The last unit in my Pre-Algebra classes involved many related topics, but a large portion of the errors and misconceptions arose as we learned slope. Students who rocked out rate of change early in the unit kept flip-flipping things when we dropped context and looked at ordered pairs or a line on the coordinate plane. Students who were easily able to catch their mistake when they wrote hours/dollar now were oblivious when they had run/rise instead of rise/run.

In some respects, I think some of the errors were related to my consistency. I kept switching my wording for slope in hopes students would get flexible thinking about slope in multiple ways. Rise/Run. Change in y-values/Change in x-values. Change in Dependent Variable/Change in Independent Variable. Steepness of a line. All of these definitions are helpful, but switching the wording of my questioning time after time is not helpful for students working with slope extensively for the first time.

As what tends to be the case in reflections, I noticed that my questioning was probably the largest contributing factor in all of confusion I witnessed in my students. After rethinking how I would approach slope in the next couple days, I decided one of the best ways to address student concerns would be to use a set of questions consistently.

The next day, students walked in the room and I shared my observations. We watched Slope Dude to revisit the more basic elements of slope (+, -, 0, Undefined), completed a couple examples, then I set students loose to practice. Throughout the lesson (and the remainder of the unit), I kept using the same battery of questions when I encountered a student exhibiting misconceptions:

- How are you reading this graph/table/set of points?
- For the graph/pair of points/table, how would you describe the slope?
- How are the y-values changing?
- How are the x-values changing?
- Did we write y-values over x-values?
- Are we forgetting anything?

I chose these questions based on the success I saw students have with rate of change (change in y-values/change in x-values is the working definition we developed), but I also wanted to stress the importance of reading before calculating and self-questioning. Thinking about the majority of mistakes students made with slope, most of the hiccups were due to students skipping the basic step of reading a graph/table/set of ordered pairs and deciding if the slope was positive, negative, etc. After making a decision about the slope, it’s really just a matter of asking how the y-values are changing as the x-values are changing. Not surprisingly, students began to make errors less frequently when I began using these questions consistently and encouraging students to adopt these questions to guide their thinking.

Deciding upon a uniform set of questions also made the remainder of the unit easier to plan. When I began planning to teach linear functions and all of the elements associated with this concept (rate of change/slope, initial value/y-intercept, tables/graphs/equations/descriptions, increasing/positive slope, decreasing/negative slope, constant function/slope of zero, linear relation/undefined slope), I was overwhelmed by the various ways mathematicians talk about singular ideas. Considering the way I kept switching the wording of my questions with slope, I recognized that I was probably causing the same amount of stress in my students. Not good. By streamlining my questions and sticking with the same wording, I was giving students the consistency they needed to get fluent with the topics we were exploring. Beyond this comfort, I was providing students with more incentive to adopt and use questions I was offering to prevent mistakes and guide thinking. Once students recognized that I was thinking about a problem in a similar fashion regardless if we were looking at a graph, set of points, table, or description, students were able to see the meaning of my exhortation that, “It’s the same idea, just presented a different way.” Basically, students were able to build the flexibility that was my initial goal.

My experience with teaching slope and my use of questions leaves me with some unresolved thoughts. People crave consistency, whether or not they are vocal about this desire. As teachers, we also know that a hallmark of problem-solving is being flexible in one’s thinking. Based on the struggles and success of my students with slope, is consistency is needed to build flexibility? What elements of my teaching need to be consistent to encourage and build flexibility? Is it a matter of consistent questions? Is it a matter of routine experiences like asking students to develop questions to ask themselves or providing questions to adopt for personal use? Do my students and I need a problem-solving framework? Since I have more experience, I often don’t consider how I am solving a problem. Am I consistent in the way I approach problems? Do I need to be consistent? Is my use of flexibility consistent? How flexible is my thinking? How does the connect to the way my students are thinking about problems? Are my students consistent in their approach to problems? Do my students ask themselves questions about problems? What questions are the most consistent for my students? Do my students recognize when they need to flex their thinking when they hit a cognitive obstacle? How consistently are my students changing their approach to a problem?

Out of all of the questions I just wrote, the question that interests me most is whether or not consistency is needed to build flexibility. I’m inclined to say yes, but I’m still uncertain about how that process looks in the classroom in relation to my teaching and student activity. I’m guessing that there’s information out there about this idea and for all I know I might have just missed the session(s) at Twitter Math Camp that were focused around this idea. If you have any suggestions, links, articles, or knowledge about using consistency to build flexibility, please leave this information in the comments!