Transcript The Components of Number Sense Final

Presented by: Margaret Adams
Melrose Public Schools
1
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Define number sense and its components.
Describe how number sense develops
through the grades.
Identify student’s misconceptions in number
sense.
Name the steps for fact fluency. Describe the
role of number sense in fact fluency.
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What is Number Sense?
Components of Number Sense
Fact Fluency and Number Sense
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Think about how you learned to compute in
elementary school.
What was the instruction like?
What sorts of things did you do?
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Definition in your own words
Examples
Facts/characteristics
Number
Sense
Visual
5
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Make a list of all the important ideas that you
think children should know about the number
8 by the time they finish first grade.
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Riddle me this and riddle me that
You can solve riddles…just like that!
I am a prime number.
I am an odd number.
I am more than 6.
My digits add up to 4.
What number am I???
7
Mystery Number
If you count by fives, I
have two left over.
I am a multiple of 7.
The sum of my digits is an
even number.
I am a composite number.
What’s the Mystery Number??
8
9
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Basic Definition: Understanding of what
numbers are and how they relate to each
other.
Example:
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6 is half of 12
It’s also 3 doubled
1/3 of 18
2 sets of 3
3 sets of 2
1 more than 5
1 less than 7
10
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“Number sense is an emerging construct that
refers to a child’s fluidity and flexibility
sense of what numbers
mean and an ability to perform mental
with numbers, the
mathematics and to look at the world and
make comparisons.”
◦ Russell Gersten, David Chard
11
Number sense as 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)
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Develops meaning for numbers and
operations
1.
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Connects numerals with situations from life
experiences.
Knows that numbers have multiple interpretations.
Understands that number size is relative.
Connects addition, subtractions, multiplication, and
division with actions arising in real-word situations.
Understands the effects of operating on numbers.
Creates appropriate representations for operations.
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2. Looks for relationships among numbers and
operations.
Decompose or breaks apart numbers in different
ways.
Knows how numbers are related to other numbers.
Understands how the operations are connected to
each other.
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Multiplication/Division-Inverse
Addition/Subtraction-Inverse
Multiplication-Repeated Addition
Division-Repeated Subtraction
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2. Looks for relationships among numbers and
operations.
Decompose or breaks apart numbers in different
ways.
Knows how numbers are related to other numbers.
Understands how the operations are connected to
each other.
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Multiplication/Division-Inverse
Addition/Subtraction-Inverse
Multiplication-Repeated Addition
Division-Repeated Subtraction
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3. Understands computation strategies and
uses them appropriately and efficiently.
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Correctly performs the steps in an algorithm and
can discuss why the algorithm works.
Makes a conscious effort to complete calculations
using prior knowledge and simpler calculations.
Often uses a variety of calculation strategies, even
when completing calculation involving the same
operation.
Chooses appropriate calculation technique to obtain
exact answers and estimate.
Calculates with accuracy and relative efficiency
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4. Makes sense of numerical and quantitative
situations.
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Expects numerical calculations to make sense.
Seeks to understand relationships among quantities
in real-world situation.
Assesses whether the result of a calculation makes
sense in the context of the numbers and real-world
quantities involved.
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Research indicates that early number sense
predicts school success more than other
measures of cognition like verbal, spatial, or
memory skills or reading ability.
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“good intuition
about numbers”
Quantity/
Algebraic
Magnitude
and
Geometric
Thinking
Proportional
Numeration
Reasoning Language
Form of
Equality
a
Number Base Ten
Components of Number Sense
© 2007 Cain/Doggett/Faulkner/Hale/NCDPI
19
“good intuition
about numbers”
Quantity
and
Magnitude
First three components of
number sense are anchors
for building a rich
understanding of math.
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Quantity - The physical amount of something.
Magnitude - Quantity in relation to other
quantities
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Math is not about numbers it is about quantity.
The physical reality of the mathematics that we
model with symbols and the number line (how
much, how far, how big, how bright, etc.).
Virtually all mathematical topics can be modeled
for students using quantity as a core
communicator.
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For instance-often times we will teach
number as memorization rather than
students deeply understanding sets.
 Can our students count to 100 rather than
understanding the quantity and
magnitude first?
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“C-A-T” isn’t really a cat.
This is not a cat either.
It is, clearly, just a picture
of a cat.
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This is the number 8. The numeral is not
eight just a C-A-T is not a cat.
Is this eight?
It is not eight. It
is a picture of
8. It represents
the quantity nature of 8.
Teach “number comprehension.”
Teach the “eightness” of eight.
