Monthly Archives: September 2012

Let THEM Figure Out the Power Rule

I have been reading and enjoying (though not fully buying everything in) Daniel T Willingham’s book Why Students Don’t Like School: A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for the Classroom. One of the ideas that I think is really useful in planning instruction is that humans are wired to enjoy learning – some scientists believe that the brain releases a little bit of dopamine every time we solve a problem. We actually physically get pleasure from solving problems.

As an example, check out these two word picture puzzles (figure out the common expression indicated by the words and their placement):

Which of the two puzzles did you enjoy more? If you’re anything like me, or most human beings, you didn’t really enjoy the one that had the answer right above it. Even if you didn’t figure out the other one, you probably at least thought about it more than the other one (though Willingham points out that the physical response only occurs when a person solves a problem). How often do we give the answers to the riddles first in math instruction?

Here is a rule and here are examples of every type of problem you will have to do with it, now do problems like those even though you kind of already know the answer.

An example of posing math as a riddle instead:

It took a few days for students to learn the power rule this year, as opposed to me just writing f(x)=x^{n} so f'(x)=nx^{n-1}, which takes about 10 seconds (if you talk while you are doing it and write very, very slowly, and have to erase something in the middle because you forgot what you were doing). Despite the time needed, I felt that the cognitive payoffs with the progression I used were great, and students really internalized the idea because THEY FIGURED IT OUT THEMSELVES. Figuring out the Power Rule is something that is totally in their reach, and I would have been robbing them of some learning pleasure had I just given them the power rule at the beginning.

PHASE 1: What is a derivative? We started just by drawing tangent lines to f(x)=x^2 at a bunch of points, estimating the slope and then making a table of values. I chose this function specifically because with the derivative of f'(x)=2x, it’s easy to see the pattern for the slopes in the numbers without graphing them (saving one level of abstraction). Yay, the slope at any point is just twice the x-value!

Then we did this two more times, once on a small sheet of paper for \sin x, and then once, in groups, on a huge sheet of butcher paper for \frac{1}{3}x^{3}. This was laborious and took a ton of time in class, but by the end I felt like students really understood well the idea of a derivative. More importantly, were ITCHING for an easier way to find it. They had all these great ideas that they were proposing, so it was easy to funnel their energy into the next phase.

PHASE 2: Finding the Rules. Then, I introduced the derivative tracer, a GeoGebra applet that does in seconds what they did in 15 minutes. I gave them a sheet of functions (below) for them to find the derivative of using the derivative tracer (kind of like collecting data in a typical canned high school science lab) and asked them to make conclusions about the derivatives that they found.

Though it was interesting to talk through with them the idea of a constant function’s derivative being 0, and a linear function’s derivative being a constant, the highlight of the lesson was seeing students figure out the power rule. When students got to that section, they seemed really proud that they could see the pattern. I had numerous students raise their hand to call me over to ask me if their idea worked, and then were so excited that it did that they immediately gave me a high-five. Students raised their hand so that I would come give them a high-five… in math class. I know the power rule is kind of easy, but I felt like they were so much more invested in the quest of learning mathematics because they figured something out themselves. Further, instead of trying to get ideas from my math notation, they had the ideas first and then I formalized it with math notation (though many students could do this no problem for themselves).

Long story short: The excitement in the room while the students were discovering something mathematical was palpable, even though that thing had been discovered many many many times before, including by their classmates sitting a few seats down. There was no “real world” motivation in this progression, no gimmicks – just the pure pleasure of mathematical discovery. So, to add to my ever lengthening list general goals for the year: I hope to avoid at all costs robbing students of the pleasure of figuring something out for themselves.

Here is my derivatives “lab” using the GeoGebra derivative tracer. Note that I’m not quite as adventurous as some and still want some structure in the classroom while “discovery” is happening. This is part of my controlling personality – tell me if you think this is too guided given my goals.

By the way, the answer to the other word picture puzzle is “mathematical induction.”


I’ve been a little quiet on the blog front for a couple of reasons, the main one being that the beginning of school is always crazy busy as you’re trying to claw back into work mode after a few months of leisure while at the same time attempting to teach students whom you don’t know and don’t know you or your routines and expectations…

But another reason I have been quiet here is that I have been steadily working on my #180blog. For those that haven’t heard of these, the idea is simple – I post a picture and a few sentences about every day of class for a whole year. I have found the process to be awesomely reflective, and I can see myself totally looking back at this in future school years.  Also, a bunch of other teachers are going the same thing, so it’s fun to get a picture of other people’s classrooms.

