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!
Education, math, project-based learning, other stuff, and possibly ninjas.
Friday, December 20, 2013
Tuesday, November 12, 2013
Twosday Things: Ingenious Responses. Also Fish.
Time again for Twosday Things!
Thing #1:
The other day, I stepped out of my classroom for a moment. When I came back, one of my students had drawn this on the board:
I took one look and figured, "what the hell, I'll tweet it." So I did:
One reply stated that this was probably a reference to Fairly Oddparents, which given the age of my current students wouldn't surprise me.
However, the prize for Most Brilliantly Mathematical Response definitely went to Gregory Taylor (@mathtans on Twitter):
I feel like if I'd gotten that kind of response from a student, I'd have just given them an A for the semester right then and there. (Okay, maybe not. But I'd be impressed.)
Thing #2:
One thing I've noticed about my teaching practice this year is that I've become more open-minded with how students respond to questions and problems.
Here's an example of what I mean. One of my students came to me today with the following solution to a problem:
Two disclaimers: (1) The student obviously took some "mathematical liberties" when drawing this diagram. (2) The student did much of their work without a calculator, but explained to me in person what was done: he used the distance formula to calculate the length of each side, then used the Pythagorean Theorem to see whether the three sides formed the sides of a right triangle.
Out of context, this seems like a perfectly reasonable way to solve to problem.
However, this actually came from a problem set focused on parallel and perpendicular lines. The solution path I was "looking for" was to calculate the slope between each pair of vertices and determine if there were two sides that were perpendicular to each other.
What's my point here?
A year or two ago, this is probably how I would have responded to the student's work: "Um... well, that's ONE way to solve it I guess, but I was really looking for [insert what I was looking for]."
But today, this is how I responded: "Whoa, that's brilliant! I hadn't actually thought of solving the problem that way, but that makes a lot of sense! This is genius!" And I followed that up with an explanation of how most other students were solving the problem by calculating slopes as I described above; but the student's mathematical reasoning was both valid and awesome.
This is a great example of how I've changed as a teacher this year. I've always been okay with students coming up with different solution paths to problems; however, I often tried to steer them toward particular solution paths, even if what my students were doing was perfectly reasonable.
Insisting on particular solutions paths isn't, in and of itself, a bad thing. There are situations where it's good to train students on solving a problem a particular way; doing so adds to their "mathematical toolbox," equipping them with a variety of skills for solving problems.
But there are times, I think, when we as math teachers need to be okay with students solving problems in unexpected ways. I think this instance was one of those times. This was a student who had been struggling with math at times this year, but today he came to me with a brilliant solution that I wasn't expecting to see. That deserved praise and recognition.
As I said, a year or two ago, I would have been "just okay" with the method my student used to solve the problem, but not all that enthusiastic because he hadn't done it the way I was trying to teach.
I shudder to think that, just a year or two ago, I wouldn't have embraced his work as enthusiastically as I did today. If I had responded with, "Well, that's one way to do it, but...", I probably would have done harm to the student's mathematical confidence. He applied previously-learned mathematical knowledge to a different type of problem. How could I have any problem with that?
Thing #1:
The other day, I stepped out of my classroom for a moment. When I came back, one of my students had drawn this on the board:
I took one look and figured, "what the hell, I'll tweet it." So I did:
One reply stated that this was probably a reference to Fairly Oddparents, which given the age of my current students wouldn't surprise me.
However, the prize for Most Brilliantly Mathematical Response definitely went to Gregory Taylor (@mathtans on Twitter):
I feel like if I'd gotten that kind of response from a student, I'd have just given them an A for the semester right then and there. (Okay, maybe not. But I'd be impressed.)
Thing #2:
One thing I've noticed about my teaching practice this year is that I've become more open-minded with how students respond to questions and problems.
Here's an example of what I mean. One of my students came to me today with the following solution to a problem:
Two disclaimers: (1) The student obviously took some "mathematical liberties" when drawing this diagram. (2) The student did much of their work without a calculator, but explained to me in person what was done: he used the distance formula to calculate the length of each side, then used the Pythagorean Theorem to see whether the three sides formed the sides of a right triangle.
Out of context, this seems like a perfectly reasonable way to solve to problem.
However, this actually came from a problem set focused on parallel and perpendicular lines. The solution path I was "looking for" was to calculate the slope between each pair of vertices and determine if there were two sides that were perpendicular to each other.
What's my point here?
A year or two ago, this is probably how I would have responded to the student's work: "Um... well, that's ONE way to solve it I guess, but I was really looking for [insert what I was looking for]."
But today, this is how I responded: "Whoa, that's brilliant! I hadn't actually thought of solving the problem that way, but that makes a lot of sense! This is genius!" And I followed that up with an explanation of how most other students were solving the problem by calculating slopes as I described above; but the student's mathematical reasoning was both valid and awesome.
This is a great example of how I've changed as a teacher this year. I've always been okay with students coming up with different solution paths to problems; however, I often tried to steer them toward particular solution paths, even if what my students were doing was perfectly reasonable.
Insisting on particular solutions paths isn't, in and of itself, a bad thing. There are situations where it's good to train students on solving a problem a particular way; doing so adds to their "mathematical toolbox," equipping them with a variety of skills for solving problems.
