Tag Archives: tmc


Reflections from TMC14, Part 1 – Steve Leinwand and NRICH

This past weekend, I had the pleasure of participating in Twitter Math Camp 2014, held in Jenks (Tulsa) Oklahoma. 150 math teachers from around the USA, Canada and England, many who had only previously shared ideas and personalities via twitter and blogs, met to share their ideas, successes, best practices, and favorite activities. Morning sessions focused on course and task-specific study groups (I participated in the Statistics group). Afternoons started off with teachers sharing “My Favorites”, followed by a keynote (Steve Leinwand, Dan Meyer, and Eli Luberoff) and a menu of teacher-led sessions. Today is the first of a few recaps I’ll share of this jam-packed learning event.

Sly Stallone

There’s a crappy 80’s movie “Over the Top” which starred Sylvester Stallone as a professional arm-wrestler who eventually battles for custody of his son (yes…this was a pretty craptastic movie). In the movie, Sly motivates himself by turning his baseball cap to the side. This action triggers some arm-wrestling adreniline receptors, a competitive “on” switch, and Sly is then prepared to kick butt (or…arm).

This is my best description of Steve Leinwand.  A self-described “math education change agent”, Steve is a mild-mannered math expert…until you place him in front of an audience, at which point the Mathmazian Devil emerges! I have seen Steve talk in person twice now (do yourself a favor and check out his Ignite talk on Youtube) and his inspirational message leaves me in a constant reflective state over my classroom practices.

In this time of debates over Common Core, “fuzzy math”, dots and standard algorithms, it’s refreshing to hear a speaker attempt to tackle the question “what is math?”.  In his presentation, Steve offers up two options for defining mathematics:

A set of rules to be learned and memorized to find answers to exercises that have limited real world value.


A set of competencies and understandings driven by sense-making and used to get solutions to problems that have real world value.

Clearly, the first definition is not correct, though I fear there are many who would find aspects of the definition acceptable.  I, and the room, gravitated towards the second definition, but is this a complete picture of mathematics?  I have 2 quarrels…

First, the phrase “problems that have real world value” bugged me quickly, conjuring images of contrived real world problems where kids factor expressions which never really occur naturally so they can find where a fake baseball which ignores some pretty important laws of physics might land.

Does “real world value” necessarily imply context? If a math problem provides insight into an abstract pattern, and the process provides some structure later to tackle real world scenarios, then by transference, the problem had real world value. so I have become ok with this aspect of Steve’s definition.  But I’d like to move beyond the perception that mathematics only adds value if it can be attached to the real world.

Working backwards in Steve’s definition, we reach the phrase  “used to get solutions”? Do all math problems have solutions? Is the primary goal of math to find a solution? Have we failed if we don’t find a solution? Some of the strongest formative mathematical experiences I have had centered around problems for which I never found a solution, or perhaps did not have a unique solution.  I prefer “used to analyze scenarios, either abstract or real-world.”

I appreciate Steve in that he challenges teachers to think about the many ways their students may approach similar problems, sieze opportunities to discuss methods, and let students determine their optimal strategy. Many of the common core math debates focus on method: there is a strange “my way or the highway” attitudes towards standard algorithms. Its refreshing to have Steve champion alternate methods so passionately, and he offers his admiration for the 3rd Standard for Mathematical Practice:

Construct viable arguments and critique the reasoning of others.

The ability to analyze, critique, and assess method is equally as important as the math being done. All of us who talk to parents, colleagues and stakeholders need to remember this and do a better job at effectively communicating the message of what math is really all about.

ProblemThe Enriching Mathematics site, NRICH, presented by Megan Schmidt in an afternoon session, provides problems with multiple entry-points which lead to argument sharing.  In the session, participants were presented with a Stage 3 and 4 problem from the site, where finding the value of the number marked with the question mark is the goal:

My PaperI chose to look at pairs of repeating symbols to craft my solution, while my tablemate dove into developing equations and forming systems. The most frustrating (but coolest AHA) moment for me when Megan offered adding sums of rows and columns as an alternate, quite obvious, possibility.  I am definiely looking foraard to exploring these problems and sharing them with my classes.

Thanks to Steve for giving us all the inspiration to think differently about classroom practices, and to Megan for the perplexing hour of sharing!



4 Engaging Ideas From Twitter Math Camp

This past week, over 100 math teachers descended upon the Drexel University campus for Twitter Math Camp 2013.  It was a fantastic opportunity to meet people I had communicated with via Twitter for some time, make new friends, and share math ideas.  It’s a real rush to hang out with colleagues who share similar ideals on math instruction, and a commitment to improve our practices.  Check out the hashtag #tmc13 on Twitter to look back on some of the action and reactions, and find new math folks to follow.

While there’s so much to share from TMC13, I know there are many math friends who couldn’t attend who are looking forward to hearing about the goings-on, so in this post I share 4 ideas from this year’s Twitter Math Camp I am eager to try in my classroom right away.

