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!

Taking a Teaching Mulligan

Sometimes, despite trying to do my best job possible as a teacher, I screw up. I'm pretty sure it's healthy to accept that it happens from time to time.

A few weeks ago, my students took a quiz that pretty much nobody did well on. Like, not even really that close. (I'm not going to go into what the subject matter was or how my lessons were designed or what scaffolding I did -- that doesn't really pertain to the message of my post today.)

Needless to say, this elicited an emotional reaction from me. I actually had to stop grading and walk away for a few minutes because I was feeling a mix of sadness and anger all at once. I reasonably sure that I was uttering curse words under my breath after I came back and continued grading.

I think, unfortunately, there are some teachers who probably would have taken that anger and directed it at their students the next class period. I've seen teachers get absolutely pissed off at their students for doing terribly on a quiz or a test as a whole group.

I'm not one of those teachers. When students don't perform well on an assessment, I blame myself. I blame myself pretty hard, actually. Maybe more than I should. I guess I can't help it.

This happened on a Friday afternoon. I thought about what to do all weekend. I came back to my students on Monday and, in each class, just laid it out for them:

"Guys, nobody did well on this quiz. I'm sorry. I blame myself for that. When nobody does well, that tells me that I probably did something wrong with my teaching. So, I'm not going to include these quizzes in your grade for now. We'll come back to it next week, I'll try to teach differently, and we'll re-take this quiz. Does that sound fair?"

And it sounded fair to everyone.

I imagine part of why my students were amenable to this is because many of them sensed that they hadn't done well. I bet many of them were afraid they'd let me down, or that I was going to be mad at them for failing one silly math quiz. They probably don't know that, when a class bombs an assessment, the first question I always ask myself is, "what did I do wrong?"

Stuff like this is a humbling reminder that, even though I work hard and try my very best as an educator, there will be times where I come up short. I try to keep those instances few and far between, but from time to time it will happen. When it does, I think the right thing is to give my students a second chance -- or, more accurately, ask my students to give me a second chance.

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.

Reflections From #precalcchat: Pre-Calculus Sequencing

I love Twitter chats with other teachers. It's a great way to make connections. It's a great way to get insight, ideas, and resources. It's also a fantastic opportunity to reflect on your own practice and to improve what you're doing in the classroom.

The Global Math Department hosts several weekly Twitter chats for math teachers on a variety of topics. Since I teach Pre-Calculus, I dropped in on the first #precalcchat of the school year last week; thanks to Mimi (I Hope This Old Train Breaks Down...) and Taoufik Nadji for hosting. Couldn't spend much time, but the topic of conversation captured my interest:

I loved that thought. It made me stop and think about how I sequence my Pre-Calculus course and why.

I start with Graphing and Functions first. To me, it's important for students to understand the basics of interpreting graphs of functions and becoming fluent with moving between different representations of a function (graphs, tables, equations). I find this to be a particularly vital theme that I want to drive home with my students, especially those who will be going on to AP Calculus or Calculus I/II in college.

Next, I follow a pretty standard sequence of Quadratics/Polynomials, Rational Functions, Exponential Functions, Logarithmic Functions, Trigonometric Functions, and Analytical Trigonometry. Again, I focus on these topics in particular to prepare my students for success in an AP Calculus course. Other topics such as Analytical Geometry, Series & Sequences, Polar Systems of Coordinates, Conics, etc. come afterward as time allows.

The other chatters all had brilliant things to say, so naturally I felt like I'm probably doing everything wrong (or maybe just some things wrong, and other things not-as-wrong).

When discussing how Pre-Calculus can seem like a re-teaching of Algebra II to students, Tina C (Drawing On Math) mentioned that her school starts with Trigonometry for that exact reason.

This was an interesting idea to quite a few of us: do Trig first semester, slowly build up conceptual understanding of the unit circle, graphing, transformations, identities, etc. Then, move into the other different functions second semester.

The more I think about doing Trig first, the more appealing it seems to me. I've always found that I never seem to have enough time to really properly teach Trig and I need to either rush a few things or cut some other stuff out. I think I probably always had the notion that Trig is "more difficult," and somehow it made sense to put the "harder stuff" at the end of the year. (That's excellent reasoning, isn't it?)

But really though, Trig is a bit of a stand-alone topic. It could go anywhere in the course sequence. There are certainly some underlying concepts that can be applied to other functions: graphing, transformations, moving fluently between representations, and so on. I usually think of these concepts as having to be taught and mastered before doing Trig, as if Trig is the "CHALLENGE MODE" of working with functions in Pre-Calculus.

Who's to say we can't use Trig to teach these concepts instead? Maybe my students would have greater success with Trig if I did it at the beginning of the year, built the concepts slowly with appropriate scaffolding, while still equipping students to be successful in working with other functions. I may have to try it out one of these years. (I already have this year mapped out -- maybe next year?)

Anyway, some great food for thought.

I'm looking forward to more of these chats this school year, and hopefully I'll find time to continue blogging & reflecting on what I take away from them.