Algebra: First Semester Lesson Plans 16-17

As I indicated in my last post, I'll be teaching a section of Algebra again next year. Please read that post for all the context, but I'll just reiterate here that I like to make an overall comprehensive plan in advance, fully realizing that I'll have to adjust almost daily based on how it goes with students.

For anyone who's interested, here are my (draft) first semester lesson plans. My attempt to blog my lesson plans didn't end up generating any feedback (so I stopped), but if you have any feedback now feel free to leave them in the comments or contact me directly.

I'm Going to Have to MTBoS the Sh*t Out of This

This year I got the opportunity to start a Computer Science program at my school, which ended up meaning I got to teach two semester-long sections of Intro to Computer Science in both the fall and spring semesters (and teaching myself Python last summer). The plan was that we were going to try to hire a real computer science teacher for 2016-17 but, since that didn't happen, I'll be teaching the Intro class again next year, along with an Advanced Python class and a Javascript class (and a colleague will thankfully be teaching a Java class) as we try to grow the program.

None of that preceding paragraph is really relevant to this post, other than to give some context as to why I'm asking for your help. Not that it needs context, because asking for your help is a good thing in and of itself, but I'm hoping to make you feel a little bit sorry for me and guilt you into helping. (Hey, it's worth a shot.) Because in addition to learning some more Python, and teaching myself Javascript, I'll also get the opportunity to teach a section of Algebra 1 again next year.

Now, the good news is, I don't have to teach myself Algebra 1, I sorta, kinda already know Algebra 1. Long-time readers of this blog may remember about 6 years ago when I got the opportunity to teach Algebra 1 after many years of being completely out of the classroom in my role as technology coordinator for my building. At that time I tried to blog my planning and did great for a couple of weeks and then petered out. This time I hope to do better, and to hopefully do a better job of tapping into the online community of teachers to get your help.

Six years ago I experimented with what's now almost universally called the "flipped classroom" (at the time there was still some debate about what to call it). While there are strong opinions on both sides of the flipped debate, I still feel like it was a worthwhile approach for me at the time because it allowed me to free up class time to do better things than I would be able to do otherwise if I had to "cover" the material in class. I was trying to bridge the gap between the expectations of my school and the rest of the math department and where I was hoping to take my Algebra class, and flipping allowed me to do that to some extent.

Well, now it's six years later, and I haven't taught Algebra the last two, and some things have changed. My district has transitioned completely to a common-core based Algebra course, using Agile Mind as the "textbook." But, more importantly, I'm more willing to go even further away from the mainstream (at least what the mainstream is in my building), and the resources available to me are even better, including even more teachers blogging about math, Desmos has gotten even better, and technology in general has gotten easier for students to use in the classroom. I think I can do better than flipping and, while at least some of the lessons/activities I used in class before are still great, I think they can be improved on.

So I'm sitting here planning my summer (last official day of work is this Tuesday) and trying to figure out how best to structure learning a bit more about Python (I have a rough outline for my new advanced course, just need to make sure I know it well enough to teach it), teaching myself Javascript and planning that course, and planning my new-and-improved Algebra class. I'm worried because the easiest thing for me to do is to focus on the computer science stuff that is new to me, and to slack off on the Algebra planning, because I know I can just rely on what I've done before and do an okay job.

But I don't want to do an "okay" job, I want to do a really good job, and take advantage of all these great resources (and my additional willingness to diverge from the traditional in my building). As I was thinking about how to do this, a line from the movie The Martian kept going through my head:

I'm going to have to science the sh*t out of this.

So the way I'm framing planning for Algebra is,

I'm going to have to MTBoS the sh*t out of this.

For the uninitiated, MTBoS stands for the MathTwitterBlogosphere, which is the self-given name of all the math folks who are sharing their thinking, asking questions, challenging each other, and generally discussing ways to improve their math teaching. My hope is twofold: first, I'm going to tap into (borrow/steal) all the wonderful resources the MTBoS has collectively created and, second, that by publicly blogging my planning I will not only encourage feedback from the MTBoS, but guilt myself into not slacking off on the Algebra planning part of my summer.

We'll see how that goes. If you're interested in helping, I would appreciate it if you would follow along with the blog I set up for this (right now the only post is a cross-posting of this one, but beginning on Wednesday I will start in earnest - RSS, Email). Fair warning that there is at least a 50% chance that this will flame out but, if everything goes swimmingly, perhaps this will not only help me tremendously, but serve as a resource for other Algebra 1 teachers.

Teach This, Not That

I think I've made it pretty clear in previous postings that I'm not a fan of standardization, but I realize that most teachers don't have a lot of choice and are required to teach to certain standards. Given that, teachers still often make choices about which standards they cover (since there's never enough time to cover them all) and how in-depth they go on each standard. Since my daughter is currently taking Algebra, I'm helping a home-bound student some with Algebra, and I occasionally teach Algebra myself, I thought I would pick an example from Algebra for my first (and perhaps last) "teach this, not that" post.

I was recently helping that home-bound student with the polynomial unit in Algebra 1. She did some marginally interesting topics, but - since I get to cherry pick for this post - she also did an assignment that I'll excerpt below.

Here's a screenshot of some of the problems she had to do.

And here's a screenshot of the answers.

Later in the semester she'll get to explore exponential functions a bit, so we'll see what types of activities she gets to do for those, but one typical way they could explore exponential growth would be a compound interest type of problem.

Now, every teacher is different, but based on my experience, if an Algebra teacher has to choose one of these two things to cover in an Algebra 1 course, they often pick the first one. Why? Because polynomials seems to "fit" better in the Algebra 1 curriculum and exponential growth does not, and compound interest problems are often presented in a way that it's mostly just plugging numbers into a formula and computing an answer.

From my perspective, however, it should be exactly the opposite. There may be some reason why some people might want to know that a certain polynomial is a quartic trinomial, but I have to think that for most of our students that's not a particularly good use of their time. Compound growth, however, is something that could be life-changing for them and their families (credit cards, car loans, mortgages, savings accounts, investments - and that's just financial applications), yet even when we do teach it, we often teach it as simply "plug-n-chug."

