Determining Importance: Math Skills & Fluency or Problem Solving?
Some educators believe that students need to be solid on math skills and fluency prior to approaching problem solving, while others believe that they can be learned simultaneously. What is the best approach? Elementary students in Waukesha spend about six weeks a year learning via Fosnot problem solving units, which reinforces our belief that conceptual understanding comes by engaging students in rich problem solving tasks that lead them to conceptual understanding. The reflection portion of those lessons, or math congress, is a powerful tool in which teachers guide students to share their strategies and solidify their new understandings. This model of working concurrently on problem solving and skills & fluency is carried out through the rest of our math curriculum as well.
There is a plethora of research which supports this instructional model. In Mathematical Mindsets author Jo Boaler writes:
“When students see math as a series of short questions, they cannot see the role for their own inner growth and learning. They think that math is a fixed set of methods that either they get or they don’t. But when students see math as a broad landscape of unexplored puzzles in which they can wander around, asking questions and thinking about relationships, they understand that their role is thinking, sense making, and growing. When students see mathematics as a set of ideas and relationships, and their role as one of thinking about the ideas and making sense of them, they have a mathematical mindset.”
In Mind/Shift’s post “What if Teachers Took Computation Out of Math Class?” mathematician Conrad Wolfram argues that math can be broken down into four steps:
- Pose the right question about an issue
- Change that real world solution into a math formula
- Compute
- Take the math formulation and turn it back into a real world scenario to verify it
Wolfram argues that most math instruction in our country focuses on Step 3, Computation, which is the only step that can be automated. Instead, he and Jo Boaler agree that the focus needs to be on greater conceptual understanding of mathematics.
Perhaps you need a real world example of why we need to focus on both problem solving and skills & fluency. I recently had a conversation with a relative who is a manager in the food service industry. He told me that on a daily basis he is approached by employees with mathematical questions. The nature of these queries are never computational, as everyone carries a calculator (cell phone) in their pockets. Instead, he said they frequently struggle with setting up problems such as:
96 buns were ordered and buns come 6 to a package. How many packages of buns do I need to send to that cafe?
It’s a bit shocking that adults are unable to set up routine math problems. This illustrates the need to build conceptual understanding by working on problem solving and skills & fluency concurrently.
But how do we get students to move in this direction? One method is Numberless Word Problems. If you get frustrated when you carefully craft or choose a rich problem-solving task, and the first thing your students do is take the problem’s numbers and start plugging them into algorithms without first trying to understand what the problem is asking, then you need to give this a try. It may initially seem counterintuitive, but one way to mitigate this student practice is to have them tackle word problems without numbers. Numberless word problems are a scaffold which allows students to concentrate on the structure of word problems without worrying about computation. Don’t worry, that step will come later in the process!
Here is an excerpt of a class session tackling a Numberless Word Problem from the blog Numberless Word Problems, which has a vast array of resources. The teacher begins by presenting the problem:
Some girls entered a school art competition. Fewer boys than girls entered the competition.
She projected her screen and asked, “What math do you see in this problem?”...
“There isn’t any math. There aren’t any numbers.”
She smiles. “Sure there’s math here. Read it again and think about it.”
Finally a kid exclaims, “Oh! There are some girls. That means it’s an amount!”...
“And there were some boys, too. Fewer boys than girls,” another child adds.
When it seems like the students are ready, she makes a new slide that says:
135 girls entered a school art competition. Fewer boys than girls entered the competition.
Acting very curious, she asks, “Hmm, does this change what we know at all?”...
This is where the class starts a lively debate about how many boys there could be. At first the class thinks it could be any number from 0 up to 134. But then some students start saying that it can’t be 0 because that would mean no boys entered the competition. Since it says fewer boys than girls, they take that to mean that at least 1 boy entered the competition. This is when another student points out that actually the number needs to be at least 2 because it says boys and that is a plural noun.
Stop for a moment. Look at all this great conversation and math reasoning from a class that moments before was mindlessly adding all the numbers they could find in a word problem?
Once the class finishes their debate about the possible range for the number of boys, my co-worker shows them a slide that says:
135 girls entered a school art competition. Fifteen fewer boys than girls entered the competition.
“What new information do you see? How does it change your understanding of the situation?”...
Even though most students anticipated the final question to be: , “How many boys entered the art competition?”, the teacher instead asked: “How many children entered the art competition?”
Of course I am not proposing that you approach every math problem in this manner, But, if you are looking for a novel way to encourage your students to think deeply about the math problems they approach, you might want to give Numberless Word problems a try in your classroom.
*In addition to the resource listed above, you can learn more about Numberless Word Problems on Twitter by using the hashtag #numberlesswp.
