Thursday, August 28, 2014

A Brief Look at Phrasing Descriptions of Functions

During a lesson this morning, students were asked to describe this function rule in their own words:

One student came up with this:

We had a brief group discussion about whether this phrase made sense or not. One student said that they might interpret this phrase differently:

Some students agreed, saying that the wording could possibly suggest that the sum was performed first, and then squared after. I asked them to think of a better way to say what was trying to be said, and they came up with this:

Not terribly different, but the order of the words made a lot of difference. The students agreed that this phrasing was clearer than the original.

Admittedly, this was a pretty simple problem, but the students brought up some good points in our conversation around it. It was a good opportunity to explore the nuances of being precise when talking about mathematics. Even one turn of phrase can be misleading; hopefully the students learned a bit about being careful about their wording while still being succinct.

Wednesday, August 27, 2014

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!

Friday, December 20, 2013

Leaving It All On The Field

As far as the school year goes, winter break is halftime. I'm exhausted. I feel like I'm going into the locker room having left it all on the field.

After what felt like 12,383,908,786,358,213 years (give or take a few), the long wait for winter break is finally over. In a few minutes, the dismissal bell will ring and we'll all be running out the door for a glorious two weeks away from work.

I've been feeling that really prolonged kind of tired these past couple of weeks. The kind of tired that teachers feel after a few solid months of establishing classroom routines, figuring out the best ways to help students learn, continually assessing & giving feedback, communicating with parents, collaborating with other teachers, and getting involved with other facets of the school community behind the scenes.

It's been a crazy-busy first semester for me, and I haven't had time to blog about all of it. I'm on a district-level committee working on developing a new evaluation tool for the teachers in our district, necessary because of changes made to our state's laws. I've given up most of my lunches for Anime Club and to help students with re-taking quizzes, catching up on missing work, or getting extra help. For the past month, I've also been involved in a secret project (shh!) that I'm pretty excited to be a part of. I also had the pleasure of bringing my old math ed professor from MSU into my classroom to check out what I've been doing; and in turn, I had the pleasure of being on a guest teacher panel for his class of up-and-coming teacher candidates. We had great conversations about how to reach students and build relationships, and it was great to see him in action with a few of my own students. I'm still learning from him.

As the calendar year winds down, I find myself thankful for a break. I'm looking forward to spending time with family and friends, going home for Christmas, celebrating the new year with my wife, and getting the chance to relax and re-energize.

To all of my teacher friends, colleagues, and acquaintances, I wish all of you a Merry Christmas and a Happy New Year!