Monthly Archives: March 2012

Even Great Presentations Have Their Moments….

Recently, I attended a talk where the circle graph below was used to help emphasize the many online tools our students utilize.  To be fair, the presentation was otherwise fantastic, but sometimes my stats-abuse-radar is on full alert.  Use it as an opener for class discussion, and see if your students notice the inherent problem with this graph:


Some questions for a class discussion:

  • Does this graph portray the data accurately?
  • Is a circle graph appropriate here? Why or why not?
  • How can we re-display the same information effectively using a new circle grpah, or a different type of graph?

In moments like this, sometimes it is best to draw energy from inspirational quotes.  I leave you with this, from the Simpsons:

Hypnotist: You are all very good players
Team: We are all very good players.
Hypnotist: You will beat Shelbyville.
Team: We will beat Shelbyville.
Hypnotist: You will give 110 percent.
Team: That’s impossible no one can give more than 100 percent. By definition that’s the most any one can give.

ASCD 2012 – Monday

Monday has arrived. Many more bodies on the train today. My day started with a realization that I left my conference badge on my dining room table. But thanks to the badge recycle bins scattered throughout the convention center, I will survive the day.

Uploaded by Photobucket Mobile for BlackBerry

Many thanks to Amy from Virginia for being green.

Developing Social Media Policies for Schools. Facilitated by Steve Anderson and Sam Walker

If you wouldn’t say it to your momma standing on a table in the middle of a mall, don’t say it online.

Biggest concerns for social media in schools reported by the group were cyber bullying and inappropriate relations by both students and staff.  But the number of kids online and the number of messages sent continues to increase.  Over 73% of teens participate in social networks, yet we close the door on them when they come to school.  And it is starting earlier….Club Penguin is the largest social media site for younger kids.

Do went want kids collaborating, sharing, connecting?  Of course, but we need to train teachers to facilitate differently.  The use of social media tools means that students can learn at different rates, at the times they choose.  One of Steve’s colleagues does a social media project where students choose what they would like to learn and report out on it.  We need to raise a generation of kids who are comfortable with collaborating, and they are doing it already on their own.  How do we facilitate that in schools?  Many schools want to embrace social media, but under rules they define.

As part of an improvement plan, Steve and Sam’s district developed a goal for social media which includes students, parents and educators.  They have also embraced problem-based learning to allow students to explore ideas.  In one interesting project given to third graders, the challenge was to develop a list of Internet usage rules, which would then be presented to parents.  Their district does not have a social media policy for teachers, except to say that teachers are “strongly discouraged” from contacting students after-hours via social media, and that teachers are expected to follow the same general ethical standards policies of all employees.

I appreciate Steve’s message: “Why continue to put in place restrictive policies that handcuff the ability of educators to do their jobs?”.  We need to embrace the possibilities and the wealth of collaboration which social media provide.

Check out the presentation, and more from Steve’s blog: Blogging About the Web 2.0 Connected Classroom

You can follow Steve on twitter @web20classroom, where he frequently posts ideas and articles.

Developing a Framework for Science and Math Instructional Coaching

Given that this is my first year as a math instructional coach, I am eager to absorb ideas for maximizing my effect as a coach.  The team of 4 today I heard speak are from the Chicago area, largely DePaul University.

The team has developed a concerns-based adoption model (CBAM) which helps facilitate change in the questions teachers ask, and identifies ways to assess 7 stages of concern, based on the adoption model from Hall and Hord in “Change in Schools, Facilitating the Process”.  The model measures teacher progress from the awareness / informational level, and moves them towards management and collaboration.  Charlotte Danielson’s Framework for Teaching was also incorporated into the model to identify and measure teacher proficiency in the implementation of curriculum.  The team has identified components of successful science and math classrooms, and developed rubrics which incorporate the Danielson models for basic / proficient / distinguished practice.

I appreciated a comment from one of the moderators, who noted that we need to differentiate for teachers in the same manner we need to differentiate teaching strategies with our students.  A needs inventory is utilized in Chicago in order to identify needs on both the teacher and coach end, and have teachers reflect upon their classroom practice.  For example, the question “how confident do you feel pushing student thinking through the use of questioning and wait time” may open a door for a teacher to seek help with a coach, but may also cause a teacher to honestly evaluate and reflect upon their current classroom practice even if coaching is not utilized.

In order to facilitate effective lesson planning, the team developed a checklist for orchestrating productive math discussions: anticipate – monitor – select – sequence – make connections.  The coaches then address proficiency and misconceptions in each area through pre-conference planning and post-lesson reflection.

