My Memories of Memorization in Math Class

When I reflect back on my experiences as a student in the math classroom, flashbacks of constant memorization fill my mind. As a primary ELL (English Language Learner), I struggled to retain math concepts from grade to grade and became very anxious during math class. I became a silent student that would rarely participate and quickly learned that if I stayed silent long enough, teachers would eventually give up and stop asking me math questions. However, whatever I lacked in comprehension, I made up for in effort and memorization.

I found a way to cope in the math classroom and realized that if I memorized hard enough and followed the steps demonstrated by my teacher, I would be able to do the math. Sure enough, as I approached the junior grades, I would show up to school in September and would still know my “multiplication facts”. I remembered to ‘carry’ the number when adding, ‘borrowing’ from number to the left when subtracting, and writing a zero when multiplying multi-digit numbers. I didn’t know why I had to carry or borrow or write a zero but it didn’t matter because I could do the math. That just drove me to memorize even more and found myself studying hours in preparation for math unit tests. I eventually  evolved into a master of memorization and memorized anything that I couldn’t understand (cross multiplication, dividing fractions, integer operations, area formulas etc). By the time I reached the intermediate grades, I was getting A’s in math and was considered a great math student by my teachers. I thought I was a pretty darn good math student too. However, my perception of math at the time may have been a bit skewed. I thought math was all about studying hard, memorizing the facts, procedures, and formulas. In grades nine and ten, I was on top of the math world and I equated success in math to achieving math marks in the 90’s. By that time, I had mastered the art of deciphering the high school math textbook. If there was a word problem that I was confused with, I would just find the similar sample problem in the textbook with the different numbers or I would look to the answer key at the back of the textbook, find the solution to the problem and try a variety of procedures or formulas hoping that one of them would eventually lead to the correct answer. I knew how to play the game and was I winning… until I reached senior math.

As I was introduced to the world of trigonometry, calculus, derivatives, vectors, logarithms, I could feel the old but familiar sense of anxiety that I experienced as a primary student slowly creeping back into my mind and eating away at my confidence. I became desperate when my A average fell to a C average and did the only thing I knew how, I memorized harder, pulled all nighters and took summer school. I was still able to graduate from secondary school with an A average with great effort and memorization but as I remember and reflect on my entire math education I realized that I wasn’t really “doing” math, I was “memorizing” math.

I’m not trying to imply that I never learned or understood the math that I was taught. I’m saying that the way I was taught math and learned math was very inefficient. My perception of math was repeating and applying standard algorithms and formulas that I never really understood. I was never asked to try and solve a problem using my own invented algorithms. I was never exposed to mental math strategies for the basic operations. I never used tools like the open number line or an array to add or multiply. Problem solving wasn’t embedded in the curriculum when I learned math but rather a unit that was also taught very procedurally. Math solutions were expected to look identical to the teacher examples and marks were deducted for missing a step or forgetting an equal sign. It took me a long time to realize and accept the fact that my perception of math was false and that I was a product of procedural teaching and it was a hard pill to swallow. Memorizing is not a mathematical process and not an ideal way to acquire an understanding of math concepts and skills. Math in the classroom is about problem solving, reasoning and proving, reflecting, selecting a variety of tools and strategies, making connections, representing , and communicating.

