Saturday, September 14, 2013

The Exponential Curve, Phase II

I haven't posted anything for a couple of years now, and I think I'm finally able to start up again.  I'm at a very different place in my life now, compared to when I first started this blog, and, thankfully, compared to when I stopped writing.

I am really excited by the move toward Common Core, and the abandonment of the CST tests.  I've felt my spirit and teaching practice withering and dying with each new modification I made to try and cram more standards in faster and more efficiently.  When students would ask why they need to learn math, I would give a standard reply about problem solving abilities and critical thinking in all areas of life, and this felt more and more cynical as I compared what I was saying to the actual content I was delivering.

This year, in my intensive algebra 1 class (for students who are well below grade level), I've quit worrying about skills lists.  I'm focusing on using high-quality problems and resources.  I'm trying to actually do what I say, and engage them with activities that require thinking, explaining, justification, problem solving, and persistence.  And since the new Smarter Balanced tests, according to David Foster, will be only 31% material at levels 1 and 2 of Webb's DOK, and 69% at levels 3 and 4, I feel like I can justify my new approach to any skeptic.

Just yesterday, I spent nearly 30 minutes playing the Game of 21 with them, a quick and easy misere game.  (The first person says 1, 2, or 3.  The next person increases by 1, 2, or 3.  Alternate turns.  You can't go past 21, and the first person to say 21 loses).  It was great to watch them struggle with it.  Of course, I made them start, and so I won game after game.  But then some students started realizing that I was going to win as soon as I said 16, and after a while longer, they realized I would win when I said 12.  Some students were frustrated, some continued to challenge me blindly, and some were clearly paying close attention and trying to develop a strategy.  I love the moment when the first kid tells me that, no, *I* need to go first.  And then when they finally beat me, it's a great moment!  At this point, I stopped the game without discussing the strategy, and told them to play it against a friend or family member for homework.  We'll pick it up again on Monday and see if more can beat me.

We've been working on finding out the rule for patterns, using only visual examples, building them with toothpicks, etc.  So I decided to start them on the non-linear Growing Staircases POM from SVMI for the second half of the class (A pattern where you start with one square, then add two, then three, etc).  First, I spoke with them about what perseverance means, and why it's so important.  I told them that we were going to all get through level C of the POM, but we were not going to finish today, or even by the next class.  That real math problems take longer than an hour to solve (or the 15 seconds that they are accustomed to).  They mostly completed level B (figuring out how many squares it would take to build a 10-step staircase), and they used counting and other patterns to do so.  Level C (find a rule for the number of squares in an n-step staircase) is going to be a lot harder, and I'm looking forward to seeing what they come up with.

I decided to do a lot of patterning work with them before we even talk about solving equations, using their visual experience with developing a rule as leverage for understanding.  When they see something like 4n + 3 = 51, I want them to think something like "That's 4 groups of n-blocks, and then 3 blocks more, giving a total of 51 blocks.  So the 4 n's would have to make 48 blocks.  So each one would be 12!".   Right now, their pattern work ends with coming up with a rule.  Soon, I'll start asking them to figure out what step number it would have to be, given a specific total.  I'll treat this as a problem-solving exercise now, and eventually formalize the steps into the standard solving equations algorithm.

I've also been drawing heavily upon the excellent work found in the Formative Assessment Lessons at the Mathematics Assessment Project .  These are really meaty lessons that incorporate excellent group tasks, and really help push students to use the CCSS Math Practice standards.  So far, in this algebra class, I have used Positive and Negative Numbers in Context and Interpreting Algebraic Expressions.  The one caveat I have is that the amount of time it takes to run one of these lessons is well underestimated in their notes.  The expressions FAL took my intensive class four lessons to complete, not the optimistic 1 lesson with a 20 minute follow-up!  But of course this will vary from class to class.

Aside from this, I am also teaching Algebra 2, and I am using CPM instead of whatever NCLB book we have sitting in our library.  This will be the subject of other posts, as my students and I wrestle with a completely different way of teaching and learning math.

And, as if this weren't enough, I am teaching the first year of IB Math SL (which is roughly equivalent to the first year in a pre-calculus/calculus sequence).  This class is filled with really positive, engaged, motivated students, and it's a joy to teach, so I shouldn't complain too much about the extra prep!

So hopefully this post will be the first of many more to come.   Being part of the mathtwitterblogosphere is invaluable, and I want to start adding my voice to the mix again.


Kristin said...

Welcome Back! We are doing "Positive and Negative Integers in Context" on Monday. I am hoping to keep it to 1 1/2 class periods, but it never works out that way.

Sorry if I have sent this numerous times, I don't know if it is working!

Dan Wekselgreene said...

I enjoyed that FAL. However, it wasn't nearly enough for my students to master integer addition and subtraction. They have so many misremembered rules from middle school clogging their heads and they same-change-change everything. I made additional practice problems, and when I forced them to draw a diagram, they could almost always get the right answer. But when I don't explicitly tell them to do so, they fall back on old habits and start getting the operations mixed up again. I am going to have to continue practicing integer operations all year, I think. But they will get it at some point! Please let me know how it goes in your class.

Anonymous said...

It's great to see your post! We began Algebra 1 this year with the FAL Laws of Arithmetic. It took us about 4 days and was a great way to start the year. I'm somewhat procedural myself, so it really helps us all to work with visual representations of math.

I look forward to hearing more from you.
(Gill Sans is my font, too.)


Anonymous said...

Welcome back. I just found your blog this summer, and have linked to several of the resources here. I love this 21 game and the way you describe it. I'm hoping you manage to keep updating it with the things you are doing this year.

untilnextstop said...

Wow! Welcome back!! And let me know if you have questions about IB SL at any point. I look forward to reading about your many ideas!

Dan Wekselgreene said...

