Would the real “real world” please stand up.

I really appreciate Dan Meyer’s post on real world vs fake world and getting the MTBOS to help define what makes a problem “real world.”  He put a call out to Mathalicious (specifically), which is a team of highly-skilled designers of “real world” math problems.  Mathalicious states in a  post concerning the purpose of two types of tasks problem solving tasks and application tasks:

While the purpose of problem solving tasks is to help students develop a deeper understanding of math, the purpose of application tasks is to use math to develop a deeper understanding of how the world works. It’s pretty Radiolab like that. [Link to Radiolab added.]


As math teachers, we need to be able to integrate both types of tasks into our courses.  I am making an effort to use both by using Dan’s 3-Act tasks (which have a more problem-solving flavor) and Mathalicious.com (which have the ever-expanding real-world tasks). My battle is with how much time I devote to these types of tasks, which, as you can imagine, take a lot of class time.  Mathalicious rightly states this toward the end of the post:

Instead of discussing which type of activity – procedural, conceptual, or applied – we should use, a more constructive conversation would be about how often and when. For instance, is it more effective to start a unit on linear functions with an open-ended problem (e.g. Dan’s Groceries), an application activity (e.g. our Domino Effect), or a procedural task (e.g. Khan Academy’s activity on  slope-intercept form)? Also, within the unit, how much time should we allocate to each type of activity, i.e. how much time should we spend in each world?

I highly encourage everyone to read Dan’s original post (including the comments!) and Mathalicious’s reply.




Desmos–Function Carnival and Des-man

The team at Desmos has two really good activities:

1.  Function Carnival–Students graph the motion on three rides.  They can do this in class on computers or iPads. The teacher dashboard rocks too.  I made a quick handout here.


2.  Des-man–students make faces using functions.  This will also help reinforce domain and range.  I have made a quick handout for that one too.


Updated 3/20:  Here is a quick video of the teacher dashboard for both the Function carnival (0:00-2:10) and Des-Man (2:10-End) that we did in our workshop with about 20 participants on using tablets in the classroom.  There is a little down time that I failed to edit out. I also want to give a big shout-out to Desmos for responding rapidly to a small issue we were having.

Order of Operations/Distributive Property Reflection.

This term has been crazy busy. I have neglected to follow up on my Explore-Flip-Apply journey this term in an Elementary Algebra-type class. I will attempt to summarize the activities we did during the first half of the term I will start with the first activities we did: order of operations and the distributive property

  1. The Explore part of the order of operations activity took longer than expected. Many of them struggled with the 2nd part of the first 6 problems where they are asked to insert parentheses to make it true. They handled #9 better than I thought. They seem to do better when there is more than one path to the answer.  The Explore part of the distributive property lesson was not concrete enough for them. The first problem was good.  However, I did not make a strong enough connection between the two methods of arriving at an answer and the property later on.  The geometric interpretation seemed completely irrelevant to them.  Again, in the past, I would normally eschew these constructions.  I attribute much of this to my own inexperience of conducting such activities. (I did not get to the input/output table and never revisited it.)
  2. The Apply part for order of operations was not as successful as I had hoped. Not all of them had watched the video lecture that I posted (for the flipped part).  I will redo this video to make it considerably shorter as it is the longest video so far (~20 mins). The students seemed a bit off with problems 5-10. And no one even tried the extensions. I should build those up more in class (praise for those who get it…or even get close.)  The Apply part for the distributive property was much quicker and went well, with the exception of problems where they had to distribute the “understood” -1.  Again, no one tried the extension.  I may have said do them for homework. I neglected to check up on it.

I always feel that I do a lousy job of teaching “Chapter 1” in any course I have taught (e.g. algebra, trig, Applied Calc, Applied Stat, Math for “Liberal Arts,” Finite math).  It seems as though all of the “exciting” material starts with chapter 2 FOR EVERY ONE OF THESE COURSES. I will consider skipping chapter 1 in the future and build those learning outcomes into other lessons.

Next week I will begin to post on other lessons we have done: Solving 1-step linear equations, Solving multi-step linear equations, Formulas and problem solving (3-Act tasks / 101qs), solving linear inequalities, and loads of good stuff on graphing with Mathalicious tasks galore.

First Day of Algebra Class

I’ve always thought the first day of class was anticlimatic.  Students come buzzing in, excited to begin college for the first time (most are incoming freshmen). What did I usually do to fuel their excitement?

