Activating the student-learner (emphasis on active)

I always found it tricky to get students up and out of their seats meaningfully. I know that cooperative learning manuals are full of ways to get students meandering around the classroom, but I always liked to try to make sure that the movement was meaningful as a tool to help them learn the content.

In the past week, I’ve seen two examples of meaningful active learning in the realm of Geometry. And as it happens, both are from my neck of the world.

One comes from Tara Maynard who teaches middle schoolers in Zeeland. Her post on “Dance, Dance, Transversal” plays off the mechanics of Dance, Dance Revolution while putting the students in an experience of having to know the different angle pairs coming out of parallel lines cut by a transversal.

Photo credit: Tara Maynard

Photo credit: Tara Maynard

The thing I like about this is that in the end, this is a vocabulary exercise, but Tara has found a way to use movement and activity to add some life to it in a way that fits pretty naturally. This is a nice pairing. And, as she states in her post about the activity, she pairs the students up so that one can watch the feet of the other to make sure they aren’t making mistakes and reinforcing poor understanding. She also includes the file so that your students can play, too!

The other comes from a pre-service teacher at Central Michigan University. Tod Carnish used Twitter to share a nice idea to help students explore geometric transformations. And as much as it hurts for me to say nice things about Central folk (Go Broncos! #RowTheBoat), this seems like a pretty solid idea.

Transformation Tweet

Sticking to the idea that transformations are really just the organized movement of vertices, why not have those movements represented with the students acting as the vertices? Think of the ways that one could use this as an introduction to function notation of the transformations? Characterizing the movement by it’s vertical and horizontal components?

Nice thing about these two fine educators is that they love to share. If you have questions about their work, or want to drop them some props, I’m sure they’d love to hear from you!

The Role of Reading in Math Class

My more recent endeavors have moved me away from exclusively math and allowed me to enter into the world of literacy, particularly elementary literacy. (Math teachers out there, if you haven’t gotten a chance to sit and participate in a conversation among elementary folks talking about kids learning to read, the essential components of curricula and classroom activities, the different types of assessment and support for struggling learners, etc. then do it. Such interesting conversations…)

But my math background led me to look at the new literacy experiences I’ve been having through the lens of the math classroom. Specifically, what are the elements of literacy education in a problem like this:

Volume Geometry problem

taken from Holt’s Geometry, 2009 Edition, Page 702 #25

What math content knowledge is this problem asking students to use? Volume of cylinders, volume of rectangular prisms, and simple probability.

But we need to keep in mind that the ability to read is a significant skill to being able to apply math to this problem. Especially when you consider that the typical high school student probably doesn’t read quite as well as we’d like them to. I want to be clear that I’m not saying that this is an excuse for us to eliminate reading as a way to accommodate this potential weakness in our students, but it is important that we recognize that we are asking students to exercise that skill when solving this problem.

When we are asking our students to read as part of a math experience, we should do so deliberately. There are times when one of our stated learning outcomes is helping students learn to read math. There is a lot of value in that skill. There are also times when we are much more interested in helping the students set-up and solve math problems in ways that may not require as much reading.

This requires us to be strategic about what we are expecting our students to be able to do by the end of a lesson. Consider the math problem from earlier in the post. Literacy doesn’t have to be a barrier to that math content.

Andrew Stadel provides an example of how volume of multiple solids can be integrated into a math problem in video form.

Dan Meyer integrates a fairly similar set of content in a much different (and more delicious) scenario.

Using video in your class is one way to provide access to a math problem without having student having to tackle the reading part, which can be a pretty significant barrier to some students. In the past, I certainly haven’t been nearly conscious enough of the literacy demands I was putting on the students when I asked them to explore math content. Here’s a perfect example of a handout from my geometry course. How many students would benefit from a video introduction to this handout? (Especially when you consider how many students aren’t real strong using a protractor…)

It just all goes back to us, as educators, having a clear vision for what we want our students to learn by the end of their time with us and being willing to do what it takes to help them get there.

The Role of Feedback

I had to learn this lesson the hard way.

In the fall of 2012, we had to part ways with our Geometry textbook. We replaced it piece-by-piece by what became our Geometry Course It was definitely a net positive for everyone involved, but as you might expect, it didn’t come without some growing pains.

As it turns out, in order write a decent course, it helps to know a little something about how young people learn. When I built in the activities that made up the course, I tried my best to pay attention to what the students would be doing. What was my target content and how was I giving the students a valuable experience with that content that would help reduce as many barriers as I could to the learning. I tried to pay attention to foundation knowledge, reviewing old content and building in new understanding in reasonable chunks with practice.

What would the handouts look like? What should the wording be? What sequence will I ask the questions? How many practice problems should they have? What type of product should they produce?

