Teaching number sense sounds simple…
…but it is not as simple as we once taught it. Before, if students understood 1+2=3, then our job was done. In 21st century learning, math has become more of a problem-solving science and an art, not just simply give the correct answer. The counting on addition strategy is one of these concepts that students need to master.
It all started with the Common Core State Standards, which states in 1.OA.C.6 that one way students can demonstrate math fact fluency is to use strategies, one of them being counting on. What does it mean to use the counting on addition strategy?
What is the count on method?
Counting on is when students, ideally, take the larger of the two addends and “count on” with the other addend to get the answer, or sum. For example, if the number sentence is 7+2, students will identify the 7 as the larger number and then count on two more–“7…eight, nine. The answer is nine.”
Every year, there are always at least a handful of students who count all, even though there are shortcuts and strategies we teach them. “Put the big number in your head and count on.” They may do it with us, but not always independently. We teachers need to know more than just understanding the counting on strategy, but also student misconceptions to keep them from using the strategy on their own.
How does counting on help?
Math fact fluency opens the door for students to allow their working memory to do the taxing job of problem-solving. If students cannot recall their math facts quickly, it is taking them even longer to complete and solve multi-step problems. This tends to lead to an increase in learned helplessness, and we hear things like, “This is too hard!”, from the student. This leads down the slippery slope of avoidance to finally noncompliance. Not good! Not fun for the student or the teacher.
Before learning counting on, of course students should be able to count all. From the time they are able to understand counting as a toddler up to kindergarten, students are given many counting all opportunities. Counting all is the only way to count before kindergarten. Counting all the objects is a trusted practice that is hard to change.
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Research on teaching the counting on strategy
I’m going to share what I’ve gleaned, and if you’re a data geek, keep reading. If you want the nitty-gritty and don’t want to geek out on the research right now, then scroll down check out the freebie at the end.
According to the research of Secada, Fuson and Hall, there are 3 stages of learning when teaching counting on. First, students must know how to count one-to-one to find the whole set. After students can count one-to-one, they move on to knowing the total of one part and then counting the other part to find the whole. When students become fluent with counting on, introduce the strategy of starting with the biggest part and count on the rest, or other part, to find the whole.
It is important to give lots of time teaching students to count on from any number before teaching the strategy to start with the larger part before counting on. It is developmental and an extra process students tend to overlook if not given enough practice of counting on whether starting with the larger or smaller part.
One way to do this is to show students a number as one part and giving them objects to count on from that number, which is the other part, to find the whole. If students see objects to manipulate for both parts, they may be likely to count all instead of count on. To push students towards counting on, they need to see numerals and objects to count.
If students are struggling with counting on…
…it just may be that they have the misconception that to count on we always start with “one”. They may need lots of teacher-modeled practice to understand this isn’t true in order to move from counting all to counting on. Let’s look at a breakdown of what a student must do to successfully count on.
Working memory is busy with counting on strategies, whether with the larger number or not. First, a very concrete concept is the simple understanding of counting as one-to-one correspondence. For students to move from concrete to abstract, they need to count up from the first addend to the second addend precisely and correctly in sequence, without any mistakes. Third, the actual work of counting from beginning to end in the correct order choosing the larger addend to start. The last two are more abstract concepts, with students usually using their fingers as a bridge back to concrete.
In order to count on successfully, students must say one addend from the number sentence. Then with the second addend in their mind, count “that many more” to get to the sum. This can be tough work for students and that is why modeling is pertinent at this stage.
So, what can we as teachers do to help students learn these count on strategies? Well, I’ve created some easy to follow mini-lessons. They follow the sequence of successfully learning and retaining the counting on method, as determined by Secada, Fuson, and Hall (1983).
For these interventions, show students the number and objects or dots that represent the amount for numbers up to 20. There are 4 lessons that are super quick. You teach and repeat each lesson until students are successful in so many trials before moving onto the next one. Each lesson only takes a couple of minutes at a time.
Mini-lessons included are:
- believe the number
- interrupting teacher
- the “plus one”
- start with the big one
Notice that starting with the bigger number isn’t until the very last mini lesson. There are 3 other lessons students need to learn before even bothering with finding the greater addend.
What do others students do while you are teaching small groups or individual students these lessons? There are some great math centers that lend themselves to the counting on addition strategy. They are easy prep and simple for students to understand. Perfect for the busy teacher!
Why is counting on important?
Counting on is the next step after the counting all method. Counting on in addition builds on number sense. When a student is counting all, they will always start counting at “one”. This makes for tedious work when adding larger numbers. So in order to add larger numbers, learning to count on in addition is the next logical step in mental math.
Secada, W., Fuson, K., & Hall, J. (1983). The Transition from Counting-All to Counting-on in Addition. Journal for Research in Mathematics Education, 14(1), 47-57. doi:10.2307/748796
You can do it all with this researched-based intervention kit for assessing and teaching students, to the easy to prep counting on math centers. I will share the behaviors typical for each strategy and the four mini-lessons important to grow your students to become more fluent and fast in their addition math fact fluency.