Hello, World. ¬…

Hello, World. ¬†I was recently asked where I’ve been lately (thanks, Jason!), so I thought I should stop by with an update.

Let’s see. ¬†Just over four months ago, I gave birth to my second little girl. ¬†Before that, I was pregnant, believe it or not. ¬†Tired a lot, sick a lot, and at the very end, hospitalized for a week due to pre-eclampsia. ¬†Not the most fun thing ever, but I got a decent prize at the end (really, she’s awesome – I thought the first one was easy, but Girl 2 makes Girl 1 look like a lot of work).

While I was on maternity leave, my husband and I reached a decision: After this school year is over, I will be doing the stay-at-home mom thing. ¬†I still love mathematics and mathematics education, and I plan to tutor if I can; a couple of people have already expressed interest in that. ¬†I also still hope to earn a Ph.D (still waiting on the university to approve the program I’m hoping to do). ¬†And I’m working with my master’s advisor on a book on using history in the teaching of mathematics. ¬†But I will not be in the classroom any longer. ¬†I’ll be at home with my girls.

I am really excited about being able to stay home with them. ¬†I’m a little terrified, but very excited. ¬†I look forward to spending time teaching them academic things – Girl 1 is three years old and eager to learn EVERYTHING. ¬†It’s so much fun, and I can’t wait to do more of that. ¬†(I’ve taught her the word “sinusoidal,” and she can point out curves that are sinusoidal. ¬†Don’t you just feel sorry for my poor children? ¬†And they’re going to be stuck with me all the time.) ¬†I look forward to teaching them to be generous and to give of themselves as we work together with ministries that are important to our family. ¬†I look forward to seeing Girl 2 do lots of firsts over the coming year or so. ¬†I look forward to being able to go and visit out-of-town family without having to worry about sub plans.

Anyway, so I fell behind on blogging, and I usually find Twitter to be overwhelming to catch up on. ¬†But I’m doing well, intent on finishing this year strongly. ¬†I’ve done some good things with my students and some things that had much room for improvement. ;) ¬†I’ve continued to use SBG, even through maternity leave subs (two different people filled in for me…well, actually three, but that’s a long story). ¬†That was interesting, and hearing from the students upon my return, I think one of the subs didn’t quite get it after all. ¬†But it’s hard to step into someone else’s classroom and system in general, and I do think she did a good job teaching the students, so it’s okay.

I feel like I’m rambling, so I’m going to wrap this up. ¬†I don’t know if I’ll be posting here again, maybe if I return to teaching or have other contributions to mathematics education to share. ¬†But I do get email notifications when someone replies to a post here, so feel free to do that if you’d like to get in touch with me. :)

SBG Year 2: Initial Thoughts

Okay, time for some brainstorming! I mentioned in my last post that I have a LOT of areas for improvement in my implementation of SBG. I got all inspired reading the comments to Shawn’s post where people shared how they make reassessing work, and here are my initial thoughts, very much subject to change.

  1. I want to start each day with either a warm-up (over new material, from the last day or so) or a quiz (over material I expect the students to understand). It could be either one on any given day, depending on what I feel like the class needs.
  2. Warm-ups are only given comments, no scores.
  3. Quizzes are given comments and scores.
  4. Quizzes cover multiple Learning Targets. Students are told when a Learning Target will be quizzed for the first time, but older Learning Targets are always a possibility.
  5. Students may use their notes & textbooks during warm-ups, but not during quizzes. (Consideration: Maybe if students feel completely lost on a quiz, they can pull out book & notes and just make a note of that before turning it in? It would go down as a score of 1, but at least they would have the ability to work on learning it better rather than staring at it and then turning in something blank.)
  6. Occasionally, Learning Targets may be assessed via a project (in or out of class, group or individual), or through demonstration of mastery in class discussion.
  7. I think I’ll no longer be using the “most recent score wins” as my way of determining grades. ActiveGrade has several options for how to determine a student’s current score on a Learning Target, and I will play around with those some more before deciding on a method.
  8. For individual reassessments, each class will have an assigned day on which students can come in at lunch to reassess. In order to reassess that week, the student must respond to the following questions at least 2 (3?) school days ahead of their class’s day.
    1. What went wrong before?
    2. What have you done to improve your understanding? Provide evidence of the work you’ve done.
    3. How do you know that your understanding is better now?
    4. How would you explain this concept to a friend?