I thought of all of these things. The first year, there were lots of positive signs, but I still noticed students struggling in areas that I wasn’t expecting. The students were getting better at verbalizing their thoughts both in writing and speaking. They were using the vocab fairly well. But they struggled in ways that after a while, got pretty predictable. It started to seem like the more extensive the paper-and-pencil process, the more likely the students were to struggle, and not just clumsy-style, like in ways that seemed to suggest that they didn’t really know how to complete the process.

What I never thought of: How am I going to build in opportunities for the students to receive feedback on the products they were producing?

As the year progressed, the seemed to get better and better at discussing the math and using the words, sometimes with some wonderful results, but that was because the whole time they were talking, I was walking around talking with them. Interacting, correcting verbiage, modeling effective uses of the essential vocab and connecting it back to the problem. I was responding to their curious looks and asking them to clarify their confusion. These were my favorite times in math class. So I did them as often as I could.

No wonder they showed growth in this way.

But what was I doing once they completed a handout like this? That first year, a whole lot of nothing. We moved on. Or we would discuss the broad topics and I would pose a problem like this with a mind to formatively assess their progress.

But I wouldn’t simply show them the answers. I forgot about the feedback. And it got in the way.

A few quotes from Dr. John Hattie on this topic:

“In short, receiving appropriate feedback is incredibly empowering. Why? Because it enables the individual to move forwards, to plot, plan, adjust, rethink, and thus exercise self-regulation in realistic and balanced ways.”

“Imagine a group of students who are about to embark on a series of lesson, and during these early experiences we pause and show them the various way they will be successful at the end of the lessons, or tell them how they will know when they have been successful in these lessons. The is a relatively cheap investment, takes little time, but of course, it provides teachers with a challenge – that of working alongside the students to maximize the number who reach the success criteria.”

“When we interview students on what they understand by feedback and why it is important to them, on theme emerges almost universally: they want to know how to improve their work so that they can do better next time.”

“The feedback you offer your students provides the tools they need to be able to perceive the immediate path ahead, and so decide that it is really worth effort.”

Second year we determined that we were going to have an answer key (and if possible two or three) so that the feedback could start by them checking their answers. I instructed them to ask anytime they had questions (even if the question was “mine looks different than yours, but I still think it’s right.” I actually rather like that question.) Dr. Hattie is indeed right that that was a cheap investment. My goal was for every student to get to compare his/her solutions to the correct ones. It slowed progress a bit through a lesson the first time a bit. But I had to do significantly less reteaching, so I got that time back.

When the students were left to practice and then draw their own conclusions about their own comfort level of their progress, I had forgotten one rather important understanding that both myself and the students were largely aware of. They mostly lacked the expertise to properly determine whether or not they should feel comfortable with the work they just completed.

But given the appropriate feedback, they can see where they were successful, were their focus needs to be for future success and how much room their is between the two.

Quotes taken from Chapter 8 of Visible Learning and the Science of How We Learn by John Hattie and Gregory Yates. Routledge Publishing – 2014

A word about teaching to the test…

It’s testing season here in Michigan, the dawn of a new age! Last fall’s paper and pencil MEAP is replaced with the this spring’s computer-driven M-Step.

And the M-Step is putting a lot of pressure on districts to reconsider what “test readiness” even means. What skills are required of a student who is having to transition from a question like this:

MEAP Grade 8 Practice

to something like this?

MStep Practice Grade 8

The “Performance Task” is another new type of test item. Teachers lead a scripted discussion through a content exploration. Students get time to discuss, explore, ask questions. Then they go off to answer a question based on the experience they just had. This all combines to create a startling amount of new experiences all in the same year.

Preparing educators for this newness was a large chunk of my work as my colleagues and I provided workshops to help explore this testing experience that is new for all of our students. As I had discussions with genuinely and appropriately concerned teachers and administrators, there were a few recurring concerns that bubbled to the surface.

1. What if a student knows the right answer, but has computer/technical skills that prohibit him/her from answering that question correctly (or in a reasonable amount of time)?

2. How are the non-traditional items going to be graded? (Non-traditional… rubric scored items, items with more than one correct answer, the drag-n-drop question like the number line one above, etc.)

Certainly there were lots of other concerns, but these two kept coming back again and again with good reason. We want to make sure that our students are getting recognized for the knowledge and skill they possess. This new test presents a new set of potential barriers.

So, how do you get around those barriers? Well, one way is to teach to the test.

“Teaching to the test” as a phrase is usually not seen as a good thing. Although, I can remember a conversation with a district leader a while back who was quite proud of the fact that curriculum decisions for their math department were based almost solely on what was on the ACT… he refrained from using the term “teaching-to-the-test”.

This pride fits in with recent statement from a colleague. “Teaching to the test is only a bad thing if the test is no good.”