  1. Call roll (ensuring everyone’s name was pronounced correctly)
  2. Go over the syllabus and expectations
  3. Show how to log on to MyMathLab (our LMS)
  4. Let them go early.

All of the energy that previously existed was gone upon exiting.  Sure, I’d do the name game and let them tell a little about themselves (or introduce a partner).  I’d even go around the room and attempt to call everyone by name, occasionally eliciting a few laughs.  Are you sure your not Bob? You sure look like a Bob. But I was not able to model the learning environment that I wanted in subsequent classes.

Until now.

I flat out stole Dan Meyer’s styrofoam activity (he details it more here).

What I did:

I.  Group into 3-4 students per group. I rearanged the furniture so thay had to look at each other.

II.  You say, how many stacked cups would it take to reach the top of my head? You hold one up.

III.  Ask for them to guess.  I had them write the guesses on the whiteboard and hold it up. Ask for a realistic minimum; ask for a realistic maximum.  I then wrote these on the board. They took a lot of time with this.  They did not seem to be comfortable estimating.  A lot of them were really close, and 1 was correct to the nearest cup (84).

IV.  I passed out 3-4 cups and a ruler per group. I walked around and ask the groups how they are going to figure it out.  They will,  of course, need to know how tall you are.  Using centimeters will help keep the numbers fraction free.  In my case I am about 183 cm tall.  My cups had a lip height of 2 cm and a base height of 10 cm.  I ask them what they need to know if order to solve it. One group honed in on the common sense solution really fast (number of cup lips x 2cm + base height of 10cm gives us the stack height).


V.  Most groups thought they new what was going on.  One was really strugling on knowing where to even begin.  Asked: “How would you determine the height of 3 cups.” Answer: “I’d measure it”  Fair enough.  “OK, how would you find the height of the cups if you did not have a ruler, but only the measurements of the cup (and lip)?”  I then left them to think about it.  It is important to withhold the actual “solution,” as there are many ways to get there.  I want them to see that there is often more than one pathway to the answer.

VI.  Then I had them put their new (and hopefully improved) guesses on their whiteboard and hold it up.  I wrote these down on the whiteboard in front of the class.  Then the moment of truth.  How would their model stack up (no pun intended) to the actual stacking of cups. All but 1 were within 3 cups.


What will I do next time to improve it?  I did not do a good job of getting them to tell me how they modeled it.  We were near the end of the 55 minute class and I had to make sure we got the cups stacked up.  I did not mention the pattern or formula to them, and, perhaps I shouldn’t. But I should have pushed for them to give me their perceived pattern.  We will most likely revisit this problem when we learn of linear equations in two variables.

Explore-Flip-Apply First Lesson

Here is my first attempt at creating an Explore-Flip-Apply type lesson.  These are basically all in draft form.  I would like to be able to do this EVERY DAY.  Not 100% sure that is going to happen.  So far, I have created the first unit, which consist of about 5-6 lessons on solving equations.  Any feedback would be greatly appreciated.  I plan on posting 1 lesson per day for the next few days while I work on Unit 2 (graphing).  I have tried to source pages that inspired me.

Class: Elementary Algebra
Class Time: 55 min, 3 days a week for 15  weeks
Topic: Order of operations, exponents, & the distributive property

Explore: Order of Operations (key) |Dist. Property (key)

Flip: Order of Operations (MP4) | Dist. Property (MP4)

Apply: Order of Operations (key) | Dist. Property (key)

Comments:  I struggled with this one.  Sometimes when the concepts are very simple, I am tempted to just TELL THEM.  But I know from experience that they always mess up the order of operations.  So I want them to realize that we need to have some sort of order, else we get different answers.  I want them to see how this relates to evaluating polynomials (degree > 1).  I want to point out the tricky problems that are actually just poorly worded problems (e.g. Evaluate  5÷xy for x = 2, y = 3).  No self-respecting mathematician would every write something like this due to the ambiguity it presents.    I decided against providing a broader application context.  I did not want to obfuscate the simplicty of how we combine terms mathematically.

With regard to the distributive property, I am trying the “Area of a rectangle approach for the first time. I am including “Extension problems” for some of the lessons.  This is for groups who finish early.  I try to show where  this topic is used later in this course.   NCTM Illuminations has a lesson on this that extends to factoring that might help. Factoring is the last unit of the term and  I may consider going in this direction later.

