Number Necessities: Number Sense & Fluency

subtitle: considering the elusiveness and importance of number sense - NOW.

Francis (Skip) Fennell

McDaniel College

Westminster, MD

www.mathspecialists.org

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Now – he tweets…

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F. (Skip) Fennell ��@SkipFennell

A sense is number is acquired - it's developed. YOU don't teach number sense, you teach to nurture its development.

Some History!

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Curriculum and Evaluation Standards, 1989

Children with good number sense

1. Have well understood number meanings,

2. Have developed multiple relationships among numbers,

3. Recognize the relative magnitude of numbers,

4. Know the relative effect of operating on numbers, and

5. Develop referents for measures of common objects and situations in their environments.

7NCTM, 1989, page 38; Hilde Howden

Others

• Bob and Barbara Reys

• Jack Hope

• Judy Sowder

• Ed Rathmell and Chuck Thompson

• Bob Siegler

• Paul Trafton - A sense of

number unfolds

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More recently…

• …poor number sense interferes with learning algorithms and number facts and prevents use of strategies to verify if solutions to problems are reasonable.

9NMAP, page 27, March 2008

And for us (adults)

• Number sense is implicated in virtually any quantitative task, from deciding whether a price is reasonable to interpreting statistical information about the risks associated with a medical treatment (Curry, et al., 1996; Steen, 2001).

• Really? Take a look…10

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• University of Missouri researchers recently tested 180 seventh grade students in the Columbia, Missouri school system and found that kids who lacked math fluency in middle school were the same kids who struggled with number concepts in first grade (Geary, 2013).

• Parents can do plenty to help kids gain number sense.

Not just for parents…

• Introduce children to numbers as soon as possible.

• Present math activities with the same enthusiasm that you display for language acquisition and reading.

• Go beyond counting! (5 cookies is the amount five and we use 5 to represent that amount).

• Play math games that informally engage operations.

• Measure – pretty much everything

• Talk about time

• And on and on…

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Do you have a sense of number?

Is 4 x 12 closer to 40 or 50?

How many paper clips can you hold in your hand?

If the restaurant bill was $119.23, how much of a tip should you leave?

How long will it take to make the 50 mile drive to Washington, D.C.?

If a 10-year old is 5’ tall, how tall will the child be at age 20?

Contributors to Number

Sense Development

• Conceptual Understanding

– Counting

• Subitizing

– Place Value; Comparing and Ordering• Composing and decomposing

• Mental Math Opportunities

• Estimation

– Magnitude

– Computational Estimation

• Reasonableness of Results

• Questioning to promote a sense of number…

• Numbers in context22

All students should leave elementary school computationally fluent AND with a

strong sense of number

What does that mean?

Why is that important?

How do we do that?

AND, does the CCSS-M help?

Consider…

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Number Sense

Foundations

Practices

Fluency

from the CCSS-M

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K 1 2 3 4 5

Counting, cardinality,

comparing

Extend the counting

sequence

Understand addition as

putting together and

adding to, and

understand subtraction

as taking apart and

taking from

Understand and apply

properties of operations

and the relationship

between addition and

subtraction

Understand properties

of multiplication and

the relationship

between multiplication

and division

Work with numbers 11-

19 to gain foundation

for place value

Understand place value

Use place value

understanding and

properties of operations

to add and subtract

Understand place

value

Use place value

understanding and

properties of

operations to add and

subtract

Use place value

understanding and

properties of operations

to perform multi-digit

arithmetic

Generalize place value

understanding for

multi-digit numbers

Use place value

understanding and

properties of

operations to perform

multi-digit arithmetic

Understand the place

value system

Perform operations with

multi-digit whole

numbers and with

decimals to hundredths

Develop understanding

of fractions as numbers

Extend understanding

of fraction equivalence

and ordering

Use equivalent fractions

as a strategy to add and

subtract numbers

Apply and extend

previous understandings

of multiplication and

division to multiply and

divide fractions.

15 – why is this number important????

Subitizing

Sun Mon Tues Wed Thur Fri Sat

1 2 3 4 5 6

7 8 9 10 11 12 13

14 15 16 17 18 19 20

21 22 23 24 25 26 27

28 29 30 31

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100 Chart Puzzles

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Hmm…think about this

• Comparison time increases as the numbers to be compared become close to the reference number.

• It takes more time to determine that 71 > 65 than 79 > 65, and comparing 69 to 65 takes more time than comparing 71 to 65.

• Just one example of the importance of place value as a mathematical foundation in work with whole numbers. (but what about fractions…more later!)

29Dehaene, The Number Sense, p. 75

• For students in grades K-2, learning to see the relationships between addition and subtraction is one of their most important accomplishments in arithmetic.

Resnick, L.B. (1983)30

Math Wall Activities

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Today’s Date

Composing and Decomposing Number is Critical!

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100

• 100 is a big number when it’s:

• 100 is a small number when it’s:

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Number Sense Language

• bunch

• pile

• flock

• herd

• stack

• handful

• basket

• cord

• crowd

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Favorites

• Write 3 numbers that have some significance to your life.

