# When Errors Lead To Understanding

I gave a test on equations and inequalities recently. From earlier posts, you may remember I am not a huge fan of tests. Certainly I understand that this is a controversial opinion, but testing seems arbitrary to me. We decide that by a certain date students should master material we teach them. The truth is that some students will master concepts by a specific date and some might not be there yet. Regardless, instead of simply offering support and troubleshooting without judgment, grades speak the words teachers do not wish to say. Well, at least teachers like me don’t wish to say because I know they are often statements riddled with falsehoods.

On a recent test, my students did not get the “right” answers at a percentage that assured me that they mastered the material. Due to a curriculum calendar, I knew I should move on with the content, but I just couldn’t. I scored each question as either right or wrong, took copious notes to the errors that abounded and quietly reflected. The truth was, most students clearly understood the process and concepts of equations and inequalities. They dropped a negative sign, they accidentally performed the same operation in one step, but correctly used the inverse operation in others. They made sense of the story problems and set up tables to organize their information, they translated the English to Algebra, but then, forgot to distribute a factor in their solving. In other words, what I saw was clear understanding of concepts marred by precision errors.

How do you get students motivated to improve precision?  How do you even make them aware that they are making the errors? There is an excellent reason precision is one of the mathematical practice standards we emphasize.  I remembered reading a post by Andrew Stadel about how much students love to find other people’s mistakes and how valuable it is in their learning. In addition, there was a wonderful Teaching Channel clip my entire math department watched (and then I re-watched multiple times) that highlighted a math teacher showcasing her favorite mistakes to her students so that they could all learn from them. I adopted this practice long ago, but after these test results thought, I need to do more.

Since I had corrected the tests and left my error notes on my desk, I decided to get to my classroom at around 6 a.m. the following day. I recreated similar problems completed with the common errors. I told students about the precision errors on their tests and that I knew were not a reflection of their total understanding. Their task for the day was to prove me right. After distributing the common error sheets, they were challenged to work together to discover what was wrong on each question. I circulated around the room and students pulled me over to say, “Oh my gosh, I do this too. I think I did this on my test.” I also heard, “I feel so sorry for this student, look at all the great understanding all the way through this problem and to get it wrong at the last step, ugh, heartbreaking.” Exactly my sentiments, I thought.

After this exercise, I saw my class again during a rotation period and gave them back their tests. There were no notes on their tests, just circles of wrong answers. Students were challenged to now find their errors.

The results:  Amazing. Every single student was able to find and correct mistakes without any specific feedback on the individual tests.

The next day, I asked the students what they thought about the exercise. The response was overwhelming. Every single student raised a hand when asked if they found it helpful. Students were speaking over each to explain why it had been such a productive exercise.  One in particular mentioned how much easier it is to find someone else’s mistakes than their own. Another talked about how when you can find an error, that means you truly understand the math enough and it strengthens understanding (seriously, those were her words). It was student comment after comment that made me say, gee, I should do this type of exercise more often.

The question remaining is, how often?

# What Makes A Circle, A Circle?

In the seventh grade curriculum, geometry studies are few and far between. Sometimes students will stumble upon a textbook question regarding angles, triangles, area, or perimeter, but there is little exposure in isolation of geometric properties in our grade level. (Of course, all mathematical concepts are connected, but I digress). In our school, 6th grade is the year for circles, so we were advised to insert a review or to backfill for any or all students who possibly did not receive exposure to the concept. This makes it tricky when some students have lots of experience and other students have none or simply no memory of it. And although this might sound like an unusual problem to someone without 25 or more different personalities in class, it is often par for the course for every teacher and student each day. Some concepts are completely mastered by some students, others are sort of remembered, a few have surpassed curriculum expectations through independent study, and some have no recollection, regardless of the topic at hand. The reasons are plentiful. We as teachers all do our best to expose students to topics that they will claim they never saw, even if we previously taught it to them in the same year! The question is: how do we as teachers prevent this scenario?

We are supposed to be able to differentiate for all levels and reach all students in every lesson. Although I try via various modalities and methods, I have yet to figure out how to accomplish this with fidelity each day. However, one method that I find differentiation is embedded, is in well-developed three act tasks. Today, I used Dan Meyer’s Best Circle task. Here is a link: http://threeacts.mrmeyer.com/bestcircle/.

