What it Means to Slow Down a Problem

We did a really ambitious activity early this year in Geometry. We slowed down an optimization problem for start-of-year Grade 9 students, in order to get every single kid to understand the process. It worked brilliantly, and we are trying now to help them extend the idea to other optimization problems.

I want to document and share this journey, because I think the experiment that we have started is SO challenging and SO worthwhile. We are trying to get the kids to think like mathematicians. By slowing them down.

Here was the first time we formally introduced optimization (after having the kids play around with building their own popcorn containers). If you read it carefully, you might notice that we tried to emphasize a few things: 1. Tactile learning. 2. Justifying their thoughts. 3. Understanding what x and y represent. 4. Understanding how to analyze the domain. 5. Resourceful use of technology. 6. Interpretation of results back in context.

A little while after going through this, we gave the kids an open-notes test, using a different sheet of paper to start. They did well, which showed us that they really understood the different pieces. Then, on the actual closed-notes exam, we worked backwards by giving them a factored cubic equation, and asking them some relevant questions: 1. What is the dimension of the piece of paper that we had started with? 2. What is the domain of this problem, and why? 3. What is the largest box that can be built here, and what are its dimensions? 4. How many different boxes can we build that would have a volume of _____, and what are their dimensions? Again, the kids did brilliantly!

It has been amazing and humbling to see how far along these 9th-graders have come.

The next question that we gave them, which they worked through in small groups, was an optimization problem involving a known perimeter of a rectangle and in trying to maximize its area. They needed to take the problem from start to finish, in writing a system, combining it into a single function, and doing domain and graphical analysis to find the maximum. Then, as usual, interpreting back in context.

Now, the next problem they are tackling has to do with maximizing the area of an isosceles triangle whose perimeter is 30 units. Not easy. Some kids figured out right away that this follows the patterns of other similar problems, where they will try to write an area equation. Some kids started to make tables -- another really great habit of a mathematician! The class, as a whole, needed a nudge to help them figure out how to get the height of the triangles. We stopped discussions today at writing a general height equation as a class, in terms of x, the length of the two congruent sides, and I asked the kids to keep thinking about the rest of the problem.

A worthwhile experiment, indeed! If by the end of the term, they can do even half of these problems completely independently of us, we will be so thrilled. I think the key is to slow them down. By feeding them the understanding in pieces and then giving them another similar problem, we are building the foundation that it takes to transfer the knowledge.

Stay tuned! 

First Term Reflection

This is a rough brain dump after the recent end of school term, but I am going to try to stay coherent and helpful.


1. This past school term, I decided to be tougher-than-usual and to cut off accepting make-up work a day before the end of the term. It seems like a small change, but it really saved my sanity by so much! I got to get all of that make-up work graded and returned by the last day of the term, and to just focus on the bigger assignments (like tests) once the kids went on term break and I was working on finalizing their grades. It probably didn't save me much time, but it saved me lots in terms of sanity and focus. I wasn't trying to grade a thousand different assignments all at once while trying to re-calculate their grades.

2. Another thing that really saved me is that I created a learning rubric, asking kids to rate themselves on their growth mindset, reflectiveness, responsibility, resourcefulness, and organization on the last day of the term. It was tremendously helpful to me in writing comments for them, looking at how they rate themselves in each category of the rubric! At this point of the year, it allowed me to really incorporate their own self-assessment into their evaluation, while keeping it somewhat objective (action-based, as my rubric was formatted to be, rather than opinion-based, like it would probably be if I gave them an open-form self-reflection). Again, in the end I don't think that I necessarily saved time in writing their evaluations, but I think that I wrote really detailed and comprehensive ones, considering that it is only the first term and we have only had 6 or so weeks of school.