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Help children “see” numbers in a way that
will help them understand the
compositions and decompositions of
number.
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“instantly seeing how many”
This is a critical skill and may lay underneath
early math number sense difficulties with
addition and subtraction.
Most humans can only subitize
collections of 4 or 5.
After 5, we must combine smaller
numbers to comprehend the collection.
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8-5=8
7-4=7
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-
=
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Number Worlds Griffin
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In reading, we develop context for students
through experiences, practice, and text
analysis. In early math, the number line is
the context.
It is particularly important for the student to
associate magnitude to the NUMBER LINE.
The idea that moving in a certain direction
implies an increase in quantity. (Griffin)
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How large is eight?
Where does it live on the number line?
How much larger is 8 than 7?
How does 8 relate to 5 or 10?
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Makes
Ten
Left Over
lesson: valerie faulkner
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8+5
8
+ (2 + 3)
(8 + 2) + 3
1 ten and 3 ones (13)
Makes
Ten
Left Over
lesson: valerie faulkner
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8+5
8
+ (2 + 3)
(8 + 2) + 3
1 ten and 3 ones (13)
Makes
Ten
Left Over
lesson: valerie faulkner
36
8+5
8
+ (2 + 3)
(8 + 2) + 3
1 ten and 3 ones (13)
Makes
Ten
Left Over
lesson: valerie faulkner
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https://www.teachingchannel.org/videos/min
gle-count-a-game-of-number-sense
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“good intuition
about numbers”
Numeration
First three components of
number sense are anchors
for building a rich
understanding of math.
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It is the word, symbol, and the visualization.
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numeration (noun)
◦ the action or process of calculating or assigning a
number to something.
◦ a method or process of numbering, counting, or
computing.
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Gellman and Gallistel’s (1978)
Counting Principles
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1-1 Correspondence
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Stable Order
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Cardinality
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Abstraction
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Order-Irrelevance
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Knowing the counting sequence is a rote
procedure.
The meaning attached to counting is the key
conceptual idea on which all other number
concepts are attached.
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Verbal Counting has two separate skills
1. Child must be able to produce the standard
list of counting words in order: “One, two,
three, four…”
2. A child must be able to connect this sequence
in a one-to-one correspondence with objects
in a the set being counted. Each object must
get one and only one count.
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The cardinality principle is an understanding
that the last count word indicates the amount
of the set.
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The ability to count on is a “landmark” on the
patch to number sense.
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Three relationships students must
understand:
1. One and two more, one and two less
2. Anchors or “benchmarks” of 5 and 10-Because
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10 plays such a large role in our numeration system and because
two fives make up 10, it is very useful to develop relationships for
the numbers 1 to 10 connected to the anchors of 5 and 10.
3. Part-part-whole relationships-to conceptualize a
number as being made up of two or more parts is the most
important relationship that can be developed about numbers.
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In Pre-K, the student should know how to:
◦ Count to 8 (know the number words and their
order).
◦ Count 8 objects and know the last word tells how
many.
◦ Write the numeral 8.
◦ Recognize and read the numeral 8.
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Using their number sense, the student should be
know…
◦ More and less by 1 and 2-8 is more than 7, one less
than 9, two more that 6, and two less than 10
◦ Patterned set for 8
◦ Anchors to 5 and 10: 8 is 3 more than 5 and 2 away
from 10
◦ Part-whole relationships: 8 is 5 and 3, 2 and 6, 7 and
1, and so on (This includes knowing the missing part
of 8 when some are hidden)
◦ Doubles: double 4 is 8
◦ Relationships to the real world: my brother is 8 years
old; my reading book is 8 inches wide.
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In kindergarten, students will begin to
understand 16 as a set of six and a set of 10.
This work with decomposing numbers from
11 through 20 in kindergarten as seen as an
essential foundation for place value.
Students should continue to extend one more
than, two more than, one less than, and two
less than to numbers in the teens.
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Developmental Level Conversion Chart
Raw Score
Developmental Level
Score
C.A. Equivalents
1-3
-0.5
2-3 years
4-6
0.0
3-4 years
7-8
0.5
4-5 years
9-14
1.0
5-6 years
15-19
1.5
6-7 years
20-25
2.0
7-8 years
26-28
2.5
8-9 years
29-32
3.0
9-10 years
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“good intuition
about numbers”
Equality
First three components of
number sense are anchors
for building a rich
understanding of math.
52
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Equality is a mathematical statement of
equivalence of two quantities and nothing
more.
