Until learning about VTL I hadn’t really thought about how I could / should use contrast in my lessons. Now, I’m really seeing how it is a powerful tool.

Today in class we were looking at patterns and equations. Specifically we were looking at the affect of a changing coefficient, while keeping the constant the same for all equations.

With the class I modeled y = 5x + 3 on the board. We walked through this example with proper language. We looked for similarities and differences between our drawings for Figure 1 and 2, then used this to infer Figure 3.

Following our discussion students were randomly put into pairs and assigned an equation (keeping the constant the same, but varying the coefficient). The equations used are listed below:

y = x + 3 | y = 2x + 3 | y = 4x + 3 | y = 6x + 3 |

y = 0.25x + 3 | y = 0.5x + 3 | y = 0.75x + 3 |

I purposely did not use y = 3x + 3, so that there was no confusion between coefficient and constant. The second row was used to enrich for certain pairs. Students represented their work with a diagram, manipulatives, a table of values and ordered pairs. Once they finished this, each pair identified a partner as A or B. The students then engaged in a gallery walk. All of the A’s stayed at their station for the first round and the B’s moved around the room. Each B was to go to each group and ask the A person to explain their diagrams, manipulative representation and t-table. Once the B partner had moved through each group, we switched and partner B stayed at their station as A circulated throughout the room.

Finally, to wrap up the lesson we brought certain groups to the front to discuss the effectiveness of their manipulative representations. Some groups used one colour to consistently represent their “x number of groups” and a different to represent their constant, while others were haphazard with their use of colour. We discussed as a class how one model visually communicated something different than the other. Students were able to identify that using two colours consistently made it easier to identify the constant and how the groups of x were increasing in each figure.

The power of using contrast was evident when comparing the various models made by the groups. Students were able to see similarities between all of the models, re-affirming each groups work, but more importantly were able to notice differences and how those differences affected the clarity of the model. This allowed for the final “aha’s” to occur. I heard some students say “Oh, we did that. We need to make sure we are using the same colours throughout,” then getting up, getting new blocks and fixing it.

I really like the discussion by Durkin, Star and Rittle-Johnson (2017) around questioning and how the use of questions can lead to students seeing differences. I liked the questions provided in the reading:

- Which is better?
- Why does it work?
- Which is correct?
- How do they differ?

I like these questions when comparing two pieces of work that I’ve provided. They feel a little to-the-point when comparing work of students in front of their peers. I worry that asking students, “which is better?” when comparing their own work will lead to social dynamic problems within the classroom. I think that, “how do they differ?” Is along the same lines as, “what is similar? What is different?” and will ultimately lead to students giving constructive criticism to their peers, but I would anticipate that this feedback would be received differently than the feedback given to the question, “which is better?”

Until next time,

Mr. B

Durkin, K., Star, J., & Rittle-Johnson, B. (2017). Using comparison of multiple strategies in the mathematics classroom: lessons learned and next steps. ZDM Mathematics 49, 585-597.