3. At the end of the term, I really liked ending Algebra 2 with choice assignments and ending Calculus with no-requiz-option exams. For Algebra 2, the choice assignments were all regression activities, and all groups ended up learning/practicing the same skills, but offering them the choice meant that we would potentially have richer discussions in coming weeks, and their interest level was also very high during the task. For Calculus, giving them a quiz that does not have any re-quizzing options was a great way for me to ask the students to step up to a "college-level" challenge after a term of slowly ramping them up to my expectations, and to show them that they could still do very well if they would commit to preparing and asking questions in advance. It worked! Although in general, I am a believer of re-quizzes, I think giving one no-requiz assessment every term will actually reinforce their confidence over time, even for the students who initially don't do well on them. It will also make for a more realistic preparation for college next year.

4. In Grade 9 Geometry, we had given an open-notes exam half-way through the first term, and then a closed-notes final exam at the end of the first term. These tests were super helpful, in combination. The open-notes exam was a great informal check-in on how responsible they are as learners, in asking clarifying questions in class and making sure that they had understood a quite complex task. (Their open-notes task was to take a sheet of paper and to write a cubic equation modeling the volume that could be built by folding up the corners into a box. The 9th-graders needed to do domain analysis and to use Desmos to optimize the volume, and then to construct the box accordingly, individually.) The closed-notes exam at the end of the term, then, assessed how well they are practicing/preparing for exams. I loved this combination, particularly in Grade 9. I thought it was a very developmentally appropriate way to introduce them to high-school expectations.

5. Math journals. Love them, but they're so much work! I am still looking for ways to cut down the work load that is to grade these concept journals all the time (often twice, if the kids are doing revisions on their entries). Any tips?

6. Overall, a very exciting, albeit hectic, first term!

Wordle as a Tool to Respond to Feedback

This year, particularly after reading Thanks for the Feedback, I am making a conscious effort to gather on-going feedback from my students, in order to address them in real time and to engage my students in a two-way communication. But, let's be honest, the thing that takes the time in class is not to gather feedback, but to go over it. It always feels tedious to go over kids' concerns and appreciation bullet point by bullet point, so this year I am going to try using Wordle (or one of the alternatives) to make the discussion take up less time.

Here is an example of my Algebra 2 wordle, based on feedback for what is working well in the class thus far. I couldn't get the Java interface for Wordle creation to work either on my laptop or on the school's laptop, so I used TagXedo to make this in the end. I like TagXedo, because you have choice over both font and orientation of text. If I wanted to, I could have orthogonally-oriented text. As you can see, kids mostly thought that group work and their classmates were very helpful, as well as handouts that had some basic examples on them. Lots of kids mentioned fixing their errors as being very helpful as well, and the idea of growth mindset being weaved into that. With this picture, I can hopefully whittle down this part of the discussion into a couple of minutes, and then focus on discussing what isn't working well yet. (It didn't make sense for me to create a wordle for what isn't working well in this class, because they are only individual concerns and no real repeats.)




Incidentally, we had a great open house this week, which also weighs in as a source of feedback for me. I tried out an exercise where I asked my students' parents to write down on an index card one hope or excitement they have towards their child's math class this year, and one anxiety that they have towards the class this year. My student parents blew me away on this task! Their responses continue to reinforce my belief that our approach to teaching mathematics needs to consider, in every step, how we are impacting our students' mindset and attitude towards math.

Here are their hopes or excitement. Believe me, their concerns are equally insightful!!

I would love it if [my child] developed a sense of the Beauty of Mathematics.

I am hopeful that [my child] can regain [their] confidence and enthusiasm for math.

Feel like a mathematician and enjoy math.

Hoping that [my child] sees the beauty of method and that it becomes a great mix of method, understanding, and simplicity as a meditative experience/tool.

[My child] continues to love learning.

I hope [my child] grows [their] confidence in Math and is able to solve problems with numbers.

Confident enough to not fear mistakes and attempts and iteration.

Learn algebra

I hope for [my child] not to hate Math.

To gain confidence by asking questions and talking more in class.
 
Excited that [my child] is taking Calculus in high school.

