Online Gallery Walk Using Lino It

In one of my previous posts about the 3 part problem based lesson, I referred to the third part (reflect and connect) as the most important part of the math lesson since the learning comes from the student work. Students are given the opportunity to explain their math thinking, pose questions, defend their ideas, and make connections with other solutions. The teacher is the facilitator of the discussion as well as a participant. The learning comes from the math community.

The following article describes three approaches to the Reflect and Connect: 1)Math Congress, 2) Bansho, and 3) Gallery Walk.

In this post, I wanted to focus on the gallery walk because to be honest, this is an approach that I don’t use very often when consolidating a math lesson. This is an approach that requires a lot of movement of students in the classroom and could possibly take some time and good management skills to facilitate. In most gallery walks, student solutions are posted around the classroom, and students circulate with sticky notes writing down questions and comments and placing it on the solutions. The idea is for students to read the comments and questions on the stickies about their own group’s solutions and use that feedback to help them prepare the explanations of their work to the class. Problems can arise if students aren’t given enough “wait” time to think of questions or comments to write down on their sticky notes. This can result in comments like, “You spelled multiply wrong.” or “I like how you used the colour red to write your solution.” The goal is to promote higher order thinking and questioning but if students aren’t familiar with a gallery walk or need more time to reflect on the student solutions then perhaps an online gallery walk could be a worthwhile alternative.

Lino it is a great web application that provides you with an online canvas and allows you to post online stickies, pictures, videos, and attachments. You can also share your online canvas with others by sharing its URL. Sharing the URL would allow others to post stickies as well. You can probably see how Lino it could be used to conduct an online gallery walk. To give you an idea of what it could look like, I inserted an screenshot of an online canvas with pictures of four multiplication solutions and stickies with questions and comments posted around them.

The comments/questions were generated by a team of math facilitators when I shared the link with them, which should also give you a good idea of how Lino it could be used for teacher moderation of student work.

By using an online gallery walk, students could post their comments/questions in class or at home and have time to reflect on student work and pose thoughtful questions and comments. Groups could review the feedback the next day and prepare the presentation of their solution to the class.

I would love to hear your thoughts about this alternative approach to communicating in the classroom.

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 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.

A Closer Look at the Three Part Problem Based Lesson

John Van deWalle, Cathy Fosnot, Marian Small, and Marilyn burns are all key researchers when it comes to mathematics in education. According to these researchers, an ideal math lesson consists of three parts. The HWDSB math facilitation team refers to the three parts as: 1) Getting Started 2) Working On It 3) Reflect and Connect.

Let’s just say we are going to teach a grade 5/6 initial three part problem based lesson on multiplication and the goal of this lesson is simply to see what multiplication strategies students are bringing to the table. The “Getting Started” part of the lesson would involve some sort of activation of students’ prior knowledge related to multiplication (simple problem). During the “Working On It” part of the lesson, the following problem could be presented to the class.

29 students are going on a field trip to a museum. The field trip will cost $20 per student. How much will it cost for 29 students to go on the field trip?

This is an example of an open routed question. There is only one answer but there are multiple strategies to get the answer. Therefore, we ask the students to solve the problem in groups and in more than one way. The first strategy will come naturally for some students however, the second strategy may be more difficult to come up with. Again, the goal of the lesson is to see all the multiplication strategies that the students will use solve the problem. As the students “work on it” we would circulate around the classroom asking questions about students’ strategies, guiding students through the process, and allowing mistakes to occur (these will be addressed during the “reflect and connect”). It is also important to note that not every group needs to be finished before moving on to the “reflect and connect”. Sometimes incomplete solutions provide good starting points for classroom discussion.

The third part of the math lesson is the most important part of the lesson but often the part that gets left out by teachers. It is also considered by many teachers as the most difficult part of the math lesson to facilitate. The “reflect and connect” is when the learning of the math concepts really occurs because the learning comes from the student work. This is the part of the lesson where students are given an opportunity to explain their strategies and solutions and where teachers are given an opportunity to focus on key strategies and concepts by guiding a math discussion through strategic questioning. This math discussion is very important because the conversation is less teacher centred and more student centred. Students ask each other questions about their solutions, make connections between their solutions, and defend their math solutions. The goal of the “reflect and connect” is to create the culture of a math community that allows students to take risks and where mistakes are considered to be opportunities for new learning. Ideally, this is what the “reflect and connect” should and could be like however, it takes time to get there. Students need time to learn how to ask appropriate questions, give constructive feedback, and receive constructive feedback. Teachers need time to learn how to ask probing and guiding questions and look for connections between student work.

There are a few ways to conduct a “reflect and connect”. The following article titled, Communication in the Mathematics Classroom explains three different approaches of math communication that can be implemented during a “reflect and connect”:  1) Gallery Walk, 2) Math Congress, and 3) Bansho.

My next post will focus on the gallery walk and a possible way to enhance math communication via Lino it using the following student solutions:

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Math is beautiful but do our students know?

I recently attended the Ontario Math Coordinator’s Association (OMCA) 2011 coference, Math More Than Magic and had the pleasure of listening to Dr. Nathalie Sinclair, assistant professor at Simon Fraser University’s Faculty of Education and author of Mathematics and Beauty: Aesthetic Approaches to Teaching Children. She began her presentation by sharing a math doodle video clip by Vi Hart. Following the very cool video clip, Nathalie went on to speak about math in a way that I’m not accustomed to hearing. She called math elegant and beautiful and asked us when the last time we heard or used those words in the same sentence with the word “math” and also challenged the audience (a room full of math coaches and coordinators) to tell students that math is elegant and present them with opportunities to learn math not only visually but dynamically. This really got me thinking about math in education in general.