Here are two problems that I think would be interesting for every high school student to explore (and probably most of the high school staff, for that matter).

(Please note that while the math in these examples works no matter what, the feasibility of these scenarios is much more likely in a middle class or higher household. Those happen to be the students I work with, but I understand and empathize with folks who might be frustrated with these examples because they work with students in poverty.)

Scenario 1: Save for your retirement . . . before you graduate from high school.

Many students in my school get a job in high school, often over the summer after their sophomore year. If they work full-time over the course of that summer, they could easily gross $3000. Now, being the teenagers that they are, they are most likely going to want to spend a fair amount of that money. And they should. But I would suggest that by exploring the mathematics a bit, they - and their parents - might also want to invest it.

So, if this were my daughter (we'll see if she chooses to get a job after her sophomore year or not), I'd suggest she invest at least a bit of that money in a Roth IRA. And then I would contribute the rest up to whatever her gross earnings were for the year (we'll say $3000 for this example). Here's why:

1. She won't owe any income tax on that low of earnings, so even though Roth IRA contributions are "after tax" contributions, this would effectively be "no tax" contributions for her, and all earnings will be tax free.
2. I would suggest she invest that money 100% in a low-cost equity index fund, reinvest dividends, and never touch it again until retirement. (No reason not to be 100% in equities for this type of investment and time horizon.)
3. Current assumptions (which I think will change, but we'll go with it), is that a current 16-year old might retire at age 67 or so, so we're looking at a 50-year + investment horizon. What will $3000 grow to in those 50 years? And there's the exponential growth question.

So, what will $3000 grow to in 50 years? Well, to be sure, no one can answer that question, but we can estimate based on a lot of data from past experience. (This is assuming that the way economies and capital markets work will not dramatically change, which I think is perhaps not a good assumption, but for estimating purposes it's the best we've got.) Since 1930, the long term annualized return of the S&P 500 is about 9.7%. If our 16-year old would achieve that kind of return over 50 years, she'd have about $307,000 at retirement. Just from that one summer's investment. If she works after her junior and senior years and puts in an additional $3000 each summer, she'd be looking at over $900,000.

But since we're talking about 50 years, I think we should at least consider investing in riskier equities that - over time - are likely to achieve a higher return. Since 1930 Large Cap Value has returned 11.2%, Small Cap has returned 12.7%, and Small Cap Value has returned 14.4%. Now, most folks would look at that and say that certainly the amount she'd have in the end would be higher, but I'm not sure they'd realize how much higher.

For the $3000 investment, the total after 50 years for Large Cap Value would be over $600,000, for Small Cap would be over $1.1 million, and for Small Cap Value it would be over $2.5 million. For $9000 investment (3 summers), triple those numbers. Keep in mind, that's all tax free, and all with not contributing any money to her retirement account after graduating from high school. (With the assumption that even if tax laws change, they will grandfather in existing accounts.)

Of course now would be a good time to talk with our student about inflation, and how that $7.5 million ($9000 for 50 years in Small Cap Value) in 2067 won't buy the same amount as $7.5 million today. So let's assume an average annual inflation increase of 3.5%. Lots of interesting discussions to have here about how students could use that information to calculate the end result but, simply discounting our returns by that amount turns that $7.5 million into about $1.99 million in today's dollars, which translates to being able to spend about $80,000 a year - (today's dollars) using the 4% rule. Still pretty darn good, which is why I think this is a worthwhile scenario to explore with students and why I think this might be a better use of time than learning about quartic trinomials.

Scenario 2: Don't go to college . . . and retire much earlier.

Yes, it's provocative, but that's part of what makes it interesting. I've written before about our assumption that college is the default goal for our students, but let's look a bit closer at the mathematics.

Like most parents, I've paid attention to the tremendous increase in the cost of attending college. We also started saving for college even before we adopted our daughter, using a tax-advantaged 529 plan. We invested in Colorado's plan because, in addition to earnings and withdrawals being tax free, contributions are exempt from Colorado state taxes (which is like earning 4.63% right off the bat). Due to our diligent saving and investing, and the benefits of compound growth (even with 2008), we have about $120,000 set aside in our 529 for our daughter's college expenses.

Well, that sets up an interesting scenario for a problem about exponential growth. What if she didn't go to college and, instead, invested that money now (we'll take a tax hit since it's not being used for college, but I'm willing to cover that), immediately got a job that didn't require a college degree, and continued to add to that investment over the years? Lots and lots of messy details here, which is why it's such a good problem situation to work through with students, but let's look at a simplified version with lots of assumptions just to get the feel for it.

We'll use the same investment return information from Scenario 1, including investing in index funds with 100% in equities, since she's young and has a long investment horizon. We'll assume that she'll get a job paying at least $25,000 per year to start off with, and that each year she'll get a raise that's at least equal to inflation. We'll also assume that she'll be able to save and invest an additional $3000 each year. I realize that can be tough when she's starting at $25,000 per year, but that works out to a reasonable 12% of her income, and perhaps we'll let her live at home for the four years she would've been in college to help her start off. I'm going to make one more assumption, which is that she could retire comfortably on $40,000 per year. That's for just her, if she gets married she would obviously have additional income, additional investments, and additional expenses that would complicate it a bit; but as a family of three we are currently spending about that much (when you take away what we're saving for retirement), so I don't think it's an outrageous assumption for one person.

Well, the numbers are pretty interesting to play with, especially with the excellent FIRECalc tool. Lots of choices to make here as well, but on the first tab (Start Here) I put in $120,000 portfolio to start, with anticipated spending needs of $40,000 per year (today's dollars), and wanting it to last for 80 years (50 years after she retires). I left the second tab blank, meaning I'm assuming no social security or pension income (there probably would be some, but we'll leave it at 0 for now). On the third tab (Not Retired?), I put in a retirement year of 2045 (so that's assuming working for 30 years, starting now), and that she'll add $3000 to her portfolio each year (adjusted for inflation). For the fourth tab (Spending Models), I chose Bernicke's Reality Retirement Plan. The fifth tab (Portfolio), I adjusted to 100% equities. When I do all that, it gives me this. (You'll have to click submit if you follow that link to see the results page yourself, but here's some of the verbiage):

Following the "Reality Retirement Plan" as described by Ty Bernicke, withdrawals after age 55 are reduced by 2-3% per year until age 76.