Hope [my child] continues to enjoy math.

Deeply interested in math/Calculus both as theory and application.

I want [my child] to enjoy Calculus and math and to want to do more of it in college, to embrace quantitative theory and analysis.

Excited for [my child] to learn Calculus!!

[My child] is totally turned on by math. [They have] not expressed any anxiety.

Understanding the basics of differential equations and still keeping [their] interest/love/self-confidence in math.

Understanding the basics of Calculus.

I never took Calculus so I am excited for [my child] to learn something beyond what I took.

Glad [my child] has got classes and teachers that [they like].

I am excited that my student is in Calculus and I'm looking forward to [their] development of understanding derivatives.

Excited about [my child] building confidence with Calculus to ease the college course experience.

[My child] seems to love it.

My hope is for [my child] to appreciate that Calculus will be applied to Sciences, and necessary for [them] to succeed in.

Excited for [my child] about [their] continued exploration of new math concepts.

[My child] LOVES math and I'm excited that [they are] able to do 2 classes this year.

I am excited that no matter what [my child] learns, [they] will know more than me.

[My child] is taking charge of [their] work and meeting with you regularly.

Excited for [my child] to continue advancing in math as it is one of [their] favorite subjects.

That [they] can apply [their] learning to real life situations.

Thinking about incorporating some of this parental input into my discussion with the students about how class is going, where we are headed, and why.

How do you manage gathering and addressing feedback in your classes, on an on-going basis?

Changes to Class Structure

This year, I have made some exciting changes to my classes.

1. I thought long and hard about how I do math journals. Last year, I got raving reviews of the math journal setup from my thoughtful and reflective students, but I didn't think that the way the math journal was run actually benefited the other students. It was a hard sell. They always procrastinated at the end of the term with revising all entries, and thereby it lost its value as a reflective tool throughout the term. Also, many students complained that I was "looking for something very specific" in their revisions, so it was very stressful. This year, I decided that more than accuracy in their math journals in reflecting upon the big ideas, I just want them to engage in the process of reflection. I also decided that the stress of trying to get the most accurate answer was turning some of them off from the process of reflection, which was having the opposite effect of what I was trying to achieve. So, this year I am de-coupling feedback from their grade. The first time they turn in a journal entry, I grade them based on a simple 6-point rubric that looks like this:

So, at that point, they have some score out of 6 that I write down. I hand back their journal with both comments scribbled in the margins and a printed rubric with their score. They are then asked to revise it before I collect their journal again in a few days. (I go over in class the common errors, and I also post a couple of example entries on my wall, to help them with this revision.) The second time that they turn it in, although I still return a 6-point rubric to give them feedback, I don't penalize them for their errors. I give them full credit in my grade book for just engaging in the process of trying to revise their understanding.

I have noticed three things from this: One, that most students get the revisions correct anyhow. Two, they are proactive about revising in order to get the points back. Three, they are turning the journal entries in more or less on time each time, which allows me to dialogue with them about their misconceptions in real time. The journal is serving its purpose!!

2. Last year, I gave short one- or two-problem practice quizzes as we were learning the material, and I would grade them on a green, yellow, red sticker system. My intention was that it would be a low-pressure way to give students early feedback, without affecting their grade. But, I wasn't happy about how the kids with the red stickers would put off meeting with me, because they weren't feeling the urgency until the real, big quizzes rolled around. So, this year I still give practice quizzes, but with a few points attached. On the returned practice quizzes, I would write down a similar problem and ask all kids to show me via doing the similar problems that they now understand their error and the concept. I find that this is more productive towards encouraging improvement and a growth mindset, because in order for kids to practice a growth mindset, they have to actually do some work towards making progress!

Thus far, I like the system a lot better. It's still reasonably low stress, but the expectation is clear that kids need to work on their areas of weakness, in order to make back those lost points.