Now, that having been said, there are a lot of people right now in Michigan who are feeling like the M-Step is no good and they might be right. As I write this, we’ve just completed our second day of testing, so I’m going to reserve judgement until we’re more than 48 hours into this new experience.

But regardless, that sample question from above…

MStep Practice Grade 8

… isn’t a ridiculous question. It lends itself to considering if teaching strategically in such a way that students could do that (which is motivated primarily by the fact that a high-stakes, state-level standardized summative assessment is going to ask them to do it) is a bad thing? I would suggest that this is a task that most 8th grade math teachers would want their students to be able to do. Where the conflict arises is when teachers or districts aren’t inclined (or able) to provide exploration or assessment of this skill in a computerized way. The test becomes a motivation to have to change course. So we find ourselves faced with “teaching to the test.”

So, if we’re going to be “teaching to the test” anyway, what situations can we design for our students that while we are probably only doing ten because they will be on the test, are still, in the end, high-quality experiences for the students to have?

One suggestion comes out of this blog post from Fawn Nguyen who is trying to help the students interpret a scoring rubric. That was one of her goals, because her second goal was to help the students focus on a set of skills they will need to be successful on the test they are going to take in May.

The skills she’s referring to: “For them to attend to the same thoroughness and precision in their own solution writing when it’s their turn…”

While Ms. Nguyen is openly “teaching to the test”, that skill she’s highlighting is a valuable one. I encourage you to read the post. The tone of it doesn’t sound like someone who begrudgingly throws test prep items at her students, but rather a teacher who saw an opportunity to weave test prep into an experience that ultimately led to student feedback like this:

I believe this was helpful because when I take the test, I will be more aware of the questions and what is expected of me. I will make sure to always back up my answers with evidence.

I’ve talked to a lot of teachers (math, ELA, social studies) who’ve expressed frustration that their students won’t “always back up their answers with evidence”. Ms. Nguyen provides us an example of how a test prep experience can be used to further broader goals.

And I think, in the end, given the stress and strain that these testing situations put on all of us, it’s nice to see an example that reminds us that teachers still have ability to make decisions that end up as net positives for our students.

Vocab: The Foundations of Math Talk

I want to talk about vocabulary for a minute.

Specifically, I want to describe one way that can support students struggling to learn vocabulary that is necessary for effective math talk. Often times, struggling math students are willing math talkers, but the math talk is filled with pronouns (that thingy right there, you know what I mean?), hand gestures, and rough sketches.

This can make communication a bit of a chore, especially if the student is talking to another student who is also still in the beginning stages of developing understanding in that topic.

So, here’s a way that you can change the conversation a bit.

Consider this handout combined with this Google Form.

In short, the students would take a few minutes exploring different formal definitions for the vocab words you are exploring. There are always differences if you go to different sources, so by forcing them to explore a variety of interpretations, you can really help the students see that this vocabulary is about describing an idea, not memorizing a specific wording. This is important, at least to me.

Pair the students up (or group them in 3’s) and have them consider the big ideas and develop their “own-words” definitions for each vocab word. Then they plug them into the Google Form.

At that point, you’ve got a nice collection of what your class currently has taken away from their time. (And by “time”, I mean about 20 minutes from start to submission.)

As you wander around listening to the conversations, you’ll likely notice a few definitions that are coming together more slowly than others. This is inevitable.

At that point, I’d bring in Wordle. Wordle is a free tool that build word clouds. Word clouds can have a nice visual appeal and change the way a block of text looks by changing the physical size of a word based on its frequency in the text. For confusing definitions, this tool has the potential to help the students hone in on the big ideas. For my students, it was often “circle.” Most kindergartners could draw a decent circle (or at least a shape that you would guess quite quickly was supposed to be a circle.) But, that visual understanding is often as far as students get.

Want to see how well your students understand what a circle really is? Ask them why this is a lousy attempt to draw a circle. See what they say.

bad free hand circle

If your students are struggling with an idea, though, go into the Google Form responses, copy all of their definitions and paste them into a Wordle.

My third hour Geometry last year, produced this word cloud.


This isn’t going to solve all of your problems by any means, but with a visual like this, your students will have the opportunity to see that the major idea of this concept of circle revolve around the notion of a center point, a distance, and points in a curved shape. Those are major steps in the right direction when we are moving students away from a purely visual understanding of what a circle is and helping them understand the geometric properties of a circle.

Remember, the goal of learning vocabulary is to facilitate student understanding of the content. Memorizing a definition only goes so far. We have to continue to develop the means to push students to internalize the bigger ideas within the vocab so that those words then become foundations to build more sophisticated ideas.

If you feel like you’d like a tutorial of how to use Wordle, I went ahead and made one.