Category Archives: teaching

Lesson Closure & Exit Polls – Images

I’ve received a couple of requests for some larger images from the last post on Lesson Closure. Here’s my attempt at providing them.

First, the process map.

exit poll process map

A few exit poll examples.

I have a few other exit polls, but you get the idea. One question to rate the day and one question to elaborate a little bit.

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Lesson Closure & Exit Polls

At the end of June I wrote about Continuous Improvement and promised that I would share updates throughout the upcoming school year. Well, here’s update #1, thanks to a great post by @druinok about Closure and Exit Slips.

Just like the post says, I, too, have always struggled with wrapping up lessons before the bell rings. Okay, we don’t have bells, but there comes a time when the students have to move on to another class. Too often, it seems like we are all so involved that the time just creeps up on us and off we go. That means that I have to rely on my gut instincts to plan for the next day. After so many years of teaching, it seems to work, at least from my perspective, but am I really serving my students in the best possible way that I can?

As a member of the Better Math Teaching Network, I had to come up with a plan – something in my practice that I can tweak, test, and adjust with ease. So, I decided to focus on class closure. Since I don’t have an actual process for this, I had to think intentionally about what I might be able to do. I created this process map:

exit poll process map

I focused on the final 10 minutes of class. Who knows if this is appropriate or not. That will be one of the adjustments that I will have to make, I’m sure. But, I have created a set of Google forms that are designed to solicit some focused feedback that I’ve designated as “process” or “content” oriented. Here is a sampling of “process” Exit Polls I’ve created:

And for “content”:

 

What I like about the Google Form is that I anticipate it will be easy for students to access (most have smart phones, all have laptops) and I can post a link on the Google Classroom.

I am hopeful that this process, this structure, will push me to gather deliberate and intentional data from my students so that I am able to plan better each day. Time will tell.

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Continuous Improvement

How do teachers improve their practice? This is a question I have been asking for my entire career (over 25 years). During the past year, I was involved with a group of high school teachers, coaches, administrators, and researchers working on how to scientifically study how to improve. In our case, the focus was on improving student engagement, specifically in Algebra 1. Since this is seen as such a gateway into high school mathematics, if we cannot help students to engage, we are narrowing their future opportunities. So we tried this new (to me) approach called a PDSA (Plan, Do, Study, Act). You set a goal, decide how you will measure your progress toward the goal, make some predictions, collect the data and analyze it, then revise. These are meant to happen in short cycles, 1 to 2 weeks.

What did we do?

My small group focused on student communication. Students often seem reluctant to share their thinking, so we devised a protocol called “Structured Math Talk” during which students were given a task to work on individually for a few minutes and then turn and talk with a partner. The partner talk was by turn and timed. One partner talked and the other listened and then they switched. This is our first PDSA form. It turned out to be quite challenging to gather this data. We were teaching under different circumstances: some of us had 55 minute classes that met every day, some had 80 minute classes that met every other day, and others had 90 minute classes that met every day. Trying to figure out the right amount of time that constituted that 1 to 2 week cycle was a challenge. (Plus, I often forgot to have students complete the exit slips.) But, it was clear that our students were compliant. We asked them to talk about math and they did. We were concerned, however, that they were only talking to each other because of the structure we imposed. Would they continue to share their thinking with each other even when we weren’t watching? This was our revision for PDSA cycle 2.

Our data was showing so much success that we questioned our entire process. Are we asking the right questions on the exit slip? Do our students understand the questions on the exit slip?  Are we using the right kinds of tasks? Are we asking our students to engage in meaningful mathematics? So, we paused. We went to the ATMNE 2015 Fall Conference together. We read. We learned. We regrouped and refocused on the idea of productive struggle. That would feed the conversations, get our students to persevere, and push us to make sure that we were providing meaningful mathematical tasks.

What did I learn from this experience?

  • It’s difficult to document the small adjustments that teachers make every day, all the time. It’s difficult to be scientific about those small changes that happen in the moment. It’s important to develop a mindset of doing this, however, because that is how we can help each other improve.
  • I’m not sure we were asking the right questions. Not the right question to study, not the right questions of our students, and not the right questions to help us learn.
  • My students are generally willing to engage in whatever task I throw at them. It was never a problem for me to get them to talk to each other or to try something that they had never done before.
  • This process is an adaptation of Edward Deming‘s process cycle. My brother has done this work for 30+ years and is an expert in Lean management techniques.