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Presenting 7 as a collection of objects
strung out one by one is appropriate
initially while students are working on one
to one correspondence and counting skills.
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Once students can subitize numbers up to four and
three, teachers work with students to develop their
ability to combine numbers into larger numbers.
Three and four together combine to make seven.
This is a much more powerful understanding of
seven than the one-by-one correspondence.
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3 + 4=7
7=3+4
Students understand
that seven has many
compositions.
What are some other
compositions of 7?
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11
3
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“good intuition
about numbers”
Base Ten
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Bridging the gap - moving
children from counting to
breaking up numbers
Students are counting by ones.
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Student counts
“One, two, three,
four bunches of
ten, and one,
two, three, four,
five, six, seven,
eight singles.”
Think about how
novel this is. The
student has
never thought of
counting a group
of objects as a
single item.
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8+5
8
Moving from ones work
to tens work
+ (2 + 3)
(8 + 2) + 3
1 ten and 3 ones (13)
Makes
Ten
Left
Over
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Reflect on how strange it must sound to say
“seven ones.” Students don’t say they are
“seven ones” years old.
“Ten” then becomes a singular noun.
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Unitizing is the place value understanding
that ten can be represented and thought of as
one group of ten or ten individual units.
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When working with numbers, you can take an
amount from one set and add it to another
set, the total amount does not change.
Referred to as “compose and decompose”
numbers.
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The key idea is that students practice
bundling and connecting this to symbols. For
example, 17 is read as seventeen, and one
ten and seven ones, and modeled as one ten
and seven and also seventeen singletons.
Students need to group materials so they
eventually understand that one ten and ten
ones are the same.
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Use Digit Correspondence Task.
Next, write 342. Have student read the
number. Have the student read one more.
What is the number that is 10 more? What is
the number that is 10 less? Watch to see
whether the student is counting on or back or
if they immediately know that ten more is
352.
Next, ask student to write the number that
represents 5 tens, 2 ones, and 3 hundreds.
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The Common Core State Standards in
mathematics were built on progressions:
narrative documents describing the
progression of a topic across a number of
grade levels, informed both by research on
children's cognitive development and by the
logical structure of mathematics.
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Read your grade level.
Annotate the text considering…
◦
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the mathematical language
Standards for Mathematical Practices
Methods or strategies
Common mistakes or misconceptions
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https://www.teachingchannel.org/videos/cou
nting-by-ten-lesson
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Identify one activity that supports teaching
the base ten system for your grade level.
Make sure you are able to describe the
activity and how the activity develops
students’ number sense.
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“good intuition
about numbers”
Form of a
Number
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Form of the number can be defined as
multiple representations of quantity, ratios,
and mathematical information.
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Number Worlds
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“good intuition
about numbers”
Proportional
Reasoning
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Proportional reasoning involves a
multiplicative relationship between two
quantities.
Proportional reasoning is one of the skills a
child acquires when progressing from the
stage of concrete operations to the stage of
formal operations.
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We view proportional reasoning as a pivotal
concept. On the one hand, it is the capstone
of children’s elementary school arithmetic;
on the other hand, it is the cornerstone of all
that is to follow.” (Lesh, R., Post, T., & Behr,
M. (1988)
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A fourth grade class needs five leaves each
day to feed its two caterpillars.
How many leaves would it take to feed twelve
caterpillars?
Solve this problem using a picture model and
then multiplication/division.
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“good intuition
about numbers”
Geometric
and
Algebraic
Thinking
It is human to seek and build relations. The mind cannot
process the multitude of stimuli in our surroundings and
make meaning of them without developing a network of
relations.
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Algebraic thinking or algebraic reasoning
involves
◦ forming generalizations from experiences with
numbers and computation,
◦ formalizing these ideas with the use of a
meaningful symbol system,
◦ and exploring the concepts of patterns and
functions.
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How are things related physically, what
information can we derive from patterns, how
are the numeric patterns and the physical world
related.
It is not a coincidence that Geometric and
Algebraic Thinking are right next to Quantity and
Magnitude.
Algebraic and Geometric concepts describe,
explain and predict quantity and magnitude in
the real world.
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14
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63
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45
n+1+1-1-1=n+(1-1)=(1-1)=n
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When skip counting, which numbers make
diagonal patterns? Which makes column
patterns? Can you describe a rule for
explaining when a number will have a
diagonal or column pattern?
If you move down two and over one on the
hundreds chart, what is the relationship
between the original number and the new
number?
These examples extend number concepts
to algebraic thinking.
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“When will this be true?” and “Why does this
work?” questions require students to
generalize (and strengthen) the number
concepts they are learning.
