Monthly Archives: April 2012

NCTM – Saturday

Much quieter here today, as the math folk move out and an antiques show moves in.  But still time for some sessions!

Essential Understandings in Grades 9 – 12 Statistics: Preparing for the Common Core.

Very excited for this session today, as one of the speakers is Roxy Peck, former Chief Reader of the AP Stats exam.  Also, looking for some ideas for our freshman year prob/stat course.

Big ideas –

1.  Data consists of structure + variability: look at math models but explore the big picture

2.  Hypothesis tests answer the question, “do I think this could have occurred by chance?” – what hypothesis is more plausible?

3.  To evaluate an estimator, you need to consider bias and the sampling method.

4.  Describe variability – distinguish different ways distributions are used (population, sample, sampling distributions) and be able to compare them. As more teachers are compelled to teach stats in HS, we need to train for the abstraction of sampling distributions.

How much do sample distributions tend to look like population distributions, and how can samples differ based on randomization?

5.  The way in which data are collected matters.  The is a risk of error that needs to be acknowledged and quantified.  Also, the collection method determines the type of inferential conclusions that can be made.  If the sample is not representative of the population, we should be suspicious of generalization.

“as statistics people, we are OK with being wrong 5% of the time” – Roxy Peck

All HS math teachers will find themselves in the dual role of being teachers of math and teachers of statistics. This is a bit scary to me.  While many resources are coming out which will assist teachers in presenting statistical ideas, I wonder how many math teachers are prepared to facilitate a discussion, perhaps over many class days, centering on one scenario and its many statistical concepts.  A colleague, at the end of the session ,offered that his department is “petrified” of the prospect of being made to teach stats courses.  Many great math teachers I know are like musicians playing classical music, adept and expert in the rhythm and complexity of math.  How many of my colleagues are prepared to become jazz musicians, and have conversations meander in new and exciting directions?

After attending 2 conference in the last month, and moving through meeting rooms, with various degrees of “fullness”, I have begin to develop the “Large Ballroom Theory”

Large ballroom theory:  given a large ballroom with many empty seats, people will

  • Find somebody familiar and sit with them
  • Sit in the back, for easy escapability from intolerable sessions
  • Sit in a location which will maximize the apparent fullness of the room, looking for bare spots

Philadelphia-20120428-00195, Uploaded by Photobucket Mobile for BlackBerry
It is the last bullet which I believe could be the start of a full-blown thesis.  In the photo above, think about where you would sit?  On the end?  Which row?  Someplace equidistant to others?  Or would you sidle up next to a stranger?

NCTM – Friday

Back on the train.  Jealous of the students and coaches from my school headed off to Penn Relays on the train, while I get off a few stops earlier for NCTM.  Looking forward to the day…

Supporting the Mathematical Practices: What’s In Your Coaching Toolkit?

This session had an interesting opening activity.  Seven statements about classroom culture were provided, and teachers were asked to assess themselves on a continuum?  Some examples

  • From focus on correct answer towards focus on explanation and understanding.
  • From mathematical authority coming from the teacher or textbook to mathematical authority coming from sound student reasoning.

Already, the activity caused me to reflect upon the textbook-driven culture in schools, and how we approach classroom practice.

How can we provide optimal learning opportunities for students to become mathematically proficient?  The team create a check list of “look-fors” which cover each of the 8 Common Core standards, and shifts in classroom practice which attempt to match the new standards.

This was one of the best sessions I have attended, as I have some great ideas here for guiding the teachers I work with through the increased rigor of the Common Core.

Unfortunately, I am off to a track meet tonight, so today’s post is a quickie.  A late night tonight, then an early train tomorrow for more sessions.

NCTM – Thursday

Very exciting day….quite a buzz as I arrived today.  Looking forward to some interesting sessions in the next few days.  The hard part is deciding which sessions to choose.

Teaching Proof: Lessons From an Action Research Study.  Pete Johnson, Eastern Connecticut State University

Yesterday, I attended a research session on proof,where’s a 5-person panel each explained research they had done on encouraging justification and proof in both middle and high school math courses.  From that session, there were two takeaways for me:

  • When we provide a proof, for whom is the justification intended?  Who is the audience for the communication? Let’s focus on the end user.
  • An interesting activity is to provide a number of givens, then having students work to  support the “strongest claim”.  We often tailor arguments to fit our pre-determined conclusions.  But what are the possibilities, given provided information?

My first session today continues my focus on proof.

“writing proofs” is not a topic or a bit of content, it is a a process, a way of thinking that evolves over time

Much of the discussion in this session focused on the following challenge:

Prove that if n is an odd positive integer, then n squared is an odd positive integer.

A few approaches emerged.  Let n = 2k + 1, then simplify n-squared.  Are we guaranteed that the result is odd?