What’s next?

The small group has expanded and we’re now known as the Better Math Teaching Network. Our first meeting is in July, a 4-day institute where I hope to share my new learning with others and learn better techniques for meaningful data collection. The trick, I think, will be to ask the right questions.

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More 3D Geometry

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After my last post, Mike Lawler gave me all of these awesome ideas for my 3D geometry class. Considering that my class has been working on nets, I was most fascinated by the dodecahedron that folds into a cube, which came from Simon Gregg.

When I first watched the gif animation, I just couldn’t figure out what was going on. I thought, “I’ve got to show this to my students!” Thursday was that day. I tasked them with a build challenge. Of course 55 minutes wasn’t enough time to complete anything, but students had drawings (which gave us insight into the construction)

a CAD rendering (completed during a snow day)

and a previously constructed dodecahedron that had been re-purposed (completed during lunch).

So, thanks Mike, for the inspiration, and thanks #MTBoS for being there helping us to support each other.

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“How Do We Know That?”

questions

I’m teaching this 12 week geometry class focusing on 3-dimensional figures. It’s a brand new class, like many at Baxter Academy, so I get to make it up as I go. Since our focus is on 3-dimensional figures, I thought I would begin with some Platonic solids. So I found some nets of the solids that my students could cut and fold. Once they had them constructed, there was a lot of recognition of the different shapes and, even though I was calling them tetrahedron, octahedron, and so on, many of my students began referring to them as if they were dice: D4, D8, D12, D20. Anyway, I must have made some statement about there only being 5 Platonic solids, and they now had the complete set. One student asked, “How do we know that? How do we know that there are only 5?” Great question, right?

I really felt that before we could go down the road of answering that question, my students needed a bit more knowledge and exploration around these shapes, and maybe some thinking around tiling the plane would help, too. So we spent some time trying to draw them, counting faces, edges, and vertices, visualizing what they might look like with vertices cut off, unfolding them into nets, and wondering why regular hexagons tiled a plane, but regular pentagons did not. We played around with the sides – a lot – and even talked about this thing called vertex angle defect. Then we returned to the question of why only five. Students were able to connect the need for some defect (angles totaling less than 360 degrees) and the ability to create a 3-dimensional figure. Through the investigation, they were able to see that the only combinations of regular polygons that worked (by sharing a vertex) would be 3, 4, and 5 equilateral triangles, 3 squares, and 3 regular pentagons. They could give solid reasons why 6 triangles, 4 squares, 4 pentagons, and any number of other regular polygons could not be used to create a new Platonic solid.

I had not anticipated this question, and had not included it in my plans. But, because it was asked, thankfully, by a student, it pushed us into thinking more deeply about these shapes (and their definition). And, ultimately, my students were able to answer the “why only five” question for themselves.

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Stacking Pennies

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There are lots of things I love about my classes – my students are at the top, but I also am enjoying teaching a course for the 2nd or 3rd time. This activity is for a statistics course called “Designing Experiments and Studies.” It’s a course that lasts for one trimester (about 12 weeks) and we have just completed week 4. The Penny Stacking activity is an introduction to experiments. I ran this lesson last week (Jan 12 & 13).

The rules of penny stacking are simple:

  • You can only touch one penny at a time.
  • Once the penny is placed on the stack, you cannot move it.
  • Stack as many pennies as you can without having the stack fall over.
  • Each student stacks pennies once, with either their dominant hand or their non-dominant hand.
  • Students are randomly selected to stack pennies with either their dominant or non-dominant hand.

Before embarking on the experiment, we made the following predictions:

  • More pennies would be stacked with the dominant hand (although a few students disagreed and thought the results would be the same for both hands).
  • A few students thought the ratio of pennies stacked with dominant hand to non-dominant hand would be 3 to 1.
  • The range of pennies stacked with dominant hand would be 15-45 pennies, while the penny stacks from the non-dominant hand would range from 10-35 pennies.

Then it’s off to conduct the experiment. This is pretty tricky given that up to four students sit at a table and our building is so old that if someone walks across the floor upstairs, our floor will shake. Here are the results:

penny stacksHow would you interpret these results?