But there are times, I think, when we as math teachers need to be okay with students solving problems in unexpected ways. I think this instance was one of those times. This was a student who had been struggling with math at times this year, but today he came to me with a brilliant solution that I wasn't expecting to see. That deserved praise and recognition.
As I said, a year or two ago, I would have been "just okay" with the method my student used to solve the problem, but not all that enthusiastic because he hadn't done it the way I was trying to teach.
I shudder to think that, just a year or two ago, I wouldn't have embraced his work as enthusiastically as I did today. If I had responded with, "Well, that's one way to do it, but...", I probably would have done harm to the student's mathematical confidence. He applied previously-learned mathematical knowledge to a different type of problem. How could I have any problem with that?
Tuesday, November 5, 2013
Twosday Things: Solution Methods and Student Initiative
It's Tuesday, which means it's once again time for Twosday Things!
In my dauntless endeavor to blog regularly, I am continuing to write about two (big or small, mostly small) things that happened in my teaching world over the previous week. This makes the third week in a row. Not bad.
Before you read on, be sure to open up Geoff Krall's awesome PrBL starter kit in a new tab; this should be your next bit of reading after you're done here. You're welcome!
Thing #1:
One of my students (we'll call her Susie) was having trouble working through the following problem:
"Find the value of two numbers if half the larger number plus two equals the smaller number and their sum is 44."
Setting up a system of equations to represent the problem wasn't terribly much of an issue; Susie was able to do this on her own with a bit of questioning from me to prompt her thinking.
After we had set up the system, Susie seemed stuck on what to do next (though we had been working on systems of equations all week and I'd seen her succeed in completing similar problems).
Before I even said anything, another student (let's call her Nadia) offered to help explain what to do next, and I gladly obliged. Nadia used the elimination method to solve the problem while explaining her steps to Susie:
After Nadia finished, Susie seemed confused. She understood that 28 and 16 had to be the numbers described in the problem, but she wasn't clear on the elimination method that Nadia had used. "I actually thought you were supposed to plug the equation for b into the second equation," she said. Susie proceeded to solve the problem using the substitution method:
I found what happened next to be interesting: Nadia seemed confused about the method that Susie had used to get her answer, even though they both came up with the same thing! I said to them, "so Susie, it sounds like you were confused when Nadia used elimination to solve this problem, and Nadia, it sounds like you were confused when Susie used substitution." We talked about it, and both girls said the methods they each used just made more sense to them. I stressed to them that it was important to understand both of these methods (as well as solving by graphing), but also that it was great to see that each of them had their own way of figuring out this problem. As has happened in my class before, students are seeing there can be more than one path to a solution.
Thing #2:
The above situation touches on something that has been developing among my students in my classes over the past few weeks: they're starting to regularly help each other out on their own.
Stuff like the above has been happening with greater regularity in my classes. It's happened faster among my advanced students, but my other students are starting to do it as well.
A lot of classroom time is spent allowing the students to work through problems at their own pace, with me providing one-on-one or small-group assistance as needed. This can be challenging to manage, particularly with having my "regular" and "advanced" classes in my room at the same time every period.
That's part of the reason why I love it when students start to take the initiative and help each other out. I also love this kind of initiative because it's so important for students to be able to take agency of their own learning. It's an important life skill (at least I think so).
One of my classes has really figured this out. Every day, they come in and they all get with their usual groups (I did no grouping; they formed these groups on their own and they work really well). They figure out what their assignments are. If I don't have a new workshop or learning module for them, they get to work and ask me for help whenever they have questions. They're also getting very good at helping each other out, checking each other's work, and asking each other questions (Thing #1, above, happened with this group).
When students figure out how to take charge of their own learning, good things happen. One student (let's dub this one Marie) had been struggling all year with math. Last week, the small group of friends Marie regularly works with really focused on helping her understand how to solve systems of equations. They were able to give her a greater amount of attention and assistance than I was able to by myself. After a while, Marie started to be able to solve systems of equations on her own; she even got so excited about getting a problem right, that she wanted to do it on the board! AWESOME!
It's not like this every day, and it's not like this in every class. But it's starting to happen more, and it's great to have one class that's really taken off with helping each other out.
In my dauntless endeavor to blog regularly, I am continuing to write about two (big or small, mostly small) things that happened in my teaching world over the previous week. This makes the third week in a row. Not bad.
Before you read on, be sure to open up Geoff Krall's awesome PrBL starter kit in a new tab; this should be your next bit of reading after you're done here. You're welcome!
Thing #1:
One of my students (we'll call her Susie) was having trouble working through the following problem:
"Find the value of two numbers if half the larger number plus two equals the smaller number and their sum is 44."
Setting up a system of equations to represent the problem wasn't terribly much of an issue; Susie was able to do this on her own with a bit of questioning from me to prompt her thinking.
After we had set up the system, Susie seemed stuck on what to do next (though we had been working on systems of equations all week and I'd seen her succeed in completing similar problems).