• Exchange lists. Provide random clues for the numbers.

• Guess which numbers fit the clues.

Computational fluency - having and using efficient and

accurate methods for computing. Fluency might be

manifested in using a combination of mental strategies and

jottings on paper or using an algorithm with paper and pencil,

particularly when the numbers are large, to produce accurate

results quickly. Regardless of the particular method used,

students should be able to explain their method, understand

that many exist, and see the usefulness of methods that are

efficient, accurate, and general. (PSSM, 2000, page 32).

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Procedural fluency refers to knowledge of

procedures, knowledge of when and how to use

them appropriately, and skill in performing them

flexibly, accurately and efficiently (Adding it Up,

2001, page 121).

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…the Panel proposes three clusters of concepts and skills – called the Critical Foundations of Algebra –reflecting their judgment about the most essential mathematics for students to learn thoroughly prior to algebra course work.– Fluency with Whole Numbers – By the end of Grade 5 or 6,

children should have a robust sense of number…

– Fluency with Fractions – Before they begin algebra course work, middle school students should have a thorough understanding of positive as well as negative fractions…(National Mathematics Advisory Panel, pages 17, 18, 2008).

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Fluency: Simply Fast and Accurate? I Think Not!

By NCTM President Linda M. GojakNCTM Summing Up, November 1, 2012

…a frequently asked question is, “What does it mean to be fluent in mathematics?” The answer, more often than not, is, “Fast and accurate.” Building fluency should involve more than speed

and accuracy. It must reach beyond procedures

and computation.

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The fluency thing…

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K 1 2 3 4 5 6

Facts

Fluently add and

subtract within 5

Add and subtract

within 20,

demonstrating

fluency for addition

and subtraction

within 10. Use

strategies…

Fluently add and

subtract within 20

using mental

strategies. By the

end of grade 2, know

from memory all

sums of two one-

digit numbers.

Fluently multiply

and divide within

100, using

strategies…By the

end of Grade 3,

know from memory

all products of two

one-digit numbers.

Operations, Algorithms

Fluently add and

subtract within 100

using strategies

based on place

value, properties of

operations, and/or

the relationship

between addition

and subtraction

Fluently add and

subtract within

1000 using

strategies based on

place value,

properties of

operations, and/or

the relationship

between addition

and subtraction

Fluently add and

subtract multi-digit

whole numbers

using the standard

algorithm.

(capstone)

Fluently multiply

multi-digit whole

numbers using the

standard algorithm.

(capstone)

Fluently divide

multi-digit whole

numbers using the

standard algorithm.

(capstone)

Fluently add,

subtract, multiply,

and divide multi-

digit decimals using

the standard

algorithm for each

operation.

(capstone)

Fractions? What price fluency?

How Does Computational Fluency

Develop?

• Experience along the continuum enables the student to better determine the reasonableness of an answer.

• Students move along the continuum at individual rates.

• Often it is the difficulty of the problem that determines the strategies the student will use.

Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children’s

Mathematics. Portsmouth, NH: Heinemann

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Computational Fluency, Algorithms, and Mathematical

Proficiency: One Mathematician’s Perspective

Hyman Bass, TCM, February, 2003.

“Understanding algorithms is central to developing computational fluency. Teachers must combine a renewed appreciation of the contributions algorithms make to mathematical proficiency with a design of approaches to teaching and learning that can develop both understanding and skill. They also must appreciate what efficiency affords and respect what it takes to use compact methods sensibly,

flexibly, and appropriately.” 42

Well, of course…

The Practices and Number Sense

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Make sense of problems and

persevere in solving them.

…and they continually ask themselves, “Does this make sense?”

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Reason abstractly and

quantitatively

“…considering the units involved; attending to the meaning of quantities, not just how to

compute them; and knowing and flexibly

using different properties of operations and objects.”

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Construct viable arguments and

critique the reasoning of others

“…Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades…Students

at all grades can listen or read the arguments of others, decide whether they make sense,

and ask useful questions to clarify or improve the arguments.” 46

Model with mathematics

“…routinely interpret their mathematical results in the context of

the situation and reflect on whether the results make sense, possibly improving

the model if it has not served its purpose.”

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Use appropriate tools strategically

“…They detect possible errors by strategically using estimation and other

mathematical knowledge.”

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Attend to precision

“…They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate

for the problem context. In the elementary grades, students give

carefully formulated explanations to each other.” 49

Look for and make use of structure

“Mathematically proficient students look closely to discern a pattern or

structure. …will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the

distributive property…” 50

Look for and express regularity in

repeated reasoning

“Mathematically proficient students notice if calculations are repeated, and look both for

general methods and for shortcuts. …As they work to solve a problem, mathematically

proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of

their intermediate results.” 51

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Boxes to multiply…

• Draw a rectangle to show 46 x 7 = 322

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40 6280

4246 x 7 = (40 x 7) + (6 x 7) =

280 + 42 = 322

The importance of representations…

Blame it on your brain…

• We become so skills focused with the how’s and procedural “stuff” related to computation (mental or paper/pencil) that the “work” starts automatically…

• A farmer has eight cows. All but five die. How many cows remain?