Because you indicated a future retirement date (2045), the withdrawals won't start until that year. Your contributions will continue until then. The tested period is 30 years of preretirement plus 50 years of retirement, or 80 years.

FIRECalc looked at the 64 possible 80 year periods in the available data, starting with a portfolio of $120,000 and spending your specified amounts each year thereafter.

Here is how your portfolio would have fared in each of the 64 cycles. The lowest and highest portfolio balance throughout your retirement was $120,000 to $56,587,349, with an average of $15,925,319. (Note: values are in terms of the dollars as of the beginning of the retirement period for each cycle.)

For our purposes, failure means the portfolio was depleted before the end of the 80 years. FIRECalc found that 0 cycles failed, for a success rate of 100.0%.

You really should explore FIRECalc some more but, based on a lot of baked-in (but not half-baked) assumptions, it tells me that for the 64 possible 80 year periods that the historical data supports, not once would she have run out of money (and usually would leave quite an estate). Note that has her retiring at the age of 48 and living until 98. (If you want to change it to constant spending power instead of Bernicke's Reality, then you still have an 81.3% success rate. But working just 3 more years, so retiring at 51, would have had a 100% success rate.) Keep in mind all of this is assuming no pension or social security income, which you definitely would have if you worked for 30+ years. After playing around, you can even discover that she could retire in 2036 - so at age 39 - with a 97% chance of success (and with 59 years of retirement, and usually a sizable estate). So, at an age when some college graduates are still paying off their college loans, she could be retired. Provocative enough?

FIRECalc even lets you download a spreadsheet based on your inputs that you could analyze with students to examine (and perhaps manipulate) the formulas. Again, I would suggest this is not only more interesting than quartic trinomials mathematically, but also practically for students. And, of course, there's nothing preventing our student from doing both Scenario 1 and Scenario 2.

That's just two examples. Lots and lots more you could do with debt (credit cards, car loans, mortgages), governmental policy (budget, entitlements, social security, medicare), and on and on and on. But I don't know anyone that really does, because there's always one more standard we need to cover, and students just might get asked to name a quartic trinomial on some test sometime. It's probably a good thing, though, since we wouldn't want our students to be financially independent and able to retire before we can, would we?

FaceTime + Stoodle + Desmos = Virtual Algebra Help

Nothing earth-shattering in this post, but I thought I'd share in case it might give someone an idea.

A friend of ours is a 15-year-old freshman at a nearby high school. Unfortunately, she's been dealing with some serious medical issues that have kept her home-bound almost all of this school year. She has a home-bound tutor provided by the school district and is doing her best to try to keep up, but it's tough. Her medical condition causes her to be in pain and often extremely fatigued, so it's very difficult for her.

For the last couple of months I've been trying to go over on the weekends to help her with Algebra, which allows her to concentrate on her other subjects with her district-provided tutor. Because of her fatigue, however, she often has to cancel or cut short our sessions. In an effort to provide more of an "on-demand" option for her, so that I can help her on short notice when she does have some energy (and cutting out the drive time to get to her house and back), I cobbled together a technology solution that, so far, seems to work reasonably well.

Our district is using Agile Mind as the textbook for Algebra, so the text is online. The teacher emails her all the worksheets from both the in-class work and the homework, as well as which parts of Agile Mind they go with when applicable. (I'm not a particularly huge fan of Agile Mind or worksheets but, as worksheets go, these are better than average, with lots of material based on the work of the Charles Dana Center.) She tries to figure out the concepts on her own (she's a talented math student), but it's tough without being in class, so I end up doing a lot of questioning to help with teaching/explaining, as well as just overseeing her procedural work.

This is what we have set up for the virtual option.

• She FaceTime's me from her iPad to my Mac. She rests her iPad on a stack of books and uses the rear-facing camera to stream video of the worksheet she's working on as she's working on it. I see a reasonably large version on my Mac, and we can talk back and forth. (She can also see me if she wants/needs to.)
  
• Since I'm mostly questioning, she does most of the writing, but at times I want her to be able to see something I'm writing (either working out an example, drawing a picture, creating a table, etc.). Since I'm using the Mac's camera for FaceTime, it's not particularly convenient to use paper and then hold it up in front of the iSight camera. So for my writing we use Stoodle. It gives us a simple, shared online whiteboard in real time. (She can write on the Stoodle as well and I can see it, but I didn't want her to then have to copy it to the worksheets that she has to turn in.) I use my wife's or my daughter's iPad to make the writing/drawing easier, and she has the Stoodle open on her Macbook.
  
• Since this is Algebra, Desmos is very helpful as well. We've used it a lot in our face-to-face sessions, and now we can use it virtually as well by sharing a link or copying and pasting into the Stoodle. (The Desmos link is not "live" to both of us simultaneously, but still works pretty well).

I think face-to-face is still better and more productive but, given the unpredictability of when and how long we can work together, this has worked really well to make the most of the times when she does have energy.

The Textbook Is Not The Curriculum

I thought I'd give a little update on two of my previous posts. Despite overwhelming opposition by the professionals that we pay to teach our students mathematics, our district went ahead and adopted the Agile Mind materials. The rollout this fall has been less than smooth, including many technical issues that have been slow to be addressed (students not being able to login, students not being able to load the materials, some of the materials loading but not others, etc.). We are diligently working through those technical issues and I imagine we will solve them reasonably soon.

But the other concerns the professional mathematics educators expressed regarding the materials are not so easily addressed, and many of them are coming up in the day-to-day use of these materials. The Algebra team at my school (Algebra being the only course where they are required to use the Agile Mind materials at this point, although Geometry and Algebra II will be required in the future) has been extremely frustrated up to this point. Being the dedicated teachers that they are, they aren't giving up, they are still trying their best to make this work, but they have been burdened by the expectation that they must use these materials. (And, as stated previously, Agile Mind seems to be designed to work as a script, not a resource.)