Thanks all for the encouraging words. I do hope to keep posting, and I think I can make it happen!

Untilnextstop, I'm sure I will have questions about SL. Do you have any suggestions or things I should be thinking about from the beginning?

untilnextstop said...


One of the things that really threw me off about that IB SL curriculum is how fast-paced it needs to be in order for you to cover everything. Realistically, you have only 1.5 years to teach all 6 math topics, because during the last half year of the IB program, your students are going to be SO stressed out from all their other classes' due dates for big projects and papers that even the best students will be in melt-down mode and unable to absorb new material effectively. At that point (starting around early February 2015), hopefully you've already finished teaching all the topics and you're just reviewing and helping them with review strategies.

Also, calculator use cannot be over-emphasized. On the IB exams, the calculator section can be way trickier than the other section in terms of complexity and content, which means that your students must have the instinct to go to the calculator whenever it is available, to help get through parts of a problem. You should always teach/practice calculator skills prior to introducing the manual calculation component, else the kids will not develop that instinct to rely on the calculator. They must learn to approach the two exams (calculator vs. manual) with entirely different mindsets.

Also, all of your assessments from the start should be using old IB problems as much as possible, because the IBO tries to make the language in the questions as mathy as possible, which means that the kids will need significant ramp-up time to learn to read and decipher those problems comfortably. You can buy an IB Questions Bank CD from IBO to help with planning those assessments.

Another thing that I recommend definitely is talking to your kids early on about the three elements of successful test prep:

1. Reading questions and understanding what's being asked.

2. Connecting questions to key concepts (I teach my kids to create flash cards for this to help reinforce those connections)

3. Basic skills for executing the solution (of which there are many, and you shouldn't hope to review all of them during class... some of this will have to be done at home)

If you keep in mind that the IB is taught worldwide, it touches upon the cultural differences to education: American kids are used to their teachers reviewing all necessary skills during class. European and Asian kids do the bulk of their learning independently at home (and class time is diminished during high school compared to before), so your kids will HAVE to be more responsible than the average American student in order to excel on this test, because the sheer coverage is not possible to do entirely in class including review of basic skills.

In terms of pacing, I recommend that you get through all of Algebra, Functions, Trig, and differential Calculus by the end of the first year, as well as finishing the internal assessment project before the summer. This leaves you with a few months to do integral Calculus, Prob/Stats, and Vectors in the second year. Leave Vectors till last because that's one of the more straight-forward topics as your students are starting to lose their minds over stress from all classes...

I wish I could say that the IB is a good curriculum. It has some advantages in integrating math topics, but the amount of stress really takes its toll on students and teachers, and so you have to approach it with that long-term goal in mind. Good luck!

Dan Wekselgreene said...

Thanks for taking the time to write! I know exactly what you mean about the stress level of the students. I taught IB Math Studies for the last two years. They were great students who were loaded down with IB classes, and very often Math Studies took a back seat to whatever major assessment they were working on. And it's understandable, because the students who chose Math Studies instead of Math SL are typically either weaker in or less interested in math, or both. They take it so they can have a 4th year of math, or if they need it for diploma. So I tried to be pretty flexible with them, as they were often coming in after staying up all night, bleary-eyed, sometimes crying.

Right now, at the beginning of year one of SL, everything is fine, but I can just imagine what it is going to be like for them next year.

I agree with you about the calculator skills. That is something I will definitely focus on this year. I'm also going to incorporate using desmos, because I think it is much easier to use, see, and understand what is going on. Too bad they can't use it during exams! But I think that the specific TI skills are easier to teach than the actual concepts of graphical analysis. We've got a copy of the test bank CD, although it's hard to find questions this early on that cover only the skills they already know. Right now, I find I still have to modify them to an extent, but I really like using them. It does give them practice with the IB testing vocabulary. So far, we've done a lot of work with "show that", and how you can't use what you're trying to show as part of your explanation.

Do you have suggestions for what concepts students are able to successfully learn on their own outside of class? I often misjudge, and something I think should be easy for them turns out to be very difficult... and vice versa.

untilnextstop said...

Something that worked very well for me to was to have spiraling quizzes right from the start. This way, I am not making assumptions about what they can do on their own outside of class. One of the things I quizzed kids early on in Grade 11, for example, is to turn any standard-form quadratic equation into both vertex and factored forms, and then to graph info from ALL 3 forms on one graph. We'd start each class doing 1 such practice problem, or I'd assign 1 problem as homework to review at the start of next class, and after doing this a few days in a row, they'd say they're ready for a quiz on this skill, and we'd take a quiz.

So, instead of re-teaching everything, just highlight what you think is MOST important and do that repetitively for a few class-openers and then hold them accountable to it. Over time, the quizzes should incorporate multiple topics to deepen their understanding of connections. For example, by 2nd year of SL IB, they should be able to find a wave equation given a graph, then graph a quadratic on top of it, then find area between the curves.

untilnextstop said...

oh and I always remind them to check their answers with a calculator. So for example, they should know how to check that their formulas are factored correctly, or that their vertex formula is still the same as the original formula. Or if they are finding the integral by hand, require them to also find it by calculator. If you build this into every quiz, they'll get used to using the calculator maximally, especially if you emphasize the same in those practice quiz rounds.

untilnextstop said...

Oh and I think I should clarify that I never expected kids to learn a new topic on their own, but I don't waste class time reviewing all the pre-requisite skills they should know already. All of those I just go over quickly in class and then assign as homework for them to catch up on on their own time. If they need extra help with factoring, for example, then they need to be looking up resources online or seeing me after class, because we don't have time to re-teach it during class.

Amy zimmer said...

Just wanted to say Welcome back. Good to read you again. You were the first MTBoS guru I found and have revisited every since.

Have a great year!