Calculus Standards 2011-2012: Feedback Requested

I’ve been toying around with my learning objectives for Standards Based Grading in Calculus for three years now, and I want to get some other people to weigh in on what I have. Please, take a look, tell me what you think!

Some notes:

  1. I love the first person language, which is an idea I think I stole from @kellyoshea.
  2. The physics modelers all have crazy acronyms for their standards like CVPM and UBFPM and ERMAHGERD. These seemed confusing to me at first, but then I thought that students might really benefit from this. The standards aren’t organized around chapter numbers, or something else arbitrary, but rather BIG DEEP IDEAS (models!). I wanted to do something similar for Calculus, so I organized mine around Local Linearity, Slope Functions, Proportional Rates and Accumulating Change (with short, simply worded descriptions in the document below). I don’t know how well this worked last year, but one goal for me is to try to always relate the standards back to their big ideas.
  3. I didn’t do the standards like this fully in order, and this year I am totally changing the order. But just to give you an idea of how I did things, I did all the IP and LL (limits) standards, then SF.a through SF.g (basic derivatives), then PR.a (optimization), then SF.h through SF.n (graph sketching), then PR.b-PR.h (exponential functions), then SF.o/PR.i (implicit and related rated), then all the AC standards. It was a bit confusing to go back and forth, but organizing the standards like that made it make so much more sense to me. Tell me what you think about that…
  4. I struggle with how general/specific to make the standards, and how to include both calculation and interpretation into the standards. Sometimes I split the two, sometimes I kept them together. This is the hardest thing for me!

Anyway, any thoughts are necessary! These are my standards from last year, the second time I taught Calculus.

Calculus Standards 2011-2012

Math Blogger Initiation Week 4

And the fourth and final installment of the New Blogger Initiation. Some great new blogs popped up, ones that I definitely will be adding to my Google Reader. Please, click below and comment away!


Making Paper Airplanes | Making Paper Airplanes

Making Paper Airplanes @makingairplanes has a blog named Making Paper Airplanes. The fourth post for the Blogging Initiation is titled “Change is in the air” and the author sums it up as follows: “Faced with a schedule change resulting in taking on a new, mixed-grade class a week into the school, I have to re-think my plan for this year’s math support class! It sure pays to be flexible…” A memorable quotation from the post is: “This isn’t quite what I signed up for, but it will be an adventure!”

–> My take: So many things about teaching feel so out of our control (the schedule, the students we get, the room we’re put in etc) that changes like this can be so frustrating! This blogger has quite a challenge ahead of shim (I don’t know if it’s a woman or man) and seems a bit pessimistic – the online math teacher community to the rescue! People have already given some great advice already. My advice: have the older students teach topics to the younger students. 

Bruno Reddy | Mr Reddy’s Maths Blog

“Bruno Reddy @mrreddymaths has a blog named Mr Reddy’s Maths Blog. The fourth post for the Blogging Initiation is titled “Language Revelation” and the author sums it up as follows: “I attend a real eye-opening training session on speech and language difficulties. There are 3 very short video clips of the training to help demonstrate what was going on.
I came to realise, through a very innocent activity that the trainer had us do, that I was getting it wrong for my pupils. Wrong in the way I interpreted their behaviour and wrong in the way I posed questions.” A memorable quotation from the post is: “Suddenly my mind was racing through the faces of my pupils who do exactly the same – they find it hard to look you in the eye, their movements are pronounced and they look pained when stuck for words.”

–> My take: It is great to see someone get excited about Professional Development –  a great experience seems to be more rare than it should. I really like some of the conclusions Bruno makes from this activity, as language is a something I am intensely interested in as someone who teacher 95% students for whom English is not their first language. It just makes me appreciate how important communication is in math. Random question that the British vocab in the blog title reminded me of: some of my students here say “factorize” instead of “factor.” What’s that all about? Is that a British thang?

Nathan Kraft | Out Rockin’ Constantly

Nathan Kraft @nathankraft1 has a blog named Out Rockin’ Constantly. The fourth post for the Blogging Initiation is titled “Exploiting My Son for Math” and the author sums it up as follows: “I use my son in pictures and videos to teach 7th/8th grade math.” A memorable quotation from the post is: “Over the last year I’ve been using him for all sorts of math lessons – many times under the guise that I’m spending quality time with him.”

–> My take: You have to watch some of the videos in this post. This kid is so cute! And the problems that Nathan poses are really interesting problems, totally fitting in the whole 3-acts type of lesson design. I am so intrigued by the first one I want to go try it out!

