- The class was honors algebra 2 – a class with mostly freshmen and sophomores designed to get students through BC (or sometimes MV) calculus before leaving high school. These students are usually self-motivated, intrinsically passionate about mathematics and many are working towards careers in the STEM fields.
- The topic I was presenting was word problems involving systems of two linear equations – a skill students are expected to have mastered coming in to class on day 1.

In their defense, the problem they had was one of the most contrived, psuedocontextual questions out of any textbook.

John is collecting stamps. He can buy cheap stamps for 70 cents or expensive stamps for $1.30. He buys 15 stamps all together and spends a total of $13.80. How many of each stamp did John buy?

Yeah – I know, it’s a stupid question to begin with; but remember, I’m not trying to hook these kids on math – I’m just (what I thought was) reviewing a fairly straightforward skill. My first reaction to their complaints was to ask why I had to teach them something they were supposed to already know coming in to my class, but knowing the futility of such a comment, I quickly squelched that retort. Instead I took a new approach to the old question. It went something like this:

You’re right. You’re never going to have to answer a silly question about different priced stamps. So instead of that question perhaps I should pull data from the Keplar Space Telescope and ask you to determine whether or not a planet that is 100 light years away might be habitable. But then again, you are freshmen and sophomores in high school and not post doctoral students studying astrophysics, so you wouldn’t know where to begin on such a question. So I could create a problem about determining whether or not Core-Shell Nanodiamond Geometrics could be used to design a safer and sturdier helmet, but then I remembered you’re not post doctoral students studying mechanical engineering and have no idea what Core-Shell Nanodiamond Geometrics even is.

So I here is a stupid problem about collecting stamps, which I only give you because 1) you know what stamps are, and 2) it gets specifically at the skill of solving systems of two linear equations. When are you going to use this problem? Never. When are you going to use this skill? I can’t answer that since I don’t know your future. Of course I can tell you many places it is used, but in reality it’s just a single footnote in the fields of math and science. Is it an essential, stand-alone skill? Probably not. But it’s a lot like writing the letter “e.” It’s pretty useless on its own, but an instrumental part of understanding the vast landscape of mathematics. And as the great physicist, Richard Feynman, once put it, “to those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature … if you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.”

So here’s a stupid, made up, problem about stamps. Because it gets at just one of the millions of underlying skills in the world of mathematics. Here is one small puzzle piece to one of the most amazing things I have ever witnessed – the thing that made me decided to spend my career trying to show you what I see in mathematics. So I am going to ask you to do this stupid problem about stamps today and hope that one day you know enough math to see the true beauty of the mathematics we study.

Is this the answer I will use any time someone asks that age old question? Most certainly not. But for this group of students at this particular time, it really seemed to get their attention. They were certainly more focused that day than they have been in the past.

]]>But every once in a while, when I have the opportunity to see my students involved in various extracurricular activities; I get a rare glimpse of my students outside of this math world. In these instances I find that I bare witness to just how little of an impact I have on their day to day life. Regardless of whether or not my students experience success in math, they are all involved and successful in their own areas of expertise. It is something we as teachers all know to be true (and even rely on regularly), but nonetheless it never ceases to surprise me when I am confronted with the image of my students as more than just mathematical thinkers. In these moments I am reminded of how short an hour of math really is.

And then there’s the other times. The moments when students come to me after school in tears because, even though they are trying every day, they feel like they just aren’t getting it. There are the students with the voice in their head telling them they are an idiot because they don’t understand numbers (and no matter what I say, I can’t seem to convince them otherwise). There are the students for whom their biggest source of anxiety is the feeling of stepping into a math classroom, and the idea of sitting one on one with their math teacher is mortifying. In these moments when I am sitting down, dishing out tissues, I am reminded of how long an hour of math really can be.

And there’s the third type of students; the students who come into class, do math, and leave, without me ever knowing if my class is an hour of joy, an hour of indifference, or an hour of torture. When a kid gets an A on a test does it make them happy? When a kid gets an F does it affect them outside of the math classroom? When I send home a bad report card, or call parents, what happens next. I find that I rarely know the day to day impact (and much less, the long term impact) I have on the majority of my students. For the most part, all I get to see is my students through the distorted lens of mathematics. And I find that this feeling scares me.

]]>**Option 1: Just tell them what to do**

It’s easy enough to just tell students to use open brackets every time. Most students like this and they feel like they learned something. They can just tell themselves, “whenever I am writing if a function is increasing, decreasing, or constant, I should use open brackets.” Students see success on assessments as a result, but rarely understand why they are doing what they are doing. As I improve my repertoire of teaching tools and strategies, I find that I use this method less and less, only occasionally reverting to it out of desperation or an extreme shortage of time.