I am sure there are lots of different ways to utilize this lesson, which I would love to hear about from you if this fact applies. All I can share is what I did with it.

In the first act, four different gentlemen draw a circle. I paused the video clip and asked students to vote on which of the four circles was the best circle. I wrote the names of the circle creators (Chris, Timon, Andrew, and Nathan) on the board. Immediately, Chris received most of the votes. Students let me know it looked the least like an oval. I did not agree or disagree with their comment, but simply repeated, “Ok, less like an oval than the rest, got it.” I then asked, “Is that a true defense?” Students shrugged and expressed that they thought it was as good a reason as they could collectively argue at that point in time. Other than that, students did not have much to sustain their reasoning for choosing Chris. Although they would find out that the majority of their initial guesses was the correct one, a good math teacher always asks, “but why?” A student excited about math wants to be able to answer that question. And boy, were they interested.

Next, I assigned a new task to the students by asking a question: What makes a circle, a circle? I gave them about 10 minutes to determine the answer to this question. They were permitted to use their Chromebooks, textbooks, and each other to come up with research and data to support their opinions.  Students were allowed to revise their original hypothesis based on any information they found.

Work options were offered. Some students worked independently, others partnered with one student, and some collaborated with a group of three or four peers. One student spotted compasses on my desk and asked if he could use it. He went up to the board where the frozen screen shot with the circles remained. He slowly used the compass on each circle. As students researched and revisited words like equidistant, radius, and diameter, I was asked if they could go to the board and use a ruler to take measurements of the circles. When one student witnessed another student trying it out, she commented, “Oh, that is a good idea.” She joined the student, and then another and before my eyes, students formed a small cohort within the class in front of the screen shot with the four circles. Students began debating each other what part of the circle they should measure and whether or not they could make determinations from their measurements. Some were discussing the midpoint, others radius, hemispheres, quadrants, circumference, etc. A few were cheering themselves for their initial guess and others were disappointed that new information changed their previous prediction. When the 10 minutes was up, I shared act 2 with the additional information provided by the lesson. Students were asked whether the area, circumference and coordinate points of the circle were offered for each circle would provide a clue to help solve the problem. This is where it came out that students recalled very little in the way of formulas and the inner workings of a circle from previous learning. I showed them the resolution (the third act), but I quickly realized we weren’t done with the task today. There is so much left to dissect, I did not even want to enter into the explanation Dan provided in the teacher guide. They weren’t ready.
I do have a bit of formal review to do with students, perhaps a day or two of lessons, but we will return to this three act lesson to see if students can in fact determine why the best circle was in fact the best circle. The quick review will come…ahem…full circle. Maybe, just maybe at this time in the next school year, students will still remember some details about what makes a circle a circle.

# Stealing from Dan Meyer…again!

Recently, the great Dan Meyer started a strand on his blog called Pseudo-Context Saturday. He shows an image from a math textbook and invites his readers to guess the extremely contrived math context behind the question. It is a challenge right up my alley because I cannot stand contrived word problems that try to convince students that the math I teach them will be useful. All problem solving is useful because life is all about problem solving, so there.

I have enjoyed trying to guess the context, so I thought, if I like to do this, maybe my students would like to as well. Recently, I tried an altered version of this activity where I simply ask student in big bold letters to “Guess the question.” It is not exactly like Dan Meyer’s activity because students already know the context of the day. However, I displayed an image and asked them to guess the question that matched the image. I offered the closest guesser a prize. It went over very well, so I tried other versions. Students have been shown a calculation and an image side-by-side. In addition, I have provided them a context and background (without the image) and asked them to guess the exact question and/or the image. This small tweak is so much more powerful than simply giving students the contrived problems from the book. This easily adaptable exercise promotes student thinking and empowers them to make their own connection between context and procedure. Dan Meyer, you have done it again!

So go ahead, you try it. Guess the context and/or question that goes with the image displayed in this entry from our textbook. Don’t worry, I will provide the answer later. Just another piece of evidence that proves that math is fun!

