I was so proud. I had created a great technology activity to use in my Algebra 2 class, complete with a well-thought out lab sheet for students and their partners to work through and document their learning. It was an exploration of slopes of parallel and perpendicular lines, with students being guided to “discover” the concepts involved.*

Questions for assessment involved different levels of cognitive demand, including creating their own sets of equations, paying attention to mathematical structure, and writing an explanation of their process. The graphing calculators were ready and the students worked diligently through the class period. The lesson was a success—everyone demonstrated their understanding of the mathematical objectives.

So what was the problem? I asked a few students on their way out of class if they enjoyed the calculator lab activity since it was different from our “regular” routine. They told me: “It was fine. But Mrs. Campe, we all already knew about slopes of parallel and perpendicular lines.”

I had failed to properly pre-assess my students’ understanding of the concepts. I had wasted a full class period to cover something they had already mastered, when, instead, I could have been moving forward or exploring some other problem more deeply. I didn’t check where my students were in their understanding before launching into my “great” activity.

Similar things can happen in my one-on-one work with students. Since I am not in their classroom with them, when students arrive for a work session, I have to rely on them to tell me what their lesson and unit topics are. Sometimes I go down a path that veers away from what they have done in class. Some students resist conceptual explanations, wanting only the quickest route to the answer. I have to push them to realize that learning the “why” behind a procedure helps them understand when and how to use it, and the conceptual background makes their learning more durable and leads to more success in math class.**

So what have I learned from these situations?

1. It is vitally important to pre-assess and utilize formative assessment to know where my students are. Class time is at a premium and I want to use it wisely.

2. Don’t rely on students’ self-report of their understanding; require them to demonstrate their capabilities by doing problems, explaining a process, and answering “why” questions.

3. Don’t use technology just because I have it. It must further the lesson objectives and enhance student understanding. The same warning goes for “fun” or “cool” lesson activities.

4. Reflect on your lessons: ask yourself what went well and what needs improving so mistakes don’t get repeated. And discuss with your colleagues, local and virtual. You will find lots of support in the MTBoS; one teacher commented to another on Twitter just last night: “Thx! I am always looking to improve my teaching!”

**Mistakes, obviously, show us what needs improving. Without mistakes, how would we know what we had to work on? **

**Notes & Resources:**

The technology lab activity on Parallel and Perpendicular Lines is here. It was written for the TI-84+ family of calculators, but any graphing technology may be used.

*This lab activity is a “Type 1” investigation structure in that it guides students toward the desired mathematical knowledge, in contrast to a “Type 2” inquiry which encourages more open exploration. Both types of lesson structures are effective, so match the level of exploration with your objectives. More about this in McGraw, R. & Grant, M. (2005). Investigating Mathematics with Technology: Lesson Structures That Encourage a Range of Methods and Solutions. In W. J. Masalski & P. C. Elliott (Eds.), __Technology-Supported Mathematics Learning Environments: 67 ^{th} Yearbook__ (303-318). Reston, VA: NCTM.

Another dimension useful in analyzing a lesson is type of teacher questioning. “Funneling” questions guide students through a math activity to a predetermined solution strategy, while in “Focusing” interactions, the teacher listens to students’ reasoning and guides them based on where they are and what strategies they are employing, rather than how the teacher might solve the problem. More in Herbel-Eisenmann, B. A. & Breyfogle, M. L. (2005). Questioning Our Patterns of Questioning. *Mathematics Teaching in the Middle School,* 10(9): 484-489.

**Connecting new knowledge to what you already know (elaboration), building conceptual structures (mental models) and practicing what to do when (discrimination skills) are among the strategies for successful learning discussed in __Make It Stick__ (Brown, Roediger & Mc Daniel, 2014). See this website for more.

As a final thought, my title is misleading, because I don’t just want to meet the students where they are and stay there, I want to plan for appropriate challenges to take them beyond their current understanding. There is great value in productive struggle, and choosing lesson components within the students’ “Zone of Proximal Development”.

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