First Day: Gathering Students' Impressions of Math

If there's one thing about teaching I'm not very great at (and there are many such things), it's the first day of school. I always struggle with it. I find myself so busy preparing for the year at large, or getting my classroom ready, or whatever else is demanding my attention, that I never really take the time to plan out a really great first day.

In part, I ended up doing what I described (in tongue-in-cheek fashion) to my students as the "time-honored tradition" of going over the syllabus for the first day of class. At one point, one of my administrators walked in to watch my class for a bit, and all they saw was me going over the syllabus. It was one of those "please just kill me now" moments for me.

I'm being over-dramatic, though. It really wasn't so bad. I'm really looking forward to working with the group of seniors I have this year, and I enjoyed meeting them today. I definitely did a lot of talking, which I never prefer to do, but it'll be different tomorrow.

As part of the first day of class, I had my students fill out a survey about how confident they feel about their math skills, what "doing math" means to them, and what they hope they'll have learned by the end of the course. The first three questions were Likert scale items. Here are some of the numbers:

1. How confident are you in your ability to "do math"?

Completely confident: 11/78
Mostly confident: 31/78
Somewhat confident: 22/78
A little confident: 6/78
Not at all confident: 8/78

2. How confident are you in your ability to talk about math verbally using mathematical reasoning and vocabulary?

Completely confident: 5/78
Mostly confident: 16/78
Somewhat confident: 31/78
A little confident: 14/78
Not at all confident: 12/78

3. How confident are you in your ability to communicate about math in writing?

Completely confident: 7/78
Mostly confident: 14/78
Somewhat confident: 36/78
A little confident: 18/78
Not at all confident: 5/78

Overall, my students this year seem to be carrying a healthy level of confidence in their ability to "do math." (Of course, that depends on their definition of what it means to "do math," which I asked later on.)

There's a considerable split in confidence with my students as far as communicating mathematically. Those are two areas I intend to focus on this year: I want my students to speak and write confidently about mathematics. I want them to be well-versed in the Math Practice Standards by the end of the course.

There were some other short answer questions. There are too many responses to list, so I just picked a few examples that I think give the general view of the students:

4. What do you think it means to "do math?"

"I think it means solving problems with numbers. Doing math is when you work out a math problem. Also taking time to make sure your answer is right."

"To do math is understanding the logic behind a problem. It is the ability to explain problems to others verbally and on paper. Doing math is using more than one technique to find the correct answers."

"I believe that 'doing math' is thinking about a problem critically and using certain formulas to find out the answer to something."

"To do math is to find the answer to a problem that involves numbers, distances, functions, or any form of measurement. A math problem usually has a set number of answers that have to be found through use of mathematical functions or equations. But to do math is to use logic to solve something."

"Doing math means to completely understand it, and for me that comes in 3 parts. Before you can properly plug in numbers to equations, you must first know what those equations mean, and what answer(s) they are trying to achieve. After knowing that, you must know how to correctly plug in the numbers in the equation to get your answer. The final thing that you need to know how to do when 'doing math' is being able to explain what you did, and why. If you are not able to explain how or why you did what you have done, then there is no way to tell if you were right in your thinking."

"'Do math' to me means to solve a puzzle. You need to find all the pieces of the puzzle in order to solve the problem."

"To 'do math' is to have an answer to the problem presented. However, I think that 'doing math' also includes the full understanding of the problem. Also being confident in the answer that you have."

5. What does it mean to be a "good mathematician?"

"Math is easy to learn but hard to master. Given enough time, anyone can solve any problem. Being a good mathematician means being able to solve equations in a quick manner."

"Being a good mathematician means that you can easily identify and solve problems quickly and correctly."

"A good mathematician doesn't give up easily, but keeps trying different methods until the problem can be solved. A good mathematician learns to apply conclusions to the world surrounding him or her."

"A good mathematician is not necessarily someone that finds answers quickly, but rather one that finds answers effectively."