3. This year, I am working actively on building community in my classes. One thing that I did for the first few weeks was to always assign seats. Everyday, the kids would sit with a new partner. They would take the first couple of minutes of class to ask their partner for a fact that they didn't already know about that person, and they would write it down on their own name tag. This helps me to keep track of whom they have already sat with, and they started collecting lots of facts about lots of peers! I loved it! I have also started randomizing partners now that they have sat with a majority of the class. Each day, they come in and they pick a name randomly from the pile. If it's their own name, then they put it back and pick another. Otherwise, they sit with that person. (If they pick out names of people who already have picked a partner, then they give me those name tags back and pick another.) This is a low stress way of making sure that they rotate partners all the time.

4. I am using Microsoft OneNote for two of my classes! Algebra 2 and Calculus. I kind of love it. I can lesson plan very far in advance, and make modifications in real time as I see fit. The OneNote websites serve as my course webpage, where kids can look up both notes and homework assignments. (For a long time I didn't believe in posting homework assignments, but this year I am doing the opposite after reading Switch, because I want to shift the entire classroom culture towards completing homework as a norm, rather than just something that is done by already-responsible students. Thus far, it seems to be working great!) I also upload video links for the kids who just want to use additional resources. Thus far, I absolutely love it.

You can check them out, too! Please don't modify any of the content though, just look.
http://bit.ly/calcNoteboook and http://bit.ly/alg2Notebook are the two links I am using.

That's it for now. Ciao!

The Growth Mindset Experiment

Hi, blogosphere! I thought I'd check in quickly. This school year, I really wanted to make the focus to be on helping kids to transition to my class. One of the things I wanted to teach them about is growth mindset, and at the same time to be crystal-clear about why we do the things that we do. For example, why do a pre-test? Why do we have practice tests when not everyone is ready? Why play around and intuit things before we discuss them formally?

I did a brain talk on Day 1, which had elements of the growth and fixed mindset as part of the talk, including asking the kids to practice re-framing certain negative sentiments as growth mindset statements. For homework, I asked the kids to write about an example of growth mindset in their lives, outside of math. I learned so much about their lives through this first assignment! I learned about their interests and also how they respond to setbacks. Subsequently, I have shared my own story of a growth mindset with them, and continue to think daily in terms of what I need to do or say in my class in order to generate and sustain more growth mindset. Now every time a kid asks to meet with me, I thank them for having a great growth mindset. When a kid fails an assignment, I ask them to keep working on it to revise their understanding, and I write, "Growth mindset!!" on their paper. In our Geometry class, on the day when we had individual algebra tasks, I asked the kids to write on a post-it one thing that they had learned during that class. (Some kids said, "I didn't learn anything new! I had to review..." and it was a great opportunity for us to chat about how learning new things sometimes requires re-learning old stuff first.) I am thinking about doing a growth mindset check-in, where each kid is asked to write down one thing that they recently learned or improved on and one thing that they are still working on or challenged by. This would hopefully remind them that learning is a spectrum, in order to reinforce their confidence in the process.

Anyway, all this thinking about growth mindser is directly feeding into the classroom culture, I think!! Thus far, I am loving my classes and find the kids to be actively engaged everyday.

Next time, I will talk about actual changes I have made to my classes...Some good ones, I think!

Summer Reading #2: Switch

Wow, this week has passed rather quickly. It's the last week of holidays before we head back to school to prep for students to return. (But, don't be jealous if you have already returned to work; in order to make up for the late start of school this year, we are shaving off holidays during the school year, ie. the buffer days that used to exist in between terms, which I would have probably rather kept.) Since I am nursing a foot injury (plantar fasciitis on both feet, sigh...), I did some planning, read some books, and hung out with the hubby*. It is so luxurious to be able to have lunch and coffee dates with my husband during the week! I love it.

*The hubby works from home.