As a result of the technical problems with Agile Mind, I have had a fair amount of interaction with the Algebra teachers around Agile Mind (full disclosure: I'm not teaching Algebra this year so am not experiencing this myself), and I've mentioned several times that they don't have to use these materials. Like any textbook or other approved materials, it is simply a resource for them to use. The only expectation of them is to help students learn the mathematics curriculum as decided by the Board of Education, and they can use their professional judgement on how best to do that. Every time I've brought this up they look at me and say, essentially, they been told they must use these materials. (Apparently it was even mentioned that the district can "track" how often students log in, and therefore how often teachers are using the materials, although it's somewhat unclear as to which half of this the emphasis was being placed on.)

Which brings me to why I'm writing this post. I have a daughter who is in ninth grade and is taking Algebra at Arapahoe, so therefore is using the Agile Mind materials. She recently brought home a letter from the district saying that all the students would be surveyed three times throughout the course of the year to help determine the impact on mathematics instruction and achievement of the use of the Agile Mind materials. The purpose of the letter was to allow us to opt-out of the survey if we wished, per School Board policy.

Normally we wouldn't have any issue with our daughter taking a survey such as this but, for the first time, we are choosing to opt her out. It's not just because of the 45 minutes of mathematics learning she will miss out on while they are taking this survey, although that's certainly part of it. But it's because of the letter itself, and how it reflects on the above discussion and my previous blog posts. Here's the full text of the letter.

Do you notice anything about this letter? Here's what I noticed. It refers to the Agile Mind Curriculum in the header and three separate times in the the first paragraph. It's not a curriculum, it's materials that have been adopted to support the curriculum. This may indeed just be a slip of the keyboard (although four times in one paragraph is a whole lot of slippin'), but the problem is that - whether it's a slip or not - that is exactly how it's being implemented by the district. It's being treated as a curriculum, not in support of the curriculum. Not only is this exactly what the professional teachers of mathematics feared back in the spring when we were discussing this, it is in direct violation of the curriculum adoption process in our school district.

I've stated before that I think the materials adoption process in our district is deeply flawed, but at least it was a process that was more-or-less followed, even though the results of that process were not ideal. But clearly the process for curriculum adoption was not followed, and so for the district to now be referring to this as the Agile Mind Curriculum in a formal communication home to parents is stunning.

So, as a professional educator, as someone who has a fair amount of experience teaching students mathematics, as someone who is fairly well connected to the on-going discussions around learning and what's best for our students, as a staff member in Littleton Public Schools and at Arapahoe High School, and as a citizen and taxpayer, I have a question. But let's ignore all those roles and I'm just going to ask this question from one perspective:

As a parent of a ninth grader enrolled in Algebra at Arapahoe High School in Littleton Public Schools, I'd like to know what the LPS School Board is going to do about this?

An Open Letter to the Littleton Public Schools Board of Education

This post is kind of an addendum to my last post. On Tuesday of last week the district asked our Math Department for a list of our concerns over the materials that were being recommended. Since the Board meeting was on Thursday, the department met after school on Wednesday and generated the letter below. The letter was shared with the district and, to their credit, they shared some of the bullet points with the Board when they made their presentation. To the best of my knowledge* (see update at bottom), the entire letter was not shared, so I thought I would share it here.

There are two other high schools in our district, plus a small, alternative setting. At the Board meeting the Math Department Chair of the next largest high school (we are the largest, 15 teachers in the department) spoke against the materials adoption, stating that 10 out of the 11 math teachers in her building opposed it (you might recall from my previous post that we had been told something different). The third high school is apparently in support of the materials although, again, we're hearing second hand from at least some of those teachers that, and I paraphrase, "we felt like we didn't have a choice so why fight it."

The School Board will decide on the materials adoption at their next meeting on Thursday, May 8th.

---

To: Littleton Public Schools School Board

From: Arapahoe High School Mathematics Department

Re: Concerns over materials adoption

As you know, LPS is in the process of adopting new materials in order to implement the newly revised Colorado Content Standards in Mathematics (which, in turn, align with the Common Core State Standards - Mathematics, or CCSS-M). As is the usual process, a committee was formed to preview what materials were available, then review selected materials, then make a recommendation to the Board on what materials to adopt. While this process has worked reasonably well in the past, we have concerns that the materials that are being recommended this time are not in the best interests of our students.

First, a bit of context. While there has been a lot of discussion both locally and nationally around the Common Core State Standards (not just the Mathematics ones, but the Language Arts ones as well), it’s important to keep in mind that these standards are still relatively new, especially in the context of textbook development cycles. Consequently, many of the choices that are currently available from textbook publishers are not (yet) of the quality we would like.

In addition, we are very much in a transition period between print resources and digital resources. While we clearly are headed toward digital resources, textbook publishers have not yet figured out the best way to utilize this new medium. (There are also questions of how the publishers will maintain revenues and profits, but we will not go there as part of this discussion.) Many publishers initially just tried to port their existing print resources to an online format, usually as some combination of non-editable PDF’s and non-editable web pages behind a login. While that was a natural first step, it really didn’t provide any advantages for the end user over a print textbook (in fact, it was probably more difficult to use). (It did, however, provide a cost-savings to the publishers as digital is much cheaper than physical; a cost savings that sometimes was passed along to the customer, but often was not.) This is an example of “Substitution” phase in the SAMR model that LPS uses for 21st Century Literacy.

Over time publishers began receiving feedback that customers did not like this and began to investigate better ways to take advantage of the affordances of digital platforms. The Agile Mind materials that the committee is recommending we adopt is an attempt to both address the new CCSS-M Standards and take advantage of those digital affordances. In our opinion, however, they have not successfully addressed either one.

Here’s a brief - although not comprehensive - list of some of the concerns we developed in a department meeting.

• Much of it is not editable (exams, lessons, activities in PDF form or online) - can’t be easily customized. A physical text would not be editable either, but we feel like other digital resources are much more customizable (our own fledgling efforts on ck12 (still very much in alpha form), test generator-type software from previous publishers, Google Sites/Drive, etc - all have the ability for us to modify).
  
  
• Not adaptable across our level of instruction such as remedial, on track, and advanced. This ranges from concerns from our Learning Support Services folks about reading level of the text (not adaptable to lower reading levels) to concerns about the ability to extend for our more advanced students. This also includes concerns about how we implement the various intervention models we have developed in our PLCs over the last few years using these materials, including how our Study Center personnel will utilize it.
  