"A good mathematician is someone who can answer the problem that they have set in front of him or her. They can execute the best possible method of doing a problem, in the quickest way possible. They also understand all of the math behind it."

"A good mathematician would use... nothing other than your brain. Wouldn't use a calculator and know every function in math. Be like Albert Einstein."

"Being good at math means being able to remember formulas and solve problems quickly. I also think it means being able to help anyone when they need help during a certain area they don't quite understand."

"To be a good mathematician means you have a brain like a computer. If someone asks you a difficult math question you should be able to answer it in a matter of seconds."

"A good mathematician would know how to recognize a math problem. A good mathematician would actively seek answers to things he/she doesn't understand. Finally a good mathematician knows and studies deeply the subject of math."


6. What do you hope you will have LEARNED in Pre-Calculus by the end of the school year?

"I want to learn how to solve math problems in the quickest ways possible. I also want to explore different forms of calculators and their functions."

"I hope at the end of the year I learn how to solve my problems, without errors or depending on anyone for help."

"Pre-Calculus should teach students more advanced forms of mathematics, past the formulas and equations. Pre-Calc is a dreaded class by some, but can be helpful in certain professions."

"I really want to know how some advanced math could be used to solve everyday problems, so if it is just the same old stuff revisited from last year at least show us how it applies to real life."

"I honestly just hope to learn something new in Pre-Calculus. I want more challenging problems so I can have more math skills."

"A way to understand Calculus without being a mindless zombie to the textbook. Well, understand enough to understand college Calculus."

"I hope that I have learned new formulas and learned them well."

"Hopefully I will be able to pass."

"I hope that I will have learned to explain my reasoning with most of my math problems, thus broadening my horizon on how to be a good teacher."


7. What do you hope you will have EXPERIENCED in Pre-Calculus by the end of the school year?

"I hope to experience an even greater understanding of math as well as enjoy it more. It's currently my favorite subject, so I believe that most, if not all, of my experiences will be positive in this class."

"I hope that I will experience how to speak mathematics in a different kind of language than what I usually use when I explain a solution to a problem."

"Uhm, what am I SUPPOSED to have experienced? I don't really have any hope for anything in this class."

"By the end of the year I hope to have experienced how to deal with stress when it comes to math. Math has always been my worst subject and I get stressed a lot while doing math."

"I hope to experience new things and different ways of solving problems."

"I really don't know. Surprise me."

"I hope to experience what it will be like to use math in the real world, such as: taxes, sales, etc."

"I hope to have experienced the questions that make you sweat, and look back in your notes to figure out. I love puzzles and math and I love a challenge so I want to experience a good challenge in a math course. I want to be able to help others with their homework and also be able to say I had the best Pre-Calc teacher in high school history." (Geez, no pressure there, right?)


While I definitely don't think this first day of school was the greatest, I did end up getting a lot of really thoughtful responses to these questions (again, way too many to list). The attitudes and views of my students towards math definitely cover a wide spectrum this year. I'm really encouraged by the number of students who said they're craving challenge. I love it. I hope I can deliver.

We're starting a group task by the end of the week. I'm going to try grouping students so that each group member has a certain level of confidence in talking about math, writing about math, or just doing math. I may also group them by how they responded to the written questions. We'll see how it goes.

It will be interesting to see how the students answer these questions in May. I hope that more of them will see "doing math" in terms of problem-solving, constructing arguments, modeling, looking for structure, and so on.

And so a new school year begins. Allons-y!

Two Things From a Tuesday

Or maybe I should title this post "Twosday Things." Because I like portmanteaus.

Thing #1:
Today, I was talking one-on-one with a student about functions. We were talking about the relationship between domain and range, and how to tell if two sets of values make up the domain and range of a function. We talked about how values in the domain are each assigned to one and only one value in the range by the function. I chimed in with the "mailbox analogy" to further explain the relationship: say you're mailing a bunch of letters. The stack of letters is like the domain, and the houses the letters are being mailed to are like the range. You can mail multiple letters to the same house, but you can't mail the same letter to multiple houses. "So you can't mail the same letter to Chicago, New York, and San Francisco simultaneously," I said to the student.

