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!

Coffee Spills! Sales Sheets! Math!

I like to think I have good taste in music. When I was a kid, I played a lot of video games. Video games are super fun. The best part about video games, arguably, is the music. I will always hold the opinion that the Super Nintendo era gave us some of the best video game tunes in the history of ever. EVER.

So, these days I listen to a lot of video game music (VGM) cover bands. One of my current favorites is a recently-formed act, The Returners. They're based in Austin, TX and they totally rock.

But they don't totally rock just because of their music. They totally rock because the band's founder, Lauren Liebowitz, recently helped me out with putting together a math task that involved coffee spills and band shirts.


Background
Over the past couple of years, the math team at my school worked together to put a four-year curriculum in place that's closely aligned to the ACT college readiness standards in mathematics. The skill that my students are currently working on is XEI 602:

"Write expressions, equations, and inequalities for common algebra settings."

We wrote a ton of problems related to each skill. For this particular skill, we wrote problems such as the following:

(And actually, that should be 46 cakes, not 44. Typo. Oops.)

To supplement this skill, I thought of a different way to present this type of problem. Instead of spelling out the necessary mathematical information in a word problem, I wanted to present a more realistic situation and have the students work a little bit more to dig up the mathematics of what was happening.

So I thought of the following scenario: Suppose you were selling a few different items and keeping track of your sales on a sheet, such as this:

And then, suppose you accidentally spilled coffee all over it:

Some of the information is lost! How could we figure out the information that was ruined by the coffee spill? (Obviously, there isn't enough mathematical information in the above example, which is purely for show. But given the right info, this becomes a challenging math task. Also, as it turns out, it's pretty challenging to simulate a coffee spill. And ink is pretty resilient these days.)


The Task
I spent some time thinking about what product(s) to include on the sales sheet that I was going to spill coffee on. One night, I was folding laundry and I came across my official "The Returners" t-shirt. My brain was like, "BAM. T-shirts!"

I messaged Lauren to pretty much say, "Hey, I'm using your band in a math problem!" And she basically replied, "Cool! Can I do anything to help?" And actually involving her hadn't really occurred to me, but she was totally willing to assist and I couldn't turn that chance down.

I put together a "sales sheet" listing three different types of Returners shirts, with information about prices and inventory. Then I spilled coffee on it:

Lauren included the following e-mail, and also sent along a picture of a few of her band's shirts (rolled up neatly into little shirt burritos):

Armed with the above information, my students were set to the task of helping Lauren figure out how many shirts were sold at her most recent show.


How The Task Went
I put the students in groups of three for this task. They were each given a copy of the e-mail, the ruined sales sheet, and the photo of the remaining shirts. I also gave them a worksheet with a few questions that each group member needed to contribute to.

My first period jumped into this task right away. There were a lot of good conversations going on at the start: they were looking through the documents, figuring out what information was important, and talking about how they were going to represent each type of shirt as variables in equations.

There was debate over how to represent "blue" versus "black," since both started with the same letter. Some students decided to use b for one and bl for another, but soon found that there was still no distinction (since both colors start with the letters bl). One student finally suggested using k for black, which certainly helped things.

Students were able to figure out important bits and pieces of information needed to set up an equation to represent the total sales: from the sales sheet, they found the total money made and the cost of each shirt. From the picture, they were able to determine that there were 5 blue shirts and 3 gray shirts that went unsold from the original inventory.

Where we ran into trouble was figuring out how to actually set up equations representing the total sales. Groups during my first period class initially set up their equation as:

5.25k + 4.75b + 4.50g = 397.25

where k represents black shirts, b represents blue shirts, and g represents gray shirts. The above equation was close, but incorrect; students from my first period continued working through the problem using this equation and ended up getting answers that didn't make sense in the context of the problem (e.g. they got non-integer values when they solved for k, b, and g).

They were on the right track, but they didn't account for the fact that a few shirts went unsold; this was kind of the "tricky" part of the task and led to a lot of frustration among my first period students. I let them have some time to try and sort things out on their own and dropped a few hints to try and point them in the right direction. Eventually, I saw it was going to be best to stop and have a quick whole-class discussion about the unsold shirts.

We talked about thinking of k, b, and g as the number of shirts that were originally in the inventory as opposed to the number of shirts that were sold. I asked the students to look through their documents again and tell me what they could find about the number of shirts that were sold and the number of shirts that were still left. We talked about what expressions we should write to represent the number of shirts that were sold. Eventually, we came up with the following:

• The black shirts were sold out, so k black shirts were sold.
• There were 5 blue shirts remaining, so b - 5 blue shirts were sold.
• There were 3 gray shirts remaining, so g - 3 gray shirts were sold.