  
• There are not enough examples and daily practice provided without considerable supplementation.
  
  
• There are concerns about being able to utilize the Smart Board environment like we currently do while simultaneously accessing the Agile Mind materials on screen. Many of us have developed many digital resources that we use with students in class. We understand that you don’t have to follow the Agile Mind script exactly, but that calls into question what the advantages are of buying this resource.
  
  
• These materials are built to be delivered in a fairly particular way. While the publisher argues that the teacher has great flexibility, the materials themselves only work for a while if you follow the “script”. While teachers are under no obligation to follow that script, if they do not, then these materials are not of much use to them. In addition, the Agile Mind script requires more days of instruction to complete the curriculum than we currently have.
  
  
• There is a concern with all students having access to the Internet at home in order to fully utilize this program. While that is certainly a goal of ours and we are heading that direction (both at AHS and in LPS), the district currently cannot guarantee this. If we are going to adopt materials that are only available with an online connection, then we would have to guarantee (and provide for those who cannot afford it) both equipment and high-speed Internet access for all of our students. (In comparison, the ck12 book we are creating is online, but can also be downloaded in PDF, ePub, or mobi formats for use offline without an Internet connection if necessary).
  
  
• While we haven’t had time to explore the materials fully, we have already discovered some technical issues (for example, there are issues on pages with scroll bars and the “interactive” dragging and dropping). In addition, the user interface is not particularly well designed. There are issues with the size of the print (if you’re projecting and kids are very far back in the classroom). While there are zoom options available on browsers, the interface itself doesn’t adjust well making it not very usable with a class.
  
  
• Cost (including additional cost of printing the student activity sheets). While we have not been told an exact cost, we’ve heard numbers like $500,000 thrown around. Whatever the final number is, we think that would be a reasonable investment in materials that would help our students become better learners, but we don’t feel it is a reasonable investment given the quality of these materials. Especially when you consider the following budgetary concerns (these are just three recent examples that come to mind):
  
  
• Our Deaf and Hard of Hearing Teacher is being let go due to budgetary reasons, despite the fact that we still have students in need of those services, and the fact that we have two sections of students who are taking ASL as their World Language and will now not be able to continue with that.
  
  
• Apparently spending $15,000 on blinds to cover the windows next to our classroom doors is too much, even though we need them in a lockdown situation in order to prevent intruders from seeing into our classrooms and targeting our students.
  
  
• Apparently we can no longer as a district support individual student logins to the network. Instead, all students at Arapahoe will use a single login. This affects all students and teachers, but particularly affects instruction in Technology Education, Business and Journalism.

While not a comprehensive list, we feel these concerns are more than enough already to question the adoption of these materials. We want to be clear that we are not resistant to change, nor are we unwilling to look at new approaches, we just don’t feel like the materials we have seen so far (and, specifically, the Agile Mind materials), meets the needs of our students. Instead of adopting - and spending the money on - materials that are not up to our quality standards, we would propose the following.

• Don’t adopt anything at this point. Perhaps some outstanding materials will come along in the future that will be worth adopting but, at the moment, these are not outstanding materials.
  
  
• Instead, let's use a small part of the money that would’ve gone to this adoption and invest in professional development. Since we’re beginning this transition with Algebra, why don’t we get a group of Algebra teachers together and develop materials and come up with the types of activities we want to do with our students? We feel that we could come up with materials that were at least as good as those proposed for adoption, and probably better for our students, for substantially less money.
  
  
• In addition, that professional development is much more likely to impact our students in a positive fashion than simply purchasing these materials. Learning theory tells us that humans actively construct their own knowledge and are active meaning-makers. This is not only true for our students, but for our teachers as well. Adopting canned, pre-scripted materials is unlikely to actually impact classroom practice or student learning. If we want to actually impact classroom practice, then professional development - with teachers co-creating materials and activities - is the way to do that.
  

We feel that there has never been a better time to be a teacher or a learner. We fully agree that the affordances of digital technologies and resources can improve our instruction and our students’ learning. We simply disagree that the Agile Mind materials - or any of the materials that were previewed - will actually do that. We feel that investing in professional development - investing in us - would not only be less expensive, but much, much more helpful for our students.

You trust us with your children, please trust us with this.

Sincerely,

Arapahoe High School Mathematics Department

Update 5-1-14: We've been told that the entire letter was shared with the Board of Education.

Burden of Proof: A Textbook Example

My district, like many I imagine, is in the process of making the transition from our existing math curriculum to one aligned with the updated Colorado Math Standards (pdf), which in turn are aligned with the Common Core State Standards - Mathematics. This post is not going to be about the Common Core State Standards themselves (you can thank me now), but about the "Materials Selection Process."

Traditionally in my district (as in many), when new curriculum is adopted a committee is formed to select new materials to support that curriculum. (Although, interestingly, I recently found out that in my district there is no dedicated budget for that, they just "find" the money each time they need to do this.) Once the committee researches, previews, and reviews the various materials available, they make a recommendation to the Board of Education. After a period of time for public comment, the Board then decides whether to adopt the materials.

In many areas, particularly Math, this has traditionally been a textbook-selection process. I was not part of this committee, but the process this year was a little different for a couple of reasons. First, because the Common Core State Standards are still fairly new (at least in terms of textbook publishing cycles), there are not a lot of good choices out there. Second, we are clearly in a transition period between the traditional print-based textbook and online "techbooks".

The committee ended up deciding on Agile Mind. (Well, sort of. Apparently some folks on the committee weren't entirely thrilled with the choice, and others felt like they really didn't have much choice so didn't say anything. But, in any case, that's the recommendation that's going forward.) The math teachers at my school were then asked to review the materials briefly before a webinar from the company and to share our thoughts and concerns. Here are some of my thoughts.

To summarize those thoughts, my feeling is that this isn't a good choice. While I like some of what Agile Mind is doing (I've used some ideas from the Dana Center in my Algebra class), overall I wasn't really impressed with their online techbook (with the caveat that I haven't spent enough time with it to do a fair and thorough review). It just doesn't seem to leverage much of the affordances of digital over print (see the thoughts for more on that).

