# 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.

# How an Average 2, 000 Calories a Day Diet Inspired a Math Lesson

All of the seventh grade math teachers have been in a room lamenting about the content in our curriculum. One topic of conversation was equations. How, do you make fantastic, hands-on lessons with equations? There is no shortage of such lessons if your topics are geometry or statistics, but equations, rational numbers, inequalities? Everything is so contrived.

Ok, so perhaps I might have contributed to the complaining, I won’t confirm or deny. Regardless, I was motivated to find or create something better. Within the context of rational numbers, I had used Dan Meyer’s age activity, I had even made my own for a few, but equations and expressions? I was stumped. I tweeted out to Dan Meyer and Andrew Stadel and the world asking, no begging, for ideas. Granted, I only recently began tweeting about math and have a total of 3 followers, but that is not the point.

The angels in the twitter universe answered my math prayers and Andrew Stadel recommended Robert Kaplinsky’s lesson idea for inequalities. Since I had already spent time creating an inequality lesson based on Mr. Stadel’s sweet snacks activity, I didn’t think I wanted to throw out all of my work before even trying it. As I analyzed the 2, 000 calorie lesson, I noticed an option to use it for equations. Eureka, I thought. Now I have a great lesson for inequalities and equations!

For those who have never seen the 2, 000 calorie clip, I implore that you view it. The funny coincidence is that I had stumbled upon it during my summer searching for all things math, saved the link to a folder, and promptly forgot about it. Thankfully, Andrew Stadel reminded me of its existence.

Robert Kaplinsky offered up a video that showcases the amount of food it takes to reach the daily recommended 2, 000 calorie consumption. Some of the foods featured include McDonald’s menu items, carrots, eggs, bacon, bagels, pizza, and even M&M’s. It is fascinating for someone of any age to watch.

For a brief introduction, I reminded students that the daily recommendation for an average person is a 2, 000 calorie diet. We quickly discussed if the average American consumed more or less and one of my students shared that he once read that the average American consumed 3200 calories a day. I don’t know if he was right, but it captivated the rest of the students as they started to discuss what this overeating would lead to for the average person.

Before I showed my students Mr. Kaplinsky’s amazing video, I created a slide on a Google Spreadsheet listing all of the foods that would appear in the video. I asked them to consider the quantity of each food needed to yield 2,000 calories, and in that regard, to write a number that was deemed too high and too low for each. The stipulation was that the too high and too low guesses couldn’t be extreme; they couldn’t guess that 3 million M&M’s were too high, for example.  As Dan Meyer has pointed out, having students do this instead of asking them to just guess the exact number removes the pressure of having to be “right.”  In addition, it forces students to think beyond one number and analyze the situation in a big picture sort of way.

What this estimation process also inspires students to do is become invested in the lesson. They paid close attention because once they generated all of their guesses; they want to know their degree of accuracy. I believe curiosity is one of the greatest motivators in the math classroom.

As students were mulling over their guesses, I was asked, “Dr. Polak, aren’t avocados super fattening?” Before I even tried to respond another student interjected, “Yes, but it is the good kind of fat.” The comments and questions ran the gamut from, “I love Chipotle to what is a Cobb salad?” Basically, the students were IN.

After enough time had been provided, I played Robert Kaplinsky’s video. The reactions were priceless. Many were high fiving each other if their guesses had been close and others were giggling at just how far off they had been. A very brief discussion about nutrition emerged and then students were diving into algebraic equations. The directions were simple. The students were instructed to create an equation that would help them solve the questions about to be asked on upcoming slides. They were also directed to perform substitution to check their solutions.

The first slide, displayed a clipped image from the video of bagels. Students were asked to write an equation and determine how many calories there were per bagel. Students came up with 2000/x=7 and 7x=2000.

The next question asked was how many slices of bacon were equal to one donut. This question presented a challenge for them and many struggled. Students got out of their seats and went to consult other students across the room with their interpretations. Energy rose, anxiety increased, and anticipation mounted. At the end, there were three equations shared that all worked, but the voted-on favorite was (2000/50)x=(2000/6.6).

The scenarios increased in complexity and students were grappling, laughing, complaining, and collaborating to solve. A few wanted me to just give them equations to solve; others felt it was just too difficult, while many were eager for the next question at the next level. Without exception, they all wanted to know whether or not they were right. Naturally, I asked them to use substitution and their math sense to make that determination…Although I eventually confirmed with solutions presented on the slide.

When asked for their takeaways from the day, students’ comments included, “I never realized how quickly calories add up and the types of combinations that might make us overweight.” Perhaps that comment is not exactly related to solving algebraic equations, but it was a good point. Another added, “I learned that I prefer to solve an equation, not create one myself.” (Laughter ensued) Still, someone else said, “I understand equations better now. They are not just questions from a book, but there is meaning behind them.” Someone else added, “It shows the math serves a purpose.”

