Current research helps us understand what high achieving math students do and it is most interesting, in fact Jo Boaler tells us it is worth shouting from the rooftops so here it is:
High achieving math students use flexible thinking, are able to easily decompose and recompose numbers and naturally compress ideas to move on to harder concepts.
What does all this mean? Let’s look her explanation.
Consider a simple computation such as this:
5 + 14
There are several possible strategies to use.
Counting all: First count all the blue counters to 5. Next, count the purple counters to confirm that there are indeed 14. Lastly, proceed to count all the counters from 1 to 19.
Counting on: Count the first set of blue counters to 5 and count on to 19.
Known facts: A student may simply know that 5 + 14 equals 19.
Derived facts: Students use what is known about numbers and facts to complete the computation. Fourteen is also 10 + 4 so, since 4 + 5 equals 9, add 10 and solve the problem to get 19. Although this may seem obvious to an adult reader it is important to note the process of decomposing and recomposing the number to make the problem easier.
“People who are good at mathematics decompose and recompose numbers all the time.” Jo Baoler What’s Math Got to Do with It? p.148
Interestingly enough students who are low achieving at math use approaches that are more difficult. For example, imagine using a counting back strategy for subtraction.
25 – 14 = ___.
There are many steps involved in counting back from 25. This is a complex task and one where a student can easily become confused. Students who find math difficult often apply a ‘follow the rules’, problem solving process, lacking the understanding to make sense of numbers in flexible ways. How much easier this very problem would be, if the student worked flexibly with numbers as was done in the earlier example. Fourteen becomes ten and four. Now the problem becomes 25 – 10 = 15 and step two, 15-4 = 11.
Open Cones or Long Ladders?
High achieving students compress mathematical ideas. What does this mean? Think about learning multiplication. Initially students struggle, work through the process and practice examples. Once students understand what multiplication is, and how to use it, the concept is compressed and easily used in new settings.
Jo Boaler uses an image of an inverted cone to show what is meant by compression and how it helps students as they learn. Learning is compressed as students begin to apply understanding efficiently. New learning is built on compressed ideas and understanding grows.
Low achieving students who work at trying to remember rules, methods and procedures have a different view of mathematics, much like an endless ladder to be climbed and a long series of steps to be remembered. These students need to be guided to develop the skills of working flexibly with numbers, and to develop a deeper understanding of number sense.
This post is a summary of the information presented in Jo Boaler’s book: What’s Math Got To Do With It? Key Strategies and Ways of Working Chapter 7. Also available at the St. Albert Library.