Monthly Archives: October 2013

A Day With Rick Wormeli – Redos and Retakes

Earlier this week, a handful of colleagues from my district and I experienced the educational whirlwind that is Rick Wormeli.  I have studied Rick’s writings for some time now, shared thoughts on redos and retakes and standards-based grading before, and incorporated some of his ideas into my own classroom procedures.  What I most enjoy about Rick is that he challenges your existing classroom practices, and breaks them down to their foundations:  if it’s not about achievement, and moving kids forward, then it’s not part of the plan.

The day began innocently:

Today will be a waste of your time.

Thanks?  But Rick’s point was that a single day of PD is simply not sufficient to synthesize these ideas.  Change only comes when we take what we learn back to our school, have discussions, think about our policies, and work as a team to do best for our kids.  Rick is correct when he states that “school is set-up to meet the needs of those who get it first”.  Let’s work on breaking down long-standing policies and  drafting new ideas which benefit all learners.

Climb That Tree

While the day was billed as a “formative assessment” seminar, the concepts really be-bopped from standards-based grading, redos and retakes, learning targets, and formative vs summative assessments.  I fear this blog post would be 10 pages long if I tried to summarize everything, so I’ll instead focus on one idea I have incorporated into my classroom routine this year: test retakes.

Rick WormelliHow do students react to the grades we give them on assessments?  How do our grading practices impede students reaching their learning targets?  Rick argues that many of our strategies cause students to wind up in “the pit”; further, many schools perpetutate practices (like losing lateness points, or not allowing corrections) disguised as “teaching responsibility” which cause students to fall deeper into the pit.  It’s our duty to lead students through strategies which will get them out of the pit, and professionally unethical to conflate evidence with compliance.

I have incorporated re-takes into every exam I give this year in AP Statistics, and have allowed re-dos in many Algebra 2 tasks.  I continue to evaluate the success of these methods, and I have been largely happy with both the results, and the attitudes of students in embracing the new procedures.  When should you allow redos?  To Rick, the answer is’s our professional responsibility to allow redos – unconditionally.  Here are some resources from Rick Wormelli which will get the conversation started:

Educational Leadership, “Redos and Retakes Done Right” – requires ASCD log-in, harass your principal!

“Fair Isn’t Always Equal” – Rick’s landmark book on assessing and grading in the differentiated classroom

Video on Redos and Retakes – Rick defends the redo/retake practice.

What’s the problem with allowing a failing grade?  Doesn’t that build character?  How do students react to a failing grade, as opposed to a different designator, like “not yet”…


This year, my colleague Joel and I wanted to incorporate retakes into our statistics classes.  But there are certainly organizational challenges to be met, and our discussions challenged our beliefs on assessment and its purpose.  Here’s what we decided on for our classes this year:

  • Each unit test has two parts: multiple-choice and free-response, graded equally like they will be on the AP Exam.
  • There are always two free-response questions.  Sometimes they are actual former items from AP Exams, sometimes they are questions we write or adapt.  A free-response question is one scneario with multiple parts
  • After the exams are handed back, students may come in to take the “replacement question” for the exam.  The replacement question is a third free-reponse question, which students take on their own time during a daily directed-study period, or after school.  The grade on the replacement question replaces the score on the lower-scoring question from the unit test.
  • We don’t have a procedure for recovery for multiple-choice.  But we are kicking some ideas around.

Here’s why this procedure has worked for us.  Unit learning doesn’t end with the chapter test.  Students need to go back, reflect upon their misunderstandings, and develop a plan for doing better on the replacement question.  It’s great to see kids really reflecting about what went wrong on their test, and coming back to clarify what went wrong….that simply didn’t happen before.

The record-keeping is awkward.  But I am getting better at it, and figuring out the best way to manage this extra level of grading.  And Rick is now whispering in my head “Don’t drop the principle because you can’t handle the logistics.”

This is the first of what I am sure will be many posts reflecting upon this special day of PD.  Looking forward to sharing more ideas, discussions, and anecdotes!

To Rick: Thanks for the great day…and The Three Amigos was underrated

My Favorite Teacher Circle: PASTA

Just got back from the fall meeting of my favorite local teacher circle, PASTA.  The Philadelphia-Area Statistics Teachers Association meets a few times each year to share best-practices in statistics teaching.  Many of this month’s presenters are AP Statistics readers, and the ideas are not specific only to stats…we just share great classroom action.  I gave a recap of our last meeting in the winter; enjoy the great ideas from our Fall meeting, and visit Beth Benzing’s website for materials from the meeting!

Daren Starnes, famous in the Stats-world as author of The Practice of Statistics, shared his first experience with Team Quizzes.  I have tried team quizzes before, mostly for quizzes where I knew students were having the most difficulties with material.  But Daren added some features I had not before considered:

  • Students are assigned to their teams at random.
  • Each team member received a copy of the quiz, and must complete the quiz.
  • In a quiz, one question is chosen randomly to be graded from each paper.  A student’s grade is a combination of the score they receive on the question, along with the average of the scores from the other papers in the team.