"Unless it's e-mail," the student replied.

HOLY CRAP. That was a really, really good point! I was utterly stunned that I hadn't thought of that. I guess the analogy kind of breaks down in that regard if you throw e-mail into the mix. I'm still pretty sure I got my point across, but it does have me thinking about the analogy I'm using to describe how functions work. Will this be an outdated analogy in the near future?

Either way, I was super-impressed by my student today.


Thing #2:
Some of my students are currently working on compound inequalities. Below is a piece of student work that I found interesting:

The left side of the compound inequality vanished! I've actually been seeing this happen with several students in my class; every time they get one side of a compound inequality equal to zero, they omit it in the rest of their work.

I've been wondering where this is coming from. I imagine it might have something to do with the fact that students are sometimes taught about the existence of an "implied" zero that isn't actually shown. (For example, what is the slope of the line y = 2? There's no x-term, but there's an implied "0x" in the equation; thus, y = 0x + 2, and the line has a slope of 0.)

Maybe it's coming from somewhere else. I don't think it's anything I've done, but I could be wrong.

Anyway, that's two things from a Tuesday. Maybe I'll try to do this weekly, so I'm blogging more often.

Week 1: Why My First Day Activity Didn't Go At All As I Had Hoped (and Why That's Awesome)

Phew, the first week has come and gone and I found myself utterly exhausted on Friday. Thank goodness for the holiday weekend; I've been able to get more sleep in the past 2-3 days than I have in quite a while.

As part of the kickoff to our school year, I had my seniors work on an "opening day" activity that I lovingly borrowed/blatantly stole from Nadji (who blogs at Physix Coolisms!) that involves grids, writing your name (a lot), and using that data to generalize a pattern.

The activity I snagged is called "What Is Math?" and is described by Nadji from 4:25 to 12:30 of this First Day of School Activities presentation from Global Math Dept. I won't re-post the entire activity here, but basically the aim of the activity is to challenge students' perceptions of what it means to do math.


The Activity
The first part of the activity has students answer the following questions:

1. What is math? What does it mean to you?
2. List 7 mathematical words or phrases that come to mind when doing math.

Often, students respond to these questions with the mindset that "doing math" means working with numbers and calculations and equations.

After answering these questions, students then fill out several square grids by writing the letters of their name over and over again.

After doing that, the students shade the first letter of their first name and then fill out a table to record the patterns that show up.

From there, students are asked to make predictions about the patterns, such as:

• Predict what pattern would appear in a 41x41 grid.
• Predict how the patterns would be affected if the second letter of each name was shaded instead.
• Predict how the patterns would be affected if students started by writing their name in the bottom right corner and filling out the grid backwards.

And so on.

At the end of the activity, students are asked the beginning two questions again; by this point, the hope is that students will start to see that math is much more than just working with numbers and calculations and equations. There is much more to mathematics: finding patterns, making generalizations, predicting unknown events, thinking critically, etc.


How Things Went:
Before I get into this, one side (yes, side, not snide) comment: I had my students fill out grids up to 10x10. If I do this activity again, I might have them go up to 12x12. I have many students with names that are 6, 7, or 8 letters long, and their patterns don't really start to become apparent until the grids get bigger. As I checked in on students and looked through the tables they were filling out, it seemed to me that the "pattern of the patterns" would be more apparent if they had more data. Something to think about for next time.

At any rate, student responses to the opening two questions went pretty much as I expected. Many students came up with responses like "math is the study of numbers," or "math is the tool of Satan," and so forth. The lists of 7 mathematical terms often included "addition, subtraction, multiplication, division, square root, equation, numbers," and the like.