  9. I think I’m not going to set a specific limit for how many Learning Targets a student can reassess at once, but I will reserve the right to say “too many” if I feel like that’s the case. Lunch is only so long, and Learning Targets do vary in complexity and time required to demonstrate understanding.
  10. I also want to allow a student to suggest an alternative method of demonstrating understanding. In theory I was doing this last year, but I didn’t regularly encourage it, so students rarely stepped up. I would love to have kids say, “Mrs. Dean, I think I really get it now. Let me show you a way that I figured out I can use it.”

So…what do you guys think? Help me work through and refine these ideas. :) Also feel free to comment on whether this sounds like a system someone else will be able to step in and use with relative ease, since I will be taking maternity leave during the third quarter.

Summertime: Recharging

School’s been out for over 3 weeks now, and I’m starting to feel like thinking about it again. :) So I’m ready to post now.

Next year will be different for me. Two of my classes will be middle school courses, which will be new to me, and one of the high school courses will change from trig/analyt to precalculus, so not new material exactly but a different emphasis and course layout. Geometry and calculus will remain in my schedule (with a new text for geometry), and statistics will not.

Also, I’m expecting baby #2 in late December or early January, so I’ll basically be taking the 3rd quarter off. That means preparing for a long-term sub. I’m a bit worried about the SBG aspect of things there.

Speaking of SBG, here are some thoughts I’ve had relating to assessment…

  • I need to get better about spiraling feedback for all students on all topics. I ended up with a “keep trying until you get it right, but then it’s okay to forget it” attitude among my students, and that’s not my goal.
  • I’m going to be using ActiveGrade this year, which I think will be great for my students. My school is going to online grade reporting as well (I know, we’re waaaay behind the times there!), but I think I can reasonably just keep the ActiveGrade updated and then enter scores in the official gradebook when I need to for report cards. ActiveGrade will give the kids (and their parents!) access to so much more information about their scores than a traditional gradebook.
  • A big problem I had last year was that as midterms or end-of-quarter approached, I would have large numbers of students in my classroom wanting to reassess. The rest of the time, not so much. It was a last-minute effort for most of the kids. (There were exceptions, naturally – this is just the general pattern.) I would also have kids who would say, “I want to reassess these three learning targets tomorrow. Can you tell me what they were over?” It was really, really painful to hear that, because it showed me clearly that it was still all about the grades. The kids were looking at their scores, finding the ones that were the lowest, and not even going back to their marked-up work to figure out what those were about, just asking me to tell them. Then they’d come in and, surprise!, not do well on the reassessment. Lather, rinse, repeat. I am working on how to remedy this, though I do think the previous two bullet points will help, but I also need to implement a better process for allowing reassessments, along the lines of Sam’s form email.
  • In the fourth quarter, I stopped keeping track of student homework completion. This was not a good idea. While I still think homework should not factor into the students’ grades, I also think that knowing they have to report what they’ve done is good accountability. And I think I need to start taking up homework for feedback purposes at least some of the time.
  • My students need to get better at knowing how to help themselves understand concepts. With some of them, I don’t know if they never bother to read my comments, or if they read them and think “okay, I’ve got it now” when they really don’t, or if they read them and think “I still have no clue what she’s talking about, or what to do from here.” I need to help them learn how to evaluate their own understanding better.

Other thoughts, not related to SBG…

  • I started out last year with some great, fun activities for the students, plenty of active uses of the mathematics they were learning. Then I got bogged down and into a rut of mostly class discussions. I want to get back to the way I started out. Discussions are great things, but not when they’re the only thing happening.
  • I want to continue working on incorporating the history of mathematics into my teaching.
  • I had a “math fair” as part of my students’ final exam grades for the second semester last year. I believe it was a worthwhile endeavor, and I plan to do it again, though it will not be part of the exam grade in the future. However, with my maternity leave planned for the 3rd quarter this year, I’m not sure if it’s something that can happen, as I don’t want to place any part of that burden on a sub. So it may be 2012-2013 before I do it again.

Seems like I had more things rolling around in my head than that, but some of them must have escaped. ;) Anyway, those are things I’m thinking about with regard to next year. Hopefully, I’ll be back with further thoughts on how I plan to improve as the summer goes on.