I found it both interesting and convenient that for the webinar Agile Mind chose "Topic 18: Modeling with Quadratic Functions" to demo their product. I had recently taught an abbreviated version of this topic (abbreviated because we are transitioning to the new curriculum this year, so we have some of the old and some of the new), so I could compare what they clearly felt was their "good stuff" with what I had just come up with on my own.

Conveniently (again), Agile Mind starts their unit with a modeling activity built around shooting a basketball. They have an animation of two players shooting a basketball, one overhand and one underhand. You really have to see the animation to get the, umm, full effect, but I'll share a screen shot here that should give you an idea.

Source: Agile Mind, Algebra 1 CCSS Edition, Topic 18, Student Activity Sheet 1

It turns out that I used a similar activity borrowed from the MTBoS (MathTwitterBlogosphere). Which version do you think makes better use of digital resources? Which version do you think is better pedagogically? I think the MTBoS version is much better, but that's certainly debatable. What's not debatable, however, is that I can modify, alter, adjust, customize, and add to the MTBoS version as I see fit, where it's difficult to do that with the Agile Mind version (their techbook is behind a login, student activity sheet is a PDF, you can't customize their techbook).

I could go on about things I don't particularly like about Agile Mind (as well as things I like - for example, it has a cohesiveness and flow that a "put-together" set of lesson plans like mine may lack), but the point of this post is not really to criticize Agile Mind. The point (I knew I would get to it eventually) is that the "materials selection process" we (and I imagine many districts) have in place is fundamentally flawed. The default assumption is that if we are revising the curriculum, then we need to purchase new materials, and those materials are going to be in the form of a textbook (either print-based or digital, but still essentially a textbook).

I think that is wrong. I think it's a fundamental misunderstanding of the context of what it's like to be a learner today. It completely misses the advantages and affordances of digital over print (or at least open digital over pre-digested, closed digital resources). I think that for all "materials selections" from here on out, the default should be to not purchase a new textbook. That doesn't mean a new textbook can't be purchased if it's decided that's the best option, but it means the burden of proof should be on those that want to purchase a new textbook to justify why we should. To use the trendy term, what's the "value add" of these materials?

We haven't been told how much this adoption is going to cost, as I don't think they've negotiated that yet, although the figure of $500,000 has been thrown around (not sure if that's an initial cost, or a 7-year cost, or what). I'm going to assume that this will cost somewhere between $50,000 and $5 million. Whatever the final figure, I think that's an egregious waste of money.

Here's what I propose instead. Don’t adopt anything at this point. Perhaps some outstanding materials will come along in the future that will be worth adopting but, at the moment, these are not outstanding materials. Instead, let's use a small part of the money that would’ve gone to this textbook adoption and invest in our teachers. Wouldn’t it be amazing to get a group of Algebra teachers together for two weeks over the summer and come up with the types of activities we want to do with our students? (Maybe $15,000 or so, depending on the number of teacher and number of days - we currently pay $150/day stipends for teachers doing curricular work.) That would give at least one full day to work on each unit in CCSS-M - wouldn’t that be a better use of our time and money? Wouldn't that end up developing materials that were at least as good as - and perhaps better - than the materials we could purchase for substantially more money? And, more importantly, wouldn't investing that money in teachers developing the activities be much, much, much (did I mention much?) more likely to impact teachers' practice?

Adopting Agile Mind (or anything else I've seen out there) isn't likely to change what happens in the classroom with kids. (Or, if it does, it will change it in a negative fashion by providing a script-like experience for students.) But give teachers time, guidance and resources (including tapping into the MTBoS), and I think you will not only develop an outstanding resource that will get implemented in the classroom, but will also influence teachers' practice, and therefore student learning.

If the burden of proof is indeed on those wishing to adopt/purchase new materials, I would suggest that they haven't fulfilled that burden in this case. And I would suggest that districts and School Boards everywhere reevaluate the processes they have in place for curriculum adoption and materials selection. If you can't justify how and why a new curriculum or set of materials is going to help your students become better learners, then you can't justify the purchase price. Instead, consider investing in your people, and their ideas. That's truly a better way to develop agile minds.

Basketball Quadratics

It's been awhile since I've had a math post on here, so I thought it might be time. At my school we are in the process of transitioning from our previous Algebra 1 course to an Algebra 1 course that aligns with the Common Core (Semester 1, Semester 2, although much of Semester 1 will go away next year as we complete the transition). As a result we are going much more in-depth on quadratics than we did previously.

To put this particular activity in context, we will have already discussed factoring, solving by factoring, graphing, completing the square, and quadratic formula. We then touch on graphing using the vertex form of a quadratic equation. As usual, I've borrowed ideas from wherever I can and, once again, this activity is from Dan Meyer.

I've previously done this activity using Geogebra, but with the recent addition of the ability to add images to Desmos, I decided to try to go that route. As is becoming a habit for me, I host the activity in a Google Doc and then link out from there.

One of my struggles is always how much direction to give the students, and I tend to fall on the side of probably giving too much scaffolding for them (compared to some other folks). My experience has been that if I don't, we don't get very far, but I still struggle with where that middle ground should be. So my compromise with myself is to give them a lot of scaffolding as we step through the first example, then turn them loose from there.

As always, I would love your constructive feedback on this.

My Second Semester Algebra 1 Lesson Plans (CCSS Edition)

If you're a faithful reader, you know that I have some concerns with the Common Core State Standards. At the moment, however, I am tasked to teach an Algebra course that is aligned to those standards, so I'm trying to make it the best experience possible for my students.

Earlier I shared my first semester lesson plans - please take a minute to read over the assumptions and caveats in that post. Over the last few days I've mapped out a rough outline of second semester. Please keep in mind that we're transitioning, so this doesn't fully align yet and there are definitely some more "advanced" topics that we aren't quite ready for yet. This year we started the year with solving and then graphing linear equations, next year we will probably quickly review and then start with Systems of Equations, which (if successful) will give us a bit more time later in the year to fold in some of those more advanced topics.