I decided to collect answers to the first two questions via Socrative, so I could quickly generate a bunch of text and then dump them into a Wordle. I thought it would be cool to generate a visual snapshot of student responses from before and after the activity so I could compare.

Here is the "before" Wordle:

As you can see, there's a great deal of "number-ish, calculation-y" stuff. I expected to see this.

Based on what I was seeing from the students as they were working on the activity and the conversations they were having (with each other and with me), I expected to see a dramatically different Wordle from the post-activity responses. After all, they were noticing patterns, making predictions about how patterns would look in grids that were far larger than they had time to fill out, and working together to describe a "rule" for making such a prediction. They weren't really doing "stuff with numbers."

So, here's how the post-activity Wordle turned out:

So uh... um... not really all that different. I mean, "patterns" showed up a lot more in this one, but there was still a bunch of "number-ish, calculation-y" stuff.

I'll admit, at first I was a little bummed that I seemingly hadn't changed very many minds or shifted very many paradigms after doing this activity.

But then I thought about it. And I became okay with it.

In fact, it's actually pretty awesome that I didn't change their minds so easily, and here's why:

This becomes a new challenge for me. This allows me to set a goal for myself. I want my students, by the end of the year, to understand that there's a lot more to mathematics than just crunching numbers and solving numerical problems.

Math is recognizing patterns and trends. Math is making use of those recognitions to make predictions. Math is critical thinking.

Math is art. Math is visual, spatial, tangible.

Math is freaking everywhere and freaking awesome.

I don't get to spend just one day trying to convince my students of this. I get to spend an entire year trying to convince my students how super-cool math is. I have a lot of convincing to do, but that's okay with me. I want to earn it.

That's one thing I learned from doing this activity. That's one thing this activity has given to me: a theme for this year: Math is freaking everywhere and freaking awesome.

It's going to be a great year.

Week Zero: Realizing I Might Actually Know Stuff

It's Week Zero. School Year Eve. The last few days of summer before I get to go back into the classroom and spend the next nine months convincing teenagers that math is freaking awesome.

I'm a teacher mentor this year, which still seems crazy to me because I'm only four years into this profession myself. On Monday, I went to an all-day mentor training session to learn about my role and responsibility as a mentor. A lot of the information was about what I had expected: the mentor wears many different hats, has to build a relationship of trust with the mentee, can learn just as much about teaching from the mentee as the mentee does from them, etc. and so on. We talked about how to have positive conversations with our mentees, how to listen and to provide feedback, and best mentoring practices in general.

We also got toys and candy, which was super cool:

One thing that struck me from the mentor training was what distinguishes a good mentor from a not-so-good mentor: the desire to keep getting better as a teacher. Good mentors know that they still have things to learn about teaching, and no matter what the difference in experience is, they can learn a lot from their mentees. (I'm pretty convinced that I'm going to learn more from my mentee than my mentee is going to learn from me.)

I was reminded of this the next day (Tuesday) when I attended the first-day morning session of new teacher orientation. I sat with my mentee throughout the morning as we introduced ourselves and learned various things about the teacher-mentor program. We had time to talk about the upcoming school year and I was able to answer some questions about curriculum and how we do things at our school.

The experience made me think back to my first Week Zero in our district, when I went through new teacher orientation. I remember feeling excited and nervous about my first year of teaching. I also remember thinking that I was probably going to make a lot of mistakes, I was going to have to learn from them, and there was so so much about teaching that I didn't know yet.

I had the same excited, nervous feeling this week. I still feel like there is so so much about teaching that I don't know. But, in the act of answering my mentee's questions, I was struck by another thought: I actually, maybe, perhaps, do know stuff about teaching now. I had never really thought about it until someone else was asking me. When I was answering my mentee's questions, I really had a lot to say. I had a place of experience to speak from. Holy crap, I have experience. And it might even be useful to someone else.

That might be my important realization from this week: There are many things about teaching I still don't fully know. But I'm also starting to understand how much I do know about teaching. Maybe I'll actually be a decent mentor.