Before I even said anything, another student (let's call her Nadia) offered to help explain what to do next, and I gladly obliged. Nadia used the elimination method to solve the problem while explaining her steps to Susie:
After Nadia finished, Susie seemed confused. She understood that 28 and 16 had to be the numbers described in the problem, but she wasn't clear on the elimination method that Nadia had used. "I actually thought you were supposed to plug the equation for b into the second equation," she said. Susie proceeded to solve the problem using the substitution method:
I found what happened next to be interesting: Nadia seemed confused about the method that Susie had used to get her answer, even though they both came up with the same thing! I said to them, "so Susie, it sounds like you were confused when Nadia used elimination to solve this problem, and Nadia, it sounds like you were confused when Susie used substitution." We talked about it, and both girls said the methods they each used just made more sense to them. I stressed to them that it was important to understand both of these methods (as well as solving by graphing), but also that it was great to see that each of them had their own way of figuring out this problem. As has happened in my class before, students are seeing there can be more than one path to a solution.
Thing #2:
The above situation touches on something that has been developing among my students in my classes over the past few weeks: they're starting to regularly help each other out on their own.
Stuff like the above has been happening with greater regularity in my classes. It's happened faster among my advanced students, but my other students are starting to do it as well.
A lot of classroom time is spent allowing the students to work through problems at their own pace, with me providing one-on-one or small-group assistance as needed. This can be challenging to manage, particularly with having my "regular" and "advanced" classes in my room at the same time every period.
That's part of the reason why I love it when students start to take the initiative and help each other out. I also love this kind of initiative because it's so important for students to be able to take agency of their own learning. It's an important life skill (at least I think so).
One of my classes has really figured this out. Every day, they come in and they all get with their usual groups (I did no grouping; they formed these groups on their own and they work really well). They figure out what their assignments are. If I don't have a new workshop or learning module for them, they get to work and ask me for help whenever they have questions. They're also getting very good at helping each other out, checking each other's work, and asking each other questions (Thing #1, above, happened with this group).
When students figure out how to take charge of their own learning, good things happen. One student (let's dub this one Marie) had been struggling all year with math. Last week, the small group of friends Marie regularly works with really focused on helping her understand how to solve systems of equations. They were able to give her a greater amount of attention and assistance than I was able to by myself. After a while, Marie started to be able to solve systems of equations on her own; she even got so excited about getting a problem right, that she wanted to do it on the board! AWESOME!
It's not like this every day, and it's not like this in every class. But it's starting to happen more, and it's great to have one class that's really taken off with helping each other out.
Tuesday, October 29, 2013
Twosday Things: Hearts, Stars, Messy Numbers
Time again for Twosday Things!
Taking a cue from last Tuesday's post, I'll discuss two teaching-related things (however big or small) that happened over the past week. I'm trying to post about two things every Tuesday throughout the school year (hence the title, "Twosday Things"). This makes the second week in a row; so far, so good.
Thing #1:
Something I've noticed that happens A LOT in my class:
This is a near-daily occurrence in my class, despite my frequent insistence that "decimals are numbers, too!" ("Fractions are numbers, too!" is similarly used often.) I cannot even count the number of times this happens in a school year.
How does this happen? How do our students reach the point where they automatically assume that "decimal answers" must be wrong? How do we let them get to high school with this assumption cemented into their mathematical psyche?
Yesterday, I took this question to my Twitter feed:
Some super-awesome math-types from the Twittersphere chimed in with their thoughts on the topic:
"Give them messiness." I love that. I feel like our students need more practice and earlier exposure to "messy numbers," because real-world math is messy and complex. Students need to learn that decimals, fractions, irrationals, etc. are all numbers, too.
At the same time, I don't think it's inherently bad that students question their answers every time they get something "messy." Sometimes (often, in fact), their answer actually is the result of a mathematical mistake, and they need to be able to figure out where the mistake was made.
I can see some potentially good habits here: stopping to think about whether the answer makes sense in the context of the problem; double-checking work for mathematical mistakes; and so forth. I just don't think that "getting a messy answer" should be the sole reason a student thinks they did something wrong. If anything, students should be trained to question "messy" answers and "clean" answers. Students should be in the habit of doubling back and re-checking their work to make sure their reasoning makes sense.
Maybe the mistrust in "messy" numbers can be a good thing; but if it is, it needs to be applied to all numbers. Equal opportunity, darn it!
Thing #2:
Today in class, I had a few students who asked for help with the following problem:
We discussed the fact that the problem mentioned "two numbers." We had no idea what those two numbers were, offhand. But, we had enough information to be able to set up a couple of equations. We just needed to pick two variables to represent the numbers first.
"We can call these two numbers anything we want," I said. "We can call them x and y. We can call them a and b, or c and d. We could even call them stuff like, 'dollar sign' and smiley face.' What do you want to call these two numbers?"
One of my students said, "heart and star."
Math, learning, and hilarity ensued:
I had a terrible time keeping a straight face, especially when I said things like, "so what expression do we plug in for heart?" or "yep, we have to simplify by combining our star terms, so star plus eight equals twenty-four," or "there we go, star equals sixteen and heart equals forty."
It was a fun little way to talk about the concept of representing unknown values with variables. Why settle for boring old x and y when you can have a bit of fun?
Taking a cue from last Tuesday's post, I'll discuss two teaching-related things (however big or small) that happened over the past week. I'm trying to post about two things every Tuesday throughout the school year (hence the title, "Twosday Things"). This makes the second week in a row; so far, so good.