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The equal sign is one of the most important
symbols in elementary arithmetic, in algebra,
and in all mathematics using numbers and
operations.
The equal sign is the principal method of
representing relationships.
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In the following expression, what number do
you think belongs in the box?
8+4=
+5
How do you think students in the early grades or in
middle school typically answer this question?
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Where does the misconception come from?
Students come to see = as signifying “and the
answer is” rather than a symbol to indicate
equivalence.
Subtle shift◦ Rather than ask students to solve a problem, ask
them to find an equivalent expression and use that
expression to write an equation.
◦ So, 45+61=40+66
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“good intuition
about numbers”
Algebraic
and
Geometric
Thinking
Proportion
al
Quantity/
Magnitude
Numeration
Reasoning
Form of
Equality
a
Number Base Ten
Components of Number
Sense
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95
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http://www.metacafe.com/watch/115094/m
a_pa_kettle_math/
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Is math-language language?
How can we use some of what we do in
reading instruction to improve our
mathematics instruction?
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Quantity/
Algebraic
Magnitude
and
Geometric
Thinking
Proportion
Numeration
al
Language
Reasoning
Form of
Equality
a
Number Base Ten
Components of Number Sense
© 2007 Cain/Doggett/Faulkner/Hale/NCDPI
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Facts with
Number Sense
Fact Mastery
Activities
Fact Fluency
10
0
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Develop knowledge and associations of
number relationships and number sense
◦ Connections to how numbers are related both
sequentially and quantitatively
◦ Number properties (commutative, identity, zero,
associative)
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Examples of relationship activities
◦ Plus 1 facts, fact families, ten frames, skip counting
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Reduces memorization load-390 Facts!
◦ 100 basic addition, subtraction, and multiplication
facts
◦ 90 division facts (division by 0 is undefined)
10
1
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Additive Identify Property of Zero
Commutative Property of Addition
Plus(or add) 1 Rule: “When you plus (or add)
1, you say the next number.”
Plus 2: Skip over 1 number; next even or next
odd number
Plus 9: add 10 and take away; take 1 away
and add 10
Doubles +1 (Associative Property)
10
2
10
3
10
4
10
5
10
6
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What number comes next?
When you plus or add 1, the answer is the
next number or
When you plus 1 you say the next number.
Let’s try it with some problems:
◦ 5+1

8+1
2+1
Let’s apply the commutative property of
addition:
◦ 1+5
1+8
1+2
10
7
10
8
10
9
11
0
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What number is 2 more than that?
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Strategies
◦ Plus 1 and plus 1 more
◦ Count 2 more
◦ It’s the next even or odd number
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Let’s try it with some problems:
◦ 5+2
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8+2
4+2
Let’s apply the commutative property of
addition:
◦ 2+5
2+8
2+4
11
1
11
2
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Think plus 10 and take away 1
Think take away 1 and plus 10
11
3
11
4
11
5
11
6
11
7
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Number families: 3 related numbers make 2
addition and subtraction facts
◦ 8,5,3: 5+3=8 and 3+5=8
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Ten frames: Visualize/recognize sums of 10
◦ 7+3 and 3+7
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6+4 and 4+6
Sums of 10 + how many more? (Associative
Property)
◦ 8+3 (8+2=10 so 1 more=11)
◦ 5+7 (5+5=10 and 2 more =12)
11
8
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Emphasize inverse relationship to addition
Minus 0
Minus 1
Minus doubles (16-8)
Minus 2
Think addition (with sums of 10 or less)
Number families
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9
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Students should move to mastery activities
ONLY after number sense activities are well
established.
Moving to mastery activities too quickly can
cause students to revert to inefficient finger
strategies.
12
0
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Ecourage students to continue to employ the
number sense activities to retrieve answers to
facts.
Encourage students to skip problems they
can’t recall quickly or with new strategy. (No
counting fingers.)
Goal is accuracy, fluency, and automaticity.
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1
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Provide massed practice on a limited set of
new facts practiced with a number sense
activity
Followed by cumulative review with mastered
facts
Control response time (2-3 seconds per fact)
20-40 problems is sufficient (1 to 2 minutes)
Set high criteria (95-10% accuracy)
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2
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Provide cumulative review during mastery
activities.
Systematically integrate the need for math
facts in daily math activities.
Encourage students to use knowledge of
math facts to solve problems in daily math
lessons.
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3
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How is math exactly like a mystery novel?
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Do we really try to see the BIG picture?
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How does our chapter in the novel fit the
whole story?
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