• Judy owns five dolls, which is two fewer than Cathy. How many dolls does Cathy have?

• 3 for each?

54Dehaene, The Number Sense, p. 75

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Estimation – Some Thoughts

• Estimating Magnitude – should begin early and occur often.

• Children are initially uncomfortable with computational estimation.

• The language of computational estimation is adult language. Children seem OK with such language as they grow –experientially.

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Estimate or Exact?

• Your school bus number.

• When to leave for school in the AM.

• When a flight will leave the airport.

• Total bill at a restaurant.

• When do you estimate?

• When must you have an exact response?

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Why Fractions?

• 50% of 8th graders could not order three fractions from least to greatest (NAEP, in NCTM, 2007).

• Less than 30% of 17-year olds correctly represented 0.029 as 29/1000 (Kloosterman, 2010).

• When asked which of two decimals, 0.274 and 0.83 is greater, most 5th and 6th graders chose 0.274 (Rittle-Johnson, Siegler, and Alibali, 2001).

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Also consider…

• Sowder (ed., 1989) noted that “number size is not a sense about numbers that once acquired for whole numbers generalizes to all kinds of numbers” (page 20).

• The National Mathematics Advisory Panel (US Deptof Education, 2008) in its final report noted that: “the most important foundational skill not presently developed (among foundational skills necessary for algebra) appears to be proficiency with fractions (including decimals, percent, and negative fractions).” (NMAP, 2008, page 18).

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Fraction beginnings…

• Which one is larger, 1/2 or 1/3?

“the size of the fractional part is relative to the size of the whole…” (NCTM, 2006)

Got Fraction Sense?

62Video here from hand2mind, 4.30.13

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Fractions are numbers too!

Developing Fraction Sense

• Representations

• Equivalence

• Comparing and Ordering Fractions

• Connections (e.g. between)

• Contexts

65Fennell, Kobett, and Wray, MTMS, April, 2014

66Fennell, Kobett, and Wray, MTMS, April, 2014

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Fennell, Kobett, and Wray, MTMS, April, 2014

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Percent Benchmarks

0%

100% 50% < 10%

~25% ~75% ~90%

> 50% < 50%

• Lefthanders in the room or class

• Once lived in New Jersey

• Been involved in education > 10 years

• People who were born in Maryland (?)

Number Sense, Addenda series

Ready??

Contexts – well maybe…

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What did he say?

117,203 attended the Preakness…• “less than .0025 of the crowd was arrested/detained…”

• Who could have possibly understood this

statement?????

• Why did he do this? I know, and you probably do too!!!!!

• Hmm – 0.0025, what do I do with that? I’ll just read

it.

• How about just saying less than 1% since it’s really ¼ of 1%? OR

• Less than 1%, fewer than 300 of the 117,000+ attendees!

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Really?

• The weather reporter on WCRB (a Boston radio station) said there was a 30% chance of rain. The host of the show asked what that meant.

• The weather reporter said, ``It will rain on 30% of the state.''

• ``What are the chances of getting wet if you are in that 30% of the state?''

• ``100%.'‘

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You can’t make this stuff up

• 50% off sale on all purchases at the Izod store. Sign indicates 50% off the all-store sale.

• Patron – “well that means it’s free.”

• Clerk – “no sir, it’s 50% off the 50% off sale.”

• Patron – “well, 50% + 50% is 100% so that means it should be free.”

• This went on for a while. AND, there was a sign indicating 70% off for some items, meaning 70% off the 50% off original sale, which our patron would interpret as the item being free and 20% in cash!

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What can you do in a minute?

Time as a context…

• Sit-ups

• Listen to a song

• Finish my homework!

• Do a chore

• Wait a minute – really?

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“Oh, about 2:45”

What is your expected finishing time?

Are you sure?

• Actual problem presented at an NCTM regional conference.

• A dog traveled 15 meters per second. How far would the dog go in: a minute, a half-hour, an hour, a day?

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Speeds of Some Animals

Cheetah 70 mph (65)*

Lion 50 mph

Zebra 40 mph

Rabbit 35 mph

“Super Dog” 33+ mph

Reindeer 32 mph

Elephant 25 mph

Chicken 8 mph

Consider…

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Number Sense

Foundations

Practices

Fluency

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It added that the wealth of the richest one percent of people in the world now amounts to $110 trillion, or 65 times the total wealth of the bottom half of the world's population.

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Concluding Thoughts

• Number sense is elusive.

• Understandings provide anchors for developing number sense.

• Number sense should be nurtured – every day! And such opportunities connect to and extend standards (e.g. estimation, mental math - reasonableness)

• A sense of number breeds confidence.

Questions?

http://www.mathspecialists.org

http://www.ffennell.com