ASCD 2012 – Sunday

2nd day of the ASCD Conference in Philly.  Today I focused mostly on best practices sessions in math.

Enhancing Concept Development and Vocabulary Proficiency in Math Classrooms, facilitated by Dr. Donna Knoell

Across the board, a focus on vocabulary increases  student proficiency rates, yet we focus little on vocab in math classrooms.  Students need to have 8 to 10 meaningful exposures to vocab before students can apply in context.  For ESL learners, the number is 12 to 15.  We want students to be able to communicate their reasoning.

For example, in elementary school, students are often taught to fold paper “hot dog style and “hamburger style”.  But this short-cut has eliminated an opportunity to discuss and reinforce horizontal and vertical as necessary vocabulary.  The human mind innately seeks meaning.  We are often in such a hurry to move on, that we often forget to provide time to think about what our students have learned.  Talking math helps us cement our understanding of math ideas.  Challenge our students to defend ideas by utilizing math vocab appropriately.

Students can personalize their experience with math vocab words by maintaining a journal of new words, with definitions, picture and contextual sentences.  This caused me to reflect upon conversations I have with teachers at my high school, where teachers become frustrated by problems involving angles of elevation or depression, bearing, or the similar terms root, intercept and zero. As we expect our students to become more adept with communication, justification, and application, helping students develop an appropriate vocabulary toolbox becomes of greater importance.

Beyond the Textbook: Math Activities to Stretch Your Students Thinking, facilitated by Dan Rosenberg

A variety of games for grades 1-8, gathered via the “CASE method (copy and steal everything)”:

Battleship: students write an algebra equation in each position to represent “hits”.  To earn the cell, students must solve the problem correctly.

Dots: play the connect the dots and square capture game, but place integer values in the cells, which become point values as squares are captured.

One game I have used in class at the start of probability units is the “card prediction game”. Start by dealing out 10 cards face up.  Students can then predict what the next card (suit and rank) will be.  Points are earned by correctly predicting characteristics of the next card:

  • If the card they predict is the same color as the next card drawn, they earn 1 point
  • If the card they predict is the same suit as the next card drawn, they earn 3 points
  • It the card they predict is the same rank (i.e. king) as the next card, they earn 5 points
  • If the card they predict is the exact card drawn, they earn 10 points.

Play the game for 10 rounds and total your score.  It’s a nice game for discussing the vocabulary of suits, face cards, and values, along with the conditional probability of events, given past information.

Dan also presented some nice hooks for class, such as one involving a “proof” that the angles in a triangle sum to 180 degrees.  Have all students cut a large triangle from a piece of paper.  Mark each of the 3 angles. Then cut the large triangle into 3 small triangles.  The 3 marked angles can then be arranged to share a vertex, adjacent to one another, and will form a linear trio.


The math games presented me remind me to do a blog post about the long-running BBC tv gameshow “Countdown”, where a numbers game is played.  Google the show on your own, or wait for my post about it next month.

Google vendor presentation


Today I also participated in a presentation by reps from Google, where a number of tools were presented, which work with their Google docs.  Flubaroo is an extension which teachers can use to quickly grade responses students submit via a Google doc.  The extension will grade the assignment, then e-mail students with a detailed report of their progress.

Mail map merge will allow you to create a distribution list, and include a map to an invited location.  Pretty snazzy!

ASCD 2012 – Saturday

Today was the first day of the ASCD conference in my hometown of Philadelphia, and I had the opportunity to attend a number of interesting sessions…

It’s Not About the Bling – Technology Through an Instructional Lens

This presentation was by the Multimedia User Group from San Juan, California, where they have embraced  technology as enhancing and transforming instruction:

technology allows for the creation of new tasks, previously inconceivable.

The group discussed the need to move to a re-difinition level, where high tech and high pedagogy are married.  Many districts, like mine, probably hover at the augmentation level, where high tech are matched with low to middle pedagogy practice.

The “It’s Not About the Bling” acronym provides a framework for student-centered technology integration.  Goals are focused on the student:

Interactions – Non-linguistic represntations – Assessments – Thinking Skills – Build

Teachers in the MMUG meet for 3 Saturdays each year for a day of sharing, collaboration and development.  The message is that it’s not about pretty background and slide transitions, but effective message delivery.

General session – Reed Timmer.  Discovery Education.  Stormchaser

Thanks to Discovery Education for sponsoring Reed’s inspirational speech, which included exciting videos of his adventures.  I was amazed by Reed’s story, and his transformation of chase videos from and old Buick to the chase mobiles he uses today.