EliELI’S BALLOONS – Followers of the blog know that I am a big fan of the Desmos online graphing calculator.  The highlight of the week for me, and I suspect for many, was having Desmos founder Eli Luberoff model a lesson using his creation.  Eli’s enthusiasm for sharing Desmos, and his sincere desire to work with teachers to improve the interface, are infectuous.  There were many “oooh” and “aah” moments from the assembled group, and a loud cheer for the “nthroot” command…yes, it’s a pretty geeky group!  (thanks to @jreulbach for tweeeting out the great picture of Eli showing off Desmos’ position when you Google “graphing calculator”)

Eli’s lesson idea has a simple and engaging premise:

  • Hand out balloons
  • Blow up the balloon.  For each breath, have a partner record the girth of the balloon
  • Consider the data set


That’s it.  No worksheet.  No convoluted instructions.  Eli walked us through an exploration of the data set using Desmos, using the table to record the data, and considered various function models: is a square root model?  Is it logarithmic?  The group eventually settled upon a cube root as the proper model – and how often in class do we encounter data best modeled by a cube root?!  Since the explanatory (air entering the balloon) is volume, and the response variable (girth) is linear, the cubic model makes perfect sense. Fun stuff.  But wait…there’s more!  Eli then analyzed the fit of curve by looking at the squares of the residuals.  Click the graph below to check out my best-shot recreation of Eli’s presentation, and play around with the fit of the curve by toying with the “a” slider.

More great new additions to Desmos are coming.  Thanks to Eli for letting us preview some of them!  Was a pleasure meeting you and hearing about your fascinating story.

GLENN’S PROBLEM POSING – Glenn Waddell is a colleague I feel I have a lot in common with, in that we have both experienced the frustrations of trying to “spread the word” to colleagues of the great new ideas, and strong need, for inquiry-based mathematics.  In this session, Glenn presented a framework for problem posing in mathemtics which can be employed equally-well with real-life problems (see the “meatball” example in Glenn’s Powerpoint, which was adapted from a Dan Meyer “math makeover” problem) or with a garden-variety drill problem.

The framework asks that teachers lead students in a discussion that goes beyond just the problem in front of us.  Think about the many attributes of a problem, list them, consider changes to them and their consequences, and generalize results.  Glenn suggests the book “The Art of Problem Posing” by Brown and Walter as a resource for getting started, which employs the problem posing framework.


Glenn led the group through an exploration of a quadratic equation, where we started by listing its many attributes.

Glenn Problem

Now we consider changes to attributes:

  • What would happen if there were a “less than” sign, rather than equals?
  • What would happen if the last sign were minus?
  • What is it were an x-cubed, rather than x-squared?

There’s no limit to the depth or number of adaptations, and that’s why I like this method of problem posing for all levels of courses.

Download Glen’s presentation on the TMC wiki, and explore the wiki to get the flavor of many of the sessions.



If you have never visited Mathalicious, go now….take a look at some of the free preview lessons, and you will become lost in the great ideas for hours.  THEN, make sure you sign up and get access to all of the engaging lessons.  Here is a company that is doing it right: lessons come with a video or visual hook, data which naturally lead to a discussion of tghe underlying mathematics, and just the right amount of structure to encourage students to contribute their thoughts and ideas.  At TMC, Mathalicious founder Karim Kai Ani led the group through two lessons.  A brief summary is given here, but I encourage you to check out the site and subscribe….you’ll be glad you did.

The “Romance Cone” – What is the appropriate age difference between two romantic partners?  Is there a general rule?  A fun lesson, “Datelines” on Mathalicious, where students explore a function and its inverse, without using those scary-looking terms.  I have been looking for an opening activity for our Algebra 2 course, which brings back ideas of function, inverses and relationships, and looking forward to trying this as a my first-day hook.  Also a great activity for Algebra 1.

PRISM = PRISN? – I have led my probability students through an exploration of false positives in medical testing for many years, and I like how this activity puts a new twist, and some great new conceptual ideas, on the theme.  “Ripped from the headlines”, this lesson challenges students to consider government snooping, and the flagging of perhaps innocent citizens.  If a citizen is flagged, what is the probability they are dangerous?  How often are we missing potentially dangerous folks in our snooping?  What I really liked here was the inclusion of Venn Diagrams, with sets representing “Flagged” and “Dangerous” people, where the group was asked to describe and compare the diagrams.  Fascinating discussions, and a good segue into Type I and Type II error for AP Stats if you want to take it that far.

This lesson does not appear to be available on the Mathalicious site yet, (update from Mathalicious – will be released in the Fall) but will be using it when it is completed!  Later that day, the TMC teachers broke into smaller groups to gain behind-the-scenes access to the Mathalicious writing formula.  Thanks to Kate and Chris for sharing, listening, and giving us all the opportunity to contribute ideas.