Monthly Archives: January 2017

Where Are You? I’ll Meet You There


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?     

Peter McWilliams


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: 67th 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”.

Exams Ahead!

The students with whom I work are heading into midterm exams.  For some of them, the mathematics concepts and procedures come easily, and others have to work harder to feel confident in their understanding.  All of them can benefit from diligent preparation, and although a few still resist, here is the advice I am giving:


  1. Do the whole review packet and check your work against the solution key.  It isn’t optional. Show your mathematical thinking so you can analyze your process.
  1. Review past tests and quizzes, looking at both the questions you got wrong and the correct ones.  Even if you got it right in earlier in the semester, make sure you remember how to do it now.  Re-do the questions, don’t just “read it over”.
  1. The more practice and review you can do BEFORE you get to the review session, the more productive the time will be.  Mark the ones you get wrong and/or don’t know how to do so you have a list of questions ready.  Don’t wait til the last minute to start studying.
  1. Keep track of important formulas, graphs, examples & concepts on a self-created study guide.  Do it as you go through the review packet.  If you need to memorize something, write it out each time you use it until you know it. If you need to be able to solve something without a calculator, practice it that way.
  1. Make use of other resources: if your teacher has a website, go back to unit review sheets and solution guides from the semester. Check another teacher’s website from your school if your teacher doesn’t have one.  Work with a classmate, but don’t merely divide up the work: make sure you both can complete the problems.  Utilize Khan Academy, YouTube & Google.
  1. Cumulative exams are challenging; scores are often somewhat lower than your typical quiz/test scores have been.  However, remember to be confident in the things you know—yes it is a big job to prepare, but you can do it!
  1.  Take care of yourself physically over these weeks:
  • eat right and choose healthy snacks (think protein/fiber not sugar)
  • stay hydrated (more water less soda)
  • stay active because it helps relieve stress and is good for your brain: go to sports practice, work out or run or shoot hoops, or even just take the dog for a brisk walk during a study break
  • wash your hands frequently, etc. so you don’t get the bugs that will inevitably be going around
  • get enough sleep: it is far more important for your brainpower to sleep an extra hour than cram an extra hour.

Good luck!