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.

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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.

Using Zoolander to Reinforce the Concept of Scale

In search of all great lessons, I stumbled across Robert Kaplinsky’s Zoolander lesson, which was created for the concept of scale.  Although teaching scale was months away when I discovered the Zoolander lesson, I immediately created slides and customized the lesson so that I would not forget about it. I waited for months excitedly anticipating the day I was ready to teach scale to try it out with my students. It looked so good!

For anyone who ever chuckled at the original Zoolander movie as my husband and I did, there is a scene where Will Ferrell’s over the top hilarious character (Mugatu) is trying to convince Ben Stiller’s character, Derek Zoolander, to model for his show. To entice Derek out of his model retirement, Mugatu shows him a scale model of a reading center Derek had previously told his manager he wanted to open for underprivileged children. Mugatu promised Derek that he could open the center if he signed on to model for his show. Derek has no concept of what a scale model is, and when Mugatu shows him the scale model of the reading center, Derek thinks it is supposed to be the actual building. He thinks he is being taken advantage of and claims that the building needs to be at least 3 times as large. Robert Kaplinsky bleeps out “3” in Derek’s retort so students have to determine what Derek said. The question students are presented with is, how many times larger should the builders make the actual school?

Currently, I teach different levels and pacing of algebra. For my advanced students, I gave them very little to go on after presenting the question and the clip. I highlighted some of the math practice standards they were expected to follow such as persevering in problem solving, constructing viable arguments, and reasoning abstractly. I showed them several Zoolander still shots (provided by Robert Kaplinsky), shared the fact that an average story was 10 feet high in a building, and that Ben Stiller was 5’7”. That was literally all the information I gave them. I didn’t know what was going to happen, but I was excited to watch.

Initially, a few students asked for more information, but when I told them that I was not providing them with additional information, they rose to the occasion and demonstrated innovative thinking. My job for the lesson essentially became flipping the still shots back and forth on demand as students worked out strategies. They grabbed my rulers from the class stash and started measuring images on the SmartBoard such as Derek’s arm span and the distance from the building to his head. They started debating whether or not the base counted as a building story or whether the giant book on top did as well. In other words, they persevered and applied mathematical reasoning to an abstract problem because they had no other choice. It stretched their thinking and mathematical prowess.

By the close of the lesson, students successfully shared several strategies with very different answers. (Their answers ranged from 54 times as big to 120 times as big). Some students counted the base of the building and the book and others didn’t. A few students compared the model people to the height of one floor and made their mathematical predictions based on that relationship. Several people concluded that there were about 12-13 stories and used the other information to make their calculations for the scale factor. Students also estimated that based on Ben Stiller’s height, the model was about 1.5 feet and made their calculations from that vantage point. What all students were doing, regardless of whether or not they knew how to get to an answer, demonstrated an understanding of the scale factor and scale model concept based on their problem solving application. One student asked me, why are we doing this? As I repeated the question back to him, he rolled his eyes, smiled, and said, “To apply the scale model concept to something beyond the math book.”  Ha! Nailed it!

I then decided to try the same lesson with my other classes, but I knew they would need additional scaffolding. Based on my observation of the advanced students, I inserted additional clues in my slide presentation. Before I showed them the stills and provided Ben Stiller’s height and the average height of a story in a building, I structured the lesson like a Dan Meyer 3-act lesson. After showing them the clip of Zoolander with the scale model, I asked students to write down any questions that came to mind. We shared with each other. Some questions were, how tall is Ben Stiller, how many feet was the model, how many students are supposed to attend the school, what will the budget be for the school, etc. At this time, I let students know that their task was to determine how many times larger the actual school had to be in relation to the scale model. I asked students to come up with a wish list (if I would grant them their desires) of tools and/or information to help them solve this problem. Students asked for the answer (naturally), the height of the model, how tall a story was, the size of the plot of land for the building, and so forth.

Disappointing, but not surprising to them, I provided them with the limited information I had provided my previous class. This included the height of Ben Stiller, the average height of a story in a building, and several still shots from the video clip. I offered them rulers (which was the one change from the other class who just asked for them) and left them with a final thought before giving them time to solve:  Now that you know your task and the limited tools you will be provided, what strategies could you use to unravel this mathematical mystery? How can you work around not having the exact information you want?

Many students who tend to struggle rose to the occasion and illustrated bravery in taking chances in solving the problem. Yet, students who tend to crave procedure and rules were taken aback (as has been a pattern for them with these types of lessons) and continually asked me for assistance. With every question these students asked, I responded with another question. The uneasiness some students experience when not knowing exactly what to do has proven to be almost debilitating.  That makes it my job to make these students feel uncomfortably comfortable. I can’t just give them answers; I have to provide them with tools to independently and confidently find ways to chase the unknowns in math. This is an ongoing challenge for me personally and I am always searching for ways to help students help themselves.

Overall, the conclusion of the lesson resulted in a very similar outcome when compared with the advanced class closure. However, I made another change for this class and used the third act. The third act was sharing the Zoolander clip (previously bleeped by Kaplinsky) where Derek says that the building has to be at least three times as big. The class laughed and commented on Derek’s terrible analysis. A few savvy students reflected, “Well, technically, he isn’t wrong. He said at least 3 times bigger.”

I was on the fence after I used this lesson as to whether or not the process helped students strengthen their understanding of the scale model concept for all levels. For the advanced students and many students in my other levels, I certainly think it did, but perhaps I am justifying a fun experience in my room. Some lessons are like that. Yes, they make a class enjoyable, and yes, they seem like they are mathematically sound, but in the end, as a teacher, you can’t help but question if the lasting impact of the concept was made. I never forget that there is more to a math lesson than being “Really, really, ridiculously good looking.”zoolander_school_large (1).jpg

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.

 

bread calorie mathcopy-of-what-does-2000-calories-look-like

 

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.