Dialogue in Math Class

Today began the real work of the year after three half-days last week.  We’re at the beginning of a 1-to-1 chromebook initiative, which comes with its own pitfalls and moments of bliss.  Last week I had students complete a ‘menu’ for reviewing previous courses.  In the past, I’ve had it on paper, where it looks something like this:


Students were asked to complete at least one problem from each column, and score a minimum of 50 points.  Digitizing this using Khan Academy, I came up with a similar set up:



I really like the principle here–students having some choice as to their review, but still starting out the first three awkward days of the year (37 minute classes) with some actual math.  This year, I definitely did not allow enough time for the material, but I’m not sure if that was a matter of students getting used to chromebooks, technical difficulties, and lack of access (some students still haven’t picked theirs up as of today…), or that I put way too much in there for students to complete in the given time.  I’m a little mixed in terms of my feelings of how the opening activity went, so I’ll take a look over the course of the next week or so at the work students completed, and comment on that a little later.

Today’s work in most of my classes was completing our first math labs of the year.  I created ‘lab documents’ guiding students through activities (some borrowed, some created) built using the teacher.desmos system spread throughout the year in Algebra II and Precalculus.  Today, one Algebra II class explored Linear Marbleslides while my co-teacher and I went around the room chatting and listening to students interactions.  My Precalculus class completed a pair of creating functions activities (Part 1, and Part 2), both created by Megan Schmidt.  I really enjoyed listening to the students arguing about math, and coming up with some interesting discoveries.

Here’s a snippet of one conversation that I particularly enjoyed, where students were trying to figure out if it were possible for an interval to be decreasing, but positive.  These are two students in a lower level precalculus class.

A:  Can I tell you what I said to see if it makes sense?

M:  OK.

A:  His statement is false.  Just because the graph is decreasing doesn’t mean that–the function can decrease even if it is not hitting negative numbers.

I’ll close with that thought today–two students having deep conversations and thoughts about mathematics, caused by their exploration using some cool activities from desmos.



Digital Learning

Next year, we’re going 1-to-1 with Chromebooks in my high school.  I’m a little worried that initiative overload will get in the way of creating meaningful experiences for our students to engage in.

My goals are to: (1) identify a limited number of specific topics in Algebra II and Precalculus that make sense to work within a digital framework, (2) find parts of my existing curriculum that make sense to digitize, (3) create meaningful activities for students to engage in digitally, and (4) share some of those experiences here.  I’d love for this to become both a place for my own reflection, and a source for other math teachers to find useful material.

I’ve spent a long time partially engaging in the math blogosphere–silently reading blogs, and occasionally trying to start my own blog, and quickly stopping.  It may be that this attempt follows suit, but my hope is that this time it sticks.

I’m writing this sitting at a PD session focused on using Google Apps, where we began with a discussion of “digital learning” vs. “digitized learning” (article here).  The authors describe “digitized learning” as essentially creating a digitized version of materials (i.e. textbooks, worksheets), and not transforming instruction or interaction with the digital medium. Textbook publishers are often guilty of marketing their exciting new digital books as groundbreaking, when most of what they have done is create a digitized version (think non-interactive) of a paper textbook.  “Digital learning” on the other hand is digital-native, that is, something created to exist and interact within the learning environment.  Look at the activities at teacher.desmos.com, where students can play and interact while discovering important mathematical ideas.

While I tend to despise jargon – I think the distinction here is an important one.

There are times in the math classroom that digitizing is absolutely the right way to go.  I’ve got portfolio checklists, that when digitized become a paperless way for students to collect evidence of learning and send it my way.  I don’t need to create some kind of digital ecosystem for these to be useful–they serve the same function, in a prettier way than their paper counterparts.  Cool, yes.  Useful, yes.  Transformative? Not really.

On the other side are worlds that open up in the digital realm.  Using Marbleslides: Periodics, I can have my students explore graphs of trig functions in a lot more natural way than having to draw them by hand (they will do this too in my classroom, but not at first), or drawing them one at a time on a graphing calculator.  The digital environment provided by the wonderful desmos faculty creates a space where you can see transformations in trig functions, and discuss what you learn with your classmates in real-time.  As a teacher, I can create a hybrid activity, using nothing but the activity, and maybe a lab document asking students to create some kind of evidence for me, filled with screenshots, and descriptions of what they learned.

