols

4 + 4 = 8

2 x 4 = 8

THE BRIDGE TO UNDERSTANDING

Representation/Pictorial “SEEING” Stage

Concrete Abstract “DOING” Stage “SYMBOLIC” Stage

WHEEL SHOP

¢ The Wheel shop sells bicycles and go-carts. Each bicycle has only one seat, and each go-cart has only one seat. There are a total of 7 seats and 18 wheels in the shop. How many are bicycles and how many are go- carts?

¢  Use pictures, words, and numbers to show your math thinking.

CLIMBING THE NUMBER SENSE LADDER

¢ Rote Counting

K.CC.1. COUNT TO 100 BY ONES AND BY TENS

Conceptual: ¢  count by ones in sequence from 1 to 10. ¢  count by ones in sequential progression from 11 to 20. ¢  count by ones in sequential progression from 21 to 100. ¢  count by tens in sequence from 10 to 100.

Procedural: ¢  rote counting by ones up to 10. ¢  rote counting by ones up to 20 and continue in sets of 10 to

100. ¢  rote counting by tens up to 100.   Representational: ¢  use kinesthetic movements to represent counting connections

(e.g., clapping, jumping, etc.). ¢   

CLIMBING THE NUMBER SENSE LADDER

•  One To One

Correspondence

¢ Rote Counting

K.CC.4 UNDERSTAND THE RELATIONSHIP BETWEEN NUMBERS AND QUANTITIES; CONNECT COUNTING TO CARDINALITY.

Conceptual: ¢  develop strategies for keeping track of counted

objects. ¢  accurately count objects with one-to-one

correspondence up to 20. ¢  count various groupings and arrays up to 20. ¢  identify “how many” objects they counted. ¢ understand quantities of “one more” up to 20 (e.g., 7 is one more than 6).

Procedural: ¢  touch, slide, tap, drop, color, etc. to accurately count

objects. ¢  identify the last number counted as the quantity of

objects. ¢  create groups of 10 and some quantity for easier

counting of teen numbers. ¢  create a given number of objects and one more. Representational: ¢  color, slide, tap, drop, and move, objects as they count. ¢ use ten frames, dot cards, domino, dice, or other

arrangements to assist counting. Students can use cards, dice, dominoes, written numerals, etc. to name quantities.

¢  demonstrate an understanding of quantities of one more.

CLIMBING THE NUMBER SENSE LADDER

•  Counting on

•  One To One

Correspondence

¢ Rote Counting

CLIMBING THE NUMBER SENSE LADDER

•  Subsitizing

•  Counting on

•  One To One

Correspondence

¢  Rote Counting

HOW MANY?

CLIMBING THE NUMBER SENSE LADDER

•  Complements of Ten

•  Subsititzing

•  Counting on

•  One To One Correspondence

¢  Rote Counting

�  Complements to five

COMPLEMENTS TO FIVE

Pre-K

HOW MANY?

K.OA.4: FOR ANY NUMBER FROM 1 TO 9, FIND THE NUMBER THAT MAKES 10 WHEN ADDED TO THE GIVEN NUMBER (E.G., BY USING OBJECTS OR DRAWINGS), AND RECORD THE ANSWER WITH A DRAWING OR EQUATION.

Conceptual: ¢  understand how to use drawings to make 10 when given a smaller number. ¢  understand how to use equations to make 10 when given a smaller number.

Procedural:

¢  add on to make 10 starting from a given number by using objects.

¢  add on to make 10 starting from a given number by using drawings.

¢  add on to make 10 starting from a given number by using equations.

REPRESENTATIONAL:

¢ Students can model adding on to make 10 by drawing or writing an equation.

IMPORTANT COUNTING STRATEGIES:

Counting on Counting back Skip counting

MCI2 Session 1

CLIMBING THE NUMBER SENSE LADDER Step 5 ¢  Counting strategies Step 4 ¢  Complements of Ten Step 3 ¢  Subsitizing Step 2 ¢  Rational Counting ¢  One to One Correspondence Step 1 ¢  Rote Counting

MC

I2 Session 1

COUNTING ON

¢ Three Stages �  count all �  count on �  count on from larger.

MCI2 Session 1

K.OA.5: FLUENTLY ADD AND SUBTRACT WITHIN 5.

Conceptual: ¢  automaticity with addition facts to 5.

¢  automaticity with subtraction facts to 5.

¢ understand when and how to use addition and

subtraction appropriately, ¢  skill in performing them flexibly, accurately and

efficiently.

CLIMBING THE NUMBER SENSE LADDER

Step 6 Conservation of numbers

Step 5 Counting Strategies

Step 4 Complements of Ten

Step 3 Subsidizing

Step 2 One to One Correspondence

Step 1 Rote Counting

MC

I2 Session 1

PROCEDURAL:

Use the following strategies to attain fluency with facts to 5:

¢ Counting on ¢ Counting back ¢ Counting up to subtract ¢ Using doubles ¢ Using commutative property ¢ Using fact families ¢ Within a given amount of time, fluently solve addition

and subtraction facts within 5, orally and or in written form.