Another attendee suggested letting n = m + 1, but then how do we know that m-squared is even?  What is the assumed toolkit of knowns and agreed upon principle in this problem?  What does it mean for a number to be odd?

Also, the group tended to focus on the oddness, but have we proven that the result is positive, or  an integer?

Findings: Teaching proof as a “separate topic” does not work.  Also, instruction in formal logic does not seem to transfer well to mathematical proof.

Engaging Activites for Your Classroom: technology in Middle School Mathematics

The main event of this session featured activities utilizing the TI Nspire CX Navigator system.  A few years ago, I acquired the Navigator system for the TI84 calcs, and had used it in some of my classes.  But, over time, I found the system cumbersome, and that the classroom payoff was not often worth the set up required.  I was eager to try this updated system, as it is now wireless, and integrates with the new color CX calculators.

My first impression is that sending files has become more intuitive, and the entire interface is cleaner and less clunky than the 84 software.  The examples demonstrated today were pulled from the TI Activity Exchange, and could easily be edited for use with TI Publish View, which mentioned in an earlier blog post.

Philadelphia-20120426-00191, Uploaded by Photobucket Mobile for BlackBerry

Looking forward to more great math discussions tomorrow!

NCTM is here!

Happy that NCTM is in my hometown in Philadelphia.  Looking forward to lots of great sessions, with a large interruption from a track and field invitational I run each year.

Look me up at these Thursday sessions.  I am the bald, handsome guy….

  • #30, Teaching Proof
  • #68, Assessing Students on the Common Core
  • #208, Engaging Activities for your Classroom
  • #300, Algebra Tasks

Say hello for a card and a free hug !

From “24”…Graduate up to Countdown.

This past week, I had the pleasure of attending a tournament for the popular school math game “24”.  You know the game…you are given 4 numbers, and can use any of the 4 basic operations (add, subtract, multiply, and divide) to arrive at an answer of 24.
At our district tournament, school winners competed to be district champion, and all of our 4th and 5th graders attended to provide energy, enthusiasm and support for their peers.  What an exciting morning, and kudos to our elementary staff for organizing a fun morning, all based on this simple math game!

While many of us are familiar with 24, did you know that our friends in the UK have been enjoying a similar game on the TV Game Show “Countdown” for the past 30 years?  In Countdown’s “Numbers Round”, contestants complete a task similar to that of 24: use the 4 operations to get as close as possible to a target number.  Here are the details:

  • There are 24 numbered tiles: each number from 1 to 10 has two tiles, and the “large” numbers 25, 50, 75 and 100 are also included.  6 tiles from the 24 are randomly chosen by the contestant.
  • A 3-digit number is randomly generated, and the contestants must use the 4 basic operations, along with the selected numbers, to reach the target.
  • You do not need to use all 6 numbers, and you may use a number twice, if it appears twice in the list of 6 chosen numbers.
  • Contestants have 30 seconds to work, and earn points based on how close they come to the target.  Note that it is possible for a game to not have a solution.

If you try a search on YouTube, you will find various episodes and clips from the show.  Below is a simulation which demonstrates the game, without having to wade through a host or commercials:

Try this challenge:  your numbers are 3, 5, 7, 7, 9 and 25.  Your target is 997.  Give yourself 30 seconds.  Solution below.

You can do an internet search and find many links to Coundown’s “Numbers Round”, but Graham Hutton’s site explains the rules clearly and provides a neat Flash app which you can use with your classes.

And that challenge I gave you earlier?  Here is a screen cap from the site I referenced above which gives a solution:

101 Questions

One of the more intriguing math-related websites I have been following this year is by Dan Meyer.  The site has a simple concept: you are presented with a picture or short video clip, and are asked to contribute the first question that comes to your mind.  I have contributed a few items to the site, and reading some of the questions posed often leads me in directions I hadn’t initially considered.  How neat!  You can also view questions which others have contributed for each item.  The pictures and videos are meant to serve as “first acts“, mathematical conversation-starters which lead to problem-solving discussions.

What I like most about this site is that there are no answers.  Rather, our focus shifts to posing interesting questions, facilitating meaningful discussions of problem solving methods, and working towards plausible solutions.

As the site became populated with more “first acts”, I recruited volunteers in my district to find a way to use this site with their classes.  I found two high school teachers, who were eager to share their Academic (our most basic) Geometry classes.  It’s a shame that we often reserve interesting, open-ended tasks for our highest achieving kids, so I was interested to see how these groups would take to the project.  And while my high school colleagues were enthusiastic about using the site to develop a task for their students, there were some natural questions about managing the task: How will kids react to having such an open-ended task?  Will they persist in completing the task?  How will we assess their work?

Note: one teacher I am working with attempted to utilize the site, after we had some discussions of a project, but found that her students were blocked from the site at school, due to its YouTube links.  I have since taken care of this snag, but you may need some coaxing with your higher-ups.