The Dance

I have the odd little habit of finding the prime factorizations of numbers that come across my path (mile markers, page numbers, etc.). I regularly factor whatever page number I’ve told my students to turn to in their textbooks. For some reason, this is awe-inspiring to them.

Yesterday I was speaking with my calculus class about a particular number being prime, and I explained why I stopped checking factors once I got to the number’s square root. “Wow,” they said. (Note that my calculus class is comprised of three students, all of whom have gifts in mathematics, but only one of whom would say he likes the subject.)

I replied, “Aren’t numbers beautiful?”

They asked if I had considered decorating my two-year-old daughter’s room in numbers. While I acknowledged the awesomeness of that suggestion, I corrected them: “Not numerals. Numbers are beautiful…the mathematics, the relationships between the numbers, the way they work with one another.”

D said, “But it’s a cold beauty.”

No, I said, it’s not cold at all; it’s a dance.

He countered that language is a dance.

I replied, “But what is mathematics if not a way of expressing ideas? Isn’t that what language is, too?”

D answered, “If only there were a way to test the temperature of a dance!”

They may, and likely will, graduate without having a love for mathematics. But they’ll know that it’s possible to have such a love, because they’ve seen it from me…and maybe they’ll be a little more likely to look for the beauty, to watch the numbers dance.

Puzzling Problems Preferred

Yesterday morning I determined that I would find something from my day to blog about, and hey! I did! :)

My geometry kids are learning about quadrilaterals. The prior day’s lesson was about properties of rectangles, rhombi, and squares. Here’s the warm-up problem I gave them:

In rhombus QRST, the diagonals QS and RT meet at point U. If QS=12 and RT=16, find the perimeter of the rhombus.

I think it’s a strong problem, because they have to recognize all of these things:

  1. that since the rhombus is a parallelogram, the diagonals bisect one another;
  2. that the diagonals are perpendicular;
  3. that the four small triangles are right triangles;
  4. that the Pythagorean theorem will allow them to find the length of one side of the rhombus; and
  5. that all four sides of the rhombus are congruent.

Almost all of my students struggled with the problem. I gave some hints – “What do you know about the diagonals of a rhombus? Okay, now think about triangles.” One student asked me if it was a “trick question;” I told her that it’s not a trick, just a puzzle.

But once they solved the problem, whether they needed hints or not, they felt proud of their work. That was awesome to me. Some of them told me that they really liked the problem. It was a challenge to them, so they had a sense of accomplishment once they had solved it. One student asked if I could give them more problems like that for their homework; the book’s problems are pretty simplistic and generally don’t require them to make connections across multiple concepts. I really need to get better about giving them more challenging problems for practice.

In other news, I am ordering sample texts for almost every mathematics course we offer for grades 6-12. The math department (all 3 of us) will be meeting at the end of March for a day of articulating who we are, and then using those decisions about curriculum. I’m really excited about this. :)

As part of that process, I’m reviewing the calculus materials available from The Worldwide Center of Math, and so far I am loving their model for the modern textbook. Check it out.

Warm-up Follow-up

We’re in week 2 of doing daily warm-ups like I mentioned in my last post. One class had 4 days of them last week, and the other four classes had 3 (we spent one day going over the exam and one day reviewing what we’d been doing just before the break). What I told the kids is that my New Year’s resolution is to give them better and more frequent feedback, so that they don’t end up surprised on the quiz when they don’t know something as well as they thought. I also told them that I want them to improve their ability to self-assess, and the way to do that is by practicing it.

Here’s an example of what they look like, the final version (split into two images – they just have blank space in the middle to work the problem):

I have it set up so that I’ve got a Word document mail-merging an Excel file where I enter the variable information. (I <3 mail merge.) The only problem I have so far is that I can't enter both text and an equation into the same cell in Excel, so I sometimes have to write an equation on the worksheet before making copies. The same thing would apply if I wanted to add a figure. But that's not a huge deal.

I feel like it's working well. I've been able to get the papers marked from the previous day with no problem (I know that's just an issue of discipline, but marking papers quickly is something I have struggled with in the past). I am seeing where the kids are making errors and addressing them sooner rather than later. I get to see how well they feel like they're doing, and they've naturally started adding comments by the faces to explain why they're confused or don't feel completely confident with something. That is GREAT, because they are thinking about their thinking, which is a skill they need to continue practicing.

Today's warm-up says this:

We’ve been doing these warm-ups for a few days now. Do you feel like they are helping you understand the concepts more clearly? Why or why not?


