NUMBER SEQUENCE
VS.
COUNTING TO ADD
EARLY COUNTING AND ADDITION
FOCUS ON THE THINKING PROCESS
• When working with children in mathematics, the focus should be
on the child’s responses and strategies rather than right/wrong
answers.
• The goal is to learn about a child’s current mathematical thinking.
UNITARY VS. COMPOSITE STRATEGIES
• Constructs 0-3 are considered “unitary” thinkers because they are your “count-by-one”
strategy children.
• Construct 3 is moving toward a composite thinker.
• Constructs 4-5 are considered “composite” thinkers because they are able to use noncount-by-one strategies to solve addition and subtraction tasks.
ADDITION AND SUBTRACTION STARTS WITH THE COUNTING
PROCESS…. WE ARE TRYING TO FIND EVIDENCE OF THEIR
CONCEPTUAL CONSTRUCT FOR ADDITION AND SUBTRACTION.
• Construct 0 (Emergent)
• These children are UNABLE to count visible items.
• Construct 1(Perceptual)
• These children are able to count visible items. They need to see, hear, or feel items
or the number to count. These children use “replacement” when they counting by
putting up fingers to represent the number.
ADDITION AND SUBTRACTION STARTS WITH THE COUNTING
PROCESS…. WE ARE TRYING TO FIND EVIDENCE OF THEIR
CONCEPTUAL CONSTRUCT FOR ADDITION AND SUBTRACTION.
• Construct 2 (Figurative)
• These children can count concealed items. They typically count from 1. They use
“re-presentation” which means they may replace one number by seeing dot pattern
in their head. They need a tracking system to know when to stop. One might say, “9
is 1, 10 is 2, 11 is 3” when solving 8+3. Some kids need a running start for a problem
such as 13+4, the child might have to start saying “8,9,10…” to count up to 13 and
then count on 4 more becuase they can’t just start at 13.
ADDITION AND SUBTRACTION STARTS WITH THE COUNTING
PROCESS…. WE ARE TRYING TO FIND EVIDENCE OF THEIR
CONCEPTUAL CONSTRUCT FOR ADDITION AND SUBTRACTION.
• Construct 3 (Initial Number Sequence)
• These children can count on rather than counting from 1. A student in this construct can also
solve missing addend tasks by counting on. He/She can also use a count-down-from strategy
to solve removed items task (17-3 as 16, 15, 14). Must be able to do add/sub by count-on and
count-down in order to be a construct 3.
• Construct 4 (Intermediate Number Sequence)
• These children can choose the more efficient of count-down-from and count-down-to
strategies.
ADDITION AND SUBTRACTION STARTS WITH THE COUNTING
PROCESS…. WE ARE TRYING TO FIND EVIDENCE OF THEIR
CONCEPTUAL CONSTRUCT FOR ADDITION AND SUBTRACTION.
• Construct 5 (Facile Number Sequence)
• These children are able to use a range of non-count-by-one strategies. These students can use
compensation, using a known result, adding to ten, commutativity, subtraction as the inverse of
addition, and awareness of the “ten” in a teen number.
GOALS FOR ADDITION/SUBTRACTION FOR K-1
• KINDERGARTEN
• Fall: 1
• Winter: 1
• Spring: 2
• FIRST GRADE
• Fall: 2
• Winter: 3
• Spring: 4
LEVELS OF COUNTING
• If teachers have a good grasp of the levels of counting, they are better able to
develop instructional approaches focused on developing more sophisticated
arithmetical strategies in students.
• Emergent: unable to count visible items
• Perceptual: able to count visible items. Can take a number of items out of a
collection
• Figurative: count-from-one counter.
• Count-on: can count on instead of starting at 1
• Non-count-by-one: can use knowledge of 5, 10, doubles, etc to help solve
tasks.
COUNT VS. FORWARD NUMBER WORD SEQUENCE
• Counting: Matching a quantity to a collection (using number within a
context)
• Forward Number Word Sequence: Saying a sequence of numbers NOT in
a context.
CHILDREN WILL REGRESS IF NOT ENCOURAGED
TO MOVE FORWARD
• Children tend to revert back to the “easy” strategy for them when trying out new
strategies. They do not see the new way as easier when first learning.
• Our job as teachers is to encourage them to use more sophisticated strategies as they
are ready (not to push them too fast though, or you will force them to skip steps that are
essential to their numeracy development)
• Children will be ready at different times during the year for different strategies.
• It is important to take a child from where they are and progress them to the next level of
mathematical thinking as they are ready.