We settled upon a structure to help kids step through the task.  In day 1, partnerships of students will:

  • Select an item to explore.
  • State your question.
  • Develop a plan of attack and list measurements you will need to consider.
  • List the math (formulas or concepts) you will need.

The partnerships will then meet with the teacher to discuss their ideas and revise, if necessary.  The task then moves on to day 2:

  • Complete the plan of attack.
  • Answer the question.
  • Reflect upon your process and state any changes or improvements.

To complete the task, students will create a presentation which steps through their question.  In order to help students understand the task and our expectations, I visited the classes, and modeled the process for one of the Top 10 pictures from the site: the Ticket Roll.

The class discussions were rich, and allowed many students to provide ideas:  How many tickets are there?  How long is the roll?  How will we find the thickness of a ticket?  How precise do we need to be?  Why are we doing this?  In both classes I visited, we discussed the dis-comfort we feel when we have a question without a known answer, and how rare it is to have this happen in math class.  To complete the ticket roll problem, I shared a Prezi I made to model our expectations:

As students complete this task, look for an update here and I will share some of the presentations.  Would love to hear all of your ideas for how to utilize this rich resource!

Bob’s Favorite Things

This year, as I have moved from a role as classroom teacher to a math coach, I find myself sharing some of favorite lessons and approaches with my colleagues.  Taking a cue from Oprah, this is the first of a series of my “Favorite Things”, hopefully without the rampant screaming or random things hidden under your seat.

The Monty Hall Problem on NLVM

The famous “stick or switch” problem is one which will generate tremendous discuss in a classroom.  The premise is simple: you are offered the choice of one of three doors, behind one of which there is a great prize.  After the host reveals a non-winning door, you are offered the chance to switch your guess.  Should you stick or switch?  The problem rose to a next level of fame with fierce debate in the “Ask Marilyn” article by Marilyn Vos Savant in Parade Magazine.  After having included the problem in my prob/stat course for any years, the discussions rose to a crest as the problem was featured in the movie “21”, with even students I did not know stopping me in the hallways to ask me about its logic (note, this video has a number of ads on it):

As an emerging stats teacher, I would play the game using cardboard doors and hand-drawn pictures of cars and goats.  In later years, I used an applet at the National Library of Virtual Manipulatives to play the game with my class.  Look under Data Analysis and Probability for “Stick or Switch”.  What I really like about this applet is that it will play the game 100 times quickly and display the results.


There are many ways to explain this game to your students.  One straight-forward approach is to have students attempt to write out the sample space (all possible outcomes) for this game.  For example:

Car behind 1.  Pick 1.  Shown 2.  Stay.  WIN

Car behind 1.  Pick 1.  Shown 2.  Switch.  LOSE

Car behind 1.  Pick 1.  Shown 3.  Stay.  WIN

Car behind 1.  Pick 1.  Shown 3.  Switch.  LOSE

Patterns in Pascal’s Triangle

In one of my favorite lessons each year, the many, many patterns in Pascal’s Triangle were discussed.  This would come after an investigation of combinations, and students  would often recognize that each entry in the triangle is actually a combination, along with other patterns like the counting numbers or perhaps the triangular numbers.  But how about the Fibonacci sequence and the “square” numbers?  What’s the “hockey stick” theorem?


Over the years, I have used a number of sites to display the triangle and invite students to share their ideas.  One nice resource which explains patterns in the triangle can be found here.

The cherry on the Pascal sundae would come when I would invite students to approach the board and circle all of the even numbers.  Often, only 15 rows or so would be visible, so the chance to make a generalization is rich and there to discuss.  How would this look if we colored 100 rows, 200 rows, 10,000 rows?  And do we only get patterns for even numbers?  How about multiples of 3, or 5, or 24?  Check out this site from Jill Britton, which includes an applet that will color multiples in Pascal’s Triangle.  Watch the expressions from your students when you reveal multiples of 5 in the first 128 rows of the triangle:


Try different multiples and observe the great symmetries.  Prime numbers like 17 and 23 provide the most surprising results.  Digging deeper, start a discussion of the modulus function with your class (it’s just the remainder, no big deal….).  What happens when we not only color multiples, but also color the remainders similarly.  From the Centre for Experimental and Constructive Mathematics, we get this great applet, which will color remainders based on specifications you provide:


Electric Teaching

While I have been using the Monty Hall problem and Pascal’s Triangle in my classrooms for many years, this last resource is fairly new to me.  Electric Teaching by David Johns is an excellent site with an effective interface which is ideal for SMART Boards.  The site challenges students to match-up equations with data tables and their graphs.


The site has problems for linear, quadratic and trigonometric functions, along with conics and even problems which include derivatives.  What a great way to not only have students participate, but verbalize their thoughts as they identify the matches.  The YouTube channel has additional resources from Dave, including a tutorial of the Electric Teaching site.