I was most interested in the unique data collection methods, which take place in relative chaos…parachutes launched from potato guns attached to hulls of vehicles.  Parachutes open and follow spirals of tornadoes, which collect data on speed and temperature every 5 seconds.  Later, my boss and curriculum director noted “It’s like the guys from Jackass melded with science”.  He’s probably not all that far off…..

Thanks to Janeeta for her gracious invitation to the Discovery Ed dinner, and best wishes to Pat, as he moves on from the teaching ranks to his exciting new job with Discovery!

Differentiated Math Strategies for Addressing the Common Core

This session included a few interesting nuggets for challenging student thought and discussion: Example for defending reasoning: Does .9 repeating equal 1? Discuss and defend.

One colleague noted…We agree that 1/3 = .3 repeating, and that 2/3 = .6 repeating…therefore .9 = 3/3 or 1

There was a intriguing response from the group which I appreciated: “if two numbers are not equal, there must be some other number between them.  What number lies between .9 repeating and 1?”

Finally, thanks to the ASCD crew for the great fun at the tweet-up this evening!  Look me up at @bobloch

Looking forward to tomorrow!

TI Publish View – Bringing Interactive Lessons Home

In the past few years, Texas Instruments has been aggressive in developing and marketing its Nspire product line.  I recall the first time I shared the (now) old blue click pad product, and the oohs and aahs from my students when I showed them how you could trick out the keyboard with an 84 keypad.  This was soon followed by the touchpad, and now we have the CX, with its thin design and color screen.


Along with the improved hardware, TI has also improved its software options, providing an opportunity for teachers to create their own lessons and demonstrations on the software.  Files can be easily traded and shared with students, or used on a whiteboard as a classroom manipulative.  Last month, I had the opportunity to attend a free morning of professional development on the TI Publish View feature.  This feature of the Nspire software allows teachers to embed some of the interactive features of Nspire CX files into documents.  The TI-Nspire document player then allows students to open these files and navigate the lesson.

In the short example I created below, the coefficients of a polynomial can be adjust using “elevator buttons”, which are sliders used to change the values.  Students can then observe the value of the discriminant and look for patterns in the values.  Click the link to join in the discriminant insanity!

Additional files to try can be found at the TI Activity Exchange.  What an interesting way to have students explore on their own.  Thanks to Mike Darden from TI for the great session, and Doyt Jones for his continued hard work in bringing these sessions to the Philly area.

Encouraging Persistence Through Contest Problems, Part 2

In my last post, we looked at an AMC-12 problem of moderate difficulty, but with an premise that could be understood by many.  This time, we’ll take a look at a problem which delves into more abstract concepts, and explore how technology can allow students to consider solutions.  The following problem was question #23 from this year’s AMC-12.

Let S be the square one of whose diagonals has endpoints (0.1, 0.7) and (-0.1, -0.7).  A point v = (x,y) is chosen uniformly at random over all real numbers x and y such that x is between 0 and 2012, inclusive, and y is between 0 and 2012, inclusive.  Let T(v) be a translated copy of S centered at v.  What is the probability that the square region determined by T(v) contains exactly two points with integer coordinates in its interior?

In this year’s contest, where over 72,000 students participated, this question was answered correctly by only 4.5% of students, and was left blank by 81.6%.

Tomorrow morning, write this question in its entirety on a side board, and observe student reactions.  How many students begin to sketch the square described in the first sentence?  How many ask questions about some of the sophisticated language?  How many shrug and turn away?  This problem presents a number of chance for students to summarize given information, summarize “scary” language, and consider possibilities.

The first sentence describes a task which all geometry students, regardless of phase or level, can pursue.  Give students the task of telling you everything they can about that square, and share out their ideas.  Let’s look at it piece-by-piece.  First, we have a diagonal with defined endpoints:


Can we find the other diagonal? This is a great opportunity to look at perpendicular bisectors, and consider slope. After the other diagonal is found, we can see the given square:


Is there anything else we might need to know about this square? Can we find its side lengths? How can we find its dimensions? Another nice connection, to our old friend the Pythagorean Theorem, emerges…


How convenient for us! We have a unit square, where all sides have length 1. Even if we don’t consider the rest or the problem, think about how rich of a discussion we have already had!

Now for the scary part….all that spooky language. But is it so bad really? What is the question really asking us to do? Challenge students to re-write the premise of this problem, so that it can be explained easily to a friend or neighbor. Here’s what we are being asked to do:

  • Pick any random point in the first quadrant, but don’t go above 2012 for x or y.  We’ll call it point v.
  • Take the square we just made, and copy it, so that v is the center.
  • How likely is it that the new square contains two points with “integer coordinates”?  This is great time to introduce the term “lattice point”.