Thing #1:
Something I've noticed that happens A LOT in my class:
- Student is working through a (typically algebraic) problem.
- Student gets a non-integer answer (i.e. a "decimal answer").
- Student immediately assumes they must be wrong. Often accompanied by asking the teacher, "am I supposed to get a decimal for my answer?"
This is a near-daily occurrence in my class, despite my frequent insistence that "decimals are numbers, too!" ("Fractions are numbers, too!" is similarly used often.) I cannot even count the number of times this happens in a school year.
How does this happen? How do our students reach the point where they automatically assume that "decimal answers" must be wrong? How do we let them get to high school with this assumption cemented into their mathematical psyche?
Yesterday, I took this question to my Twitter feed:
Some super-awesome math-types from the Twittersphere chimed in with their thoughts on the topic:
"Give them messiness." I love that. I feel like our students need more practice and earlier exposure to "messy numbers," because real-world math is messy and complex. Students need to learn that decimals, fractions, irrationals, etc. are all numbers, too.
At the same time, I don't think it's inherently bad that students question their answers every time they get something "messy." Sometimes (often, in fact), their answer actually is the result of a mathematical mistake, and they need to be able to figure out where the mistake was made.
I can see some potentially good habits here: stopping to think about whether the answer makes sense in the context of the problem; double-checking work for mathematical mistakes; and so forth. I just don't think that "getting a messy answer" should be the sole reason a student thinks they did something wrong. If anything, students should be trained to question "messy" answers and "clean" answers. Students should be in the habit of doubling back and re-checking their work to make sure their reasoning makes sense.
Maybe the mistrust in "messy" numbers can be a good thing; but if it is, it needs to be applied to all numbers. Equal opportunity, darn it!
Thing #2:
Today in class, I had a few students who asked for help with the following problem:
"We can call these two numbers anything we want," I said. "We can call them x and y. We can call them a and b, or c and d. We could even call them stuff like, 'dollar sign' and smiley face.' What do you want to call these two numbers?"
One of my students said, "heart and star."
Math, learning, and hilarity ensued:
I had a terrible time keeping a straight face, especially when I said things like, "so what expression do we plug in for heart?" or "yep, we have to simplify by combining our star terms, so star plus eight equals twenty-four," or "there we go, star equals sixteen and heart equals forty."
It was a fun little way to talk about the concept of representing unknown values with variables. Why settle for boring old x and y when you can have a bit of fun?
Friday, October 25, 2013
For Posterity: An Awesome Teaching Day
There are about 180 days in a school year. Multiply that by the number of years you've been teaching. That's a crap-ton of days.
Many of them are good, or just okay. Many of them leave you wondering whether or not you're any good at teaching at all. A few of them turn out to be terrible.
But once in a while, you have a day that makes living through all of those other days completely worth it. For me, yesterday was one of those days.
Last night, I was recognized by our district's Board of Education for "contributions to the school and the community." It was a pretty cool honor.
I was already having an awesome day. I finished grading quizzes that my students had taken this week. They did well overall, but I was particularly impressed by many of my students who had been struggling. Those students have been making a point of coming to me for help, participating more in class, and generally just turning things around. They've been busting their butts, and their work is paying off. It's thrilling to see those students experience success in my class; there have been smiles and high fives all around.
In the evening, I attended the Board meeting to be recognized along with a few other students and staff from our district. When my name was called, I went up to the front to receive my framed certificate:
After my principal said a few things about me, I had a moment to shake the hands of every Board member.
One of them, whose son graduated a couple of years ago (and whom I had taught for two years) stopped me for a moment to tell me: "I want you to know, my son has decided to switch his major to math because of you."
Wow. I mean, just wow. That pretty much describes what I felt at that moment. I didn't really know what to say. I think I almost started crying, or at least got a bit teary. It was just so awesome to hear those words.
That news, that moment, meant a hundred times more to me than any award or certificate. The feels. The feels.
As a teacher, when you have students, you don't always know whether or not you've having any kind of impact on their lives. Most of the time, I feel like I have no idea whether or not I'm having any impact. I guess sometimes you don't find out for sure until after those students graduate and move on. But however long it takes, when you do find out you had a positive impact on a student, it's completely worth it. Nothing else really matters.
Somewhere down the line, I'm going to have a terrible day and I'll need something to lift my spirits; that's why I had to write about my awesome day. I'll always have this to come back to when I need to be reminded why teaching is so rewarding and worth everything.
Many of them are good, or just okay. Many of them leave you wondering whether or not you're any good at teaching at all. A few of them turn out to be terrible.
But once in a while, you have a day that makes living through all of those other days completely worth it. For me, yesterday was one of those days.
Last night, I was recognized by our district's Board of Education for "contributions to the school and the community." It was a pretty cool honor.
I was already having an awesome day. I finished grading quizzes that my students had taken this week. They did well overall, but I was particularly impressed by many of my students who had been struggling. Those students have been making a point of coming to me for help, participating more in class, and generally just turning things around. They've been busting their butts, and their work is paying off. It's thrilling to see those students experience success in my class; there have been smiles and high fives all around.