ACTIVITIES TO LEAD CHILDREN IN THE DIRECTION
OF MORE SOPHISTICATED STRATEGIES…
• Irregular dot cards with numbers 1-6. Place out cards and ask; Which has more, which
has less, how do you know, how do you see these?
• Increase and decrease in the range of 1-6 (move up to 10 when ready): 2 items visible.
Ask how many are there. Put them in a cup. Add 2 more and ask how many. Then add 1
more and ask how many. Then, take 2 out and ask how many. Continue taking and giving
up to 2 items at a time.
ACTIVITIES TO LEAD CHILDREN IN THE DIRECTION
OF MORE SOPHISTICATED STRATEGIES…
• Domino Addition: One domino side covered and one open. “I have 4 under the cover
and 3 here for you to see. How many are there altogether?”
• Spinner game: 2 spinners; one with 3,4,5,6,7,8 and the other with 2,2,3,3,4,5. Partner 1
rolls and gets that amount of items. Partner 2 rolls and gets that amount of items. Each
puts their items under cover and they have to discuss how they can figure out how many
items there are in all.
ACTIVITIES TO LEAD CHILDREN IN THE DIRECTION
OF MORE SOPHISTICATED STRATEGIES…
• On the Mat: Put out 4-6 mats, carpet squares, hoops, OR divide the room into 4-6
sections. When you give a determined signal or stop the music (if you choose to do it
with music), kids go to a “spot”. One person will count and tell the number of children
are on their spot.
• Discussion: Which group has the most? Which group has the least? How many more does
this group have than another? How many kids are there altogether between these two
groups?
ACTIVITIES TO LEAD CHILDREN IN THE DIRECTION
OF MORE SOPHISTICATED STRATEGIES…
• Toy Box/Collection Box: Roll a die only marked with a 0 and 1 (3 of each). Roll and add
that many to your collection each time.
• Discussion between partners: How many do you have? Which one of us has the most? Which
one of us has the least? How many more/less do you have? How many more do I need to get
to have the same amount as you?
• Vary the size of the objects you play with each time (have a mixed size item collection so they
learn that the larger objects doesn’t mean you have more)
• Encourage the children to have varied arrangements: ten-frame set up, scattered, circle, etc.
ACTIVITIES TO LEAD CHILDREN IN THE DIRECTION
OF MORE SOPHISTICATED STRATEGIES…
• Chains: Give 5-6 students a colored bracelet (can just use construction paper and staple
it together). All kids move as instructed “slither like a snake, hop like a rabbit, tip toe like
a ballerina” and then freeze when they hear the magic words (you choose) or stopped
music. Everyone with bands joins hands with anyone within reach (NO moving). Count
how many in the chain. Continue as long as desired.
• Discussion: Which line is the longest? Shortest? Can you estimate how many are in this line if
there are 5 in this line? What is happening as we continue play (we want kids to realize that
as the chains get larger in number, the number of chains reduces).
ACTIVITIES TO LEAD CHILDREN IN THE DIRECTION
OF MORE SOPHISTICATED STRATEGIES…
• Pass It On: Dice 0, 1,2… Each child has 5 items in front of them. Child rolls and passes that number of
objects to the person on their left. As play continues, the teacher should be asking questions such as:
• How many do you have? How did you come to have that amount (looking for kids to say, “ I had __ and
got __ more and now I have __”)
• Who has more _____ or ______
• How many more does ____ have than ____?
• Teachers: Notice how they are counting. Are they starting at 1 each time? Counting on? Are they getting a
running start to count-on?
ACTIVITIES TO LEAD CHILDREN IN THE DIRECTION
OF MORE SOPHISTICATED STRATEGIES…
• Hide and Add:
• Partner A rolls 2 dice and then covers them with a cup as they tells Partner B.
• Partner B has to determine how many pips (or dots) are under the cups. This person tells
Partner A what they think and Partner A agrees or disagrees.
• Both partners check by looking under the cups.
• Teachers should notice how students are counting and adding. Encourage more
sophisticated strategies if you know a child can use them and are not using them.
ACTIVITIES TO LEAD CHILDREN IN THE DIRECTION
OF MORE SOPHISTICATED STRATEGIES…
• Rhythmic Patterns:
• Clap, pat knees, tap desk, hop, etc.
• Alter speed and action
• You are aiming for students to be able to remember “I clapped twice on my knees, 3 times on
my desk, and clapped my hands twice… that means 2 and 3 and 2... So that is 7 altogether.”
• The student may count on or repeat the actions and count from one.
• Teacher: Notice the way each student goes about this activity. This will be harder than
other tasks where they can see a number, set of items, etc.