*Recently, I began a MOOC (massive open online course) called How to Learn Math – For Teachers and Parents (2). More than 40,000 people took the last class – mainly teachers, parents and school administrators. 95% of people completing the end of course survey said that they would change their teaching or ways of helping as a result of the course. The course offers important new research ideas on learning, the brain, and math that can transform students’ experiences with math and is based on the work of and narrated by Jo Boaler. It’s divided into 8 sessions and each one ends with a prompt to write a one paragraph summary reflection on the ideas. My plan is to post those reflections here.*

The fifth session is titled, *Conceptual Learning, Part I: Number Sense*. There are 3 ideas that spoke to me here.

Number sense is built from the flexible use of numbers. In my experience, teachers’ most urgent need for their students is number sense or basic skills. Does that mean that students need to be able to recall multiplication facts quickly? Does that mean that they need to remember to write the ones digit down and carry the tens digit when they add two digit numbers? Does that mean they need more procedural fluency? When students perform a question like 18+5, they may find it easiest to just start at 18 and *count on* to 23 or this might just be a *known fact* to them because they’ve seen it many times. It’s impossible to know all possible addition facts and it becomes extremely difficult and requires significant cognitive function when the numbers get larger. As a result, there is enormous room for error. This is why students need to use numbers *flexibly* so they can decompose and recombine them to make *friendlier* numbers. In the case of 18+5, they need to see this as 18+2+3. So the 18+2 becomes 20 which is a *friendlier* number. Adding 20+3 is easier to see than 18+5. The beauty of seeing numbers flexibly, is that there are many ways to see the same question. Some of us may see 18+5 as 5+5+5+3+5 and add up all the 5’s and then add-on the 3. As the numbers get larger, there will be more varied opportunities to break up the numbers and then put them back together.

Studies have shown that the difference between high and low achieving students isn’t that the high achieving students learn more – it is that they learn to act upon numbers flexibly. Flexible use of numbers and shapes is critical to all higher level math learning. In fact, it is foundational for pretty much everything that comes beyond in algebra, calculus, and beyond. The problem that low achieving students face is that they fail to engage with math in this way and rely solely on their counting method. As a result, they come to believe that math is only about the “right” answer. However, counting gets more complex as problems become more difficult. These students are often identified early on and given more drill and practice but without the proper strategies, this is probably the worst thing for them.

Mathematics must be taught as a conceptual subject. When we learn anything conceptually, a process called compression occurs in our brains. New learning takes up a lot of “space” in our brains. Eventually, if conceptualized, our learning is compressed and it becomes easier to access in the future. If new learning never progresses past methods and procedures, it will not be compressed and will become more difficult to retrieve in the future. This is why many students have difficulty with a mathematical process days or weeks later. If teachers and students aren’t engaging in conceptual thinking during mathematics instruction, students won’t benefit from compression.

Number Talks are easy and effective for developing number sense. Number talks is a process where teachers lead discussions with students on the flexible use of numbers as described earlier. Generally, students work individually, then together with one or two others on a math calculation like 18+5. As a group, various solutions are displayed and discussed. Right answers, wrong answers… it doesn’t matter because the process of how an answer is arrived at is more important than the answer itself. This process of taking numbers apart and then regrouping them as *friendly* numbers helps to conceptualize students’ understanding of operations with numbers and at the same time, teaches them number fluency and automaticity. Students can really benefit from number talks because they show how math is a creative and flexible subject not just a series of rules and methods to memorize. This flexible use of numbers is a very important building block in number sense and mathematics which is the foundational base from which all other mathematics builds. Here are a couple of resources for number talks.