Category Archives: Study of Change

Week 1 – Practicing engagement

We got through a full week of Calculus AB without a droll lecture.  The first two days were mostly for introductions and just some review, and to sort of “warm-up” my engagement/facilitation skills.  They need to practice their engagement, as well.  Students are trained to believe that teaching means showing examples and giving problems like the examples.  We’ve trained them well at this and then wonder why they can’t solve a problem.

Day 1:

After quick introductions around the table we moved to graphing.  I used a Dan Mulligan technique of engaging a class while having them talk the math.   Here’s how it works.  You work in partners and they sit back-to-back, one facing the board (board partner), and one facing away at a desk with pencil and paper (drawing partner).  They can sit shoulder-to-shoulder if that works better for the classroom.

The board partner must clasp his hands in his lap (this was amazingly fun to watch and police).  An image is projected on the board and the board and started a timer.  The board partner described the image to the drawing partner who drew what he heard.

The images began with a smiley-face, then to a graph of a line, and then to more intense graphs with holes, asymptotes, cusps, max/mins, jump and point discontinuities, etc. All that “stuff” they learned in Precalculus.  The trick is to have two, or ideally, three of each “type” of graph.  One to let them completely fumble and get wrong.  Then they can talk about what they could have said to get it right.  This is the real power of the math-talk.  This is where they discuss and agree on proper vocabulary.  They try again a second time and get it right, and a third time to switch roles and do it again.

This is a powerful conversation.  A student seeing an image, putting it into words, another student hearing words and putting it into an image.   Then talking about what words they should use.  I’d sometimes stop and ask which teams used terms like increasing, decreasing, concavity, turning points, just to give them some pointers.  I really didn’t need to, they figured it out fast.  I heard them say domain and range to set up the axes, piece-wise, closed point, open point.

Exactly zero people checked out during class.  Zero heads on desk.  Zero whining.

I’m looking forward to doing this with derivatives and second derivatives later in the year. Here’s the document with the images.

Day 2:

Still in introduction phase, with no specific learning targets identified.  Although this really did work out to teach average rate of change and just the idea of getting a rate of change from something that changes.  We used the two videos I spoke of on this post .  We used this document to instruct them how to do it. We watched the video of the speedometer.  We talked about what we saw, how fast it went, how long it took, etc.  Then we watched them only the odometer and told them to make the speedometer out of it.  I gave them this rubric and instructions of how it should be completed.

I have their first drafts, and the data looks good.  This is a work in progress that I hope to have some nice projects completed.  Jury still out on this one, but so much potential to come back to when I do integral calculus and block out the odometer and show the speedometer and ask them how far the Lamborghini went.

Day 3:

Whiteboarding!  I have seen teachers online talk about their big whiteboards and I wanted to try it.  I bought an 8×4 ft board at Home Depot for $11.73 and they cut it for free giving me 6 big whiteboards.

The activity consisted of writing equations of tangent and normal lines.  I walked them through the 10 steps and they passed the whiteboard around their group of 4 taking one step each.  The steps:

  1. Draw a x- and y- axis
  2. Label axes with tick marks
  3. Draw a curvy graph (not necessarily a function)
  4. Draw a point on the graph and label its coordinates
  5. Draw a tangent line to the graph through the point
  6. Measure the slope of the line (need a 2nd point on the line)
  7. Write the equation of the tangent line
  8. Draw the normal line to the curve through the point
  9. Find the slope of the line using the tangent line
  10. Write the equation of the normal line

They did this at least 4 times until everyone had at least done each part once.

I felt the first three days went well.  The engagement aspect of this sort of teaching is astounding.  They are actively engaged for the entire block with no effort from me to get them “back on task”.   Some things I want to remind myself about active engagement:

  • Nothing is fun forever.  These activities must be short and concise and have a learning target visible for them to know why they are doing it and when they can stop
  • Get used to chaos and learn to “hear through it” or stop it.  There were times when I could stand at a group and watch them finish a round of 10 steps of drawing in 3-4 minutes.  While they were drawing they may have been talking about something non-math related.  I allowed this.  We’ll see if that has repercussions as the year goes on.   Be alert, watch for questions all the time, and walk the class constantly.   If nothing else is going on, ask them deeper questions about what they are doing.
  • They must feel success, even if I need to intervene.  It is not ok to hope they got something out of it.   If I need to jump in and give them guidance, that is better than letting them flounder hopelessly.  The first time they walk out of that class confused they’ll say “she doesn’t teach, she just walks around asking questions”.  This is not acceptable, not in the beginning.
  • Always get an exit card.  See above.  Still working towards that.
Next class, now that they understand how things go, they will perform their first skill assessment.

The first “Big Idea” of Calculus

I may have had a breakthrough finally in how to implement a modified standards-based assessment system (notice I did not use grading or reporting in that term, I still need to report with A, B, C, D, F) in my Calculus class.  Finally!  Trying to come up with the learning targets was more effort than I expected.  Reading Understanding by Design made it clear to me that I had to put much more thinking into the targets than just putting the book sections into a file.  Big Ideas and backwards design are key.

I think I really have things going in the right direction with this.  I’m putting students in charge of their learning and I’m documenting the targets out front.

A virtual thanks to Sam Shah for posting his experience with UBD that got me thinking in the right way about what a big idea is.  I haven’t done all the other parts of the backwards design yet formally, but getting the enduring understandings (“students will understand”) and learning targets in place was the first step.  Like all work, this is a draft; a draft I’m going to use for the first few weeks, anyway!