# Is The Common Core Just Misunderstood?

Please forgive me if you hate the words Common Core. I don’t try to go out of my way to write about something controversial, but I know the potential firestorm for this topic. My first question to all those that abhor the Common Core is:  Do you every wonder why the Common Core came to light? Although I have background knowledge, I quickly did an Internet search to see what explanations abounded. Terms popped up like, ‘college ready’, ‘consistent expectations for all regardless of zip code,’ ‘national standards,’ etc.

There are a lot of people, both in and out of the education field that hate that explanation, so it is not one that I will support in this entry. Preparing students for the real world, yes, obviously that is something that we focus on as much as possible, but what does that even mean? The meaning probably depends on whom you are speaking with. All I can offer is my interpretation. I want to prepare students to think critically and deeply about any problem, whether numbers are involved or not. My hope is that students analyze problems carefully and reflect seriously about all options before trying to attack any problems in the “real world.” I think the Common Core actually helps with that objective.

Please allow me to offer my classroom perspective. I have been teaching math to students for 15 years. 10 years was in an elementary setting, and the last 5 have been in the middle school.  Within that 15 year span, teaching philosophies (as well as several math programs) have come and gone. Throughout all of the math trials and tribulations, one consistency remained; students were not retaining the math. I know this is not just a phenomenon I have witnessed, because if it were, there would be no Common Core. The traditional way of teaching math would involve students learning an isolated concept. After learning it, students would study it for several weeks with lots of practice examples. The examples might be peppered with some derived textbook problems and culminate with a test. This is how I was taught and I know how many of you were taught as well.

Immediately after the test, many students would promptly forget about the past concept(s) and move on to another topic. Some of the details would re-emerge as necessary, but many students would notice that previously learned concepts drifted out of their minds after moving on to another topic. There was little transfer of knowledge from the temporary memory to long-term memory storage in the brain. Some students would retain rote procedures, and be promptly labeled as math people. Those who were unable to remember were labeled another way.

This was and continues to be a huge problem. Math concepts build on one another. They only have the opportunity to do so when students actively make connections from one concept to another in experiences where they witness the fluidity. For those who label The Common Core as fluff and not real math, please allow me to assure you that it was not designed to eliminate the algorithms. In everything I have studied, the algorithm (procedures we all learned growing up) is still the goal.  The difference between direct procedural teaching and problem based learning is that students receive the opportunity to investigate the why first.  The investigation allows students the chance to actively make mathematical connections with the ‘why’ to the procedure. Often, when students are given a problem, it creates the interest in the procedure that would never have been there if it were the only teaching point. What does this mean for our students? Instead of promptly forgetting procedural math, visual and problem based learning allows students to double down on their understanding and have the option to not only solve a specific problem in a unit, but provides students with tools to figure out how to solve all problems as efficiently as possible.

One of the largest obstacles of this philosophy is the incredible push back against it. This does not just come from parents, but also from fellow teachers. Change is hard, no doubt about it, but I have seen with my own eyes the difference between students memorizing a procedure versus deeply understanding why they are using it. The difference is stark. The reality is that the transition has not been easy and we all feel the growing pains together. But fear not…

I truly believe that I am a much better math teacher today than I was 5 years ago. I can imagine and hope I will be that much more effective in 5 years compared with the way I teach today. This means my students will be better prepared for that scary real world we love to discuss. I credit my continued improvement to the Common Core because of my virtual colleagues. Math superstars like Jo Boaler, Dan Meyer, Robert Kaplinsky, Fawn Nguyen, Yeap Ban Har, and Andrew Stadel were likely brought together by The Common Core initiative. Thanks to social media and passion, we now have resources that allow us to collectively and positively impact our students’ minds.

I accept that challenge. The question is…do all of you? If the answer is yes, please stop picking apart The Common Core or shuddering at the mere mention of the term as if it were ‘Voldemort’ from Harry Potter. The Common Core’s evolution came from student necessity. It is time that we work together to address the ongoing needs of our students, parent communities, and even the frustrations when we fall short. Two words should not undermine our purpose nor our passion that were actually developed to ignite them both.

# Rediscovering Lessons

One of the reasons math teachers often get a bad rap is because we fail to provide opportunities for students’ deep understanding of concepts. Ever since my wake-up call and recognition of just how tricky integer mastery was, I have tried finding ways to reach students at a deeper level. The algorithm is there, it is always there and usually discovered eventually by students. Nowadays, students visually see the concept by using integer tiles, the number line, and/or creating their own model that makes sense.