I have tried to link these to my Google Drive.  I have the comments function open if you want to comment directly on the document, feel free. You may, of course, leave comments below.

I find that I kind of cave on the “Flip” part of the lesson.  I bassically digress back to direct (Khan-like) instruction.  This is the longest video in the first unit. Most are under 20 min. I have embedded a quiz (using Camtasia Studio 8).  When I do this, I get a report sent to me about how they did and how much of the video they viewed.

Explore-Flip-Apply: My Initial Model


Explore-Flip-Apply: My Initial Model

Here is my initial model as I go through the process of reorienting my class to inquiry-based learning. I am sure there are some flaws. Each topic will begin with Explore.  This  should take 20-30 mins. Some Explore lessons may be significantly shorter. The intent of the explore is to spark a student’s curiosity. Some recent research suggests that students should do an in-class activity before watching the out-of-class video. Showing the mechanics (e.g. solving a linear inequality) is pushed outside of class (i.e. the Flip). These will most likely be a short (<20 min) video that gives some standard (direct) instruction. Currently I use Camtasia Studio 8.1 to do this and a Dell Latitude XT2, which is perilously close to burning out.

When students return to class, they will begin the Apply phase. Now that they have been given some of the tools, we will either finish the “Explore” problem we worked on previously (maybe a 3-Act problem) or they could just apply their skills at solving some problems (anything wrong with that?). Most all of the in-class time will be spent in groups (or at least pairs).

We use a Pearson online product (MyMathLab) as an LMS and for them to do “homework.” My current strategy to implement this is to have optional homework problems for them to practice. I will also have a very short online quiz (3-6 questions) in which they will get multiple attempts.

I will post some initial iterations for the first unit (Linear equations, Inequalities, & Problem Solving) later this week. I still have not nailed down everything. Any feedback would be greatly appreciated.

Thanks to Ramsey Musallam & Dan Meyer for all the work they do. If you are not following them on Twitter, you are really missing out.

Flip 2.0

While I have been doing the “traditional” flip since 2006, I am exploring ways to improve it.  I am going to attempt to keep a record here of my journey, including sources of inspiration, any support research, and even some sample lessons.  Dan Meyer and Ramsey Musallam are two educators who are are on bleeding edge of technology and pedagogy, yet from slightly different camps.  Dan Meyer, the rock star of high school math teachers who is now finishing his doctorate at Stanford in Math Ed, has an amazingly active blog.  His 3 Act math lessons, which are growing in number weekly, seem like the new standard in teaching math.  Act 1 is the hook.  Something to get them interested.  Perhaps its a video of someone stacking a bunch of pennies in  large pyramid, or maybe it is a video of someone filling a tank with water.  The teacher then probes the students to find out if they have any questions they want resolved (e.g. how much money is the pyramid worth, or how long will it take to fill up the tank).  The teacher can then probe a bit to get some buy in from the students.  Ask students to give a maximum and a minimum: What is the most amount of money you think there is in this pyramid?  What is the least?  How much do you *think* is there?  Even students who are timid with mathematical calculations have some sense of this. Act 2 is where the teacher gives the students the tools to solve the problem.  However, unlike the text book, the teacher asks the students what they need to solve the problem.  They might want to know the dimensions of the square base of the pyramid.  They would want to know the flow rate of water being poured into the tank. Only then does the teacher furnish them with the tools.  Act 3 is the resolution.  Hopefully this has been figured out in class and students are wanting to know if they are correct.

Ramsey is a high school chemestry teacher who has been flipping his class for a number of years, but most recently has transformed a traditional flip (just placing the lectures you would have done in class online and having those done as homework, then working what would have been done at home during class) into a type of pedagogy where the simple word “flip” does not do it justice. His method, while certainly not new, comes in a nice package: Explore-Flip-Apply.  So, rather than the students’ first exposure to the material coming outside of class as a full 20-30 minute video lecture (which is what I did), the teacher first has students explore some topic.  They explore this topic without any new knowledge–only prior experience.  Most, if not all, students should struggle with this part–this is a good thing. The teacher should not give away too much here.  There is an intentional withholding of information.

My plan is to (somehow!) blend these two paradigms–Meyer’s 3 Act and Musallam’s Explore-Flip-Apply–into an algebra class here at MTSU.  Meyer has stated that the 3-Act approach and the flipped approach are mutually exclusive.  Perhaps with the hybridization of these two, I can make it work.