As always, these are a work in progress, and I'm open to suggestions (either in the comments or via email), particularly if you have ideas/links to additional good activities/learning experiences for students around these particular topics. While I've "created" this particular set of plans, you'll quickly see that I've borrowed ideas from the rest of the math-twitter-blogosphere so follow the links to get to the good stuff.

Your GDA Is More Important Than Your GPA

Just posted this to my Algebra class blog, thought I would share it here as well.

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My daughter is thirteen and each morning I send her off to school with some combination of the following advice.

Have a great day. Work hard. Do your best. Learn a lot. Have fun. Be happy. Be kind. Be curious.

If she achieves most of those on any given day then I think it was a good day. I was thinking about this earlier this week when I was invited in to some of our Government classes here at AHS to talk about education. The students were having a fishbowl discussion about education, and our school system, and what changes they might want to make to make it better. Mr. Escue and Mrs. Winn invited in some other teachers (including me) to participate in the fishbowl discussions and push their thinking a bit.

Several things struck me from the conversation, including how stressed many of these students appeared to be. They were mostly stressed by the pressure they perceived they were under to "perform well" and "get good grades." It was a real struggle to direct the conversation back to learning, and what they felt the purpose of high school should be, and away from grades, graduation requirements, and colleges.

As I was thinking about this last night I thought about what I tell my own daughter each day and wondered if she feels as stressed as these students do. It turns out she does feel many of the same pressures, even though she has heard her parents say many times what we feel is important (and can even repeat it back to us). That makes me think that we (as parents) and I (as a teacher) need to do a better job of stressing what I think is important to you (my students) as well as to my daughter. Even though what I say to my daughter each morning is pretty good and succinct, I tried to think of something that might sum it up a bit better in one sentence. This is what I came up with:

Your Good Day Average (or GDA) is more important than your Grade Point Average (GPA).

So I'm going to try to say that a bit more often to both my daughter and to you guys. Because I really do believe that if you work hard, do your best, learn a lot, have fun, be happy, be kind, and be curious - then you did have a good day.

This would be a great conversation to have with your parents as well to see what they think about all of this. What do they - and you - consider to be a good day?

Air Travel: Google Apps Style

Like pretty much anything good I do in my Algebra class, this is simply a slight modification of something someone else has created - in this case, Dan Meyer. Please go read his post - and the comments - for all the goodness.

But I thought it might be helpful for somebody if I shared the slight modification I've made to it to take advantage of Google Apps. Like many districts, we are a Google Apps for Education district, so I wanted to try to leverage that for this assignment and use Google Spreadsheets instead of Excel. I'm going to have the students gather the data outside of class, following these directions and using a copy of this Google Spreadsheet.

Then they'll come to class, hopefully with the printouts of their two graphs but, if not, I'll hopefully have their shared copy to pull up on screen if necessary. We'll then talk about sketching a line of best fit (they on their printouts, me on the smart board) and coming up with a rough equation for that line. We'll then discuss, hopefully teasing out the same questions that Dan talked about in his post.

Some folks might refer to this as a "flipped" lesson, as I'm having them doing the research outside of class and then we discuss in class. I don't much care what we call it, I'm just trying to maximize my face-to-face time with them and hopefully get them to engage in some interesting mathematical thinking. They'll have two nights to complete the research, which I'm hoping will only take them 20-25 minutes total.

While I'm not a huge fan of homework, given the restraints of our schedule (my Algebra class meets four days a week, 59 minute periods when no special schedules - PLC, assembly, testing, etc.) and the curriculum (still too much for even a 5-day a week, 90 minutes a day Algebra class), I feel reasonably good about 20-25 minutes with two nights to get it done. (Next year I'm hoping to give them 3 or 4 nights to do it, I just didn't figure it out in time this year.)

My interest here wasn't so much in plotting the points, which is why I've created the two graph tabs in Google Spreadsheet for them. My interest is more in the relationship between time and distance and cost and distance, and whether they can think that through. And then translating all of that to an equation to model it. I debated for a while with just giving them the flight/time/distance/cost information up front, as gathering that information isn't the essential part of this activity, but in the end I decided there was some value both in them learning about flights/cost and how to gather the information, and about how to organize it in a way that might help analyze it. I would be interested to hear your thoughts on that decision (as well as any of the other decisions I made with this lesson.)

Help Me Make This #howtolearnmath Poster Better

I'm still enjoying my How To Learn Math class. In a recent session, Professor Boaler talks about her own "Mathematical Thinking Process" and how we often don't make that very transparent to our students. (She describes her process in the first few minutes of this video.)



As a result I got to thinking that perhaps I should create a poster for my classroom that illustrates some of these ideas, both for students to refer to and to remind me to make my own thinking transparent when I glance at it.

My design ability is, ahem, less than awesome. So I thought I'd ask for your help. Here's my first attempt. It's just text right now, but I think it could be improved in a lot of ways. Perhaps change the wording, or the font, or the colors, or the layout, or add pictures or illustrations. I'd like to have it on one page so that I can create a large poster of it and perhaps print it out for students to put in their notebooks.

It's in Google Docs, so go ahead and make a copy, improve it, and then post a link in the comments. Or, if you'd prefer to use a more robust piece of design software, go ahead and do that and then leave a link in the comments. I'd really appreciate your time and talent if you'd like to spend a few minutes making this better. As with everything on this blog, it will be freely available to everyone, so please don't include any copyrighted images.

Mindset

My How to Learn Math class is going very well - I'm halfway through the sessions and feel like it has been very worthwhile. A good portion of the first part of the class was built around the ideas in Carol Dweck's Mindset. As a result, I decided to create an "assignment" for the parents and students of my incoming freshmen. Here's the email I just sent to them:

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Algebra Parents and Students,

I hope your summer is going well. We'll be starting school in just under three weeks (woo-hoo!) and in a few days I'll be emailing some more information about our class and some tasks for the students to do. In the meantime, however, I wanted to give both the parents and the students a joint "assignment." Now, just to be perfectly clear, this assignment is optional, but I really, really, really (did I mention "really"?) think it would be helpful for each one of you to do this assignment between now and the start of school.