Daren also commented on the roles of introverts and extroverts in the teams, and how this method could empower introverted students to self-advocate.  He suggest the book Quiet: The Power of Introverts as a resource.

AdamAdam Shrager, famous as the social director and man-about-town at the AP readings, shared his movie-correlations activity.  This has become one of my favorite activities during the stats year.  Students are asked to fill out a movie-preference survey, which Adam then uses to compute peer-to-peer correltations in Excel.  (look for “correlation” in excel…you may need to activate the Stat Pack) Discussions regarding the interpretation of positive and negative correlations then occur.  Most importantly, mis-conceptions of the meaning of low or zero r-values are discussed with a context easily understood by students.


Leigh Nataro shared her “Pacing a Normal Distance” activity, where students walked between 3 different campus buildings using “meter-long” steps.  The data is then entered into Fathom, and is used to discuss variability, the 68-95 rule, and normal probability plots.  Fun discussions of outliers and error as well!


Our host, Beth Benzing from Strath Haven High School, shared a family income Fathom file which draws samples of various sizes from a clearly skewed distribution.  In addition to to having students record observations and work towards generalizations, Beth has worked to increase the rigor in her associated questions, using past AP items as her framework.  Some examples:

  • What is the probability that a sample of 5 families will have a combined income of over $500,000?
  • What is more likely: a sample of size 5 having a mean income of over $80,000, or a sample of size 25 having a mean income over $80,000?  You may recall a similar AP question from a few years ago regarding samples of fish.


Brian Forney shared ideas for bringing concepts from Sustainability to the AP Stats classroom.  In one example, Brian shared data on depths of ice sheets over time, with excellent opportunities to discuss cause and effect from scatterplots.  Check out Brian’s presentation on Beth’s website.

Finally, I was happy to share my recent lesson on Rock, Paper, Scissors and two-way tables.

The meeting concluded with some great ideas for making multiple-choice assessments more fair and effective.  There were a number of excellent ideas here, but I think I’ll look up some more info on alternate assessment methods and save it for another post…so stay tuned!

A Math Teacher Ventures Into a Western Civ Class

During my prep period last week, I came back to my classroom after a trip to the main office and overheard some familiar language: Euclid, Pythagoras, The Elements.  What’s intriguing here is that my next-door neighbor isn’t a geometry teacher; rather, my colleague Glen is a social studies teacher, with 3 sections of Western Civ each day.  Excited, I popped my head into his classroom.  And after some good-batured ribbing out how he was advancing on my math turf, I went back to my prep.  But Glen and I later talked about our shared interest in the Greeks, which ended with an invitation to come into his class to share a brief math history lesson.  I’m no stranger to the occassional cross-curriculur lesson, so this represented a fun opportunity.

One on my favorite courses from my time at Muhlenberg College was “Landmarks in Greek Mathematics”, where I was fortunate to have William Dunham as a professor.  His enthusiasm for math storytelling has shaped my approach as a teacher, and his book, Journey Through Genius, was not only used in the course, but is a book I often come back to for inspiration and contextual reminders of math concepts.  The book both walks you through the mathematical landmarks (like Euclid’s proof of the infinitude of primes) and provides a backdrop of the places and people (like the fascinating battling Bernoulli brothers) which shaped the surrounding culture.  It’s a great resource for any math teacher.

For Glen’s classes, I chose an example which 11th graders could easily understand and which would provide a glimpse into the genius of the greek mathematicians: Eratosthenes’ approximation of the Earth’s circumference.

Eratosthenes observed that on the longest day of the year, sunlight would shine directly into a well, so that the bottom of the well could be seen.  But that farther from the well, in other towns, this did not occur.  The well was located on the town of Syene, which we now lies directly on the tropic of cancer.

Syene Well

In Alexandria, a known distance away from Syene, Erotosthenes measured the angle produced by the sun’s rays off a post in the ground.


Taking this further, we can use alternate-interior angles to use this same measured angle as one coming from the center of the earth.


This central angle, along with the known distance from Syene to Alexandria, yielded an estimate of about 25,000 miles (or the Greek stadia equlivalent), an estiamate with an error of less than 1% of the actual circumference!  Both classes I visited seemed to enjoy this math diversion in the Western Civ class, with one student wanting to know more about how the Greeks approximated pi.

So find your local Social Studies teacher, and offer to bring in a little math!  There are some fascinating stories to tell.


Excerpt from String, Straightedge and Shadow

From the Mathematical Association of America

From Jochen Albrecht, CUNY

Finally, from Carl Sagan’s landmark series “Cosmos”

Exploring the MathTwitterBlogoSphere

explore MTBOSThis month, some of my Twitter Math Camp friends are hosting a fun, month-long event called “Explore the MathTwitterBlogoSphere”.  You can check out the website for more details, and each week promises a new task designed to encourage math teachers to reach out via blogs and twitter.

For the first weekly challenge, Sam Shah has asked participants to share their favorite rich task.  Even with having taught for 17 years, it was not easy to come up with one task which I felt summarized my philosophies, but here is what I feel is my best question.  It is one I have given many times in algebra 2, and our freshman-year prob/stat course:

How many zeroes are there at the end of 200! (200 factorial)?