I think that there are places for this conceptualization of digitized learning, digital learning, and lots of paper and pencil (or pen) work in the math classroom.  It’s important to understand when you’re doing each, and figure out when it makes sense to create digitized/digital activities for student learning.

I do believe that math education can be enhanced and transformed by technology.

I don’t believe that the facelifts given to ‘traditional’ methods (reading textbooks, listening to lectures) are the epitome of what it means to be a modern student.

I hope my work and writing here can explore and expand the boundaries in my own thinking of math education.


Each person completing this for credit gave a brief explanation of a lesson.

(1)  FTC redesign of Definite Integral Unit

(2)  Graphs of f, f’, f”:  given a function find first, second derivative, review some precalc polynomial sketching.  Complete table after check-in for where graphs are concave up, increasing, positive, concave down, decreasing, negative.  Helps them find intervals where things are true.  Like f concave up with f” positive.  Use vertically aligned graphs and highlight same color, connected concepts.

(3) a.  FTC, working on getting kids to chain rule the upper limit using example we did this week.  b.  inverse functions working on getting kids to not memorize, but instead derive it each time–easy and not too bad to remember.  c.  volumes in cross-sections, how do we get kids to keep the pi out?

(4)  Passed out playdough and forks.  Graph a pair of functions on paper, use playdough to create a solid with semicircular, or triangular cross-sections.

(5) Exploration activity for starting integrals.  Find the area based on a graph.  Counting boxes and slicing up things using rectangular or trapezoidal approximation.

(6)  Overestimates and Underestimates based on concavity.  Using calculus in motion.  How do we know if a tangent line gives an over/under estimate based on curve.

FTC Lesson Brainstorming

1.  Predictions.  Let students use what they know to try to predict accumulation based on simple functions.  displacement for velocity; water in pool based on rate, et cetera.  See if students can come up with some kind of system for accurately approaching an answer.

Introducing the FTC is really a problem of introducing definite integrals more than one of a discrete topic.  So we start at the beginning–I’ve got a couple of activities that I like for this, both from AP Central.  One based on finding areas under curves with over and underestimates, the other based on velocity and displacement.  The overall goal is to realize that smaller slices give us better estimates, and move into some different types of estimates (TABLE PROBLEMS!!)  I’d like to add a focus on estimation–that if we know a starting point, and how much something has changed, we’re in good shape to find out pieces of the function.  Run some AP Graph based integral problems with defined areas.  Work on formalization of integral notation, math practice and predicting values.

After teasing out the definition of a definite integral, we can begin talking about how they are connected to derivatives, and what happens when they bump up against each other.  We would have already talked about undoing basic derivatives (finding a function that gives a derivative of f'(x)) for basic polynomials, trig, et cetera, but now can begin thinking about them in a more formal sense…

and!  Begin talking about functions defined by integrals.  I like the idea of a starter problem getting students to explain the meaning of int(f'(x),0,x), and try to put into words what it means to take the derivative of that.  I like the guided practice in the ‘Functions Defined by Integrals” module.  Here we can work a couple more free response problems involving functions defined by integrals–particularly focusing on getting students to interpret their meaning, and figure out what the derivative of such a function would be.

At some point in here, we need to formalize the FTC in both parts, so might be a good time for some brief lecture on the topic.


1.  AP Central FTC Focus


Mother Function!

Reflection for Thursday, 8/1/2013

Today, we opened with a graph matching activity that I found particularly interesting, because I have learned that my students have a few particular weaknesses heading into my calculus class:  (1) anything involving exponential or logarithmic functions, (2) problem solving involving trigonometry, (3) algebraic manipulation including factoring, solving, rationalizing, manipulating expressions, (4) graphing (or having a feel for the shape of a graph without a calculator).

In my school, we have a six period, seven day rotating schedule, which means I see classes for six days in a row (with one long period during lunch), and then drop the class for a day.  We also, for AP Classes, have an extra flex-block period after school, once every seven days.  This year, I’m thinking about using those flex-blocks (at least at the start of the year) to reinforce some of these skills, and I think that working with graphs of functions might be a good way to start that.

