Student blogging has me thinking… (reaching out for help once again.)

I think I want to try student blogging next year in my Algebra II classes. I’ve only ever taught Algebra II once and I didn’t do a particularly wonderful job.

It was the sense-making that really got to me. My students were pretty good at learn procedures and algorithms, but the long-term retention was remarkably low. I have seen several examples of student blogging and feel like if I framed the discussion questions properly and encouraged the students to read each other’s posts, and comment. That could… COULD… open up a different mathematical thinking experience for the students.

If that were used to supplement the number-crunching practice, and the group problem-solving and exploration, that could potentially act as a way to deepen (or at least broaden) the thinking that the students were being asked to do. In addition, the opportunity for the entire internet to read and respond can add an extra-level of interaction. The students wouldn’t have to apply their real name if they didn’t want to. There is a chance for creative anonymity.

All of that being said, if you have your students blog, will you please comment on this so that I can pick your brain on what’s worked, what hasn’t, what to watch out for and what to definitely do! Links to other blog post would be much appreciated. E-mail this post to people you know who do this. I would love a rich, challenging comment section on this one. And trust me, if you don’t help me, I will make my own idea and learn this the hard way!



NPR wants to advise your pizza order…

Quoctrung Bui from NPR says that there are at least 74476 reasons that you should always get the bigger pizza. (The article has an awesome interactive graph, too!)

If we could mix the article with a math exploration, we could provide an awesome opportunity for a math-literacy activity that can combine reasoning, reading, writing, and some number-crunching all in the same experience. That’s a nice combination. Also I suspect the content hits close to home for most students. (The leadership in our district is often looking for opportunities to increase authentic reading and writing in math classes. This seems to fit the bill quite well.)

Here’s an activity:

Although without fail, the menus from a variety of local pizza joints will probably be a bit more engaging. (Look for an update coming soon…)

Shauverino Pizziano

But the big question is why?

According to Bui: “The math of why bigger pizzas are such a good deal is simple: A pizza is a circle, and the area of a circle increases with the square of the radius.” 

Yup… that’s pretty much it.

Trig Curiosity

There is something about a question not being graded that makes the students aggressive and risky. That can create the conditions for some of the best thinking. There are many days when I think grades and points and the division between problems that I will “collect and grade” versus the ones that I will not.

In the system in which I exist, sometimes bonus questions on formative assessments are the only way to really perplex a student – to push them at the risk of pushing each student beyond their current ability to reason, but still get a solid effort.

On today’s quiz, I added the following question as a bonus:

If you type “Tan 90″ into a calculator, you will get an error message. Knowing what you know about trig, discuss the possible reasons that taking the tangent of a right angle in a triangle would make your calculator show an error message.”

This isn’t something that has come up in any of our discussions. I would like to share with you some of my student’s answers.

From James: “It has an opposite which is the hypotenuse, but it has two adjacents so you wouldn’t know which one to use unless you put it in the calculator.”

From Tyler: “There is no such thing because when you plug in Cos 90 you get 0 and when you plug in Sin 90 you get 1. Maybe it is because since Tangent is TOA, it tries to add up to 90, so like opposite is 30 degrees and adjacent is 60 degrees.”

From Brianna: “Because Tan 90 would be opposite/adjacent, but the opposite side of the 90-degree angle is the hypotenuse and you can’t have the hypotenuse on top.”

From Jeremy: “It shows an error message because the right angle on a triangle doesn’t have a defined opposite or adjacent side length because the angle is touching both legs.”

From Lauren: “With tangent, you are finding opposite/adjacent. Those are the legs, and that 90-degree angle is being made by the legs.”

From Dayna: ” There could be an error because the opposite of the right angle is also the hypotenuse of the triangle.”

Quiz Bonus 3

From Ally: It’s not clear where the negative idea comes from, but it is curious that in a Trig world of decimals and fractions, 90 in the other functions gives 1 and 0.

Quiz Bonus 2

From Victor


Quiz Bonus 1

Perhaps Josh’s picture says it all.


Now, the next question: If the “two-adjacent-sides-so-the-calculator-doesn’t-know-which-you-mean…” explanation wins out…

…then why don’t we get an error message for Cos 90?

“Truly wonderful, the mind of a child is”

photo credit: Flickr user "sw77" - used under Creative Commons

photo credit: Flickr user “sw77” – used under Creative Commons

I agree with Yoda. (The title of this piece is a quote of his.)

This morning, as she was finishing her cereal, my daughter was completely taken by the reflection of her cup on our table. She was passing her hand under the table and above the table and, as is her custom, talking the whole time.

Then she fired out a wonderful question.