Their feedback is almost entirely positive, and they’ve made some thoughtful comments that are helping me consider how I can improve things. If you’d like to see their responses, I’ve typed them up here (yes, I have small classes, though I was missing 3 kids in 1st period).

Moving toward more formative assessment

It’s been a while since I’ve had a chance to post! I’ve been doing some thinking about my classes in general, how SBG is going, changes I want to make in the coming semester.

A few people have asked me, now that I’ve got a semester of it under my belt, how do I like standards-based grading? The answer: I love it. I still believe what I did at the start – that SBG is a fabulous tool both for my students and for me. Students are aware of exactly where they stand on specific concepts. There’s no question of “but you didn’t say this would be on the test!”, because the Learning Targets are clearly identified for them. They seem to see more clearly that they are the ones in charge of their grades, because they are the ones in charge of their learning. I get to make sure kids demonstrate clearly that they understand the concepts, and I can hold them to a high standard without worrying about what happens if some people haven’t met it yet when we have the quiz. Parents are loving this system – I’ve had a number of them say that they might have enjoyed math in school had they had a teacher who graded like this.

All that said, I definitely have places to improve how my classes work, and I’ve been working on how I can better use this fabulous tool to promote student learning. My biggest concern is that I’m not giving my students enough feedback before they reach a quiz. I’ve been doing whole-class feedback, but not individualized, and that’s just not working. I don’t like waiting until the quiz to discover that the quiet half of the class all had the same misconception about a particular idea. I still want homework to be something that the students are in charge of (meaning it’s practice, not graded, and the students determine how much of it they need to do for their own personal learning).

I’m pretty sure I’m heading in the direction of a daily warm-up. My thoughts about how this will look: I’ll have half-sheets of paper ready for each class, and students walk in and pick one up, immediately getting started on it as they reach their seats. Each warm-up will look something like this (with the parts in bold being different each time):

When they finish, they turn it in, and then they look back over their homework so that once everyone’s done the warm-up we can review it. (This will have the added benefit of helping me structure my class time better. I have a tendency to let time be wasted, and I’m really not okay with that about my teaching.)

I’ll review their work each day. The homework completion goes in an ungraded category in my gradebook, so that I have a record of what they’re getting done (the warm-up sheets will also save class time in that they’ll just write it down and I’ll record it later, rather than asking each person at the start of class). I’ll write comments on their work so that they have individualized feedback, and this way I will have a better idea of whether there are some widespread misconceptions going on, so I can address them in my teaching. I want to use more formative assessment, and this is a step in that direction.

Things that worry me: I tend to get into a what-am-I-doing-next-period sort of place, especially given that this is my first year teaching any of these classes. This daily warm-up thing will require me to stay on top of things much better, given that I’ll need to write one for each class each day, but I’ll also need to read and respond to each student’s work each day. This scares me. I’m afraid I won’t be able to sustain it. But I can’t not try, because I really think this is something my students need.

I’m also worried about the self-check aspect of it, as I’m not entirely sure what I’m going to do with that besides making them think about where they are. I know that they need to be able to self-assess so that they’ll be able to identify their own areas of strength and weakness so that they can best use their study time. I guess the self-check part of the warm-up sheet is a way of starting a conversation? I don’t know. I don’t want it to be just something they fill out and then nothing happens.

Anyway, I’m out of blogging time for today. I have more ideas for the coming semester, but this is the biggest one, so it’s a start.

Trig Functions – a w00t and a help request

First off, a “hey this went well,” and then a “help me out.” :) These are both about my trig class.

Hey, This Went Well
I started today talking about amplitude and period of sine functions. They’d already understood that the period of sin(x) is 2pi, and they got pretty easily that the amount it goes above and below the x-axis is 1; I just had to add the word “amplitude” to their vocabulary. Check.

So then I put a GeoGebra window up on the projector and entered f(x)=sin(x), and I asked them what I should change about the equation in order to make the amplitude different. They weren’t quite sure what I meant, so I said, “Should I multiply sine x by something? Should I multiply the x by something before I take the sine? Should I divide instead? Should I add something at the end? Should I add something to the x before I take the sine? Should I subtract instead?”