Anyhow, I thought I'd jot down some notes about Switch, which is written by Chip and Dan Heath of Made to Stick fame. Switch is a book about how to initiate change in other people or in a system, when people's natural inertia is to resist change. I wanted to read this book because I think that as teachers, we are constantly trying to change our students' approaches and attitudes towards their learning. In our minds, there is a vision of what an ideal student does, and we are striving to move all of our students a bit closer to that model. For example, for me, an ideal student is always actively engaged and reflective of their learning. They are always asking questions and trying to draw connections between topics. They are risk-takers, communicators, and they have a deeply rooted growth mindset. The ideal student does not necessarily always enter the class with all of the skills from previous classes (...in fact, sometimes they can be missing significant skills...), but they have a big, open heart, ready to take on feedback and to re-tool their learning processes as needed.

The book outlines a deceptively simple framework for initiating change, and then it illustrates the use of this framework through a variety of almost unbelievably successful stories. The framework is basically this:

* Motivate people emotionally. In order for someone to change, they have to want to change. You don't motivate people using numbers or research, because our intellect is not what causes us to change; you have to find a way to give them a vision and trigger their emotions. Sometimes, this can mean helping them to envision or build a new identity, because we tend to behave in a way that we wish to see ourselves. (This is a silly example, but have you ever wanted to buy something from a store that looks really stylish, but whose clothes are not very flattering on you? You're subconsciously trying to assume the identity of the type of person who would shop at that store, and you're shifting your behavior to match that identity. Another example is how Geoff and I started composting. In Seattle, it's part of our collective identity because the city provides infrastructure and even a small financial incentive to compost, as compost-collection costs less than regular garbage. Now that we have grown to see ourselves as composters, we can't seem to break the habit of looking to compost everywhere, even when we're away traveling. It has changed our behavior permanently!)

* Give clear directives and "shrink the change". Once you get people on board emotionally to change (which, I know, isn't easy), you have to give them clear, achievable, and black-and-white goals in order for them to get started and feeling successful. The example given in the book is that instead of telling someone to "eat healthy", to say instead to "drink 1% milk." This clear health directive has completely shifted the way America now consumes milk. When you script these directives, think in terms of something immediately achievable, although you can link it to a longer-term vision that it will hopefully pave way to. Give the brain a reason to follow the change and give it also a clear path to follow.

* Follow the bright spots. Change is hard and change takes time. Instead of focusing on what isn't working, look for what is working and highlight those consistently. Find ways to duplicate the success until it creates a positive momentum. For example, are 70% of the kids turning in their projects on time? Praise those kids and ask them to share what strategies are working for them to help them stay ahead of schedule.

* Shape the path. Are you making it as easy as possible for people to create and maintain the change? For example, if your students are not in the habit of doing homework, are you posting homework online and giving them organizers in order to help them ease into the habit? Are you finding ways to create a positive group culture wherein the norm is to do homework? If the change you wish to see is kids volunteering to speak in class, are you providing a structure wherein it's the norm to speak up?

I liked the book, but what I would like to hear is what other teachers think are challenging changes to institute in their classroom, and how we can use this framework as a lens to help us think about transitioning students into more successful learners over time. For example, one incentive that I want to try this year is the ability for the class to earn "homework passes" by showing consistency in completing homework. It's not so much the homework that I care about, as much as I care about them doing and thinking about math consistently outside of the classroom and really attempting problems when I am not there by their side. I figure that if I start with an incentive system, over time the learning will hopefully become its own reward, and I can wean them off of "homework passes." But, in the mean time, it can help me to shrink the change from "be a more proactive learner" to "try your homework", to help the kids who don't already have good study habits to start to build them.

Thoughts??

Summer Reading #1: Thanks for the Feedback

OK, web, I am back! I had a fabulous summer. My husband and I did a big trip! We started in New Zealand and ended in Iceland, covering a total of 12 countries in 60 days, so it was quite a sprint.