The students adjusted their equations to reflect an accurate model of the total sales:

After my first period class, I adjusted my lesson plan so that we talked about the correct expressions for shirts sold toward the beginning of the activity. This adjustment made things go more smoothly in my other three classes. Although, in my second period class, students were having trouble with question #2, which required them to re-write their equation from question #1 in terms of one variable. We stopped to have another conversation about what to do.

I asked students to re-read their e-mails and to look specifically for relationships between the different types of shirts and how many of each kind there were. The students noticed the following info:

• There were twice as many black shirts as either the blue or gray shirts.
• The number of blue shirts was the same as the number of gray shirts.

From here, we talked about how to use this information to write a few small equations that would help us with question #2. The students came up with the following equations:

Once we were armed with these equations, we were able to go back to our total sales equation from question #1 and use substitution to re-write it in terms of one variable:

I made another adjustment to my lesson plan for my third and fourth period classes to include a discussion about representing these relationships toward the beginning of the task as well. With this guidance, students were able to successfully determine how many shirts had been sold:

What I Would Change Next Time
This task was given to the students while we were in the middle of our mini-unit on writing equations and expressions based on information from word problems. After doing this task and reflecting on how things went, I think this task has a lot of merit as one of two things:

1. A guided task at the beginning of, or during, the unit, with appropriate scaffolding included; or
2. A performance task at the end of the unit.

My students were resilient and we were able to have a lot of good mathematical conversations during this task. I could also tell there was frustration stemming from confusion about what to do at each problem. I think this might have been the result of not yet having enough practice with the skill, and I also think I didn't provide an appropriate amount of scaffolding, originally. In my later classes, I made adjustments and discussed some important aspects of the problem at the beginning of the task; this seemed to help students successfully solve the problem.

In the future, I would probably embed further scaffolding and questions into this activity. For instance, I would probably ask the students:

• What is the problem you are being asked to solve? What information are you supposed to determine?
• Choose a variable to represent each different type of shirt. Write expressions to represent the amount of each shirt that was sold.
• From the e-mail, what information can you determine about the number of shirts that Lauren originally had? Write equations that represent these relationships.

By having the students doing this work first, the rest of the task would probably go more smoothly as is.

I might also change the prices and the types of merchandise on the sales sheet in the future; a few students looked at the prices and thought, "why didn't they just sell shirts for $5? It'd be easier to make change." In reality, the shirt prices are actually more like that. I made each shirt a different price; otherwise, the task would have been pretty easy to solve. Next time, I'd probably include other merchandise such as CDs, buttons, etc. and let the prices reflect something that would actually be charged at a show.

Overall, I was pleased with this task. I certainly learned a lot, and I hope to come up with even better tasks in the future and I continue trying to figure out how to do PrBL!

"When Am I Ever Going to Use This?" ...Sometimes I Don't Know the Answer

For some reason, the question of "when am I ever going to use this in real life?" seems to pop up at a disproportionately higher rate in math class than in any other subject. I'm willing to bet that's the case.

Do I have any scholarly research or statistics to back up this claim? No.

But I did go to Google and type in the phrase, "when am I ever," to see what popped up:

See? Algebra and Calculus! Obvious proof that this question is asked more in math class than in any other class! And if you aren't convinced by this, then... uh... um... well, then you probably think critically about your info sources and have good judgment.

At any rate, the new school year is around the corner. I've enjoyed the time away from the classroom and have spent many hours mapping curriculum, trying to keep up with the happenings in the MTBoS and preparing some new tasks to try out this year.

As I was reflecting on my first four years of teaching and looking ahead to year five, I kept thinking about what I'm going to do when, inevitably, the question is asked:

"When am I ever going to use this kind of math in real life?"

This question nearly always evokes some kind of emotional reaction from me. One of two types, in fact:

(1) UNADULTERATED, ABSOLUTE JOY, because I have an answer to the question that is totally satisfactory, underscores the relevance of the current mathematical topic, lets me talk about math (which is super-cool because I love talking about math) and helps the kid to see just how motherfreaking awesome math is,

or

(2) MURDEROUS RAGE that someone, who's half my age, who hasn't even learned as much math as I've forgotten, would have the impudence to ask me that question. Not just to ask me that question, but to ask me that question when I have no idea whatsoever how to answer in a way that isn't complete bullcrap.

Okay, I don't actually get mad at students for asking me that question. Or any question. Not ever. I like having curious students. And a job.

I do try my best to be prepared to answer the question of "when am I going to use this?" for any mathematical topic that comes up in my classroom. But sometimes I do feel annoyed when I don't really have a good answer. Not annoyed at the student (not much, anyway), but more at myself for not being prepared with a brilliant, insightful response. After all, I'm the math teacher, right? I should know, in great detail, when the hell someone would ever use an inverse tangent function when they get out into the real world. I should be able to spell out the exact situation in which one would need to know everything there is to know about the latus rectum. (Tee-hee.)