CONSERVATION OF NUMBER

¢ A “number” means an “amount” and that amount does not change no matter how you arrange the objects.

6

MCI2 Session 1

MC

I2 Session 1

HIERARCHAL INCLUSION

¢ An understanding that 19 is inside of twenty, the numbers are nested inside each other and that the numbers grow one each time.

¢ “1” “2” “3”

¢ Child is able to “see” the number as a unit, while at the same time “seeing it made up of it’s parts”.

MCI2 Session 1

CLIMBING THE NUMBER SENSE LADDER

Step 7 Compensation

Step 6 Conservation of numbers

Step 5 Counting Strategies

Step 4 Complements of Ten

Step 3 Subsitizing

Step 2 One to One Correspondence

Step 1 Rote Counting

MC

I2 Session 1

COMPENSATION ¢ When working with numbers you can take an

amount from one set and add it to another set, the total amount does not change.

MCI2 Session 1

COMPENSATION

¢  IMPORTANT conceptual skill. ¢ Referred to as “compose and decompose”

numbers. ¢ Flexibility with numbers

MCI2 Session 1

COMPENSATION

¢ Suppose the problem is 44 - 28. Many problems with give us the same answer. 43 - 27; 36 - 20 42 - 26; 41 - 25; 40 - 24; 39 - 23; 38 - 22; 37 - 21;

MCI2 Session 1

COMPENSATION STRATEGY

¢ Shift both numbers to amounts that don’t require regrouping. (46-30)

¢ Students MUST understand a strategy to be competent with it.

MCI2 Session 1

CLIMBING THE NUMBER SENSE LADDER

Step 8 Fact Families

Step 7 Compensation

Step 6 Conservation of numbers

Step 5 Counting strategies

Step 4 Complements of Ten

Step 3 Subsitizing

Step 2 One to One Correspondence

Step 1 Rote Counting

MC

I2 Session 1

In reality, no one can teach mathematics.

Effective teachers are those who can

stimulate students to learn mathematics. Educational research offers compelling

evidence that students learn mathematics well only when they construct their own mathematical

understanding

Everybody Counts National Research Council, 1989

WHAT IS NUMBER SENSE?

A “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).

MCI2 Session 1

A PROBLEM

¢ My monkey had 12 grapes in his bowl in the morning. I added 5 more grapes before I went to school. My monkey ate some grapes while I was at school. If there are 3 grapes left in the bowl when I get home, how many grapes did my monkey eat?

¢ How many ways can you find the answer?

Number Sense

Problem Solving

Computational Fluency

Conceptual Understanding

9/12/14

PROCEDURAL / CONCEPTUAL KNOWLEDGE? ¢  Procedural Knowledge-skillful use of mathematical rules or

algorithms ¢  Conceptual Knowledge-understanding meaning of

mathematical concepts

Add then subtract…

Adding is putting together

Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

VALUE MULTIPLE REPRESENTATIONS…

concrete or pictorial

tabular

verbal

symbolic

graphical

DRILL AND PRACTICE- THERE IS A DIFFERENCE!! Practice means we are doing meaningful mathematics that

we have recently learned in an attempt to achieve proficiency.

Drill is activity designed to promote automaticity and speed

in recall. (Timed tests or quizzes are considered drill and often cause more anxiety in students who do not work well under pressure).

ASB MCI2 Problem Solving

CLIMBING THE NUMBER SENSE LADDER

Step 9 Place Value and operations

Step 8 Fact Families

Step 7 Compensation

Step 6 Conservation of numbers

Step 5 Counting strategies

Step 4 Complements of Ten

Step 3 Subsitizing

Step 2 One to One Correspondence

Step 1 Rote Counting

mci2

9/12/14

CLIMBING THE NUMBER SENSE LADDER

Step 9 Place Value and operations

Step 8 Fact Families

Step 7 Compensation

Step 6 Conservation of numbers

Step 5 Counting strategies

Step 4 Complements of Ten

Step 3 Subsitizing

Step 2 One to One Correspondence

Step 1 Rote Counting

mci2

9/12/14

CLIMBING THE NUMBER SENSE LADDER

Step 9 Place Value and operations

Step 8 Fact Families

Step 7 Compensation

Step 6 Conservation of numbers

Step 5 Counting strategies

Step 4 Complements of Ten

Step 3 Subsitizing

Step 2 One to One Correspondence

Step 1 Rote Counting

mci2

9/12/14

CLIMBING THE NUMBER SENSE LADDER

Step 9 Place Value and operations

Step 8 Fact Families

Step 7 Compensation

Step 6 Conservation of numbers

Step 5 Counting strategies

Step 4 Complements of Ten

Step 3 Subsitizing

Step 2 One to One Correspondence

Step 1 Rote Counting

mci2

9/12/14