All in all, the students were animated and lively. The lesson was fun, but I was unsure whether or not I had truly met the objective of helping them with understanding two-step algebraic equations. To find out, I followed up on two separate days with (what I called) calorie math warm ups.  One of those questions was directly offered to me from Robert Kaplinsky himself after I tweeted him a request for a better tie-in to two step equations. That question was, “What is the maximum number of carrots or eggs (I let them choose) you could eat if you had already eaten 720 calories and wanted to eat exactly 1800 calories? Their responses that afternoon let me know the objective was met. Very quickly, the majority of students demonstrated how to interpret real information, come up with an equation to represent a situation, solve the problem, and interpret the information. Don’t get me wrong, there were a few who still needed scaffolding, but by the end of the review, it was clear that the lesson itself had been time well spent.

Robert Kaplinsky, Andrew Stadel, Dan Meyer and so many other mathematicians have changed the teaching game. These wonderful professionals selflessly share their resources with the world to use. The looming question for me after any lesson is always, did I do enough? If I didn’t, what can I improve for the next time? Sometimes, after lessons like these, I cannot think of any improvements, even if I know I can somehow do better. Granted, I already made small changes in my slides to make a clearer presentation, but overall, there wasn’t much I could think to revise. Although there is always room for improvement, as of this teaching moment, I am reveling in gratitude for the opportunity provided to me by Mr. Kaplinsky.

# In Defense or Offense of Teaching Procedural Math? An Open Letter to Everyone.

Dear Mathematicians, Parents, Students, Educators, and All Interested Parties,

Is it a sign of weakness for a teacher to admit perpetual confusion on the best way(s) to administer instruction? Although I have been teaching for about 15 years, only a few of them have been spent teaching math at the middle school level. Since making the glorious move to middle school, the distinct advantage of pouring all of my extra time and energy into one subject has both reinvigorated my purpose and sent me down a path of wonder.

In my quest to prevent any student from truly thinking he or she does not have the math brain, the amount of articles consumed by me is, to say the least, staggering. Can I remember who wrote most of them? Not usually. I peruse for content. Only after multiple exposures from the same author do I start to take notice. This is why a few names have made their ways to the corners of my cerebrum  where the long term storage of my memory lives (Thank you Sousa). I usually refer to the information annoyingly as, “I read an article that stated…” Yes, I have turned into one of those people.

My favorite pastime is to research lesson structure ideas as this is my professional focus. Some of the names that continually pop up in my consumption within that topic are Jo Boaler, Dan Meyer, Andrew Stadel, and Yeap ban Har. Each of these math gurus share a common thread, which is that mathematics is a subject that spans beyond mere procedure. Although I could not agree more that math is not strictly procedural, each time I read an article I find myself asking, is there still a place, and furthermore, a need to teach procedure(s) in a math class?

If it is true that the best teachers steal from the best, in some small way, that categorizes me as the best. I have “stolen” lessons from my teaching counterparts, Dan Meyer, and Andrew Stadel.  The stolen lessons have been glorious experiences.  However, I do not believe any of the stolen lessons would have been successful if students had not possessed the background knowledge on procedures as well. Now I wonder, did I enhance their conceptual learning or detract from it with that viewpoint?

Our district was blessed by the personal teachings of Yeap ban Har. I spent a good month after that momentous training opportunity trying to design my lessons just like him. This was not easy to do with only one real half-day of training, but I really gave it my all. Some lessons went astonishingly well, others, not so much.

What I do know is my goal is to do better every single day. This is where I feel as if I am on the giant hamster wheel of math instruction.

In my mind, if students do not learn the concepts behind the math, the procedures for any and all algorithms will be meaningless. They will learn a series of steps, study them for a quiz or test, regurgitate them, and then quickly dump the total experience from their memory. Obviously, this reality is not true for all students. Those students who are excellent at rote memorization might remember the steps, but will they have any idea why they are performing them? If they don’t, can that be considered effective math teaching or learning? On the other side of this paradigm, sits many students who demonstrate conceptual learning but struggle with the rote procedures. For example, several students in my class this year forgot how to subtract opposite signed numbers using an algorithm, but when I placed a number line or integer tiles in front of them, they knew how to solve the problem immediately and could explain their thinking. Is their learning inferior because they cannot demonstrate their understanding in an algorithm?

The articles I have been reading lately push my questioning even further. I believe Jo Boaler flat out posited whether or not it is necessary for students to memorize their times tables. Is this type of thinking correct for educators, and more importantly, for students beyond the classroom?

Here is where I flat out ask the community for feedback.  Is there an appropriate balance needed in our classroom between concepts and procedures? Are procedures completely out of date or still necessary? Do we need to argue the opposite ends of the spectrum, or consider that the ideas are not opposing but supporting of one another? I ask you, in a growth mindset sort of way, to reflect carefully. Perhaps someone out there can inspire me to jump off of the hamster wheel, if only for a moment.

Sincerely,

A math teacher looking for answers.