In the evening, I attended the Board meeting to be recognized along with a few other students and staff from our district. When my name was called, I went up to the front to receive my framed certificate:
After my principal said a few things about me, I had a moment to shake the hands of every Board member.
One of them, whose son graduated a couple of years ago (and whom I had taught for two years) stopped me for a moment to tell me: "I want you to know, my son has decided to switch his major to math because of you."
Wow. I mean, just wow. That pretty much describes what I felt at that moment. I didn't really know what to say. I think I almost started crying, or at least got a bit teary. It was just so awesome to hear those words.
That news, that moment, meant a hundred times more to me than any award or certificate. The feels. The feels.
As a teacher, when you have students, you don't always know whether or not you've having any kind of impact on their lives. Most of the time, I feel like I have no idea whether or not I'm having any impact. I guess sometimes you don't find out for sure until after those students graduate and move on. But however long it takes, when you do find out you had a positive impact on a student, it's completely worth it. Nothing else really matters.
Somewhere down the line, I'm going to have a terrible day and I'll need something to lift my spirits; that's why I had to write about my awesome day. I'll always have this to come back to when I need to be reminded why teaching is so rewarding and worth everything.
Tuesday, October 22, 2013
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.
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.
Tuesday, October 8, 2013
Multiple Solutions (A follow-up to "When Is the Right Answer the Right Answer?")
A couple of weeks ago, I wrote this post about how I wanted my students to determine equations of lines, given certain information. The broader point, I think, was realizing that my students had more than one option for determining answers to the problems they were working on, and being okay with that. (Why wouldn't I be?)
I had another "when is the right answer the right answer?" moment in class yesterday that I thought was really super-cool.
Two students were working together on the same problem. They came up with what they thought were different answers, so they were wondering who was correct. Their work is shown below:
So both students used point-slope form for their equations, and came up with two answers that looked different. This peculiarity made them wonder who was right and who was wrong. (Which, in turn, makes me realize that I still have a lot of work to do with teaching them about making sense versus being right.) They called me over to ask me who had the correct equation.
I must have been really busy at that moment and not really thinking, because I looked at their answers and said, "actually, you're both right." Not that I was wrong in saying so; but I regret that I didn't recognize the teachable moment that had presented itself. This would have been a great opportunity to ask each of them what they thought about their equations, how they came up with them, why they thought their answers made sense, why the other person got something different, and whether or not it made a difference which point they used for point-slope form. Still, it was a really cool moment: two students have a spirited debate over who had the "right" equation, when really they were both right. It was my favorite moment of class from yesterday.
Fortunately, the same thing happened today, on the same problem, with the same work as shown above, between a different pair of students. Grateful for a second chance, I was able to stop and facilitate an awesome math discussion between the two of them.
One student was adamant that the "first" point, (-4, 3), had to be plugged in for point-slope form instead of the "second" point, "because they're X1 and Y1," she reasoned. She said this because she had labeled the coordinates as such when using the slope formula to determine the slope:
And point-slope form was written on the board as Y - Y1 = (X - X1). So I could see where she was coming from.
I asked her, "so, how would you label these points if the order was swapped?" In other words, what if the problem listed the points "(6, 1) and (-4, 3)" instead of the order they were given? She responded that she would have labeled (6, 1) as (X1, Y1) and (-4, 3) as (X2, Y2).
My next question was, "So would that change things? Would you get a different slope, for instance?" The student initially thought that yes, she would get a different slope. The other student, who was working with her, said that the slope should be the same. I had both of them determine the slope of the line with the different designations for the coordinates; naturally, the slopes turned out to be the same as in their original work.
I asked, "how did changing the order of the points affect the slope?" The student replied that the order of the points didn't change the slope at all. "Cool," I said. "So what about the two different equations you guys came up with? What difference does choosing one point over the other [when plugging a point into point-slope form] make?" The first student still wasn't quite convinced that it didn't matter what point she chose; her partner said it didn't matter what point was chosen for the point-slope form of the equation.
We decided to have each of them solve their equations for y, so they'd both be in slope-intercept form. When they did so, they came up with the same equation, and the first student was finally convinced that it didn't matter which of the two points she chose. Both students were convinced that they'd both determined correct equations for the line described in the problem. "Why doesn't it matter which point you choose?" I asked. The first student wasn't quite sure. The second student guessed, "because both points are on the same line?" I replied, "that sounds like it makes sense."
I love when students find different (yet equally valid) solutions to problems like this. It makes for some great discussion. I need to keep myself aware that it's more important to ask my students to make sense of their work instead of telling them that they're right; I missed out on having a great conversation with two students yesterday, but I'm glad I had another chance at it today.
I had another "when is the right answer the right answer?" moment in class yesterday that I thought was really super-cool.
Two students were working together on the same problem. They came up with what they thought were different answers, so they were wondering who was correct. Their work is shown below:
So both students used point-slope form for their equations, and came up with two answers that looked different. This peculiarity made them wonder who was right and who was wrong. (Which, in turn, makes me realize that I still have a lot of work to do with teaching them about making sense versus being right.) They called me over to ask me who had the correct equation.