School Improvement Plans: What’s Good For Students is Good For Teachers

For my previous module of my course, we were asked to reflect on School Improvement Plans (SIP) and make connections to student learning. I began to reflect on my district school board’s strategic directions (Achievement Matters, Engagement Matters, Equity Matters). Student engagement, achievement, and equity should always be at the heart of of every SIP and it should be linked to the Ontario School Effectiveness Framework since its main purpose is to “function as a tool for schools to identify areas of strength and areas requiring improvement in order to reach all students and improve student achievement”.
However, as I continued to reflect on the implementation and continuation of school improvement planning I began to realize that in order for school administrators and staff to achieve the goals of any SIP, they must take a closer look at their classrooms’ best practices that increase student achievement, engagement, and equity and apply it at the professional level. In other words, I realized that what’s good for the students is also good for the teachers. For instance, we know that learning in the classroom must be authentic and that students are more likely to be engaged if they are active participants in their own learning. This same thinking can be applied to the development of a SIP which must also be authentic to staff and parents in order to be a living document. It cannot be perceived to be a top-down initiative or a model replicated from another school. Additionally, if we know from research and teaching experience that differentiated instruction which focuses on student readiness, interest, and learning profile allows more students to be successful, then staff (who are at different levels of professional learning) would also benefit from differentiated professional learning opportunities related to the SIP as well.
I strongly believe that the importance of student voice, collaboration and making student thinking visible is equally as important to teachers with respect to school improvement planning. Teachers can feel very isolated in their classroom (especially if they`re in a portable!) and they need to be provided with opportunities to network and collaborate with each other in both physical and virtual environments. Teaching practice needs to be deprivatized and teacher thinking needs to be visible and shared with their colleagues. Often, it is through teacher dialogue and discussion that great ideas come to fruition. SIPs are rarely set in stone and require tweaks along the way and in order for SIPs progress and evolve. Therefore, reflective practice must be a habitual behaviour with staff and administrators. I often see large percentages of release time devoted to planning which is definitely important for any positive change to occur. However, I think that an equal amount of attention should be focused on reflective practice where teacher reflection and moderation can occur as well. Every school improvement plan should find ways to create the conditions for teacher reflection and sharing of best practice that occurs in their classrooms, grade levels, or divisions.
The goal of every school improvement plan should be to reach every student. In order to accomplish this, school administrators and staff must focus on student learning and the best practices that they wish to see in their classrooms and implement these best practices at the professional level.

Keeping the Genius

This is my daughter Ava teaching me a very important lesson (yet again) with her child-like genius.

Last week, I was in the kitchen with Ava sitting at the counter having our usual father/daughter morning conversation. Ava started off sitting in the chair on the left as I was getting breakfast ready. However, the chair on the left is not her usual spot at the counter and she explained to me that she didn’t want to sit in “that” chair and wanted to switch. I automatically said, “Just climbed down that chair and climb back up the other one.” but Ava didn’t seem satisfied with my solution. Instead of taking my advice, she climbed up on the counter and crawled across the countertop to the chair on the right and sat down. My instant reaction was to remind Ava of the dangers of crawling across high places but afterwards, I thought about what she did and began to look at her actions in a different perspective.

Ava is three years old. So to her, climbing up on the counter is a perfectly reasonable solution to switching chairs. In fact, for a petite three-year old, it is probably easier to crawl across the counter than it is to climb down her chair and climb back up the other chair.

As an adult, I would have never thought to climb up on the counter and crawl across it to switch chairs. Why is that? Maybe it’s because I’d probably fall off and smoke my head off the edge OR maybe it’s because I’m an adult and I have preprogrammed perceptions in my subconscious mind telling me that crawling on countertops is inappropriate and dangerous. Do these preprogrammed perceptions embedded in my subconscious limit my ability to think creatively and outside the box?

Ava is not limited by these perceptions (yet). To me, this was an excellent example of creativity and thinking outside the box. To her, this was a perfectly normal and reasonable solution to her problem. The reason why are children are entering kindergarten with genius-like skills is because they aren’t held back by downloaded perceptions that us grown-ups have of right/wrong, appropriate/inappropriate, good/bad, possible/impossible which are pretty well established. Children’s perceptions are not set in stone. For them,  anything and everything is possible.

Malcolm Gladwell, author of Blink, writes about the power of our subconscious and the impact that it has on our thinking and behaviour. Bruce Lipton, biologist and co-author of Spontaneous Evolution, states that 95% of our thinking and behaviours are controlled by our subconscious mind which is already established by age six. However, he also argues that the perceptions that are programmed in our subconscious can be deprogrammed. Lipton’s argument is very encouraging for the students that are currently in our education system.

So what does this mean for us educators? How do we keep and develop the innovation, the creativity, the imagination in our students who are moving through our education system? We give them a voice and listen. We allow them to think and use their own invented strategies to solve problems. We guide them to think critically about their strategies and other students’ strategies. We let them be active-participants in their own learning. We learn with them as opposed to teach to them. We look for limiting perceptions that hinder and handcuff our students’ genius and guide them to the realization that they can succeed and that anything is possible.