Every summer vacation I dedicate most of my time to researching the latest and greatest in math instruction. This past summer was no exception. Sometimes in my research, I rediscover a lesson I had seen before and then promptly forgot about. The task I just completed with my students is one such lesson.

The Mathematics Assessment Project offers some wonderful lessons and tasks. Students really benefit from the structure of the lessons themselves, and the built-in peer collaboration. The lesson I used can be found here: http://map.mathshell.org/lessons.php?unit=7105&collection=8

In a nutshell, students consider temperature changes that result from traveling from one city to another.  The collaboration occurs when students work with others to connect one city to another through temperature changing arrows. In some cases, the destination city’s temperature is provided, in others, the change in temperature is provided, and in the last scenario, the departing temperature’s city is provided.  In a lot of ways, it works like a crossword puzzle where students will figure out one answer, which will provide them the ability to find the next. Students also organically begin to discover why the algorithm works the way it does.

I followed the lesson with fidelity as I started with a pre-assessment, provided feedback, completed some whole class instruction to get students ready for the group task, and even conferenced briefly with those children who still needed some additional assistance after the activity was completed. The MAP writers recommend following the lesson the way they designed it. Before sharing their work with the world, the lessons are tested to ensure that they are effective.  I would be lying if I claimed that every lesson I created that I believed would be a rewarding experience for students in my mind turned out to be so in reality.  In other words, instead of experimenting with a lesson that I hoped would be  successful, these lessons have been tested so there is no risk involved. Amazing!

There are some resources that are worth revisiting out there in our global math world. Teachers who share their ideas with the world are pure gifts to educators and most importantly, to all of our students.  This experience reminded me that sometimes we might need to rediscover these educational treasures on another day to appreciate their value.

# Number Sense Brings Happiness

Today, my objective was to teach students how to convert a fraction to a decimal or an equivalent percent. In prior years, the lessons I found were all very procedural-based. However, this year, I decided to open the lesson with a Number Talk instead. It was a simple opener. I warned students I was about to post a familiar fraction on the board and their job was to determine the equivalent decimal and/or percent. There was a catch, they could not use an algorithm or the reasoning of, “I just knew that one.”

When everyone understood the directions, I posted ¾. Their job was to put their thumbs up when they knew the equivalent decimal and additionally, had an explanation that would satisfy the requirements. At first, a few students struggled with explaining their answer without just “knowing” some form of ¾, but eventually, students rose to the task. I also shared examples of other student responses (from previous conversations I had with students) to make sure everyone could truly understand the number sense objective.

Next, I showed them the next fraction  ⅞. With the first round completed, the students were able to offer incredible explanations that touted number sense. At this point, I segwayed into the term “terminating decimal” and showed them the algorithm of the numerator divided by the denominator, inserting a decimal, etc. Now they had a choice as to how to solve the next problems, but I did not point out this fact. I simply posted another fraction and had them find its decimal equivalence. With each new fraction presented, students gravitated towards showing and thinking about the numbers and their connectedness over the algorithm. There were a few times where the algorithm was actually easier, and they noticed this too. Their energy was as extraordinary as their flexible thinking.

This was one of those days, a day where a lesson invigorated the class and their teacher. This was a day where I know that students left class thinking about numbers, procedures and the actual relationship between the two. This was a good day to be a math teacher.

# It’s Summer Vacation and I Can’t Shut Off my Teacher Brain…

This past year of teaching was revolutionary for me. After years of reading articles, books, and receiving tiny amounts of professional development that may or may not have made an impact on my brain, I evolved. No, seriously. I went from surviving the changes perpetually implemented by the powers that be to balancing those while making hundreds of my own. Granted some were successful and others, not so much…but I digress.

It seems that all of the pieces of advice that were dribbling into my brain finally congealed. Yes, I now have the big picture of what math instruction should look like, until additional research convinces me to change everything again.

I started this past year with one crazy goal. My goal was to make all of my lessons  fantastic experiences for both the students and myself. You know the ones I am talking about teachers. I am referring to class periods where you look around the room and every single student is engaged and no one looks bored. These kinds of lessons provide uninterrupted time frames where no one is staring at the clock; understanding is elevated, amazing questions and inquiry is running amok, and where students actually feel disappointment when the bell has rung. I wanted (ok, still do) every lesson to be like that.