I must have been really busy at that moment and not really thinking, because I looked at their answers and said, "actually, you're both right." Not that I was wrong in saying so; but I regret that I didn't recognize the teachable moment that had presented itself. This would have been a great opportunity to ask each of them what they thought about their equations, how they came up with them, why they thought their answers made sense, why the other person got something different, and whether or not it made a difference which point they used for point-slope form. Still, it was a really cool moment: two students have a spirited debate over who had the "right" equation, when really they were both right. It was my favorite moment of class from yesterday.
Fortunately, the same thing happened today, on the same problem, with the same work as shown above, between a different pair of students. Grateful for a second chance, I was able to stop and facilitate an awesome math discussion between the two of them.
One student was adamant that the "first" point, (-4, 3), had to be plugged in for point-slope form instead of the "second" point, "because they're X1 and Y1," she reasoned. She said this because she had labeled the coordinates as such when using the slope formula to determine the slope:
And point-slope form was written on the board as Y - Y1 = (X - X1). So I could see where she was coming from.
I asked her, "so, how would you label these points if the order was swapped?" In other words, what if the problem listed the points "(6, 1) and (-4, 3)" instead of the order they were given? She responded that she would have labeled (6, 1) as (X1, Y1) and (-4, 3) as (X2, Y2).
My next question was, "So would that change things? Would you get a different slope, for instance?" The student initially thought that yes, she would get a different slope. The other student, who was working with her, said that the slope should be the same. I had both of them determine the slope of the line with the different designations for the coordinates; naturally, the slopes turned out to be the same as in their original work.
I asked, "how did changing the order of the points affect the slope?" The student replied that the order of the points didn't change the slope at all. "Cool," I said. "So what about the two different equations you guys came up with? What difference does choosing one point over the other [when plugging a point into point-slope form] make?" The first student still wasn't quite convinced that it didn't matter what point she chose; her partner said it didn't matter what point was chosen for the point-slope form of the equation.
We decided to have each of them solve their equations for y, so they'd both be in slope-intercept form. When they did so, they came up with the same equation, and the first student was finally convinced that it didn't matter which of the two points she chose. Both students were convinced that they'd both determined correct equations for the line described in the problem. "Why doesn't it matter which point you choose?" I asked. The first student wasn't quite sure. The second student guessed, "because both points are on the same line?" I replied, "that sounds like it makes sense."
I love when students find different (yet equally valid) solutions to problems like this. It makes for some great discussion. I need to keep myself aware that it's more important to ask my students to make sense of their work instead of telling them that they're right; I missed out on having a great conversation with two students yesterday, but I'm glad I had another chance at it today.
Monday, October 7, 2013
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.
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.
Sunday, September 29, 2013
When Is the Right Answer the Right Answer?
This week, my students have been working on determining equations of a line based on properties of parallel and perpendicular lines (GRE 604 from the ACT College Readiness Standards for Mathematics), which involves problems like this one:
Several concepts popped up throughout the week while working on this skill: determining slope, slope-intercept form, point-slope form, and the relationships of slopes between lines that are either parallel or perpendicular to each other.
Throughout the week, I have been insisting that my students give their solutions to these problems in slope-intercept form, as shown in this student's work:
Perfectly reasonable solution method, isn't it? Put the original line equation in slope-intercept form, determine the slope, use point-slope form to get the equation of the parallel line, and then solve for y to put that equation in slope-intercept form.
This morning, I found myself wondering why I was insisting on having my students put their answer in slope-intercept form.
Is it really necessary? I mean, couldn't the student have just stopped at point-slope form and still been correct? I mean, plug a few things into Desmos and it's hard to argue otherwise:
I've been thinking about this and struggling with this all morning. The focus of this particular ACT skill isn't necessarily for students to determine the equation of line and put it in slope-intercept form; the skill is just to determine the equation of a line based on properties of parallel and perpendicular lines.
In the problem above, the student is given the equation of a line and a point on another line that is parallel. The student knew to look for the slope of the original line, knowing that the parallel line they were looking for would have the same slope. After determining the slope, the student created the equation of the parallel line using point-slope form.
Should it stop there? After all, the student correctly applied the properties of parallel lines and determined a correct equation. That's what the skill is all about, right? Why was I insisting that the student put their answer in slope-intercept form? I'm not sure it's necessary, and I think it also creates a situation where the student can make a simple algebra mistake and come up with an equation that is no longer "correct." On the other hand, expecting students to be able to put the equation in slope-intercept form isn't all that unreasonable, is it? After all, the student did just that with the equation of the original line in the problem, in order to determine the slope of the parallel line. Is that a good enough reason to insist on it, though?
This is just one specific case. I know this isn't the only instance in mathematics where something like this happens. When is the right answer the right answer?
Throughout the week, I have been insisting that my students give their solutions to these problems in slope-intercept form, as shown in this student's work:
(There are some other things going on here that would also be interesting to talk about, but that will have to wait for another day.)
Perfectly reasonable solution method, isn't it? Put the original line equation in slope-intercept form, determine the slope, use point-slope form to get the equation of the parallel line, and then solve for y to put that equation in slope-intercept form.
This morning, I found myself wondering why I was insisting on having my students put their answer in slope-intercept form.