We calculated the means: dominant -> 30.9 pennies; non-dominant -> 26.25 pennies. It’s clear that, on average, the number of pennies stacked with the dominant hand is greater than the number of pennies stacked with the non-dominant hand. But is that difference in the means (of 4.65 pennies) significant? Is it unusually large? Is it more than what we might expect from randomizing the results?

To check this out, we randomize the results. Each pair of students received a stack of cards. Each card had a result. The partners shuffled up the cards and dealt them out in a stack of 10 (for the dominant hand) and a stack of 8 (for the non-dominant hand), calculated the means, and then subtracted (dominant – non-dominant). They each did this a couple of times and we made a histogram from the results of the randomization test.

pennies

(It’s a Google Sheets histogram – I don’t know how to get rid of the space between the bars)

If you look at our difference of 4.65 compared to these randomized results, it looks pretty common – not at all unusual – to get such a result. If you think that our randomization test was too small (with only 24 randomizations), then you can use the Randomization Distribution tool from Core Math Tools, a free suite of tools available from NCTM. And it’s the only tool that I know that runs this test effectively. Here are the results from 1000 runs, just like the card shuffling but faster.

more pennies

penny summaryAnd you can even get summary statistics that show that our result was within 1 standard deviation of the mean of the results that we got from randomizing the data. Not a very unusual result at all.

We followed this up on Thursday with an experiment inspired by an example from NCTM’s Focus in High School Mathematics: Reasoning and Sense Making – memorizing three letter “words.” Based on the experiment described in the book, I created random lists of three letter words and three letter “words.” The lists of words were meaningful, like cat, dog, act, tap, while the lists of “words” were nonsense, like nbg, rji, pxe, ghl. Students were randomly assigned to receive either a list of meaningful words or a list of nonsense words. They were then given 60 seconds to memorize as many words as possible. Like with the penny stacking, I made them predict what they thought the results might be. What would your predictions be?

 

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Intro to Statistics (Unit 2)

During the first few days of the new unit students explore relationships in the data we had collected about the class.

Back to the Class Data

Looking at class data gives us the chance to ask questions about relationships among the variables. Here are some questions my students came up with:

  • Is arm span really equal to height?

The easiest way to dig into this question is to look at a scatter plot of the data. So, we plotted the variables, along with the line height = arm span.

11-22-2015 Image001

We noted that two people were on the line, two others were very close, and the rest were either above or below the line. What do those points above the line mean about height and arm span for those people? What about the points below the line?

  • If my hand span is longer than my wrist circumference, then shouldn’t I be able to wrap my hand around my wrist and touch my pinky to my thumb?

11-22-2015 Image009

One hundred percent of students had longer hand spans than wrist circumference, but only a couple of students could wrap their hands around their wrists.

  • Is the age (in months) related to any other measure?

11-22-2015 Image010

It would seem that none of the other variables is a good predictor for age in months. It also seems as if age vs height has a negative association. Huh?

Digging a Little Deeper

If the line height = arm span doesn’t describe, or predict, that relationship well, then what would do a better job? We added a “movable line” and adjusted it until it looked about right.

movable line

 

Our line predicted that height = 0.85 * arm span + 26 cm. Wait, what? Height is 85% of arm span? And what is that +26 cm all about? It made for an interesting conversation, especially this question from a student: “How can a person who has an arm span of 0 cm be 26 cm tall?” Which prompted: “What does an arm span of 0 cm even mean?” I certainly don’t have definitive answers to these questions. What I can do is encourage the curiosity, the conversation, and point out that the relationship we discovered is for these measurements. Does it make much sense to use our calculated relationship to make predictions about heights for arm spans that are relatively far away from the data we collected?

Correlation, Causation, Outliers, Influential Points

All of these topics follow from this initial discussion about the class data. Ultimately, students once again find their own variables of interest and complete an analysis demonstrating what they’ve learned. This time topics included unemployment rates, marriage rates, divorce rates, distances & temperatures of celestial objects, height & weight, obesity rate & life expectancy, and mean snowfall & mean low temperature.

Once again, the variety of topics that interested my students is greater than what I could have come up with. More importantly, because they chose their own variables, they were interested in analyzing the data and answering their own questions.

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