Tag Archives: math

Perpendicular Bisectors: A Recap

At my dyad teaching placement in a middle school, a colleague and I wound up teaching a lesson this week.  It was the first time I’ve ever attempted to teach middle schoolers, and I fear I was a bit nervous.  That and some conceptual errors in the planning process led to a lesson which was a bit off target; yet much was learned, and I’m looking forward to the next attempt.

The topic was “perpendicular bisectors”, taught to a small class of eighth-graders.  I believe our central error lay in our over-emphasis on a learning activity, rather than a focus on learning objectives – what we wanted them to know or be able to do by the end of the lesson.  It was the best support for the validity of the backwards-design philosophy we all learned about in our lesson-planning class that I have seen so far – I’m confident that if we’d followed that process, our lesson would have made a lot more sense, to both us and the kids.

Here is the labor of love I designed as a learning activity:

Unicorn, etc.


It’s a set of three problems which require construction of perpendicular bisectors to solve.  The intent was to show kids that problems like this are not confined to theory, or to goofy problems from the textbook involving post offices on fictional islands.  The irony is that our focus on this worksheet as a learning activity, combined with our over-ambition regarding what we had time to teach, left us preparing the kids very poorly for what they actually had to do: use compasses to construct intersecting arcs and thereby find the perpendicular bisector of a line segment.  Instead, we found many students folding the pages of their journals in mimicry of what we had done with paddy-paper, or assigning arbitrary center-points to line segments.

It was almost textbook – the students were working with what we’d taught them, and what we hadn’t taught them.  The result was that we got a very clear and instant picture of how effective our lesson had been.

The two take-aways:

1.  Backwards design!  Too much focus was put on a nifty worksheet with pretty drawings.  Proud as I am of my art, it doesn’t mean a thing if the kids don’t learn.

2.  Time.  I’m going to make it a practice in the coming weeks to plan activities which take up approximately half of the allotted time.  My justification for this is multi-pronged: I need to train myself to be more realistic about what I have time to teach, I need to incorporate student participation and student construction of knowledge into my process – and leave lots of room for the inevitable questions and course-changes – instead of talking at them.  Additionally, I need to remember that is very easy to teach too much – having recent experience of my own in the act of being overwhelmed, I need to make sure not to overwhelm my students – to pursue depth of knowledge, in cases where depth is more important than breadth.


Dyad Moon Science Reflections

This week, the middle-schoolers I work with had a science test.  The test covered a space-science unit which touched mainly on the Solar System, the Sun, the Moon, and the concept of scaling – i.e., representing sizes and distances at a different scale.  The kids were also challenged to make ample use of scientific notation in their answers: i.e., representing “1,000,000,000” as 1×10^9.

I started observing this class only a few weeks ago, and have only been in the class effectively once a week during that time.  I’ve observed my master-teacher’s science instruction, which included a lab on the orbital mechanics and phases of the moon which I thought was engaging and clear – students got their hands on flashlights, model moons, and small globes, and got to experiment of different “Earth-Moon-Sun” configurations to see how the position of the moon in its orbit affected how it would be seen by observers on Earth.  There was also a really cool performance task in which students were given a geometric shape on paper and tasked with measuring it, scaling it up by a factor of 2, and then building a wooden replica in the school’s well-equipped shop, painting the replica and then adding it to a tessellating puzzle of all the other student’s pieces.

Tessellation – not my class’s.

Here’s the problem: after all the above engaging, hands-on learning activities, the test showed that a pretty large portion of the students weren’t getting core concepts.  The question “Why do we on Earth only ever see one side of the Moon?” yielded results like the following: “Because the Moon doesn’t rotate,” or “Because only one half of the Moon is lighted by the Sun” (which is technically correct, but happens to be the answer to a different question).  The clearest indicator to me was the first answer, and those like it – “Because the Moon doesn’t rotate”.  (To clarify, only one side of the Moon is ever pointed toward Earth because it completes 1 rotation per month and one orbit per month – thus, its near side is constantly turning toward us as it moves around us.)  Clearly this concept wasn’t explained well.  Also, I didn’t get a chance to observe my master-teacher’s instruction on scientific notation – but it’s clear from the almost universal arbitrariness of students’ renderings of numbers in this way that this, too, is a concept which will need to be revisited.

My master-teacher was clearly frustrated, and the kids were clearly stressed.  Which is all perfectly natural, of course.  It left me with a very clear picture of experiences I’m likely to have in the not-too-distant future: seeing what hasn’t sunk in during the course of a unit, and then trying to figure out how to revisit it in a different way.  We re-start the unit with a different group of kids in 2 weeks – I’m going to get the opportunity to help teach parts of it, which I’m very excited for – and I’m very curious to see the changes my master-teacher might make to the curriculum, in order to better communicate these concepts and skills to this group of kids.

Just for fun, I’ll throw out one or two ideas of what I might do, had I infinite time and money:

1. Actual astronomical observation of the Moon.  Getting to see, as a class, the Moon’s progression through its phases over the course of weeks might make more impact than a lab in a classroom.  Too many people simply don’t look up at night, or make an ordered study of astronomy – that’s why it’s easy for average people to get such notions as “the Moon doesn’t rotate”.  The difficulty of this, of course, is the fact that we can’t exactly ask kids to come to school after sundown.  Or can we – maybe as a once-a-week activity for the course of the unit?

2.  Incorporation of truly huge numbers into scientific-notation instruction – for instance, the average distance from Earth to Alpha Centauri (our closest interstellar neighbor at 4.13×10^13 kilometers), or the breadth of the Milky Way galaxy (about 1.04×10^18 kilometers) – and then comparing them to each other.  Teach them how to add, subtract, multiply and divide in scientific notation to show kids how useful this notation can be – and to feed into the notions of scale which are the core of this unit, being able to know one quantity in terms of another.  I’m not sure the best way to package these ideas, but going through them in a more in-depth and, at the same time, playful fashion might help some students.