After reading Mathematical Mindsets by Jo Boaler and taking two of her courses this summer, it fully hit me that an engaging lesson is the tip of the iceberg. I suppose I knew this already, but my understanding became deeper. Math instruction is not only about the individual lessons, just as math itself is not about the individual concepts. What my job is really about, is to help students to see math as a fluid subject. Students need to seek out patterns and find the connections so that one lesson (as engaging as it is) does not halt the learning of a concept after the bell has rung. Instead, each lesson should enhance previous learning and build stronger conceptual knowledge and deeper understanding among the connectivity of mathematical concepts.

So, now what? I need to find a way to make that connectedness a focus this year. Now that I see the picture this clearly, I have to find a way to structure my lessons to match. Dan Meyer, Andrew Stadel, Robert Kaplinsky, Jo Boaler, I look to you for inspiration and resources. Oh, don’t get me wrong, I have lots of resources already, but I want more.

The problem is I am supposed to be on vacation.  Is it wrong to spend an entire summer vacation fine tuning my professionalism? It is not that I haven’t done summer work before, but so far, every free day I have had; has been filled up with my math passion. As I read, research, and participate in more conversations about math, I find myself unable to slow down. My thirst for additional math and educational knowledge cannot be quenched!

For now, I am going to embrace the passion and curiosity I have for my own profession. Let’s be honest, the moment I really feel like I have a handle on my profession is likely the moment that I don’t belong in it any longer.

# I Hate Tests

I hate using tests and I don’t know what to do about it.

There, I said it. I hate tests. I am not just referring to the standardized tests, which have their place, blah, blah, blah…or so we are told.

My least favorite question ever is, “Dr. Polak “Is this going to be on the test?”

My disdain for that question is not because I do not understand the anxiety. I too suffered from test anxiety, not the type where I would freeze up and my mind would go blank, but it was just as paralyzing in other ways. Like so many of my students, I was grade obsessed. If I didn’t receive a 100%, I felt like a failure. This was regardless of the subject. This obsession continued through my doctorate studies and exists to this day. In fact, every year I am required to take the Blood Borne Pathogens test and I feel the anxiety there too!

I know I am not alone. This is a very common extrinsic pressure for the students (and adults) in our country. One can almost equate it to an addiction. When you achieve a high score you feel so great and relieved and proud, but before you know it, you are right back distressing about the next test. You study even harder, you sleep even less, practice more and achieve another high score, but it is not enough. The last stellar grade is never enough.

Even though most teachers, me included, are mandated by their school district to give specific assessments and score them a certain way, it doesn’t mean we feel great about giving them or think that we should. The cycle of grade obsession is just one of the reasons for my guilty conscience; the deeper reason is what it does to those students when they do not achieve that top score. Time and time again, students deem themselves stupid or as failures the second they receive a low score. The result for many students is that they stop trying.  Year after year I witness students who tell me or show me that they no longer feel motivation to learn. They have suffered trauma from these low scores and they believe there is no reason to try because they will just fail anyway.

Although I considered myself a math brain type of a student (even though I have since learned it is not as black and white as we all believe), like so many other students, I reached a point where I felt stupid in math class. When I was in High School in the Freshman Geometry Fast track class, I might as well have worn a dunce cap. Like so many students, girls especially, I did not understand concepts as quickly as my classmates. Speed and accuracy in procedures were all that mattered. Achieving a deep conceptual understanding and connections within the mathematics field was not a goal. We were all just learning algorithms, memorizing steps, and moving on to the next scenario.

I don’t want to recreate that in my class. I have spent this year creating and adapting lessons that truly offer students the options to ask questions, think deeply, wonder, and, have a little fun. And yet during many of these adventures students ask first and foremost, “Is this going to be on the test?”

Sigh.

I want students to focus on the excitement, intricacies and fascination of math. If math class was designed to inspire problem solving and questioning, it would be done right. Students should be intrinsically motivated to look for patterns and make connections with numbers and shapes. The interconnectedness between numeric topics is something they should see based on classroom tasks. Assessment, in my perfect world, would be conversations and feedback of what is working, what isn’t working.