Anyhow, I have been doing some summer reading after returning from our trip. I had picked up two professional books prior to the start of the summer, but didn't get around to reading them until I got back. (Traveling with a smallish backpack made carrying paperback books unrealistic, and as it turned out, it was hard to find places to reliably charge our phones while staying in hostels, so I did relatively minimal digital reading also while away!) I thought I'd jot down some notes about the books I read, as the books were very useful to me!

One book I read this summer is called Thanks for the Feedback, and I had picked it up because I wanted to get better at hearing and parsing through feedback myself, as well as giving feedback effectively. I highly recommend the book! It was a great lesson in thinking about nurturing my own growth mindset, because how we receive feedback has everything to do with our own growth mindset. The book, for me, contained a lot of valuable information regarding why hearing negative feedback is challenging and how we can frame our minds around this more effectively.

To give you a small taste of why I really liked the book, it talks about how sometimes someone we really don't get along with would try to tell us what they think about us. They would probably present it at the wrong time and in the wrong manner, and you can easily, based on many legitimate reasons, write them off and think to yourself that what they have to say does not apply to you, and that they're in the wrong for X, Y, Z reasons or disqualified to give you feedback for M, N, R reasons. Well, the book encourages you to put aside all those factors and to ask them questions about why they feel this way. Dig further into the data that they're looking at. Do they have information that you don't have? Is their role giving them a reason to consider other factors that are not on your radar? Instead of deflecting negative feedback as is tempting for most people to do, embrace it actively and ask probing questions in order to start a conversation. In the end, you don't have to accept all parts of their feedback, but the first step to growing is to understand where they are coming from, particularly because the people you get along with the least are most likely to offer you an honest look at yourself.

The book also talks about examples of when someone we are in a close relationship with tries to give us feedback. While reading this, I related this in my mind to my husband, who from time to time tries to give me critical feedback about something in my personality that he thinks needs some work. The book talks about how people often react to this by essentially redirecting the conversation to include a new thread about how you are also dissatisfied about something that the other person does. I know I'm certainly guilty of this, and the book gives specific strategies about how to tackle this type of impulse / conversation trainwreck to guide it towards a productive conversation, wherein you focus on one conversation at a time and really try to hear the other person's point of view.

One of the things that I really liked also is a diagrammed model in the book about how our own behaviors are invisible to ourselves. We, as individuals, are only aware of our intentions and the impact that we intend to make, but we have no visibility into our outward behaviors (particularly our micro-movements and our body language) and their impact on others. On the flip side, the people who are experiencing our actions have no visibility into our intentions, even though both parties think that we are seeing the full picture. For example, I am only aware of what I say to my students and why I am saying it, but I'm not aware of how it is coming across in the moment and whether it has the intended effect. This is why getting feedback from our students is very, very important. It helps to close that feedback loop and to make sure that the messages that I hope to send are actually aligned with the messages that the students are receiving, particularly when it comes to my instructional choices and what I think are important aspects of their learning!

The book also helped me reflect upon how I give feedback to my students! The book distinguishes between feedback forms that are evaluation, coaching, and appreciation, and gives examples for why it can cause a lot of frustration for the receiver if they are constantly missing a certain type of feedback. It also talks about the importance of giving feedback in the manner that a person prefers to hear it. So, one of the things I will do this year is to find out, from each student, how they prefer to receive feedback. I'll have to think about how best to phrase this, but the book has some good suggestions for probing questions.

Wow! It is turning out to be a fairly hefty entry here, particularly because I have gone for so long without posting much at all. I'll come back and talk about the next book tomorrow, but if this quick summary sounds intriguing to you, I highly recommend checking out Thanks for the Feedback as written by Douglas Stone and Sheila Heen. Worth a read and worth taking notes!

Circumference of the Moon

So, as a follow-up to Erastothenes using geometry to calculate the circumference of the Earth, this week I plan to go over how we can use the ratio to Earth to calculate the circumference of the moon!