“Daddy, why is the reflection upside down?”

My daughter asked a fantastic question. Would the average geometry student be able to answer it?

My daughter asked a fantastic question. Would the average geometry student be able to answer it?

In my head, I thought that my students should definitely be able to hammer out a pretty reasonable response to that question by the end of the first unit of geometry.

And thanks to my daughter, they will.

The Value of Face-to-Face

Today I got an opportunity to facilitate at EdCamp Mid Michigan, which was just the second time I served in any manner of leadership role at a teacher professional development workshop. The format was designed to be casual and conversational. Facilitators opened the conversation and the participants contributed for 45 minutes or so asking questions, telling stories, stating concerns and helping each other.

Sam Shah, a math teacher to whom I have received tremendous support developing a calculus class for next year recently posted a fantastic piece about the power of community. I encourage you to read the piece and reflect on how powerful a member of this sharing and learning community that you are. Each contribution matters, each “like”, each comment, each bit of dissent. We impact each other and we take it back to our classrooms.

And while I can completely embrace Mr. Shah’s post, I would like to offer a bit of a “yeah, but…”

Today’s EdCamp was a great microcosm of that greater community. There weren’t as many people, but each person was expected to contribute, because each person brings value to the table. They bring experiences, questions, concerns, anecdotes, advice… all of these parts are necessary for the community to flourish. The “mathtwitterblogosphere” or (MTBoS as it has come to be known) is a similar community. Some do a lot of writing. Some a lot of reading. But it is inclusive. (Shoot, if they’ll welcome me, they’ll welcome anyone.)

But EdCamp included one part that has been missing from my experience with the MTBoS: eye contact. That’s the one missing piece. The overwhelming majority of the teachers that I have communicated with through twitter and my blog are people who I have never met face-to-face. And while the MTBoS does it’s very best to facilitate conversations among folks all over the world… (I say that as though I have forgotten how remarkable it is that such technology even exists)… I wish that more could be done to create opportunities to get a chance to break bread with so many of the fantastic folks that I am meeting through twitter handles and avatar photos.

With that eye contact today, I tried (as Mr. Shah describes similarly in the aforementioned piece) to describe the value of the MTBoS to some math teachers who hadn’t explored the community much. I’m pretty sure I did a poor job. You see, one of the most important functions of face-to-face interactions is the power of facial expressions. Truth is, I am grateful for conversation because I often don’t know how well or poorly I’m explaining something until I see the faces people make when they are listening to me explain it. As I have moved out of my 20’s and into my 30’s, I am finding that I am doing more and more explaining to other people. The ability to look someone in the face and converse is one that I find incredibly valuable.

Bottom line: Thank you for all the support that you’ve provided MTBoS and thank you EdCamp Mid-Michigan for the support you’ve provided. I am thankful to be a part of both communities and hope that I’ve been a meaningful contributor. Today simply reminded me that while it is fantastic to embrace the resources and contributions (and amazing that it is even possible to do so) it is equally important to embrace the community that exists nearby me, too.

Learning from Playing Around

These past two weeks have been an awesome time of learning for the students we’ve been working with, but I’ve also done a bit of learning myself.

I’d like to have my students love math and science and naturally be interested in it. But they’re kids. They would prefer to play. People get the most out of that which they put the most in to. If given the chance, students will put a ton into playing around.

These past two weeks I’ve been working with upper elementary-aged students. I normally teach high school students. I’m not sure if the age difference changes anything. The stuff they want to play with might be different, but not the desire to play.

And after 7 years of teaching math, there’s something appealing about a situation where students will be voluntary and enthusiastic participants.

Play Science 1

I have just spent two weeks watching students play with two activities. The first was an activity called “Table Timers” where they were challenged to design and construct an apparatus on a table top that reliably moved a marble down an inclined table in ten seconds. Second, The Helium Balloon Problem challenges students to keep a helium balloon rising, but to have it travel as slowly as possible upward. Not every group worked well and not every group achieved these goals, but the engagement level has been high. I suspect this is because they we allowed to play.

Here’s what caught my attention the most: In the midst of their play, the students demonstrated some authentic problem-solving techniques. They had to identify the major challenges to their goal, which they often did. They had to brainstorm possible ways to overcome the challenges, which usually took the form of raking through a tub of blocks or looking through the supply table. They discerned which seemed like the most realistic and then test. Following a test, they discussed what happened, why, and revised. And the students were often quite excited when they got the right answer (knowing themselves that it was right and not relying on me to tell them).

That’s a pretty good learning model. That’s something that I have a hard time getting my students to do with book work.

