I hope this email finds you happy, healthy, and loved. I think this is a great week to count, count, count, count, count.

I did not love counting collections when I was first introduced to them by Dr. Angela Chan Turrou of UCLA. I thought it was silly, something that would be good for preschool-age children. However, after working with students in grades TK – 6th, I know counting collections is necessary for students of all ages.

Organization

Lack of organization will lead to miscounts, inaccuracies, frustration, and wrong answers. In the last few months, it has become apparent that many students struggle with math due to their lack of organizational skills. Students allow their fingers to glide over objects and images as they count (unknowingly missing some) instead of touching or touching and moving each object as they count. Touching and moving objects into groups when counting then transferring those groups to paper is necessary for multiplication and division success. When I lean in and listen to a student count aloud their collections, I am amazed by the many mistakes I hear, especially in grades 3-6th. Students rarely count aloud. If you have students underperforming in math, you might want to listen while they count objects.

While students are representing their problem-solving drawings or models on paper, it is sometimes difficult to decipher the groups they have created, the final answer, or the flow of their strategy. Students often have difficulty solving or explaining their processes in an organized manner. When students count collections, counting aloud and organizing one’s groups is a fundamental focus of the lesson.

Subtraction Skills

Subtraction skills tend to be more difficult for students to develop when compared to addition. To be good at addition students, need to know the number order, e.g., 11 + 5 sounds like 11, 12, 13, 14, 15, 16.
To do well with subtraction, students need to have hierarchical inclusion. Hierarchical inclusion includes knowing what numbers are embedded within another number. Students need to know and have multiple experiences counting and decomposing 12 into  (11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1). To subtract 12 – 7 = efficiently, students must know that counting up to 12 from 7 (8, 9, 10, 11, 12) will get them to the final answer.

It is also helpful if students are comfortable counting backward from any given number. Counting collections is the perfect opportunity to develop hierarchical inclusion and the skill of counting backward. Try listening to a few of your students count out loud backward this week.

Understanding the Relationship Between Multiplication and Division

When a student counts a large group of objects and puts them into smaller equal groups, that is division. Students can write down the total amount of objects they have in their large group. Then, record the number of smaller groups embedded within the larger group (hierarchical inclusion). Students even have the opportunity to work with remainders.

When a student makes equal groups of items and counts up all of the groups, that is the foundation of multiplication or repeated addition (5 x 10) = 50.

When students determine how many groups of ten are in a number, they are working with both place value and powers of ten. Students can also connect the powers of ten to exponents.

1 x 10 = 10 = 1 cup with ten blocks equals ten = 10 to the power of 1

10 x 10 = 100 = 10 cups with ten equals 100 = 10 to the power of 2

100 x 10 = 1,000 = 100 cups with ten equals 1,000 = 10 to the power 3

Students can make better sense of these abstract concepts when they count collections for 5 – 10 days in a row versus using a standard textbook or worksheet lesson. Yes, counting collections would count as your entire math lesson for the day.

Number Sense

You know I love hearing from you…

but I will be focusing some extra attention on my family in the next few weeks. (Wish me luck. Just kidding.) I will also be practicing a lot of self-care, e.g., drinking water throughout the day, long walks, moving more slowly and thoughtfully through the day. I hope you enjoy your time off and have the opportunity to take excellent care of yourself.

In the meantime, if you are looking for a good holiday read try Math By The Book from Heinemann.

The University of San Diego also has an excellent online CGI Math course. Take up to 6 months to complete the course and earn up to 3 graduate-level units.