Early Number and Counting
What letter is elemenopee? We know that it is not a letter but the alphabet song rushes through saying “LMNOP” which has caused some students to believe it is one letter. Other students think “AND” is a letter because we say “X Y AND Z”. It is imperative we slow down and explore each letter when learning to say the alphabet and to recognize the letters. Reciting the number sequence from 1 to 100 is like saying the alphabet. Just because students can recite the sequence without error, we sometimes assume that they understand that 6 is before 7 and that 7 includes 6, and that 8 is not the same as “ate”. However, reciting the number sequence is also NOT like saying the alphabet. When counting, the order of the numbers matter. One is before two which is before three. Three includes two which includes one. However, the alphabet is written in an arbitrary order. The letter m appears before n but it wouldn’t matter if m appeared after n. The letter m is not part of the letter n.
We need to ensure that students have many opportunities to fully delve into early number and counting principles in order to build deep, conceptual understanding.
In this video, Graham Fletchy walks you through his “Progression of Early Number & Counting“, providing simple examples of activities.
Big Concepts in Early Number and Counting
Visual (non-symbolic) Magnitude: Comparing two non-numerical quantities to determine which has more. Counting is not required. Looking at two sets of dots and knowing which has more, without necessarily knowing how many are in each set.
One-to-one Correspondence: Understanding that each object in a group can be counted once and only once. You say one number as you count one object.
Cardinality: The last number used to count a group of objects represents how many are in the group. After counting out a set of objects, the last number you say tells you how many objects there are.
Order Irrelevance: The order in which items are counted is irrelevant. When you count the same objects again, you can start with a different object but you still end up with the same number of objects.
Conservation of Number: The count for a set group of objects stays the same no matter whether they are spread out or close together. You can move the objects around but the quantity stays the same.
Stable Order: Number names are said in a specific order and that order does not change.
Subitizing: Recognizing, at a glance, a familiar arrangement of up to 10 objects or dots. This includes both perceptual subitizing (know how many without thinking – typically up to 6 items) and conceptual subitizing (you see multiple groups and combine them to determine the total number. ie. a domino with 5 dots and 3 dots is 8 dots). When you see a die with dots, you know how many dots there are without counting each dot. When you see a domino, you know how many dots there are in each part and can figure out how many there are altogether without counting each dot.
Symbolic Magnitude: Comparing two numbers symbolically and knowing which is larger. Four is larger than two.
Hierarchical Inclusion: Numbers are nested inside of other numbers. When you count seven objects, you know there is also six objects in that group.
DRAFT One-on-one Interview
CESD is currently developing a one-on-one interview that could be administered to students in order to identify their conceptual understanding of numbers and counting. This is in DRAFT form and currently only available to CESD educators.
Counting Strategies for Addition
When students are first introduced to addition and subtraction, they use counting strategies rather than Additive Thinking strategies.
Count All: Students count each and every object in order to determine “how many”. For example, to solve 3 + 2, they count the first three objects and then continue on to count the next two objects for a total of five objects.
Count On: Students start with the first number as a group and then count the second amount one by one. For example, to solve 3 + 2, the student might say 3, 4, 5.
If students are struggling with one-to-one correspondence, cardinality, order irrelevance and/or hierarchical inclusion, a five frame, a ten frame, an ice cube tray, an egg carton, or a numbered ten frame might help. Have them place objects on the ten frame/ice cube tray to keep track while counting. Consider statements such as,
- “How will you keep track of the ones you’ve counted and the ones you haven’t counted?”
- After moving each item: “How many are on the sheet now? How do you know?”
- When finished counting all: “How many are there altogether?”
- When finished counting all: “Clear your board. If you start with a different object first (ie. the green bear instead of the red bear), how many will there be altogether? How do you know? Let’s check.”
If students are struggling with conservation of number, place a small number of items (such as 5 blocks) on the table in front of the student. Have them figure out how many there are all together. Move them around. Ask them how many there are now. Let them count, if needed. Ask questions like: “Did I add any blocks? Did I remove any blocks? How many are there altogether?” Squish the blocks close together. Ask again. Move them far apart. Ask again. Put some in front of you and some in front of the child. Ask again. Conservation of number is also focused on in the Equality strategy section.
If students are struggling with visual magnitude or counting, give them a set of cards with one dot, two dots, three dots, etc. and have them place it in order: smallest to largest, and largest to smallest. Use a variety of non-symbolic representations of numbers.
If students are struggling with counting on, give them a question such as 4 + 3. Have them create a pile of 4 and a pile of 3. Don’t let them count all. Instead, point to each pile and “how many?”. Cover up the 4 and ask “how many under here?” If they aren’t sure, uncover and ask again. Repeat until they are confident there are 4. Recover the pile. Tell them to count with you, point to the pile and say “4”. Point to one item from the other pile. If they don’t respond with 5, point to the pile of 4 again, say 4, point to one in the other pile and very, very slowly, say ffffffffiiiiiiivvvvvveeee….five. Point to the sixth item. If they don’t say six, point to the pile of 4, say 4, point to the 5th, wait, say five if they don’t, point to the sixth and repeat the process until all items are counted. Ask, “how many are there?”
Additional simple activities focusing on pre-counting, one-to-one counting, counting sets, counting from one to solve number problems, and counting on to solve number problems can be found at NZ Maths.
Additional activities focusing on counting, subitizing, comparing number, adding and subtracting can be found at Learning and Teaching with Learning Trajectories.
Number Paths vs Number Lines
Students should explore the use of number paths before working with number lines. Number paths focus on the counting model. Students can use one-to-one correspondence to count each rectangle or place objects on each number. Number lines are length models. They focus on the distance from 0. Students often, mistakenly, count the number of ticks rather than the distance between the ticks. Number Paths printable
Subitizing: What is It? Why Teach It? by Douglas H. Clements