What I particularly like about the graph connection activity, is the oddness of the basic graphs–not ln(x), but ln(x^2), not 1/(x-2), but (x^2-3)/(2x-4).  I like the review of what I call mother functions (they especially like the ones that are BAMFs (bad asymptote mother functions)).  I think this works in a couple ways–first, gives me a chance to see how students functional thinking (sans calculator) is working after a summer off, and perhaps focus in on specific student weaknesses without taking significant time away from class; second, it gets students ready to work together throughout the year, which is an important skill not always developed in earlier math classes.


This is a collection of notes and thoughts from the MMSI AP Summer Institute at Bridgewater State University.

1.  Graph Match Activity:  Really nice precalculus review, matching 18 equations ranging from y = lnx + 2 to y = sin^2x^2+cos^2x^2 to graphs that represent them.  Thinking about doing this as a first day activity.  Especially like the scoring:  1 point for each correct match, 2 points for each correct and complete justification.

2.  L’Hospital’s Rule:  Considering that they may be moving L’Hospital’s rule to the AP Test (AB) in the very near future, good time to review ways to make it useful to students, and more importantly to help them figure out when it’s actually appropriate to apply L’Hospital (the bought theorem).

How much is enough:  Looking at selected AP problem pieces, and some example student work, where all of the solutions are correct – but not necessarily complete enough to earn credit.  Good discussion with regards to the boundaries where a point may or may not be assigned.

Resources:  We compared resources that we use in our own classes.

http://apcentral.collegeboard.com – teachers’ guide, tests, et cetera.


http://www.umassmed.edu/mathgraphicorganizers.aspx (materials used and created by Carol Hynes for her own classes.

http://online.math.uh.edu/HoustonACT/ Powerpoints by Greg Kelly



http://www.kutasoftware.com – free worksheets for variety of calc topics

http://www.desmos.com – free online graphing calculator

Application:  Grapher to create graphs and screenshots

Application:  Cheap app by McGraw Hill, 5 steps to a five.

Application:  iPad – Graphing Calc HD


2013 Exam – We worked through student samples from the 2013 exam, grading them ourselves to see how we compare to the rubrics.


Getting Better

Reflection for Day 3:  I’m thinking about what I’d like to do for my lesson to present to the group on Friday.  Chances are, I’m working by myself, since I don’t really have time to try to coordinate with someone else between workshops, driving and tutoring (not to mention wedding stuff).  I’ve got a couple options on the table–first, I could refine a lesson I sketched out over the last year, creating a stronger activity for something I’d like to go deeper on; second, and more likely, I can put together something that will help my students with something I feel I didn’t do justice with last year.  Big Hitters for this category:  FTC, or Volumes using Cross-Sections.  Since FTC seems to be a theme for these reflections, let’s go with it.  

Here are my thoughts:  Last year in my calculus class, we focused on the FTC as a means to an end–how can we solve certain problems using the FTC–students did OK in the time we spent on it, but then we moved on.  The FTC wasn’t viewed as a thread woven throughout calculus, which it really should be.  

Let’s brainstorm what that might look like:

(1)  I want my students to understand that derivatives and antiderivatives are inverse functions.  

(2)  I want my students to connect the summation form of the FTC to what they know about linear functions.  

(3)  I want my students to be comfortable applying the FTC to totally symbolic problems.

(4)  I want my students to see the FTC in area problems. 

And to do all this, without spending weeks on a separate FTC unit.  Cool.  Now to take this thought process and articulate it in a more meaningful way.  But, you’ll have to wait for that.  

Another note–again, nicely situated in the thematic neighborhood of the rest of this post–getting better.  I think it’s easy to lose track of the big picture of mathematics, or mathematics education, or on a more biological level, mathematical processing, and it’s nice to step back and take a look at what’s out there.  I found it particularly poignant today, taking a look at the cognitive leaps we make in math from the time we begin to understand that quantity is something that we can identify and compare, to the point where our solutions are no longer numbers, but functions or families of functions in the case of general solutions to differential equations.  I think sometimes it’s also useful for our students to see how far they’ve come, even if they don’t think it’s as cool as I do.