Here’s an example of what we are looking at.  The original square remains at the origin, but a new one, with center v1, has also been introduced.  Notice that this new square captures only one lattice point.


So, how are we going to find that wicked probability? Using Geometer’s Sketchpad, I created a model of this problem, which students could then use to explore the premise. Contact me if you would like the Sketchpad file to use.  Enjoy my tinkering in the video below:

Encourage your students to break down problems into smaller, digestible pieces, and not be afraid or scary-looking language. The rich class discussions which come from allowing time for questions to stew and worth it!

So, what’s the answer? Below is a link to a great summary video by Richard Rusczyk from the excellent website It’s perfect for sharing with your classes after you have explored the problem and discussed ideas:

Encouraging Persistence Through Contest Problems, part 1

One of the many tasks I perform for my school district is serving as math club sponsor.  While I often attempt to find interesting activities and experiences for the club, many students join math club to participate in the contests we tackle each year, and this month tends to be a particularly busy period for contests.  In-house contests include the Pennsylvania Math League and the American Mathematics Competition exam series.  Today, I am writing my blog post from a lecture hall at Lehigh University, while two floors below hundreds of students are participating in an annual contest organized by Dr. Don Davis, who uses the event to recruit students for his American Regions Math League teams.  Tomorrow, two teams of students I work with will attempt the Moody’s Math Challenge, where students are given 14 hours to complete and open-ended question and submit a solution.

I recall a conversation I had with a math department head I worked with when I first became a teacher, and the conversation centered around why our school didn’t have a math club.  The veteran teacher responded that he didn’t believe in doing math problems as competition, and I suppose that I agree with the essence of his argument: that sitting alone, isolated, doing a series of problems may not be the most enriching of pursuits.  But the conversations that take place surrounding challenging problems can lead in interesting directions which often encourage collaborative thinking and build confidence in approaching “scary-looking” problems.

The problems from the AMC 10 and 12 exams, in particular, lend themselves to discussions of problem-solving approaches.  Each exam is set up with 25 multiple-choice questions which loosely go in sequence from least to most challenging.  Many students I coach can handle the first 10 to 12 questions, and may venture as high as question 20, before sensing that the questions have taken a turn towards the evil…questions with wording and symbols beyond their experiences.

We can use contest questions to encourage not only higher-level thinking from all of our students, but also develop persistence in problem solving.  Consider the following problem, which was #15 on this years AMC-12:

A 3×3 square is partitioned into 9 unit squares.  Each unit square is painted either white or black with each color being equally likely, chosen independently and at random.  The square is then rotated 90 degrees clockwise about its center, and every white square in a position formerly occupied by a black square is painted black.  The colors of all other squares are left unchanged.  What is the probability that the grid is now entirely black?

This is a problem 15 from the contest, which implies that it is bordering on the medium to hard-type of problem.  I like this question for two reasons:

  • It’s a probability question.  I know that these types of problems often appeal to me, as opposed to geometry questions, which often interest me less.
  • It has an accessible premise.  While we may have trouble down the road computing the probability, this problem can be easily de-constructed, simulated and discussed, even by middle-school students.

We can walk through this problem by giving students some 3×3 grids and a black marker.  According to the problem, each square is painted white or black at random.  Have students make their own grids, then make a copy of it, which will then be rotated 90 degrees:


Then, follow the directions to make an altered grid: every white square in a position formerly occupied by a black square becomes black.  All other squares are left unchanged.  In the example below, our grid fails, since the results is not all black.


Where can we head from here?  Depending on the maturity and sophistication of the students, there are a few paths to consider:

  • With younger students, hand out some pre-made grids, where some will become all black after the transformation.  Can students categorize those which become all black?
  • For more sophisticated students, hand out more 3×3 grids and experiment to see if they can develop one or more grids which satisfy the problem.


For some students, the problem may stop here, which is fine.  The experience of having tackled part of a complex problem is a success unto itself.  You can even let the problem stew with students for a few days to discuss with friends and parents before reaching some conclusions.  In this problem, there are 3 dependencies, 3 different aspects of the grid to consider:

  • The center square must be black
  • The corner squares (A,B,C,D, below) must meet certain arrangements
  • The non-corner squares (e,f,g,h, below) must also meet certain arrangements


Can we list arrangements for ABCD which will result in all black?  Certainly black-black-black-black works, but so does white-black-white-black.  Are there others?  This then shifts the nature of the problem from a scary-looking probability question to a more tame (but still semi-scary) counting problem.  I’ll leave the counting to you and your students.

In my next post, we’ll look at a geometry example from this year’s AMC-12.