I went back to Ava that morning and told her that she used a very creative way of getting to the other chair that I hadn’t even thought of and that she taught me something new. She was so proud and gave me the biggest smile. It was a “Kodak” moment that was captured in the picture at the beginning of this post.

I thought I’d leave you with an amazing video that also inspired this blog post.

The Fifth Way

I finally finished reading my book (which I loved), Spontaneous Evolution by Bruce Lipton and Steve Bhaerman after being sidetracked by so many fantastic blogs. In this book, Lipton and Bhaerman make reference to Johan Galtung, a Norwegian mathematician and sociologist and founder of TRANSCEND International, a peace development environment network. Galtung is most known for his ability to transcend conflicts and find what he refers to as the fifth way, or fivers. He recognizes that every conflict has five possible resolutions:

  1. I win. You lose.
  2. You win. I lose.
  3. The conflict is resolved by avoiding it completely.
  4. Compromise where all parties are dissatisfied.
  5. Transcendence where all parties feel like they win and resolution is above and beyond the problem.

After reflecting on this portion of my book, I believe educators need to implement the power of Galtung’s fiver approach in education and seek ways to solve issues with resolutions that are above and beyond the problems so that all parties (students included) are happy with the outcomes. Lipton and Bhaermann explain that the first step to creating a fiver solution is for opposing parties not to settle and meet each other halfway but to work together and progress forward towards an ideal resolution.

This notion can be applied directly to the classroom where conflicts often arise between teachers and students. Often, the labels “teacher” and “student” create a separation, a polarity in the classroom. It’s the teacher vs. student mentality which results in disengaged students, late assignments, students doing the bare minimum to get a “level 2” etc.

Here’s my fiver solution for the teacher vs student power struggle that exists in many classrooms. Get rid of the labels “Teacher” and “Student” and “classroom” replace them with “learners” and “community”. It shouldn’t be about the teacher as the holder and controller of all the knowledge and the student as the observer waiting to be educated. As Angela Maiers would say, It’s about a community of learners each with valuable knowledge and skills working collaboratively to achieve their full potential so they can make their contribution to the world.

“It’s About Time, Attention, and Value”

Last Friday, I happened to come across a webcast on edtechtalk.com via Twitter when @AngelaMaiers tweeted about it right before she went on. It was a very inspiring discussion that didn’t really focus on technology at all. In fact, the topic of conversation was more about “seeing” students and helping them find their gifts so that they can make their contributions to the world.

Towards the end of the webcast (45 minutes in), Angela recalled a conversation she had with a group of students and she asked them what they thought about technology integration in education. One of the student replied, “If I have to do another Glogster, I going to jump off a cliff…Seriously, I wish teachers would lay off this technology stuff because it’s painful to watch, they’re trying too hard…If they just saw me, If they could just let us talk, If they could just let us share…” She went on to say that integrating technology in education is not that complicated. It doesn’t have to be a fancy project or a unit that is infused with technology, it’s about time, attention and getting students to feel they are valued and seen by their teachers.

After listening to this inspiring webcast for a second time, I realize that it’s not just about integrating technology in the classroom. It’s about establishing a community in the classroom and letting students become active participants in their own learning. Technology just happens to be a great tool to make this happen.

Screencasts of Students’ Math Thinking

Last year, I came across a very interesting blog that helped changed my perception of the web in education. Stretch Your Digital Dollar by Katy Scott offers useful ideas for integrating technology into all classrooms. After reading her blog about screencasts, I became fascinated by the possible positive implications this could have in the math classroom. This year, I am looking to delve deeper into screencasting and investigate its positive impact on student learning. I’m interested to hear/see how other educators incorporate this great use of technology in their own classrooms.

I have posted a Glog containing student screencasts of the multiplication strategies that they used to solve a word problem.

 

http://edu.glogster.com/flash/flash_loader.swf?ver=1309171993