The lesson idea came from my colleague John. I fleshed it out to scaffold it for my kids. It looks like this, and it ties in nicely both with our school's Grade 9 science curriculum (which teaches astronomy for all of next term), and our current circles unit. I plan to use this lesson the day after our basic circles quiz, when a few of the students will have family members visiting our class. (At our school, this is called Grandparents' Day, even though it is quite possibly not the grandparents that are coming.)

I'm excited!! I've never done this lesson before, but I like how it revisits perpendicular bisectors and makes them seem useful in application.

Addendum 3/26/15: I prepped for this lesson today and it REALLY BOTHERED ME that I got the estimate that the Earth's circumference was about 2.5 times bigger than that of the moon, when in reality it should be about 3.6 times bigger. I did some more digging and worked out a ratio to find out how big the Earth's shadow would be by the time it reaches the moon, and I think it's 9200 km in diameter at that point! That makes my ratio make a lot more sense, because this is only 2.6 times bigger than the moon!!! Go Geometry!

Geometry to Algebra Transition

Our school is trying out a new thing this year (one that I think is fabulous). We're re-shuffling 9th-grade kids' Math classes for the last 8 weeks of school this year, depending on whether they intend on taking Algebra 2 or Precalculus after finishing Geometry this year. The classes in our last term of the school year will prep the kids for transition into their choice of algebra classes for next year, and we'll assess them at the start of the term, end of the term, and again at the end of summer to determine whether their achievement and commitment-to-hard-work together seem to predict success in their choice of classes (particularly those intending on skipping Algebra 2), in order for us to advise them and their parents about whether they should be working over the summer, and what seems to make sense for their course placement. In prepping for this transition, we are including lots of algebra into our current circle unit to help kids "warm up" in thinking about algebra skills.

Below is what I have so far. The kids are definitely hitting their edge, but I am able to motivate them by explaining that quadratics is the next logical thing for us to practice, since we have done already a lot of work with lines and systems this year. Even those who have taken Algebra 2 in Grade 8 and who are intending to take Precalc next year did not have an easy time solving for points on a circle, so this is great stuff for all of them!

Here is my intro to circular equations, which most of the class is about finished with. Following it, I plan to spend a few days doing this, which is a modified version of a worksheet that two of my colleagues had created. We want the kids to get familiar with circular vocabulary (as preparation for Calculus) and to do some algebra practice involving circles, but besides it, we're not too attached to teaching all of the circle theorems, since we only have 7 more school days left of this term. I am excited to see the kids' transition to algebra after all the work we've done with them this year in terms of problem-solving. I hope it'll pay off when they get to Algebra 2 or Precalc next year!

Some Calculus Worksheets

I took some time during the previous school term to observe some of my colleagues, in order to educate myself about the ways in which they encourage inquiry in their classrooms. Among other observations, I enjoyed seeing how the other Calculus teacher (who is about 20 years more experienced than me) structures his worksheets to always circle back to applications and interpretation of answers. Since my visit to his classroom, I have been working on modifying my handouts from the previous year in order to put in more application into every concept.

Here are two of them: I used this to help kids wrap their minds around basic integral Calculus applications, and this one reviews some algebra skills from earlier this year, plus introduces the necessity of going in between algebra and the graphing calculator sometimes. The problems are not ground-breaking, but I think they've definitely helped to break up the skills practice, so I'm happy to share them if they might be useful to someone else.

From earlier months of this year, one thing that I did that totally helped with teaching Related Rates is that I first taught implicit differentiation with respect to time, and formally assessed students on this skill, prior to starting Related Rates word problems. (Sorry if this sounds obvious; it wasn't that obvious to me last year, teaching Related Rates for the first time!) Here is how I introduced implicit differentiation, using the analysis of non-functional relationships as a premise. After this, I had the students do some pure skills practice in converting geometric formulas to differential equations with time as the domain, before introducing my scaffolding for related rates problems and the many related problems I took from Bowman last year. I felt really good about this sequence of skills this year, because I noticed that it really made the problems more accessible to ALL students (as in, by the time they got to the word problems, they were really only focused on parsing the word problems process, rather than simultaneously struggling with the algebraic skills of differentiating implicitly). I recommend trying this, if your students get baffled by Related Rates problems.