Is it really necessary? I mean, couldn't the student have just stopped at point-slope form and still been correct? I mean, plug a few things into Desmos and it's hard to argue otherwise:
I've been thinking about this and struggling with this all morning. The focus of this particular ACT skill isn't necessarily for students to determine the equation of line and put it in slope-intercept form; the skill is just to determine the equation of a line based on properties of parallel and perpendicular lines.
In the problem above, the student is given the equation of a line and a point on another line that is parallel. The student knew to look for the slope of the original line, knowing that the parallel line they were looking for would have the same slope. After determining the slope, the student created the equation of the parallel line using point-slope form.
Should it stop there? After all, the student correctly applied the properties of parallel lines and determined a correct equation. That's what the skill is all about, right? Why was I insisting that the student put their answer in slope-intercept form? I'm not sure it's necessary, and I think it also creates a situation where the student can make a simple algebra mistake and come up with an equation that is no longer "correct." On the other hand, expecting students to be able to put the equation in slope-intercept form isn't all that unreasonable, is it? After all, the student did just that with the equation of the original line in the problem, in order to determine the slope of the parallel line. Is that a good enough reason to insist on it, though?
This is just one specific case. I know this isn't the only instance in mathematics where something like this happens. When is the right answer the right answer?
Wednesday, September 18, 2013
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:
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:
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:
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:
In the future, I would probably embed further scaffolding and questions into this activity. For instance, I would probably ask the students:
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!
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.
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.
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:
- A guided task at the beginning of, or during, the unit, with appropriate scaffolding included; or
- A performance task at the end of the unit.
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.
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!
Friday, September 6, 2013
Mario: Life or Death? (A Math Task)
I love math. I also love video games. So when I find a way to combine the two, I'm like, super-mega-happy.
We're in the earlier stages of a unit on graphing, and I want my students to get used to the idea that they'll need to interpret & analyze figures, diagrams, etc. and make predictions or generalizations based on solid mathematical reasoning. This is a skill that I believe is crucial for a student's success in Pre-Calculus and Calculus (and beyond).
This week, my Advanced Pre-Calc students did a one-period math task I developed called, "Mario: Life or Death?" It wasn't anything terribly complex or flashy, but we did get at a few important ideas about graphing, symmetry, and making predictions from what we know conceptually rather than numerically.
(Also, after I came up with the idea for a Mario-based problem, I found out about this Mario problem from Nora Oswald that's way, way better than what I did. I was actually inspired to follow a similar format after finding it.)
I was taking (or trying to take) a 3-Act Math approach with this math task. It definitely still needs some tweaking all around, but considering I'd never tried anything of the kind before, I was pretty pleased. (I don't know how truly 3-Act-ish this task actually is, but, eh.)
The Opening
I played this video for the students:
The students also had the first two pages of this worksheet (I purposefully withheld pages 3 and 4 until later).
As students watched the video, I had them think about what questions came to mind. They came up with some pretty good ones:
Working On the Problem
Next, the students were asked to come up with a few ideas. What information did they think would be needed to solve the problem? I gave them a little time to think of a few ideas on their own, and then they shared with each other. A couple of examples from their work:
With some group discussion, we pulled together some important concepts to think about: the distance between platforms, the height of Mario's jump, and in a few cases, the ideas of "arcs," "parabolas," and "symmetry."
I had the students explore these ideas further by having them create sketches of what they thought the path of Mario's jump looked like:
We talked about the shape of Mario's jump. The words "semicircle" and "arc" came up a lot. One or two students already knew the word "parabola."
I asked the students to think about what they knew about arcs; what is their structure? How do they behave? Particularly, I asked them if they've ever seen any "lopsided" arcs. The students said no; arcs are made up of "mirror images," or are "symmetrical." I asked them what they meant by that. They replied that an "arc" can be cut into two halves that are mirror images of each other; particularly, the arc is cut at the "highest point" into two congruent halves.
The students were asked to draw the path of Mario's jump again, this time on a grid. They were also asked to take the observations from their discussion into account:
After looking at the gridded drawings, the students had a few moments to think about how their observations about Mario's jump could be useful in predicting the outcome. They wrote their thoughts down:
After giving them time to think about how to use their observations to predict whether or not Mario would make the jump, I gave them pages 3 and 4 of the worksheet. Using the graphs of Mario's jumps, I asked the students to make their predictions and to give their mathematical reasoning:
Everyone shared their predictions and gave their reasoning. The consensus was that Mario would make his jump, but just barely.
The Thrilling Conclusion
After everyone made their predictions, the moment of truth arrived. I played the solution video:
So the students had predicted correctly! They seemed a bit amazed at how accurate their predictions were -- not only had they figured Mario would make his jump, but many of them said specifically that Mario would barely make his jump, which is what happened. They had made a mathematical prediction using conceptual observations about the path of Mario's jump, and very minimal use of numbers or calculations.
To wrap things up, we tied our discussion back to the idea of symmetry. The type of symmetry used in this problem was just one type of symmetry (even symmetry, and arguably symmetry about the y-axis). We talked about other types of symmetry besides the type used in this problem. The students took down some notes for their problem sets they were working on from our textbook.
I hope that the students came away from the lesson with a bit of a deeper understanding of symmetry and how to use it to make predictions. In the larger picture, I hope students started to see the usefulness of applying what they know and observe in order to gleam new information. As I've said before, there's so much more to math than just calculations and number-crunching.
