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.

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?

zoolander_school_large (1)

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.

Starbursts

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.Jo Boaler's book

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?

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.