I know, I know, students are going to enter the “real world” where they will be tested.  There are many times in life that it does matter to get things right the first time. If someone is performing surgery for example, I don’t want the mentality of, oh, if I take out the wrong person’s appendix, I can just make sure I get the right person the next time.” Not everything in life has a re-do option, but not everything in life has to be perfect the first time without revision options either. I ask, what is the most important aspect of student learning? Do we want students to strive for perfection, or for perpetual self-improvement?

# Flipping my Teaching, Not Just my Classroom

My teaching approach is getting flipped upside down…repeatedly.

It all started with my on-line introduction to Yeap Ban Har’s discussion on number bonds. Here is the link for anyone interested:  Number Bonds . This was the first time my mathematical mind was blown. Throughout my years teaching elementary school, I had stumbled across multiple approaches in computation, but never had the pitfalls of memorizing procedures and algorithms without context been succinctly explained. This is literally a 2 minute 50 second video!

This one youtube video launched my researching life. Don’t get me wrong, I had always tried to search for great lessons, etc., but this was the first time I felt like I was (for lack of better explanation) doing everything wrong in my teaching.

The timing for this epiphany was not super as I was pregnant with my second child and about to take the majority of the school year off to take care of my baby. In between changing diapers, cleaning spit up, and a very snowy winter trapped in the house, any spare moment was spent investigating better ways to teach math. Fast forward through 10 months massive sleep deprivation, the trials and tribulations (and wonder) of having two children instead of one, and intermittent mathematical research, I was back in my classroom wondering what to change first.

I have written a post about my first foray into 3-act math, as the great Dan Meyer was also a new discovery to me during my maternity leave/initial research period. Not only did I “meet” Dan Meyer, I also was “virtually” introduced to Andrew Stadel, Robert Kaplinksy, Jo Boaler, and of course, the DESMOS and MTBOS communities. Although I have never actually met any of these mathematicians in person, this growing group of educators provides me with daily inspiration.

Throughout this year, many 3 act lessons have made their way into my classroom. One that I recently completed, Robert Kaplinsky’s Zoolander had me questioning if what I was doing was working. Were these lessons as amazing I thought? Did they provide students with a context that made the experience and math meaningful? Were students making connections in their brains? Was I providing enough structure? In short:  effective or not?

Whenever I try something new, it is normal for me to question myself. Acknowledging this fact, I can see that this has been a wonderful transformation for my teaching and math learning for my students. These lessons have had a major impact and I know this from events in the last few weeks. Several weeks after the Starburst lesson by Dan Meyer and the Zoolander lesson by Robert Kaplinsky, my students were referencing them in math conversations in the hallway and classroom. You read that right, the hallway!!! Apparently there was a question on the standardized test about scale models and the students were discussing how easy it was compared to the work they had to do in the Zoolander lesson. Another student commented that the Zoolander lesson helped them really understand the concept better than any book and that was why the question was so easy. I rest my case.

The Starburst lesson initiated a debate about sample space. A passionate debate! When does this happen from a textbook example? I have no reference for that. In short, these lessons make a difference.

At the same time I have felt success achieved in my teaching and by my students, it has also been an immense struggle for me professionally. This is especially poignant with my lower performing students. How do I convince them to believe in themselves and see the beauty in mathematics? If they don’t know the basics, can they still participate in these lessons with confidence? How often will they give the line of, “I don’t understand” in lieu of a rigorous debate with their peers and investigative excitement?

In all honesty, I have experienced both ends of the participation specturm from lower achieving students. Although I had read numerous and convincing articles by Jo Boaler, I only just obtained a copy of her wondrous book Mathematical Mindsets. As I am reading it, I am shouting, “YES! Oh my goodness, I agree! And then in the next minute I am asking, “How can I do this every day? When does procedural math come in to play, does it?” What does this look like lesson by lesson, day by day? Does it transform the students the way she says it does? I am so IN and can’t get enough, period.

As I was researching youcubed, I noticed an opportunity for the summer to attend a workshop with Jo Boaler in California. At this time, I cannot afford to fly to California, pay for the workshop and a hotel room, not to mention the childcare issue, but oh to dream. I am going to take Boaler’s courses through youcubed and finish her book soon. Every free moment I have is spent reading, taking notes, and rereading it. It is my current math bible.