Tim Reinheimer | Asymptotically Cool

Tim Reinheimer @timreinheimer has a blog named Asymptotically Cool. The fourth post for the Blogging Initiation is titled “abstract misconception” and the author sums it up as follows: “I believe a lot of students have difficulties with algebraic rules because they don’t have any connection on which to base the abstract. In short, I believe the real world could help this problem.” A memorable quotation from the post is: “I believe a lot of students have difficulties with algebraic rules because they don’t have any connection on which to base the abstract.”

–> My take: I like this small idea to help students with the idea of the distributive property, though the science teacher in me is aaaagck-ing at the mismatch of units. Some of the basic rules for math seem arbitrary (like order of operations) but arise out of little situations like this. I guess the trick is to find these situations to latch onto.

Paul Gitchos | Second Thoughts

Paul Gitchos has a blog named Second Thoughts. The fourth post for the Blogging Initiation is titled “Thank you, Mrs. F” and the author sums it up as follows: “I’m feeling thankful that the majority of my students have had good training in working cooperatively in groups. In the post I acknowledge a colleague’s hard work.” A memorable quotation from the post is: “The most successful parts of my first couple days were due to the math teacher down the hall.”

–> My take: I usually express the opposite sentiment (I curse the teacher who didn’t really teach them what graphing meant) so I really love this positive post thanking a previous teacher for a job well done. It also made me realized how intensely satisfying a smoothly running classroom is. It feels like a waste of time to train students in things like that, but once they are trained, it is really worth it because it really facilitates learning.

Michelle Riley | A Year of Growth

Michelle Riley @mathwithriley has a blog named A Year of Growth. The fourth post for the Blogging Initiation is titled “Foldables and My Turn to Give Back” and the author sums it up as follows: “I stole a few foldables, charts, etc. from other bloggers, and this shows the way I tweaked them to work for me. I also created a (very) simple foldable for kinds of angles and shared that as my first thing I have shared with others. This is an older post… first week of school caught up with me and I ran out of time and brainpower to post something new.” A memorable quotation from the post is: “First of all, I need to say a huge thank you to the blogging community for being so willing to share!”

–> My take: Michelle totally gets the blogosphere – steal and share, steal and share, steal and share! I have to be honest that I’m not totally sold on the idea of foldables yet, but I do teach seniors, and I would probably be far more into them if I had younger students. From what I can gather with no experience with them, these seem like great foldables to steal if you use them in your classroom!!!

Math Blogger Initiation Week 3

Week 3 of the New Blogger Initiation! After three weeks, more than 90 people are still blogging. Awesome.


Joe B | lim joe→∞

Joe B @forumjoe has a blog named lim joe→∞. The third post for the Blogging Initiation is titled “Everything is Mathematical” and the author sums it up as follows: “I link to a new mathematical puzzle site called “Everything is Mathematical” and discuss the form of the content and how it will be useful. I then post my solution to the first problem, improving my Latex skills in the process” A memorable quotation from the post is: “I’m really impressed by the way Marcus du Sautoy presents the problem in an easy-to-understand way. There’s no pseudocontext here, there’s no anyqs.”

–> My take: Joe presents a really nice, accessible problem that could easily be used in the high school classroom – the question of how many palindromic numbers there are of a certain length. I love problems like this because I am totally biased to the application end of the spectrum in math in my teaching and I am looking for ways to introduce beautiful, rich, theoretical math into my curriculum. My favorite part of the post is that Joe makes a major error in his solution (one that I actually also made when I read the question) and graciously acknowledges this in in the comments. What a great model for students.

Joe Ochiltree | Brain Open Now

Joe Ochiltree has a blog named Brain Open Now. The third post for the Blogging Initiation is titled “Which Spawned the Title, “Brain Open Now” and the author sums it up as follows: “Not sure if I’ve ever explained the name of this here blog. “Brain Open Now”, you can see it right up there. So, what does it mean? I’ll tell ya.” A memorable quotation from the post is: “This sounds suspiciously like blogging.”

–> My take: Great blog name, taken from a great Mathematician! I really like this little vignette. Side note: I have actually been subscribed to this blog for a while now, so I was a little surprised to see it pop up for this.

Ana Fox Chaney | Make Math

Ana Fox Chaney @AnaFoxC has a blog named Make Math. The third post for the Blogging Initiation is titled “Computer Multiplication” and the author sums it up as follows: “I recently saw a video demonstration of “Egyptian Multiplication” in which the presenter described how both Egyptians and modern computers multiply using binary. It seemed so easy – I couldn’t resist the urge to take the technique apart and figure out why it works. Does it work with all bases? Is there a reason we don’t do all our multiplication that way?” A memorable quotation from the post is: “I like this because I talk a lot about multiplication strategies in my 5th grade classroom, modeling how multiplication works and what it means.”

