Oh my goodness, I am about to start my teaching life over from scratch in a new role. Although it is not the first time I have changed teaching positions, it is the first time I will embark on a role that has multiple official positions in one. Officially, the role is Math Interventionist/Teacher/Coach. In the classroom teaching capacity, I will be instructing a grade level class that I have never previously taught.

As the new school year begins in a few short weeks, I feel like I did as a first year teacher.  My Assistant Principal declared that I would feel invigorated with this change. Invigoration, I have decided, is the positive viewpoint of change, and it is the reason we volunteer or seek it out. This makes my fearless leader’s assertion correct. However, let’s not neglect the fact that change is also overwhelming. What do you do in this case? Find a way to overcome!

Whenever I am in a situation of unknown terrain, I turn to research. Many days and nights have been spent watching video clips, exploring Desmos, reading articles, flipping through instructional books, and finding new mathematicians to follow on Twitter and/or Facebook. How fortunate we are to live in a time where amazing resources are often just one click away! There is an entire community of mathematics teachers just like me who geek-out upon learning of an idea, an approach, or lesson that feels revolutionary. On that note, I wanted to share (especially with my middle school friends) my most recent experience of “geeking” out. I just completed a course given by Graham Fletcher, and believe me; you need this mathematician in your life!

I learned about Graham Fletcher last summer through Christina Tondevold’s Build Math Minds Community. Although geared primarily towards elementary teachers, Christina’s website has a lot of content that speak to how students can best build their mathematical foundation. As a middle school teacher dealing with so many students’ long-standing misconceptions, this site and her emails have been a welcome life line. For years, I have been obsessed with the progression of mathematical concepts in trying to back-fill and clear up student misunderstandings. Last summer, I joined the virtual summit. The second summit is occurring in a few weeks. I encourage that you all check this event out and consider enrollment.

One of the (Christina calls it PD in your PJs) events I chose last year was hosted by Graham Fletcher, whom I mentioned above. He will be speaking again this year as well. If you are a middle school teacher, I ask of you to pause for a moment, and think of the concept that you believe blocks student understanding the most.

Did you say Fractions? I would venture a guess that you did. Please, after you are done reading this blog entry, go to Google and enter Graham Fletcher in the search engine. Explore and maybe begin with his video on the progression of fractions. After you have an idea of why I am geeking out so much, please report back to me on how I helped you change your life by sharing his name with you.

You’re welcome!


Are Procedures a Dirty Word in our Current Math Classrooms?

I accidentally created an online argument between myself and other passionate educators on a Jo Boaler dedicated Facebook page. After countless hours of endless research and sending messages to math educators I admire, I still had a question that could not seem to be answered. In my perpetual attempt to improve my craft, I innocently asked how to make those “procedure teaching days” meaningful for students who had prior knowledge. So, for example, if we have students who go to Kumon, Russian Math, or even have access to a private tutor and are taught lots of procedures, how can we challenge them on days we are having the rest of the class engage with them? Many of these students feel the discovery piece is a waste of their time since they already know the end result. As often as I remind students that procedures without a solid conceptual foundation are weak, there is still push back and I was looking for a way to engage them, not for my ego, but for their betterment. Furthermore,  if a teaching goal is still for students to be able to perform algorithms to help them “Look for and make use of structure,” what is the best way to make those days of instruction inspirational for everyone?

My post got a lot of comments, and not all were kind. Disgusted responses such as, “You should never teach your students procedures, what are you even doing as a member on this Facebook page” were peppered in along with, “Following, if you find out, please tell me.” Ok, I am exaggerating the first statement, but only a little. The message was underlying and clear. Although my post was not about the straight teaching of procedures, it got me wondering:

Is it true, are procedures a dirty word in our current math classrooms?

Now listen, I am all about trying to have students discover and conceptualize before utilizing any procedure. A circle unit my students recently completed as a review is a great example. Students  measured the distance around a circular object and then across the diameter of it to provide them an opportunity to discover the pi ratio. Before the lesson, I asked them what they knew about pi. They responded that it was 3.145…and used in circles. Many commented on “pi day” celebrations of the past. None knew it was a ratio. This was a small example of procedure without substance.

For the area of the circle, students studied images of a circle being sliced up into wedges and slowly transforming into a rectangle, which they were then able to compare to the area formula for a rectangle, hence, discovering the formula for the area of a circle.

For circumference, students engaged in Rolling Tires 3-Act Lesson by Andrew Stadel.

However, after these periods of discovery are over, students still apply the formulas in and out of context with the procedure. The conceptual piece is enhanced by the algorithm, and vice versa. My critics questioned my inclusion of this piece of teaching, so I have to question it too. However, how will isolated discovery without practice allow concepts to enter the long-term storage area of the brain? Aren’t both needed? I think yes.

I have researched (through the printed word, professional development courses, and on-line videos of educators I admire). The same message delivered in different ways is that the algorithms of the past are not meant to magically disappear. “There is no new math” is a popular phrase. In my current state of teaching, I agree. Algorithms are wonderful, important, and revolutionary methods. It is not wrong to know how to use algorithms, but it is not productive to use them and have no idea why they work. Shouldn’t we have both?

I really want to know.

An Apology Letter to My Former Students

Dear Former Students,

I owe you all an apology. You see, I always thought I was a good teacher. Turns out, I could have been so much better…

As a math teacher, I have failed so many of you. Once upon a time I focused solely on the procedures. Teaching long division with “Divide, multiply, subtract bring down,” instructing you to divide fractions with keep, change flip!” eliciting the rounding chant, “five or above, give it a shove, four or less, let it rest,” and don’t even get me started on how I used to teach slope.