I do not have a neat and tidy way to wrap up this blog post. Once again, I am asking the mathematical world for a conversation about balancing the math classroom. Have you read Jo Boaler’s books? Have you tried 3-act math? What were your successes? What were your failures? How can we work together to keep the math conversation evolving and growing? Anyone else in? Leave a comment, tweet me at @drpolakmath, or send me an e-mail at mpolak@ridgefield.org. The larger our community, the greater our collective success in helping all students achieve in mathematics. Who is with me?

# Sweet Math: Dan Meyer’s Starburst Lesson and Probability

When I first discovered that Dan Meyer’s lessons could bring math to life in a new way last summer, I took the time to investigate the three-act math options he created. One that struck me as extremely engaging was his lesson on Starbursts. I saved it in a probability folder knowing full well that probability was slated for the end of our school year. A glimpse of it was so memorable; I actually had the wherewithal to incorporate it into my lesson plan this past week, in April. That might not seem earth shattering, but trust me, it is.

The first act of the lesson launches with the opening of Starburst two packs. The pack reveals one yellow and one pink. A skull and crossbones image appears over the yellow Starburst and an audible yuck is heard in the background.  A second pack is then opened, revealing two yellow Starbursts, which leads to two skull and crossbones over the Starbursts and an even louder yuck sound. Clearly, Meyer does not seem to like the yellow Starbursts. The camera then focuses on a large pile of Starbursts two packs.

That is the end of act one. Immediately, students began to debate the merit of each flavor of Starbursts and began to wonder aloud. I let them question and debate each other for a minute. Alerting them at this time I would not provide them with additional information, I asked them to make a prediction that was both too high and too low regarding how many yellow Starbursts they believed were in the pile. We wrote several too high and too low predictions on the board, and then I asked them if I could provide them with any information to help them solve the problem, what they would like to know.

Immediately students’ hands shot up and the first student I called on asked, “How many packages of Starbursts are there in that pile?” Another asked how many flavors there were. Several students scoffed at the second question and, somewhat exasperated commented, “FOUR!!! Have you never eaten Starbursts before?”  One asked to find out how many double flavor packs there were in the pile. The rest of the students loved that idea and complimented the thinking involved behind that one. And of course, inevitably, one student just wanted the answer. Sigh; there is always the need for that request!

The next two slides I shared were images from Dan’s lesson (Act 2) that revealed that there were 287 packages in the pile and the four flavors of (not by flavor, color) yellow, red, orange, and pink.

Now that students had a bit of information at their disposal, I asked them the following questions:  “In those two-packs, how many packages do you think have two yellow Starbursts? How many do you think have one yellow Starburst? What do you believe the overall percentage of yellow Starbursts is in the pile? Use what we have learned in our probability studies to make a prediction.

Students walked around the room and worked with anyone and everyone to try to figure out the answer. I was amazed as I witnessed the thinking displayed. Many students immediately wrote the total possible outcomes of Starbursts such as yellow-yellow, yellow-red, yellow-pink, yellow-orange, etc. They then used total possible combinations to convert to favored outcomes. With that, they used ratios and came up with their predictions. They found a way to apply the procedural math we had been studying for the previous two days in class on their own accord.

Students shared their predictions and many were close to each other, a few, not so close…Funny enough, many students who had different answers, upon hearing their peers’ strategies recognized probability mistakes that they made. When it was time to reveal Act 3, students were cheering. I love to hear cheering in my class, over, yes, MATH!!! They quickly calculated their percent error and found out how very close (and far) each was in their work.

A specific feature of Dan Meyer’s lessons is that he leaves them quite open for interpretation.  In my mind, he recognizes that teachers are not robots in the classroom and deserve the flexibility to interpret and customize to our heart’s content. This gave me an idea for an extension at the end of the lesson.

I pulled out a bag of Starbursts and had each student grab two. We recorded the flavors of the Starbursts pulled from the bag and made a frequency table displaying the sample space on the board. Unfortunately, we ran out of time, but I recorded our data on a frequency table so we could do a follow up the next day. My first question I plan to ask is:  What type of questions and answers can be generated with this information?

For those who might be wondering, students were granted permission to eat the two Starbursts they selected. After all, I wanted to make sure that this math lesson left everyone with sweet memories.