Play Science 2

So, the sharing, the idea-making, the consensus-building, the authentic assessment are all good things. Obviously I am not simply advocating letting the students play around all day. But perhaps by using play, we can improve engagement and the students seem to more naturally fall into a more authentic problem-solving mindset. When I consider helping them draw out the learning, some thoughts come to mind.

First, it seems like during the whole process of exploration, design, construction, testing, revising, and demonstrating, there needs to be an abundance of contents-specific vocabulary. The marble didn’t “bonk into that block.” That block “applied a force” to the marble. Students don’t “figure out how big the shape is.” The “find the area” or “circumference” or “volume.”

Second, students don’t seem naturally inclined to take data or to keep records. In the past two weeks, it seems that students are avid experimenters and do a pretty good job of verbally analyzing the problems if the plan didn’t work. Practically NONE of them documented anything on paper. No sketches, no data, no records of updates. This is an important part of the problem-solving process that would have to be established as a norm.

Third, the activities have to be tiered. Video games are great at this. The entry point tends to be quite low. The first couple of levels are pretty manageable and then the intensity and difficulty pick up. People get locked into video games through that model and people get unlocked quite quickly once the game has been beaten. Both Table Timers and The Helium Balloon Problem worked with this model. Then entry point was low for both activities and it was easy enough to begin to approach the goal, but perfecting the design and executing the plan took much more care. Then, once they hit ten seconds, we’d challenge them to add five seconds to their timer.

Fourth, I think that the groups need to be expected to summarize and present their work to each other and to field questions from the class. Class norms should allow for questioning of each other’s work and students can learn a lot about their own design, but also about the content when they know that they are going to have questions coming from their peers. Also, it would seem like this would encourage more thoughtful designs, too. Besides this, idea sharing gives the students an opportunity to look at other designs, integrate specific vocabulary into more regular use, and get the students comfortable with collaborating.


Play Science 3

I don’t think that playing around is the answer to everything, but I know that in my own experiences, it seems to be the forgotten learning model and if I’ve learned anything these past two weeks, it’s that an environment that produces enthusiastic student participation shouldn’t be ignored.

“Useless” Math Class – Misconception #1

In my previous post, I commented on how struck I was upon reading an embittered writer’s rant about having to take a “useless” math class. I mentioned four main misconceptions that we math teachers have allowed to take root in the modern academic mindset. I will now address misconception #1:

“Math is about numbers. Writing is about words. Wordy people write. Numbery people math.”

I want to start by saying that writing and math can both be treated as stand-alone topics. You can study math or you can study writing… but most of us don’t. We are typically doing either one of those things about something else. Perhaps my situation revolves around a car, but I may need to use math and writing to deal with this situation. At that point, math and writing are vehicles (pun definitely intended), but the car is still the focus of the situation.

Now, it has been said that math is a language of its own. (In fact, here is a book about it, if you want to read it.) I understand the point behind such a statement. However, in the end, math doesn’t have it’s own language. Sure there are mathematicians who can cross cultural barriers by writing everything in set notation, but among “non-mathy” folks,  mathematics requires a common tongue. In this way, math is just like anything else.  “Два яблока добавил еще два яблока в четыре яблока” is meaningless to anyone who doesn’t speak Russian and it wouldn’t matter if that was a math statement, a religious statement or a question about sale on ground beef.

Beyond that,  consider what kinds of mathematics “non-mathy” folks are talking about in everyday conversation. Nay-sayers are correct when they say that it probably isn’t going to be formal mathematics. But, it will be important. The cop gives directions to a lost citizen. The salesman explains a payment plan to a potential buyer. The marketers discuss the exposure rates of a type of advertising. The psychologist explains the results from a study to a therapy patient. The professional athlete figures out how to invest his or her signing bonus. (By the way, these types of communication fall into “mathematical literacy” as discussed by Jan De Lange in this paper.)

Unskilled wordsmiths tying to have those conversations are going to leave with more questions than answers. Those aren’t necessarily simple conversations to have. They require patience, dialogue and WORDS. Good words. Accurate words.

And where do people get a chance to discuss complex mathematical situations? In “useless” math classes… like the ones I teach!

When done properly, math class provides an opportunity for students to struggle and stumble over the wording of math situations. To listen to an explanation and respond with a targeted question. The students develop the use of specific vocabulary and learn when to use it.

So, why do wordy people have to take my “useless” math class? Because we need you. We need you folks who are good with words to help us figure out how to explain the math to the rest of the world. Sure, there are “numbery” people there to figure the tough mathematics out, but without the “wordy” people we are left with Sheldon Cooper to explain the math to the rest of us.