There is nothing inherently wrong with algorithms. They are super efficient and are worth knowing and learning. I was not wrong in teaching you how to use them. You should know them! Where I did fail, was in inspiring you to actually understand why algorithms worked and what it was you were really doing. Because let’s face it, you have likely forgotten how to use many of them now.

Guess what, I hadn’t considered those points at that time. Although I was labeled an excellent math student for a lot of my life in school, it turns out, that was a misrepresentation of the truth. If someone had a procedure for me to memorize, I had no problem. I was one of the fastest math fact people in my grade. But, what I didn’t learn to do in school was think about math in context. It never dawned on me that numbers were adjectives, not nouns. Learning how to add 3+4 should have always elicited a context of 3 THINGS + 4 THINGS. But it didn’t. This was not my math teacher(s)’ fault, because they didn’t know that was important either!

I have evolved in my teaching because I work in a profession with wonderful envisionaries. Mathematical gurus Robert Kaplinsky and Andrew Stadel both respond to my constant barrage of email inquiries. They have never met me, but have contributed to my transformation. Jo Boaler’s spoke to me through her book, but I have yet to see her in person. Dan Meyer showed me how to make math exciting, but I have never received professional development from him in the same room. Christine Tondevold is helping me go back to the basics so I can really see where I need to begin helping my students, but we have never physically crossed paths. Graham Fletcher is helping me teach my students how to decompose numbers. The point is, I didn’t realize that what I was doing wasn’t what I should have been doing. The moment I figured that out, was the moment I wanted to invite you all back to teach you all over again. 

Oprah is famous for saying, “When you know better, you do better.” I am in a constant state of trying to learn how to know better so I can do better. To paraphrase Christine Tondevold, I did the best with what I thought was the best at the time. Now I realize, I should have done things differently. So to my former students, I am so sorry I failed you.

And to my current students, I am sure I will continue to learn newer and better ways to teach and will owe you an apology in the future. Consider this payment in advance.

Most sincerely,

Dr. Polak

Can I Do Fewer Things Better? Following the Advice of Angela Watson

Another summer has almost come to a conclusion, and what do I have to show for my time to unwind and relax? Well…not so much in the ‘Unwind and relaxing’ department. Perhaps much of this can be attributed to the fact that I am a mom to two girls under the age of six, but if I am really being honest, it is because I have spent the majority of my summer immersed in educational research. I didn’t mean to do it this year, really!

A little over a year ago, I stumbled upon Jo Boaler and my teaching world opened up. As I have stated before, her book and course synthesized so many different ideas I had uncovered in my search for those truly great lessons. I felt transformed, so when she offered a second course this summer, naturally, I was obligated to sign up for it. In an intense four-week period, I completed the course.

Lucy Math


After volunteering on a committee at school this past year to “Re-imagine the Middle school,” an opportunity to become certified in Empowering the Mind over the summer was presented to me; so clearly, I had to pursue that as well. The seminar was amazing, but I still had questions and oh my goodness, there was a book to go with it! Naturally, I had to purchase the book and read it immediately in order to solidify my learning.

While in the middle of the book, a website discovered by my amazing colleague was offering free math professional development, but only for a few days. Normally, this resource charges $39.95 a month, so obviously, I had to immerse myself in more Math PD immediately, even if it was geared towards elementary math. The progression of math skills is something I am really trying to become an expert on, so although the timing wasn’t perfect, I had to jump on the opportunity.

I gained at least one golden nugget from each hour session, and that is always worth something. I have been watching the seminars in a haphazard order, and yesterday, I watched the very first one offered by an educator named Angela Watson. Her keynote speech was about giving yourself permission to do fewer things better. In her speech, she discussed something called the Minimum Viable Product.  She explained that we as teachers often spend countless hours creating the perfect lesson, deliver the lesson, and then realize it is not perfect by any means. Then, we are faced with a choice, we can go back and tweak the lesson, or completely trash it. She stated it much more eloquently than I just did, but the point is the same. Ms. Watson challenged us to involve the students in lesson design and start with something minimally ready. The lesson may not be perfect, but few lessons ever are in their first iteration. She proposes it is better to start with something unrefined and invite the students to transform it into what they need rather than teachers spin their wheels in the creation phase when the ending result is the same.

Mind blown…

I frequently spend a ridiculous amount of hours creating a 45 minute lesson. Like most teachers involved in lesson design, even “stolen with permission lesson design,” the lesson never stays true to its original design because students always guide improvement. Ms. Watson emphasizes a need for teachers to set a timer in lesson creation, use it, invite the students to tweak it, and repeat.  She refers to this scenario as a win-win.

Although Ms. Watson absolutely has an incredible point, can I really allow myself to do this? Can I give myself permission to take an idea and just use it in (for me what would be) raw form?

I am that teacher is who is often the first one in the building and the last one out of the building. I am not saying this as a point of pride; it is a professional (and personal) weakness. In speaking with my assistant principal I shared with her that a goal of mine this year was to stop doing that. She suggested I make it my S.L.O. (Perhaps only teachers will understand that joke).

Ok, so here we are, I have a few weeks left of my summer vacation. I have yet to spend much time “vacationing” from school. The question remaining is…am I the type of person who can do fewer things better?

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?

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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?

commoncorelogo-color2Please 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.