But I don't always know the answer. When that happens -- depending on my mood/how busy I am/what's happening in class/how many cups of coffee I've consumed in the past five minutes -- I tend to go with one of the following responses (I don't recommend using any of these):

• "That's a great question." *flees without saying another word*
• "I can't tell you that; it would ruin the surprise!"
• "'Real life?' Math is real life, son."
• "You know, I find that the best answers in life are the ones we find for ourselves."
• "What're you talking about? I use it all the time!" ("But you're a math teacher," the student replies. "Yep, that's why I use it all the time!" I reply back.)
• "Uh, come see me after class and I'll be happy to talk to you more about your question." (Nobody has ever taken me up on this.)

I feel terrible when I don't have an answer for that question right away. There are times when it seems like it would be easiest just to say, "you know, there's a pretty good chance that you're not actually going to use this; but hey, gotta know it to pass, right?"

I mean, I've actually uttered those words to a student once or twice. I'm not proud of it. At the time, I felt like I was just being straight with the kid(s) who asked. I guess I figured that kids appreciate honesty and have a pretty good nose for B.S. But when I think about it, I realize that what I really did was cheat those students out of a great learning opportunity. I cheated myself as well.

I'm a math teacher, yes, but I'm also a math learner. A lifelong math learner. I shouldn't be ashamed or annoyed when I don't know exactly where or when stuff like hyperbolas or the mean value theorem are used in real life. Instead, I should be seeing an opportunity to learn something new. I should be excited that I've discovered something new to learn about a topic I absolutely love. I should be, like, absolutely jacked that there's stuff about math that I don't know but can find out about for myself. I mean, that's what I'd want my students to do, right?

So this year, I'm going to start using this response (or something similar, still a work in progress):

• "You know what? I'm not quite sure. I know there's a use for [insert mathematical concept here], but I'm still trying to figure that out. I'll try to do some research on it after class. Maybe you could also look it up, and let me know what you find. That would be really helpful."

I'm a math teacher; I shouldn't be shrinking away from that question when a student comes asking. I should be full-on body tackling that thing like it's a quarterback's blind side.

I do think we math teachers try to know where the stuff we teach can be used in the world beyond school; but we won't always know. When we don't know, we need to find out. We need to include our students in the process of finding out, because they asked the question in the first place.

We can't always know, but we can always care. We can always care enough to try and find out. We can always care enough to try and do better.

So this year, I'll try and do better.

Keepin' It Real

I'm going into my fourth year of teaching math at New Tech High @ Zion-Benton East. I love my job. I've discovered that I'm actually, perhaps, maybe, starting to get halfway decent at it. At least there's a chance that I am. There's perpetual room for improvement, and today I wanted to talk briefly about one area I hope to improve this year.

In my first few years of attempting to teach a PBL/PrBL math class, I've come up with some projects that are pretty good at simulating authentic real-world tasks: Creating a different type of Oreo cookie package for Nabisco; creating a scale drawing depicting how furniture should be laid out in a dorm room; and designing a hole for a miniature golf course.

These projects are certainly useful ways to help students see how math can be used for creating and improving products in a real-world context. On the other side of the coin, they fall short when it comes to real-world results. We didn't actually create a real Oreo package for Nabisco; we didn't actually have a real room with real furniture for our scale drawings; we didn't actually build a real mini-golf course.

They were real-world projects without real-world results.

As Dennis Littky probably would put it, I had the students doing "fake real work" instead of "real real work." Something like that.

I want to change that this year. As I continue on my journey of teaching PBL/PrBL math, I believe one of my next steps is to move my students away from the "fake real work" and into the "real real work." I want students to use math to actually create things; to innovate; to predict; to think critically; to affect their community in a positive way.

How do I do this? I haven't completely figured that out. I think I have a good start with the election-themed project idea I blogged about last time. I'm hoping my students can use their experience with this project to learn more about important issues, about making informed decisions based on available data/information, and about making defensible predictions.

A few of my students will even be voting this year; this might really help them learn about being informed voters.

And, because I want my students to produce real-world results, I need them to have a real-world audience. That's why students will be publishing their findings on our class blog (link coming soon) for the community and the rest of the interwebs to see, as well as sharing them with the Obama and Romney campaigns (fingies crossed that they'll actually take a look).

I think that's a good start in my goal to move away from "fake real work" and giving my students the chance to do "real real work."

But I need more. It probably sounds overly ambitious to the point of absurdity, but I want my students to always be using math to become better citizens and to benefit their community. I think the key to this is "real real work." I would love to have 100% of the school year consist of "real real work." (At this point, I'd be thrilled to even get 25% of the school year that way.)

So that's one of my goals this year. I want my students doing "real real work" that has a positive impact beyond the classroom. I'll certainly be scouring and engaging the blogosphere, Twittersphere, and meatspace for ways to accomplish this.