**Option 2: Use the definition**

I’ve used this strategy before with some success. I give students the textbook’s definition of increasing, decreasing, and constant and ask them to decipher it in groups.

*Definition:* a function is said to be increasing on an *open* interval *I*, if for all and in that interval, implies .

This certainly teaches students an important skill: to read and make sense of definitions in a book. In order for students to be self-sufficient, I believe it is essential for them to be able to learn concepts without me explicitly teaching them. However, assuming students can decompose this definition into plain English, I would expect that only about 20% of students would truly understand the mathematical meaning of “increasing” and none would understand why the book only talks about open intervals.

**Option 3: Connect it to the “real world”**

I used this strategy recently and was quite satisfied with the results. I began by walking forwards and asked students, “am I walking forwards or backwards.” All students agreed that I was walking forwards. I then walked backwards and asked the same question. Again, students agreed that I was walking backwards. Finally I had one student take a picture of me at some point as I alternated walking forwards and backwards. I put the picture under the document camera and asked the same question.

All of the sudden there was some disagreement. Students quickly concluded that they can’t tell if I’m walking forwards or backwards from a single picture and need to know what happened before and after the photo. I then asked them if a single point on a graph is increasing, decreasing or constant. “We need more information” they all agreed. The truth is, we don’t ever know whether something is increasing, decreasing, or constant without knowing where the points near it are. When I asked the class to write a sentence for why we use open brackets at the end of class, every student gave a (more or less) adequate response.

Now of course these three strategies are not the only three. Of course I should talk about the definitions and teach students to use their book. I may even revert to telling an individual student to only use open brackets. As any teacher knows, no single strategy is the fix-all for education, but instead, it’s a matter of matching one’s teaching methods with the complexities of the students, the content and the thousand other factors that go into day-to-day teaching.

]]>I’m no regular to these conferences. In fact, I have been to very few in my teaching career, so all of this is a new experience for me. Tonight I took my seat at a table, partaking in friendly educational banter with a group of colleagues and teachers – most whom have been teachers longer than I have been alive. As the conversation meandered from topic to topic, the concept of teacher collaboration had to come up. A smooth talker began a passionate speech extolling the virtues of conferences as an essential medium for teachers to connect and share ideas. The argument went along these lines:

*Teachers can (and should) collaborate with colleagues – it is a vital component to education – but colleagues are insulated within one’s specific school system. Teachers can go online to find lessons, but they need contact and discussion to improve upon those lessons. Conferences over the past thirty years have allowed me to build a network of teachers whose opinions I respect and value. At these conferences I have the opportunity to pick and choose ideas I like, and I can use these ideas in the classroom the next day. I can engage presenters in discussions and continue the conversations with them after the conference is over. With the relationships I build with other presenters, we can share and build upon our pedagogy. And as I attend more conferences I continue to grow and learn as a teacher. Only through the 30+ years of conferences have I become the successful teacher I am today.*

Now I know this veteran teacher’s argument was somewhat **against** the idea of the internet as a substitute for conferences, but everything that this educator described sounded like the blogging community that I have spent years following. His argument for the significance of conferences as the only real professional development tool only served to reinforce the notion that (for me) the blogging community is essential for me to be successful in teaching.

So here I am, at an annual conference with 50+ presentations and 600+ attendees. This, along with many other conferences, is the veteran teacher’s elixir – allowing him to gather in improve his teaching. My elixir is the blog-o-sphere. My professional community is bloggers. My growth does not come from the annual planned conferences, but the constant stream of ideas flowing through the internet, the feedback and comments of other qualified teachers, and the online communities that support and help teachers develop professionally. I certainly understand the value in physical conferences, and will continue to attend and present my ideas, but for the other 363 days a year the internet is my continual conference available when time permits, and omnipresent.

But I haven’t written my thoughts or ideas down in years. After all, there are IEP meetings and parent conferences. There are phone calls home and meetings with students. There’s planning tomorrow’s lesson and grading yesterday’s assessment. And there’s a whole life to live outside of teaching. But I cannot allow myself to stagnate through a self-imposed isolation. It would be wrong of me as an educator to not make use of the amazing community that is the internet. Blogging is my professional development, and it is an essential component of my teaching, not to be pushed aside for lack of time, but to have time reserved for it specifically.

In short: Hello again internet.

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