–> My take: I really liked seeing Ana’s (Ana Fox?) thought process as she worked through Egyptian multiplication, comparing it to our modern algorithms in both utility and facility, and asking the all important question “WHY DOES IT WORK?” To be honest, I haven’t fully wrapped my brain around it yet, but it’s a great example of a perplexing problem that could be used with a wide age range of kids, one that grabbed me and one that might grab your students too, especially if you frame it in a mysterious, historical context. Also, Ana has ridiculously nice handwriting, something with which unfortunately not all teacher are blessed…

Mrs. W | Mrs. W’s Math-Connection

Mrs. W has a blog named Mrs. W’s Math-Connection. The third post for the Blogging Initiation is titled “Discovering and Teaching” and the author sums it up as follows: “In this post, I write about how I let my students discover the rules for exponents and the question I used that got them thinking even more about dividing exponents!” A memorable quotation from the post is: “I’ve been using some more challenging questions and my questions are creating some incredible questions and discovery.”

–> My take: I like the idea of a parking lot for exit slips. Mrs. W has a nice way of organizing and keeping old exit slips, which might be helpful. I didn’t quite understand why it was necessary that a kid park their answer in their specific spot (as opposed to just handing it in) but I am all about all things that make the classroom run smoother, and this seems to be a routine that helps her students learn. She also has a nice aside where she talks about how she motivates the need for exponent rules.

Stephanie Macsata | High Heels in the High School

Stephanie Macsata @MsMac622 has a blog named High Heels in the High School. The third post for the Blogging Initiation is titled “When will we ever use this in real life?” and the author sums it up as follows: “I wrote this blog post about how I address the constant “When will we use this in real life?” question. It is important to me that my students find the value in what I am teaching them, or at least to try my hardest to help them see the value in math. I also wrote about how I try to handle situations when a student has been told that it is ok that they aren’t good at math because their mom (or dad or brother or sister or someone important in their life) wasn’t either. “ A memorable quotation from the post is: “It doesn’t matter what type of job you have or what is going on in your life…problems arise and you have to be adept at finding solutions to those problems.”

 –> My take: Stephanie shares a lot of the same frustrations that we all seem to when faced with cultural acceptance of “being dumb at math” and reducing math to a utilitarian affair. The only thing that I think she leaves out is the idea that we should study math because math is BEAUTIFUL! There is a nice paragraph in this post where she talks about her view that anyone can learn math. Because of that, I can speak for everyone when I say that Stephanie, we’re glad to have you in the classroom!

Katie Cook | MathTeacherByDAY

Katie Cook @kjgolickcook has a blog named MathTeacherByDAY. The third post for the Blogging Initiation is titled “Why do we have to learn this?” and the author sums it up as follows: “Why do we have to learn how to do geometry proofs? Why do we even bother teaching geometry proofs?” A memorable quotation from the post is: “No one is sitting in 9th grade English class asking their teacher, “why do we have to learn how to read and write?” (or maybe someone is…I actually wouldn’t be that surprised)”

–> My take: I think Katie’s answers to this question are adequate, but she seems to struggle with something that really bugs me too – how do we teach curricular objectives on standardized exam well if we don’t really believe in them? I think that Katie uses a few too many external reasons though for motivating the math in her course, and I think it would be better if we could comment for her on some reasons why geometric proofs are worth teaching in their own right. Come on people, answer her call for ideas!

Scott Keltner | Good for Nothing

Scott Keltner @ScottKeltner has a blog named Good for Nothing. The third post for the Blogging Initiation is titled “Remainders: Not Just The Rest of the Story” and the author sums it up as follows: “Bar codes are a peculiar oddity to me, especially those newfangled QR codes (which I’m still trying to research how to decode and encode manually without the use of a camera). This post makes examples of UPCs and ISBNs and the structure that makes them what they are, including the algorithms behind each. This post shows a real-world application for remainders when using whole numbers to compose a code structure. I’m still trying (unsuccessfully, at present) to find a real world application for remainders with polynomial long division, though.” A memorable quotation from the post is: “I created (what I felt at the time was) a good introductory worksheet on modular arithmetic, using students’ complaints about “always having the same thing for lunch” and made up a rotating set of dishes served in a neighboring school’s cafeteria.”

–> My take: This is a detailed account of where we see remainders used in UPCs and ISBNs. I found it really interesting that there is a crazy complicated algorithm for this, and I still wonder why they do it like that after reading it. It’s cool to see the math behind something that we use every day. Scott asks “I’m still trying to find a real world application for remainders with polynomial long division.” My response is “Good luck.”