Jamie Sirois, Stratham Cooperative Middle School teacher and Learning Area Leader, along with two of other colleagues from Bow Memorial School and Winnisquam Regional, gave a great workshop on the Common Core math standards at the NEA-NH Spring Instructional Conference at Bow High School on April 5th.
Among other things, she said,
“Rigor requires conceptual understanding, procedural skill, fluency and the ability to apply math to real problems. It is an equal balance of these three that creates rigor.”
Jamie responded to some follow-up questions I sent her this way:
What do you mean when you say the standards demand procedural skill and fluency?
To prepare students for higher-level math, the Common Core standards require students to do the basic calculations with speed and accuracy. Third graders, for instance, need to master core functions, or “rote skills” such as single-digit multiplication, in order to have access to more complex concepts.
Teachers can promote fluency by giving students opportunities to practice those skills in every classroom through the use of additional resources and supporting materials. Some students may need more time than others but, with enough practice, they can all begin to use those skills with automaticity – with fluency.
Say more about how the standards require students to apply that procedural skill and fluency to real problems.
The standards call for students to use their skills in situations that require mathematical knowledge. That means having both procedural fluency and a solid conceptual understanding of the math they have learned.
I’ve made big changes in how I prepare for my classroom. Instead of drilling on a specific skill and then applying it to a simple problem, I am thinking in reverse. How can I give my students an opportunity to apply what they already know to a problem that requires real thinking and then spend my teaching time working the skills I see they need help on?
For example, a problem related to grocery shopping could involve ratios and rates, decimal operations, the distributive property, etc… If I embed these questions into a complex problem and ask students, “What do we need to solve this?” they might identify the skills and begin to work towards solving the problem. Or they may not quite know what to do. This would be an opportunity to teach them how to break down a problem and identify the key information. Then we can move on to teaching the skills needed to solve it.How can I give my students an opportunity to apply what they already know to a problem that requires real thinking and then spend my teaching time working the skills I see they need help on?
I can see in the normal course of solving the problem what each student knows and doesn’t know. I don’t have to spend time drilling on skills they already have but can focus in on just where they need the help.
Say you’re teaching about the “greatest common factor” – or GCF, as we call it. That’s the greatest number that divides exactly into two or more numbers. Students will have to be fluent in finding the GCF in order in order to reduce a fraction to its simplest form.
You could ask your students,
“What is the GCF of 8 and 6?”
Or you could ask, as Deborah LaChance, math teacher at Oyster River Middle School in Durham, does:
“The GCF of 8 and some number is equal to 2. Find the 5 smallest whole number values for the missing number. Show your work and explain your process.”
Before the Common Core, a teacher might drill her students on variations of that first question and, when most students could get the right answer most of the time, she might move on.
Now, using the Common Core approach, once her students know the answer is “2,” she could move to the second version. Answering that second question quickly requires procedural skill and math fact fluency but it also requires conceptual understanding.
Students must do real thinking and show a depth of knowledge to determine that 2, 6, 10, 14, and 18 are the correct answers. And to know that 12 is not correct, because the GCF of 8 and 12 is 4. And that 16 is not correct because the GCF of 8 and 16 is 8.
By the time her students “show their work and explain their process,” the teacher will have a clear view of how thoroughly her students understand GCF and can focus her work on filling the gaps.