Okay, so here it is. As your student is about to begin high school they likely (and naturally) have a lot of fears. Many of those fears are social (we've never had freshman stuffed in a locker, although I've wanted to a couple of times), and many of those are academic (we've never had a freshman's head explode from doing too much homework, although a few have cracked a bit).

This assignment tries to address some of the academic fears that many students have. By the time they start high school, many students have decided that they are "smart" at some things, and "dumb" at other things. While this is true of all subjects, it seems to be especially true in math. The thing is, those students are wrong, and not just the ones who think they are "dumb" at something, but also the ones who think they are "smart" as well.

We know from brain research that people aren't "smart" or "dumb" at things, but that everyone can learn, and that it turns out students' attitudes about learning have a surprisingly strong effect on how well they learn. This is referred to as "mindset" in the research, and the assignment I have for you to do involves learning a bit more about how your brain works in order to perhaps change your mind(set).

Again, this is optional, but I really, really, . . . (oh, you get the idea) . . . think you should consider doing this. Ideally this assignment will be completed by a parent(s) and student working together, at the same time. I've divided it up into three "sessions" as a way to break it up a bit (perhaps even over several days or weeks), but you are welcome to do it however you'd like. Simply visit this website and work through as much or as little of it as you'd like. I think you'll find this assignment very worthwhile and I hope you consider completing it.

Either way, enjoy the next few weeks and I look forward to meeting all of you soon. Again, look for another email from me in a few days that will have additional information about our class as well as some specific tasks for students to do.

Thanks for your time,

Karl Fisch
Arapahoe High School

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It will be interesting to see if any of them take me up on it. I'd also love any feedback you might have on the mindset assignment itself.

My First Semester Algebra 1 Lesson Plans (CCSS Edition) - Draft

As I indicated in a previous post, my school is in the process of rewriting our Algebra curriculum to more closely match the Common Core. I've now completed my first run through of my (very rough) first semester lesson plans to match up with this, so I thought I'd share.

Before I do that, a few important things to keep in mind:

1. Read that previous post for the specifics of my situation (the biggest one being that I see my Algebra students fewer days/minutes than a lot of folks).
   
   
2. Here is the rough outline of our translation of the Common Core Math Standards to our Algebra course.
   
   
3. When that document refers to the "CK12 Textbook", that's the textbook alternative we are creating on the CK12 site. This is even more in draft form, so you won't find it on the CK12 site just yet. We are planning on working on it throughout the school year and, with a little luck, will "publish" it to the CK12 site for 2014-15. But here's a PDF that is the result of a very quick run through to try to modify an existing CK12 text to match CCSS-M and our Algebra course. It's a good start, but has a long way to go.
   
   (Note: Our philosophy for this textbook is that it is a resource, not our curriculum or our lesson plans. As such, it is simply a place students can go to see additional examples, explanations, and practice problem sets, and therefore does not have to be perfect or match our individual teaching styles. Our additional "philosophy" is that we hope we can do a good enough job on this to convince our district not to spends tens of thousands of dollars purchasing a new "Common Core Aligned" Algebra textbook in 2014-15.)

Okay, so now on to the lesson plans.

1. These plans are written for me, so sometimes it will not be clear to someone else what I'm talking about. I do link to some things that are linkable, but if you have questions about something, feel free to ask.
   
   
2. There are also lots of question marks where I'm still debating what I want to do and/or whether I have enough time to do it.
   
   
3. As always, these are my "big picture" plans and they will get adjusted frequently based on how things go with my students throughout the semester, so don't think of these as set in concrete. For me, though, I need a pretty specific plan to work from initially, both to make sure it flows mathematically/pedagogically, and to make sure I "cover" what I have to (as much as I dislike that word).
   
   
4. You'll notice that there are five blank days at the end of the semester (as of this writing, anyway). Those are my "cushion" as inevitably things don't go as I planned (see #3). I really would prefer more than five, but given that I only get 62 class periods (with at least 9 of those shortened periods), that's the best I can manage at the moment (subject to revision as I go through it several more times - and hopefully with some feedback from you).

I'd appreciate any feedback (or questions) you'd like to share, either in the comments or via email or other means.

Three Acts/101qs/#WCYDWT - A First Attempt

So as I'm creating my lesson plans for Algebra 1 for next year (more on that in a future post, hopefully) I'm feeling pretty bad about something. You see, there's all this great material out there developed by other math teachers and I feel bad that I haven't really contributed much to it (other than to publicize it). Not bad enough, however, to not stea . . . borrow all the material. But still, I feel bad.

So as I'm planning I came across a linear equations activity I've done the last few years that involves a story of me going to my recreation center to work out, stretching first and then running on a treadmill, and figuring out calories burned. It's a perfectly fine activity, and it has worked reasonably well, but it's completely text based.

Within the last year, however, we've purchased a treadmill for home. And this morning when I was running on it I thought that perhaps I could dip my toe into creating something useful. So this could fit into a lot of categories, but I'm guessing it might fit into Dan Meyer's three acts, 101qs or perhaps #WCYDWT.

I'm not sure if it really matters where it fits, but I'm posting it here to get some help. My problem with creating material like this is that it's not "natural" for me. I can recognize great stuff that other people create with no problem, but creating it myself is a challenge. My other problem is that I can do basic video editing in iMovie, but that's about it. With those caveats, I think this is a decent start.

In my head, this is how this would sequence.

Update 7-17-13: Split the first video into two videos to make it more obvious where Act 1 ends and the info they're likely to request in Act 2.

Treadmill 1a (Act 1?) - How fast am I running?


Treadmill 1b (Act 2?) - Information they need



Treadmill 2 (Act 3?) - The answer

Treadmill 3 (continuation) - How many calories am I burning? (Internet research based on speed, etc.)

Treadmill 4 (answer to continuation)

Treadmill 5-1 (extension) - How long have I been running?

Treadmill 5-2 (the answer)

Treadmill 6 (Second problem Act 1?) - How long did I walk and how long did I run?

So, constructive criticism/feedback is appreciated - keeping in mind the caveat of limited video editing skills and software :-).

• What would you add/remove/change?
• Does the sequence work?
• In the end, is it any better than an equivalent text-based problem?
• Anything else I should've thought of to ask.

Thanks.