There was a time when letters and sounds were taught in isolation. But now we know that children learn to read more successfully when they are enveloped in a print-rich environment (books available all the time and being read aloud, posters with text on the wall, lists, charts, etc.) Not only is this more effective, but it leads to deeper reading comprehension and effective critical reading skills.
We are in the midst of a similar revolution in math. We now know, through extensive research and practice with students, the dangers of teaching math facts and algorithms (memorized methods like borrowing and carrying) in isolation. It often leads to students who can only do distinct individual processes, but don’t really understand the mathematics they are doing.
Many of us were given a similar tip-of-the-iceberg education. Because of this we can be unequipped to solve more complex, multifaceted problems that come up in our everyday lives. We have a method, but we don’t know how or when to apply it. It may take us part of the way there, but then we’re stuck, unable to determine if our solution is correct or makes sense.
There is a better way. If we start with the understanding first, then we can give kids the kind of foundation they need to live successful mathematical lives.
Recently, our fourth graders worked on a project to count all of the beans in a one-pound bag. This is a variation of Counting Collections, a learning technique designed by researchers at UCLA. While this is a seemingly simple task, perhaps better suited to small children, there is so much more to the math here than meets the eye.
I began by asking our students to estimate what they thought the total would be. The answers ranged from 200 to 2000. Estimating is an essential real life skill that we practice often in class. Then they got to work. Students partnered up, grabbed a big handful of beans, and got counting. Right away, the conversations showed deep math thinking. There were discussions about the most efficient way to count. One group started to count by twos, and then changed to counting by fours, and then fives, and finally by tens. Another group decided to make piles of 20 beans. This kind of work requires students to express their thinking, work collaboratively, and incorporate the ideas of others.
One common stumbling block was how to keep track of the beans that had already been counted. I could see this problem was impacting many students, so we came together to look at one group’s work. They had arranged their collection in rows. One of the partners explained that this helped them to know where they were in the counting and avoid recounting or skipping. Everyone went back to their own area and considered whether they wanted to incorporate this strategy. Grappling with an authentic struggle is much more powerful than being given techniques to use. It also develops essential brain pathways.
Once all of the hands-on work was done, students drew a diagram that represented their counting, and then wrote a corresponding number sentence. Moving from physical objects to pictures to numerals allows children to form important neural connections.
Sometimes while we are working on these Counting Collections the need for new notation arises. On this day, students wanted to represent an idea that required parentheses. They had grouped their beans into a 5x5 array of rows and columns with 20 beans in each pile (with 33 extra beans).
Their first attempt at showing this was 5 x 20 + 5 x 20 + 33 = 533. I had them walk me through the math, and they saw that it only came to 233, not 533. Then one student explained that they were trying to show that it was 5 x 20 done five times. I was able to show them how to use parentheses right at the moment they were needed in a meaningful, authentic way
Other times, mathematical ideas come from student conversation. During this activity, one group decided to arrange their beans in rows and columns. When I asked them why they chose this arrangement, one student explained that by putting them in an array they could easily multiply instead of adding the piles together.
After everyone was done recording, we came together as a class to discuss all of the different methods groups had used. As students shared their strategies and ideas, I wrote the equations on the board. This is an important part of the process because students see that all work is valued and that there are many different pathways to a solution. They also learn about methods they may want to try in the future.
On another day, we came back together to combine all of the subtotals. Each student had the opportunity to work independently first, and then went with partners to figure out how they would add these five large numbers together. Again we came together as a group and discussed the different methods that were used. As a final step, we solved the problem on the board. That is how my students found that there were 1,359 beans in our one-pound bag. It was also a great opportunity for them to demonstrate their learning. And, it was a robust way for me to know what they really understand and to continue building on it throughout the year.
For us, this was a great demonstration that in spite of having “covered a topic” students often continue to nurse misconceptions - misconceptions that we can only address and correct when we allow students to test their own ideas and construct their own knowledge.