I saved the day.
The math problem had a small detail in it that if my students were not paying attention, it would mess up their work. So I darted around the room and pointed out the detail at the precise moment each group encountered it. No one got confused. No one struggled. Everyone got the right answer. It was a fantastic moment…until I assigned the next problem. After a few moments all of the groups began to glance in my direction, and a boy right next to me asked, “What do we do?”
The better question was, “What did I do?” I didn’t save the day. I wasn’t developing my students’ capacity to do mathematics. Instead, I encountered a pitfall of developing doers of mathematics.
Developing Doers of Mathematics
Recently, I head an episode of the Craig Groeschel Leadership Podcast where he discussed four pitfalls leaders can fall into in developing the people they lead.
All throughout the podcast, I kept thinking of parallels between what Pastor Craig shared about leading organizations and the complexities of teaching and learning mathematics (don’t we all?). I see the job of teaching as facilitating a productive relationship between students and mathematics so that students see themselves as doers of mathematics. In other words, the job of the teacher is to develop doers of mathematics.
The Four Pitfalls of Developing Doers of Mathematics
I present these pitfalls of developing doers of mathematics knowing I have fell into each of them multiple times. By naming and recognizing these pitfalls, we can avoid them and develop the kind of relationships we want our students/children/doers of mathematics to have with mathematics. I see these pitfalls existing for teachers, parents, tutors, even students, basically, anyone who may help someone develop as a doer of mathematics.
1. Controlling – creates compliant doers of mathematics.
This pitfall is where students are not given freedom to consider their own methods for solving a math problem. Instead of giving space to explore and make sense of the problem, students are dictated a carefully constructed algorithm for solving the problem. By definition, the problem is no longer a problem. When the student has been shown exactly how to solve a math problem, the problem has transformed into a mere exercise. Controlling has turned doing math into executing algorithms.
To avoid this pitfall students need space to explore problems. A great article that was written about this idea is call Never say Anything a Kid Can Say by Steven Reinhart. The mindset presented in this article has helped me be quiet, sit back, and trust the student(s) (through gentle prodding) to produce a solution, and then use that work as a starting point for a conversation about the problem. Also Mandy Jansen@MandyMathEd has this idea of “rough draft talk” for solving math problems. The idea being lets consider the idea of creating rough drafts for papers and use that same iterative process for creating solutions for math problems.
2. Criticizing – creates insecure doers of mathematics.
This pitfall is where students may be given freedom to consider their own methods for solving a math problem, but each method is quickly identified for how it falls short in efficiency, accuracy, elegance, or just is not the preferred method of the person providing assistance. Students are eventually leery of presenting their ideas for solving a problem given the overly critical environment in which the idea is received.
To avoid this pitfall an asset-based perspective of the work students do with mathematics needs to be developed. Instead of seeing what is wrong with the method, consider what is right. This approach of having an asset-based perspective and assigning competency to students can be seen in the work around Complex Instruction.
Two books I recommend on Complex Instruction in the math classroom are both from the National Council for Teachers of Mathematics (NCTM). One book is called Strength in Numbers: Collaborative Learning in Secondary Mathematics by Horn. The other book is Smarter Together: Collaboration and Equity in the Elementary Math Classroom by Featherstone, Crespo, Jilk, Oslund, Parks, and Wood.
3. Avoiding – creates disengaged doers of mathematics.
This pitfall is where students are given freedom to consider their own methods for solving math problems, but are not given any feedback. The person providing assistance…doesn’t. They are not engaged with what the students are doing and in turn the students see it (understandably) as a lack of caring in what they are doing.
To avoid this pitfall the answer is to simply engage. The easiest way to engage is to ask questions. Try to figure out how students are making sense of the problems and attempt to do so with no assumptions.
I remember noticing on my son once identified a rectangle as having six sides on his homework. My gut told me ask him why he got the question wrong (He knows how many sides are on a rectangle, right?). Instead, I asked him how he came to the answer of six, simply and with no judgement. He told me he used a tile in the shape of a rectangle to count all the sides and came up with six. He counted around the tile and then one on top, and one on the bottom.
I immediately realized his problem was not a rectangle problem. It was a problem identifying the difference between three dimensional shapes and two dimensional shapes. It was a problem identifying the difference between sides of a polygon and faces of a polyhedron. Asking the question, and not avoiding, resulted in a wonderful understanding for both of us.
4. Rescuing – creates helpless doers of mathematics.
This is where students are given freedom to consider their own methods for solving math problems but are given help at the smallest indication of struggle. An example of this pitfall can be seen in the story that began this blog post.
To avoid this pitfall, students need to be given permission to struggle and sometimes even fail. This does not mean disengagement but providing assistance in other ways. Asking an open-ended, probing question, encouraging them to continue on a line of thinking, creating timely partnerships between students considering the same solution path, are all ways to stay engaged but not rob students of the learning potential of a math problem. Helping students learn how to deal with struggle and to learn from failure will not only develop them as doers of mathematics but also as people.
Knowing is half the battle…
In the end, avoiding these pitfalls comes down to a balance of engagement and freedom. By knowing these pitfalls we can avoid them and help our students develop the kind of relationship we want them to have with mathematics.