I can’t take full credit for the part of the document that allows the students to track their learning progress.   This was something I read recently on my google reader, but since I only just learned about Google reader, I can’t find the post again, so thank you!

Anyway, here’s the document I came up with to give to students and to schedule and teach by.  The first Big Idea is “Derivatives as Limits”, with 10 learning targets.  Some learning targets have sub-targets, but I’m only going to assess the target as a whole.  Here’s the whole document

I plan on handing this out to the students.  Also behind it is a document for students to track their learning of each of the targets.  Behind that is a schedule of when their target will be learned in class and some suggested example problems for practice.

I also included the dates of the skill assessments.  I decided that weekly skill assessments are best, and will occur on Wednesday or Thursday.  We have block classes, so whichever of those days they come to class each week will be their assessment.  I’ll assess each skill 3 times during class.  After that, if the student wants to improve, they’ll have to show me that they’ve practiced and come in on their own time.

That’s the plan, ready to act!

Speedometer = Instantaneous Rate of Change

Here’s my idea for my introduction to rates of change:

Thanks to a comment in Shawn Cornally’s blog, I found this YouTube video that I was hoping to use to teach rates of change.  Tell me your palms don’t sweat when you watch it.

My first thought when I saw this video was how quickly it would hook the student’s to ask questions.  Was that you, Mrs. Lyon?  Who was it?  Where were they? What kind of car?  Eventually, we’d get to: How far did that car go?  We’d go through estimations and could even get to limits… but this is integral calculus.  I need to teach differential calculus first.   I need an odometer…

So I cropped that video and made this one:

The audio alone makes my palms sweat.  But now I can get them to think about rates of change.  How fast is it going?  See the jolts when it changes gears?  How many gears?  How fast was it going when he changed gears?

And I’m pretty sure that odometer is in kilometers.

Ready to start, now where do I start?

Today I opened the floodgates.  Ok, not the floodgates, maybe more like a pet door.

I tried to synthesize everything I’ve learned this summer into one tight little presentation that I gave at the County In-service today.  The prezi I used to guide me through this is called Swimming with the Current

Now the cat is out of the bag, and there’s no going back.  I knew this before the in-service.

Lots of credit (and thanks) go to everyone on my blogroll for getting me ready for the presentation.

I talked through the presentation, but would like it to stand on it’s own, so I’ll be updating it. Any suggestions?

At the Center of my Classroom

This post is submitted for the Virtual Conference on Core Values.

Is change.  I’m swimming with the current now.

6 years of teaching math, 3 years as department chair, and 2 years teaching Calculus after a 22 year career as a CPA.  That’s me.

I feel, as I’m sure many math teachers do, like I’ve been swimming upstream against everything.  The students aren’t getting it.  The county wants more.  The school wants more.  The parents want more.  It’s them against me, isn’t it?  I was starting to see why teacher burnout happens between 5 and 10 years…

…until this summer when I attended the High Tech High Summer Institute and see differently.  I should join the county, school, students and parents and start swimming with the current!

The institute was awesome, but the learning didn’t stop there and the obsession began afterward.  All of the presenters at the sessions referenced “blogs” (I put “blogs” in quotes because I felt like it was a term that might be new to you.  I’m old, it’s new to my people.)  One presenter, Allison Cuttler, encouraged everyone to get involved in the online community of math teachers.  “Okay”, I told myself (quotes this time for the fact that I said it, picture a lazy fist-pump on the “kay”), so when I returned to the humidity of Alexandria, VA, I sat in my AC and started digging.

I have been nothing short of obsessed with the idea of changing!

Three things need to change immediately in my classroom, assessments, lessons, and me.  “Um, what doesn’t change?”  (that’s you speaking, hence the quotes)  My shoes, much to the dismay of my administration.

Assessments will be standards based.  I’ll need to assign a grade, but that won’t stop me from giving shorter 2-3 concept assessments that will continue to spiral through the curriculum and require mastery.  Similar to what is brilliantly laid out by dy/Dan (in 2007) here .  I have the complete support of my county, principal and administrator, and hopefully, you.

Lessons will begin with a real-life problem that needs solving, much like is demonstrated in Sam Shah’s Continuous Everywhere Differentiable Nowhere.  I lectured, I’m sorry.  This one may take some time to get into but I have some great ideas I’ll share about Calculus lessons starting with the end in mind.  See also dy/Dan’s The Three Acts of a Mathematical Story.  (to my old friends, it’s not creepy to like someone’s blog)

As for me, this is my birth into the online world of blogging, tweeting, social bookmarking and whatever else I may discover in the months and years to come. (Resist the urge to use quotes on all that foreign stuff).  I will remain active and keep you posted on my progress and will stop lurking in the sea of tweets I have yet to figure out how to sort through and perhaps give a tweet now and then.  Please point me to more good reading, good advice, and good math, good resources and good support.  If you’re here reading this, it’s because you are part of this incredibly rich online community.  Thank you, not for just reading, but for providing me with the enlightenment you have provided to all teachers.  If you’re not yet on my blogroll, it’s probably because I’m trying to figure out how it works, but send me a tweet anyway, maybe I’ll figure out how to get it.  (quotes…must not… quote)

“What hasn’t changed?”, you ask.  My love of math, teaching, and how much I care about the students.  I believe that will show if I stop fighting the current.




“blog” “tweet” “social bookmarking” “blogroll” “widgets” “rss” Had to get it out of my system.  Getting there.