This was totally the highlight of my week. However, I know that this task is far from perfect. It could certainly go deeper and touch on quadratics, graphing equations, and so forth. I'm almost certain this task could be made into something richer than its current form. I'll definitely be taking another look at it after some time passes and I'm free enough to sit down and make revisions.
But in the meantime, I welcome any and all feedback/comments/insults from any readers out there. Don't be shy!
We're in the earlier stages of a unit on graphing, and I want my students to get used to the idea that they'll need to interpret & analyze figures, diagrams, etc. and make predictions or generalizations based on solid mathematical reasoning. This is a skill that I believe is crucial for a student's success in Pre-Calculus and Calculus (and beyond).
This week, my Advanced Pre-Calc students did a one-period math task I developed called, "Mario: Life or Death?" It wasn't anything terribly complex or flashy, but we did get at a few important ideas about graphing, symmetry, and making predictions from what we know conceptually rather than numerically.
(Also, after I came up with the idea for a Mario-based problem, I found out about this Mario problem from Nora Oswald that's way, way better than what I did. I was actually inspired to follow a similar format after finding it.)
I was taking (or trying to take) a 3-Act Math approach with this math task. It definitely still needs some tweaking all around, but considering I'd never tried anything of the kind before, I was pretty pleased. (I don't know how truly 3-Act-ish this task actually is, but, eh.)
The Opening
I played this video for the students:
The students also had the first two pages of this worksheet (I purposefully withheld pages 3 and 4 until later).
As students watched the video, I had them think about what questions came to mind. They came up with some pretty good ones:
- Will Mario make it to the other side?
- When should Mario jump to make it to the other platform?
- What is Mario doing wrong that's not letting him jump across the cliff?
- When does Mario reach the apex of his jump?
- How can we calculate what height he needs to jump to make the distance?
- Why doesn't he use the chain to make it across?
- Did the player hold down the A button long enough?
"Will Mario make it across on the 3rd jump?"
Working On the Problem
Next, the students were asked to come up with a few ideas. What information did they think would be needed to solve the problem? I gave them a little time to think of a few ideas on their own, and then they shared with each other. A couple of examples from their work:
With some group discussion, we pulled together some important concepts to think about: the distance between platforms, the height of Mario's jump, and in a few cases, the ideas of "arcs," "parabolas," and "symmetry."
I had the students explore these ideas further by having them create sketches of what they thought the path of Mario's jump looked like:
We talked about the shape of Mario's jump. The words "semicircle" and "arc" came up a lot. One or two students already knew the word "parabola."
I asked the students to think about what they knew about arcs; what is their structure? How do they behave? Particularly, I asked them if they've ever seen any "lopsided" arcs. The students said no; arcs are made up of "mirror images," or are "symmetrical." I asked them what they meant by that. They replied that an "arc" can be cut into two halves that are mirror images of each other; particularly, the arc is cut at the "highest point" into two congruent halves.
The students were asked to draw the path of Mario's jump again, this time on a grid. They were also asked to take the observations from their discussion into account:
After looking at the gridded drawings, the students had a few moments to think about how their observations about Mario's jump could be useful in predicting the outcome. They wrote their thoughts down:
After giving them time to think about how to use their observations to predict whether or not Mario would make the jump, I gave them pages 3 and 4 of the worksheet. Using the graphs of Mario's jumps, I asked the students to make their predictions and to give their mathematical reasoning:
Everyone shared their predictions and gave their reasoning. The consensus was that Mario would make his jump, but just barely.
The Thrilling Conclusion
After everyone made their predictions, the moment of truth arrived. I played the solution video:
So the students had predicted correctly! They seemed a bit amazed at how accurate their predictions were -- not only had they figured Mario would make his jump, but many of them said specifically that Mario would barely make his jump, which is what happened. They had made a mathematical prediction using conceptual observations about the path of Mario's jump, and very minimal use of numbers or calculations.
To wrap things up, we tied our discussion back to the idea of symmetry. The type of symmetry used in this problem was just one type of symmetry (even symmetry, and arguably symmetry about the y-axis). We talked about other types of symmetry besides the type used in this problem. The students took down some notes for their problem sets they were working on from our textbook.
I hope that the students came away from the lesson with a bit of a deeper understanding of symmetry and how to use it to make predictions. In the larger picture, I hope students started to see the usefulness of applying what they know and observe in order to gleam new information. As I've said before, there's so much more to math than just calculations and number-crunching.
This was totally the highlight of my week. However, I know that this task is far from perfect. It could certainly go deeper and touch on quadratics, graphing equations, and so forth. I'm almost certain this task could be made into something richer than its current form. I'll definitely be taking another look at it after some time passes and I'm free enough to sit down and make revisions.
But in the meantime, I welcome any and all feedback/comments/insults from any readers out there. Don't be shy!
Monday, September 2, 2013
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:
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:
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:
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.
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:
- What is math? What does it mean to you?
- List 7 mathematical words or phrases that come to mind when doing math.
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.
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:
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.
Wednesday, August 28, 2013
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.
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 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.
Wednesday, August 21, 2013
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!
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!
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