Understanding decomposing (breaking apart)
Understanding composing (putting together)
Cardinality - knowing the number of elements in a given set
Through the many years of learning and working with subitizing, it is evident that finger patterns play an essential role in helping young children decompose and compose numbers 1 to 10.
As students move into the deep understanding of decomposing (breaking apart) and composing (putting together) numbers 1 to 10, arithmetic fluency begins: first with the basic facts and then with the teens-digit numbers. Once connections are made with the teens-digit numbers, students move into greater number(multi-digit) arithmetic with ease.
This understanding of decomposing and composing number will become evident as students engage in the learning of the basic fact strategies.
For example, let’s look at 2+3. First, before students have cardinality, they will find answers through a ‘count all’ strategy. Once cardinality is developed, students will use a ‘count on’ strategy. For the ‘count on’ strategy, the sum will not change whether the count starts with either addend (2 or 3).
If you are uncertain as to what cardinality means it is knowing the number of elements in a given set(group).
For example, if I ask a student to use counters (manipulatives) to show me a set (group) of 8. He counts out 8. Then I ask him, "How many do you have in the set(group)?"
If he can tell me 8 without hesitation he has cardinality. If he has to do a ‘count all’ to tell me there is 8, then he doesn’t have cardinality. And research states that without cardinality, it is a challenge to develop number sense.
As deep learning occurs, the transfer will be made to the doubles plus 1 strategy.
With this, a student would say:
I know 2+3 =5 because 2+2=4 + 1 is 5.
What he or she did is decompose 3 as 2 and 1.
This, in turn, will lead to the doubles plus 2 and other efficient strategies. I am focusing on addition, but this understanding is crucial to all operations (addition, subtraction, multiplication and division).
Without the work of finger patterns, efficient strategies are less likely to be understood or may take a longer period of time. In turn, our students will either remain using less efficient strategies of ‘count all’ and ‘count on’. They may require support from counting on their fingers, a number line or a hundreds chart; and/or they will memorize the facts without any understanding of the how and why.
I have heard from many upper elementary and middle school teachers that they still have students counting on their fingers. This indicates that students have not gained a deep learning of decomposing and composing number.
To help you understand the strategies I am referring to I recommend you listen to Jo Boaler’s Youtube titled, Jo Boaler’s Clip on Number Sense:
When deep and transfer learning is happening with numbers 1 to 10, students demonstrate fluency with multi-digit numbers.
For example, a student who decomposes and composes numbers 1 to 10 would explain finding the sum of 37 and 25 as such:
I will take 5 away from the 7 leaving 2 ones and I will add the 5 to 25. Now I have 30 and 32 which is 62 or I can take the 5 from 25 and break it apart as a 2 and 3, then add 3 to 37 which is 40 and 22 more is 62.
When you are working with students on greater number fluency you can work with number strings based on basic facts. A number string would look like 5+7, 15+7, 25+7, 35+7… if a student was thinking:
I know 5+7 is 12 because I think of 7 as 5 and 2 so double 5 is 10 + 2 is 12, then 15+7 is 22 because I think of 7 as 5 and 2 so 15+5 is 20 + 2 = 22.
Number strings can be used for subtraction as well. It may look like 12-8, 22-8, 32-8… if a student was thinking:
I know 12-8 is 4 because I think of the 8 as 6 and 2 so 12-2 is 10 and take away 6 more is 4, then 22-8 is 14 because I think of 8 as 6 and 2 so 22-2 is 20 take away 6 is 14 (or a student may think of the 6 as 5 and 1 so 20-5 is 15 take away 1 is 14).
If you are wanting to learn more about number strings Cathy Fosnot’s book, Young Mathematicians At Work...Constructing Number Sense, Addition, and Subtraction is a rich resource.
The above examples require prior learning of finger patterns. Finger patterns can be easily incorporated into one of the research based subitizing activities known as Dot Flash. I have engaged my Grade 3 students with finger patterns from the beginning of September, and will continue to the end of June. It is benefitting their understanding of teens-digit and multi-digit adding and subtracting, as well as with the multiplication and division basic facts.
I did the Dot Flash activity with my class on Friday, Feb. 3rd. First, we did the activity Dot Flash which I have explained in my subitizing blog. You can also find this activity in my professional capping paper I have written and published: The Benefits of Subitizing: Helping Early Childhood Educators.
After flashing the collection of dots 3 times, I went around the room letting each of my students whisper in my ear how many dots they saw. Then, I asked them to show me on their fingers how many dots they saw in the collection. The pictures below show some of the finger patterns shown by students.
I asked each of those students to tell the parts of 8 represented by their finger patterns. The picture below shows you how I recorded the parts.
Using the correct vocabulary is important so as I was recording the students’ thinking I said,
You have just decomposed(broke apart) 8 into different parts.
Then I asked,
Have we decomposed 8 into all the possible parts?
Very quickly, I heard no and students began providing me with more answers. The picture below shows their thinking:
Try it with your students
If you teach Gr. 4, 5 or 6 you can do this with numbers beyond 10.
What I do with my Grade 3 students come April, May, and June is I have them pair up or get into groups of 3 or 4 to show me the parts of a number greater than 10. They have a lot of fun with this and I have even been asked if they can take their socks off and use their toes. That we do not do, of course!
I will also be hosting a webinar to help you further understand the research-based activity known as Dot Flash on Feb. 22nd from 7:30 to 8:00 pm. I hope you can join me for this!
I look forward to hearing how it went with your students! Leave a comment below with your questions and comments.