That’s it.

Here’s why I like this problem, and why I enjoy giving it:

  • It’s has a simple premise.  Sometimes I need to embellish with “think about multiplying out 200!  It would be a really long number.  That number has a lot of zeroes at the end.  How many are there?”  But besides having to know what factorial does, it is plain and simple in premise.
  • It requires thinking about the nature of numbers.  Brute force doesn’t work well here.  When I first started giving this problem, I think I used 25 factorial, but then technology started to catch up with me.  One year, a few students used Excel, which gave a wrong answer, as it began to konk out at bigger numbers.  Even if students can now find an “answer” through some tech means, the challenge to explain the “why” remains.
  • The answer is secondary.  Communicating your reasoning is king.  This problem present great opportunities to utilize math vocabulary: factors, commutative property, grouping, etc.  I grade this task almost exclusively on communication, and students are often surprised to find that a math task can require such a level of revision and reflection.
  • I can move towards a generalization if I need to put my foot on the gas more.  If a few students seem to have the answer and communicate a solution, I can challenge them to develop a formula which works for any number factorialed (is this a word?).

Rich problem solving experiences have always been a part of my classroom culture.  This problem is one of my favorites.

Adventures in Common Denominators

At my high school, I host “math lab” on alternate days.  The lab is an open room where students can obtain math help during their lunch or free period.  I like this assignment because I get to see a lot of different kids during the week, gain some insight into the approaches of my colleagues, and get my hands dirty in many math courses.

Last week one of my “regulars”, who often leaves class early for sports, came to the lab for Algebra 2 help.  He missed out on a adding/subtracting rational expressions lesson, but had the notesheets handy.  We wrote the first problem on the board:

Problem 1

Nothing too fancy. But in math lab, I don’t know the students as well as my own, so I need to do a quick check for some background before diving into new material. A spot check for understanding of fraction operations was in order:

Add Fracts

The student approached the board confidently and “added” the fractions:


…sigh…. sometimes there isn’t enough coffee in the world. But all is not lost, and after a stare-down, the student recognized he had acted too quickly, and completed the problem correctly. This led to another problem. This time, I asked the student to just tell what me what the common denominator would be:

Problem 4

Without hesitation, the student knew the correct denominator to be 24. But why is it 24? The student could not defend his answer, but was absolutely sure 24 was the LCD. On the one hand, I am happy that the student has achived enough fluency with his number sense to confidently find the denominator. But, on the other hand, has lack of a process is going to hurt us now when we try to apply LCD’s to algebraic expressions.


When I start my lesson on adding rational expressions, I hand out index cards to every student.  I give students 2 minutes to repond to the following prompt:

How do you find a least common denominator? Provide directions for finding an LCD to somebody who does not know how to find one.

I collect all of the cards, shuffle them, and choose a few randomly to share under the document camera.  We will discuss the validity of the explanations, and use parts of the explanations to come up with a class-wide definition of an LCD.  Here is what you can expect to get back on the cards:

  • Some students will recognize that factors play a role, but won’t recognize that the powers of the factors are imporant.
  • Many students will attempt to use an example as their definition.  This allows for a discussion of a mathematical definition.  Is one example helpful in establishing a rule?  How about 2 examples?  How are examples helpful, if the reader does not know how to find an LCD?
  • Some students will provide a hybrid of the last two bullets.
  • If a student does provide a suitable definition, it’s time for you to play dumb.  Let the class assess the language and verify that the definition is, or is not, suitable.  In one class, I “planted” a working definition in with the student cards to see if they could identify a working definition.

The Least Common Denominator of two or more fractions is the product of the factors of all denominators, raised to the highest power with which they appear in any denominator.


I am curious how math teachers approach Least Common Denominators in earlier grades, and how these approaches translate to algebra success.  Here’s how a few online math sites approach LCD’s.

First up, mathisfun (search for “Least Common Denominator):


This is the approach I suspect many teachers take to help students find LCD’s.  It works for manageable denominators, but becomes cumbersome when we have 3 or more denominators to consider, and certainly is not helpful in our algebra world.  Also, if you get too confused, you can use the “Least Common Multiple Tool” this website provides.  I suppose it’s not an inappropriate method, but a more algebra-friendly process should eventually develop.

Next up, Everyday Mathematics at Home website:


The good news – we have a formal definition!

The bad news – you have to know what an LCM is to use it.

This is a more formal version of the Mathisfun example.  We could adapt it for use in algebra, but again, a definition of LCM is required here.

Khan Academy starts by using lists of multiples and provides and example with a trio of numbers for which we want the LCM:


The factor trees and the color verification that all 6, 15, and 10 are all factors of 30 is nice, but this example conveniently leaves out any scenario where a number is a factor multiple times, and this is the only example given.

Finally, let’s check out how PurpleMath tackles LCM’s, with a non-intuitive example:


Now we are getting someplace.  Not only does this method stress the importance of factors, it shows the importance of include all powers of those factors.  And I could transfer this method easily to algebra class!

Have any insihgts into teaching LCD’s, either for a fractions unit, or in algebra?  Would enjoy hearing ideas, feedback and reflections!