I’m always amused by the choices of word problems that teachers share on social media. Often times they are fictional problems that don’t have very much substance. Problem solving is supposed to give students a context to help them make sense out of the mathematics they are learning.
According to NCTM, “Problem solving means engaging in a task for which the solution method is not known in advance. In order to find a solution, students must draw on their knowledge, and through this process, they will often develop new mathematical understandings. Solving problems is not only a goal of learning mathematics, but also a major means of doing so.” (NCTM, 2000, p. 52)
Unfortunately, problem solving in elementary is being taught as a series of rote steps and in secondary the graphing calculator has replaced using prior knowledge to solve problems. Teaching at the high school level has opened my eyes to the damage of not thinking deeply in math classrooms.
Long before the Common Core State Standards, I felt like the decline in literacy was having a disastrous effect on mathematics. Reading is the foundation for all other content areas. In fact, the thinking that is used in math is taught in reading.
When students do not read at grade level it creates a glass ceiling for math teachers. Since I used to teach reading this glass ceiling has not really been a problem for me in regards to student achievement. In fact, within the last five years I’ve felt more like a reading teacher who was teaching math. I always ask myself, where does that leave other math teachers who don’t have a reading background?
As a math teacher, I focus on creating a classroom environment that encourages independence. According to Ron Ritchart, author of Creating Cultures of Thinking; there’s a clear link between expectations for learning, understanding, and the use of deep learning strategies. For some teachers it’s nearly impossible for them to lead an autonomous classroom. District personnel and administrators want to control teaching by requiring teachers to focus on performance and work.
Creating classroom environments where instruction is controlled fosters student dependence. Some potential downsides to dependence are:
Deterioration of problem-solving strategies (Dweck & Leggett, 1988) A focus on extrinsic motivation Diminished enjoyment of learning Lack of resilience when faced with difficulties and challenges Decreased creativity and motivation (Koestner, Ryan, Bernieri, & Holt, 1984)
When teachers encourage independence some benefits include students’ openness and willingness to accept challenges, greater motivation, engagement, ownership, and drive (Pink, 2009).
Teaching Problem Solving
Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and support to engage in productive struggle as they grapple with mathematical ideas and relationships.
If you’re anything like me productive struggle was a foreign concept. Teachers who understand productive struggle see students’ struggles as opportunities for delving more deeply into understanding the mathematical structure of problems and relationships among mathematical ideas, instead of simply seeking correct solutions. An unproductive struggle occurs when students “make no progress towards sense-making, explaining, or proceeding with a problem or task at hand” (Warshauer 2011, p. 21).
For most of my career, I always came to my students’ rescue when it came to problem solving. My biggest excuse was I needed to move on because I had to keep up with the scope and sequence. I didn’t allow my students who struggled a chance to make sense of the problem or provide them with the tools to solve problems that required a high cognitive demand.
Problem solving is more than a problem solving strategy. Problem solving tasks should require students to: Build new mathematical knowledge through problem solving Solve problems that arise in mathematics and in other contexts Apply and adapt a variety of appropriate strategies to solve problems Monitor and reflect on the process of mathematical problem solving
We can’t continue to give students problems to solve that don’t require them to think at a high cognitive demand. If the expectation is that students must use their math thinking and prior knowledge to solve problems then students will rise to the occasion.
Diverse Student Populations
When it comes to problem solving expectations for students I’ve witnessed teachers from different ethnicities use deficit thinking as an excuse as to why black and hispanic students can’t think at a high level.
According to Valencia (1997), deficit-thinking assumes that students who fail in school do so because of alleged internal deficiencies (such as cognitive and/or motivational limitations). It also tends to view the poor and working class children and their families (typically children and families of color) as predominantly responsible for school failure (Valencia, 1997).
You can call me a dreamer; I believe that if teaching begins with authentic relationships with students then they will be willing to at least try to meet your expectations. However, the first step is really believing in your students’ ability to meet the goal.
I teach ninth graders at a high school that has been in corrective action for ten years and I have the same expectations for them that I would have for any other students. I know that it’s going to be an uphill battle since they are so far behind, but I believe in them and they trust me.
Problem solving is an important part of teaching math. Before you can teach problem solving to students you have to believe in your students cognitive ability and create an environment that encourages independence. This means that you can’t water down the content but rather use scaffolding to make the content easier to digest.
True problem solving doesn’t begin with the right task but the right environment for students.