Anyway, back to work! Students come back next week!

Never Be (Fully) Satisfied

The past few days, I've been reflecting on how much time I spent this summer working on writing and tweaking curriculum for the new school year. It's not exactly a new activity for me -- I pretty much write and tweak curriculum every summer -- but I think I probably got more done this summer than I've ever managed to.

I actually fleshed out two different curriculum maps with topics & aligned standards (first attempt at aligning Common Core, so probably lots of mistakes). I'd never made curriculum maps with a great level of detail before, and I'm pretty sure I'm going to be very thankful I did so this summer.

I also spent a lot of time this summer working on incorporating more Problem-Based Learning (PrBL) tasks & lessons into the curriculum (one such idea I had is detailed here; feedback is more than welcome!). I teach at a New Tech Network school, so a rigorous PrBL curriculum is my goal. I've spent hours and hours looking for ideas, researching, thinking, scribbling in my notebook (particularly for those middle-of-the-night ideas), typing pages of details, and probably making my wife very annoyed that I was spending so much time working. I hope the result is that my students do some really awesome, really meaningful learning this year.

Another goal of mine is to learn more about Common Core (I admittedly am still a novice), so when my principal e-mailed the staff earlier this summer about attending a Common Core workshop in September, I was all like "MEMEMEMEMEME!!" So, I'm excited to go, learn some more about Common Core, and hopefully take away valuable knowledge that I can incorporate into my professional practice.

And the idea of improving my professional practice is something I've been thinking about over the past few days.

I've found myself thinking a lot about all the things teachers do to try and improve their teaching. I see many teachers who I follow on Twitter talk about all the conferences they attend and share what they've learned. I have several friends who are enrolled in masters programs, learning more about educational technology, developing curriculum, or otherwise broadening their skill sets as educators. I've thought about the things I've done each summer since I started teaching: working on curriculum, participating in the professional community, working on my own masters, and constantly thinking (and often worrying) about how I can be a better teacher.

And as I thought about all of this, I realized something: I'm not sure I ever want to be satisfied with the kind of teacher that I am.

I'm sure not satisfied with my teaching right now. Frankly, I'm not that great at it. (Sure, I'm funny, handsome, irresistibly charming, and very humble; but from a pedagogical standpoint, those traits can only carry me so far.)

But I don't think I want to ever be fully satisfied with my teaching, not even after I've been teaching for thirty (forty? fifty?) years. Sure, I want to feel happy about my teaching, which I think is a different thing. But not satisfied.

I think it's probably easier to feel this way now, since I'm only going into my fifth year. I know that I have a lot more to learn about teaching. Any fool can see that. There are roughly eleventy billion areas where I can to improve my teaching. I have rather lofty goals for myself this year. I might not meet them all this year, but that just means I'll regroup next summer and try again the following year. And the following year. And the following year. And so on.

But when I've been teaching for a few decades, I don't know how easily I'll still see all of that. I don't know if I'll still be this enthusiastic about improving my craft or if I'll be like, "meh, I've been teaching for thirty (forty? fifty?) years, I'm awesome enough." I don't like that idea. I really hope instead that I'll always want to be a better teacher than I was the year before. Even if it's just a teensy bit better. My students deserve that much, I think.

I talked about this with my wife the other evening. She understood where I was coming from, and noted that this is true about many professions. I mentioned that I was (and am) nervous about meeting my new students on the first day. She said one of her past supervisors once told her that's normal; "that means you care." And my feeling nervous doesn't really stem from being scared about meeting a new group of people, but more from really really wanting to be a better teacher this year than I was last year. I don't want to let these kids down.

Summer is great. It's a time when teachers can work on improving themselves and do what they can to make the next school year better than the last one. I spent a lot of time this summer working on that. I always want to be doing that. I want to be happy with who I am as a teacher. But I also think that I want to never be satisfied. Maybe mostly satisfied. But not fully satisfied.