I am also trying to place a general focus on vocabulary and communication this year. I've been doing this in all classes by giving the kids a list of essential questions at the start of a unit, and then having them journal their responses to those essential questions throughout the term. For example, for our current term in Calculus (which is short, only about 5+ weeks), I gave the kids the following questions. The questions are a mix between related rates (which we did at the start of the term) and intro to integral Calculus.

• How are "implicit differentiation", "chain rule", and "related rates" all related? Illustrating this with a simple algebra example may help to clarify your thinking.

• Take one of our Level 2 or Level 3 Related Rate problems from class and explain/describe, step-by-step, how you are able to find the missing information.

• What is integral Calculus? Describe a couple of situations where this concept is useful.

• Choose an exponential function of the form f(x) = a*e^(x - k), by assigning values for a and k. Estimate the area underneath the curve of f, from x = 0 to x = 5, using a total of 10 rectangles. Show both left-hand sum and right-hand sum, and draw labeled diagrams to show what your numbers mean.
• Show, step-by-step, how you would calculate the enclosed area that lies between two functions f and g, where f is a quadratic function of the form f(x) = ax^2 + bx + c and g is a trigonometric function of the form g(x) = m*sin(n(x - k)) + p, where the value of n is not 1. You get to choose the parameters a, b, c, m, n, k, p to start, but make sure n is not 1.

Students have shown a varying degree of enthusiasm about the journal assignment, even though I have been doing it since the start of the school year and explaining periodically its purpose. Part of the purpose of this journal is to get them to record, in their own words, examples and explanations to important concepts, so that they can have a succinct set of notes for future years. Another purpose is for me to see what they write periodically, so that I can informally gather information about common misconceptions for the topics that we have finished learning, and clarify them with the class. As it turns out, however, the naturally reflective students are thoroughly utilizing the journal to dialogue with me about their understanding, and the rest of the kids see it as a drag to have to keep revising their explanations until the end of the term, so the work that I receive is kind of a mixed bag in terms of quality. It has been a somewhat tough sell, but one that I think is important, because from time to time, students would comment on how they notice that by answering questions in their journal while learning the concepts (instead of putting it off until the end of the term), their understanding improves in real time. Do you do something like this in your classes? How do you drum up enthusiasm for such a revision-based assignment?

That's it for now! My Geometry students are wrapping up their 3-D project, which is very interesting as per usual. They have some really neat designs this year, which I might share at some point. Algebra 2 kids are knee-deep in thinking about the domain and range of different function types, and thinking about transformations on the various functions. It is nice to hear them go, "Ooh, ahh..." as they realize that they can connect information from different types of functions. Not much to write home about, but a productive time of the year nonetheless!

Hello, World!

Sorry, web, I have been away! It has been a busy few months.

Geoff and I went to Hawaii in December, followed by some busy weeks at work for both of us while shopping in our off-time for a house. In February, I went home to visit my parents, and almost immediately afterwards, my in-laws came to stay with us for 10 days in our 700-square-foot apartment. They have just left, and Geoff and I have finally begun planning for our big summer trip. This summer, Geoff plans to take off 2 months from work and travel with me. Tentatively, we will start in New Zealand, then go through Philippines, Indonesia, South Korea, Japan, Estonia, Russia, Greece, Italy, Germany, England, Ireland, Iceland before coming home to Seattle. Our house's closing date is still set for the end of March, so in the mean time, there are just a lot of things keeping us busy. Hence, the radio silence...

But, I have been reminded recently that I have a blog! Through the grapevine, two friends of friends have mentioned this blog to me. Funny, small world. So, let me think about what I can put on the blog that is worth sharing. Stay tuned.