During the first two weeks of May, I visited every class at Byng Elementary in preparation for a professional development morning on May 16. One of the school’s goals this year is to develop their students’ computational fluency. Alongside the student goal, of course, is a professional growth piece for the teachers at the school. What is computational fluency? How is it developed? How do we assess it?
Our journey began in September with a professional development morning and since then there has been a fall performance-based assessment, professional discussions at lunch, the development of posters highlighting our curriculum’s mental math strategies and the teachers have been working with their students in an intentional manner to develop their students understanding and fluency with mental math. A resource we have used to support our work is Number Talks by Sherry Parrish.
On Friday morning, I shared photographs from each of the classroom visits from K-7 and teachers reflected on and discussed their students’ growing ability in mental math strategies and fluency with working with numbers.
Although there are not mental math strategy learning outcomes at Kindergarten, necessary concepts and skills that are being developed are subtilizing, decomposing numbers and developing five-ness and ten-ness. In the K and 1 classes, we working on finding many ways to make 7 (decomposing a quantity of seven into 2, 3, 4 + parts)
and then began to think about how to represent numbers using ten frames.
With the grades 1 and 2 students we used ten frames to provide a visual support for thinking about mental math strategies to solve 8 + 5 and 8 + 7 and then expanding on these. We were looking for ways students might count on (ie start and 8 and count on five more, 8, 9, 10, 11, 12, 13) or use doubles (i.e. I know 7+7 is 14 so 8+7 is one more than that so it is 15) and making 10 (i.e. to add 8 + 5, I can take 2 from the 5 to make the 8 a 10 and then I have 3 left from the 5 so that is 10+3 which is 13).
As students shared their strategies, we recorded their thinking on chart paper, whiteboards, chalkboards and Smartboards in their classrooms.
The grades 2 and 3 students worked with adding larger numbers, beginning with a question like 8+7 so we could assess their strategy use and then building to 18+7, 48+7 and then 48+37.
The students continued to use doubles and making ten strategies and decomposed the numbers into tens and ones as well. The open number line was introduced in these classes as a way to record the adding of numbers by counting on by 10s and making friendly numbers.
The grade 3 and 4 class worked with finding differences between two larger numbers, a slightly different way from looking at subtractions as “taking away”. The open number line or linear model is a visual tool that supports this understanding.
For students to solve 123-49, some students decomposed 49 into 40 and 9 and subtracted each from the 123, one at a time. Others, using the difference approach, began at 49 and added up to 123 to find the different. This is where the open number line was used by some students. So beginning at 49, students added 1 to make 50 (a friendly number to work with) and then added 50 to make 100 then 20 to get to 120 and then 3 to get to 123. Adding up 1 + 50 + 20 +3 is a quick mental calculation to find the answer of 74.
The intermediate classes worked on the doubling and halving strategy for use with multiplication, as well as practicing annexing the zero as a strategy. We began my creating models to show how the doubling and halving strategy works. Beginning with an array of 1×16, student halve the row, bringing 8 down under the first 8 to create two rows of 8. Halving and doubling again creates a 4×4 array and then an 8×2 and then a 16×1 array. The arrays and process was recorded for the students.
Students were then asked to visualize the 4×4 array but instead think of it as 16×16. What would halving and doubling do? Most students realized that by getting to 8×32, they could probably do that easily mentally by decomposing 32 into 30 and 2, but many wanted to go further and play with the number pattern.
We then challenged the students with a seemingly difficult question, with an odd number (that couldn’t be halved into whole numbers) but students realized they could double the 35 and halve the 16. This got the students to 8×70. By annexing the zero, students could recall the multiplication fact (again, emphasizing how important it is that students have ways to mentally calculate these facts) of 8×7 and then add the zero back to get to 560.
There were some aha moments for students with this one. 16×35 seems difficult to do in your head but that is the power of mental math strategies. If you understand them and why they work (as opposed to memorizing steps or procedures) you can play around with numbers in a way that utilizes reasoning and sense-making, very important aspects of thinking mathematically.
As an assessment checkpoint and extension task, I asked students to work with a partner to think of a challenging question that would utilize halving and doubling as well as annexing. Math graffiti filled the whiteboard as students shared their questions, giving us insight into the students’ deeper understanding of these strategies.