Then they started conjecturing. I made them explain why they thought their particular change would effect the desired transformation, made them convince each other before I would type it in so we could see what it actually did. After we saw each change, I went back to f(x)=sin(x) before making another change, so we were always comparing our changes with the plain ol’ sine function. They were really pleased with being able to see the changes immediately.

It was also cool because we’d looked at what happened when we used f(x)=2sin(x) and f(x)=0.5sin(x), and then somebody said, “What if we used a hundred?” So I changed the coefficient to 100, and a couple of them said, “Whoa! It’s just straight lines!” Others said, “No, it just looks like that because it goes so far up and down.” The magic of the zoom feature allowed us to see that it was, in fact, still curving like we expected it to, just with really skinny humps. :)

So I think that using GeoGebra like that was a good way of looking at these transformations – they can make a guess, and we can see whether it’s right immediately. We can also use crazy big (or crazy small) numbers if we want to.

Help Me Out
I’ve got an idea, but it’s only partially formed right now. I don’t want it to flop like the last one (which involved kids spinning in circles and then releasing a stuffed animal – it was supposed to be about linear velocity, but it was a Fail, though they did enjoy going outside). So…here goes.

You know how when you walk, your body moves up and down? Walk across the room now. When your legs are side by side, you’re at your full height, but when one leg is in front of the other, the distance from the ground to the top of your head is less than your height. If you hold an arm straight out with a marker held to a long piece of paper, and you walk normally, the mark you make on the paper will be sinusoidal. (I remember my instructor showing us this in the health class I took in college…I don’t have a clue what his point was, but I thought it was cool.)

So I want to let my students do this. I want them to ask questions about what will be different for the curves formed by the tallest person versus the shortest person; what will be different between a girl and a guy; what will be different if a person is walking versus running; what changes if a person is on crutches (okay, I’m not sure if we could pull that one off or not, but it’s an interesting question, right?); what will be the same in each of these cases as well.

But I don’t want to get out there (I have an idea for where we can do this) and then just be wasting time with questions we won’t be able to answer. So I need to figure out how to structure this thing. Suggestions?

Unexpected Misconceptions

My trig students are now learning about the graphs of trig functions. (Yes, I’m behind.) I started off by talking about periodicity, and then I talked about how the values of sine and cosine repeat once we get all the way around the unit circle and start going around again. I then presented the idea of f(x)=sin(x) and started getting them to tell me what values I would plot.

They were confused. I figured out two major reasons for their confusion. The first one, I think I’ve addressed, but the second is, I think, still causing befuddlement.

Reason #1: I put a graph up there and used crazy things like pi/2, pi, 3pi/2, and 2pi as my notch marks on the x-axis. They’re used to integers there, and to think that I could divide my axis so that each notch represented pi/2…well, that was just insane to them. I think it relates to the way they don’t really like to think of pi as a constant, since they can’t express its value exactly in any way other than a weird Greek symbol. But I think I dealt with this one okay, though I might mention it again today just to make sure they caught on.

Reason #2: This is best expressed by giving a quote from a student.

“Why are you saying ‘sin x’ when sine is y?”

The student was confused because, on the unit circle, she had memorized that the y-value is the sine of the angle (which we’ve been calling theta). All that relating of the values on the unit circle to right triangles? Yeah, that didn’t work for her. She just memorized that “sine is y.” I tried to show her that we had been saying that sin(theta)=y on the unit circle, and that now instead of theta, x is the angle of which we’re taking the sine for our function…but I don’t think it clicked for her. Well, I thought it had, but then when I said that we could also do a function f(x)=cos(x), she said, “But then cos(x) would be y, and I thought sine was y?”

So I’m still working on that one. But I’m making myself notes, so that next year I’ll be able to anticipate these misconceptions and hopefully prevent them by changing my earlier instruction. I’m sure new ones will pop up to surprise me then. ;)

Math Department Identity

Our principal has challenged us (me and the other two math teachers) to articulate who we are as a math department. I’m trying to consider what that means in order to work on answering it. So far, here are the questions I think we need to address:

  1. What are our overall mathematical goals for our students?
  2. In what ways does our department work toward the goals of the school as a whole?
  3. What is distinctive about our math program?
  4. What is our perspective on the nature of mathematics, and how does this perspective influence our instruction?
  5. What is our perspective on the purpose of mathematics, and how does this perspective influence our instruction?

What other questions should I add to the list?

Have you gone through an exercise like this in your own department? Any additional thoughts?

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