One question that math educators across North America probably hear most frequently from their students is, “When are we ever going to use this in real life?” How do you usually respond to this million dollar question? Sure, math can be practical and help us in our everyday lives (making change, estimating amounts, calculating HST, understanding the probabilities of poker hands when watching it on T.V. etc.) However, the truth is that some of the things that we teach in math will not help students in everyday practical situations. Try explaining to an intermediate student that they will need to know the Pythagorean Theorem so that they can know exactly how far a 5 metre ladder is from a wall if it is resting at a height of 4 metres. Is this the real reason why we teach the Pythagorean Theorem? I love the Pythagorean Theorem because of it’s simplicity and the fact it can be proved in so many ways other than using the formula. If you have a right triangle, the square on the biggest side of the triangle is the same as the sum of the squares of the other two sides. I love the theorem even more when students discover it through investigation and get excited to share how they proved it. Here are some beautiful examples of math proofs of the Pythagorean Theorem:

Paul Lockhart wrote an incredibly thought provoking math essay titled, “A Mathematician’s Lament” (a must read for any math educator). He refers to math as an art and the “art of explanation”. In fact, he states, “Mathematics is the purest of the arts, as well as the most misunderstood”. Lockhart is very much an advocate of bringing the beauty of mathematics into the curriculum. The following excerpt is a playful dialogue that was taken from Lockhart’s Lament:

SIMPLICIO: Are you really trying to claim that mathematics offers no useful or practical applications to society?

SALVIATI: Of course not. I’m merely suggesting that just because something happens to have practical consequences, doesn’t mean that’s what it is about. Music can lead armies into battle, but that’s not why people write symphonies. Michelangelo decorated a ceiling, but I’m sure he had loftier things on his mind.

SIMPLICIO: But don’t we need people to learn those useful consequences of math? Don’t we need accountants and carpenters and such?

SALVIATI: How many people actually use any of this “practical math” they supposedly learn in school? Do you think carpenters are out there using trigonometry? How many adults remember how to divide fractions, or solve a quadratic equation? Obviously the current practical training program isn’t working, and for good reason: it is excruciatingly boring, and nobody ever uses it anyway. So why do people think it’s so important? I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkersthe kind of thing a real mathematical education might provide.

SIMPLICIO: But people need to be able to balance their checkbooks, don’t they?

SALVIATI: I’m sure most people use a calculator for everyday arithmetic. And why not? It’s certainly easier and more reliable. But my point is not just that the current system is so terribly bad, it’s that what it’s missing is so wonderfully good! Mathematics should be taught as art for art’s sake. These mundane “useful” aspects would follow naturally as a trivial by-product. Beethoven could easily write an advertising jingle, but his motivation for learning music was to create something beautiful.

Lockhart makes some very bold statements but I think he definitely get’s his point across. Sometimes we have to be honest with our students and when they ask us when they will ever use the Pythagorean Theorem in their lives, we tell them, “You probably will never use the Pythagorean Theorem in your everyday life but  it’s simple and beautiful and we’re going to have some fun trying to prove it.

Developing 21st Century Skills in the Math Classroom

Math plays an important role in developing 21st century learners. The Ontario Math Curriculum states, “An information- and technology-based society requires individuals, who are able to think critically about complex issues, analyse and adapt to new situations, solve problems of various kinds, and communicate their thinking effectively. The study of mathematics equips students with knowledge, skills, and habits of mind that are essential for successful and rewarding participation in such a society”. I believe the habits of mind that the curriculum refers to are the seven mathematical processes: problem solving, reasoning and proving, selecting tools and strategies, reflecting, making connections, representing, and communicating. These processes are not only essential to the acquisition of math but are  also significant in preparing students to be successful in a 21st century society. They promote collaboration, sharing of ideas, risk taking, discovery and allow opportunities to argue and defend solutions and strategies. Teaching through the mathematical processes would not only deepen students’ knowledge and understanding of math but also develop a community of critical thinkers, problem solvers, risk takers, and collaborators.

Below are some links to resources for teaching through the math processes:

Does efficiency = comprehension in mathematics?

Last year, I co-taught in a grade 6 classroom and gave the students the following problem as a part of an initial lesson to begin a unit on division,

“165 parents will be attending a family math night in your school gym. They will sit at tables in groups of 6. How many tables need to be set up?”

We asked the students to work collaboratively to solve the problem in more than one way and to show their math thinking. Below are two solutions from two different groups.

Which group has a better understanding of division? I showed these two solutions to many teachers and at first glance, many chose the group that solved the problem using long division. Why? The main reasons were efficiency and a percieved higher level of thinking. Many teachers viewed long division as a faster strategy and the latter, a time consuming low level strategy. However, is efficiency the ultimate goal in mathematics and does it mean that students that use very efficient standard algorithms to solve problems have a good comprehension of the mathematics?

The standard algorithm for division was invented long before the calculator and was viewed as the most efficient way to divide. However, I don’t think it was viewed as the most efficient way to teach division. Yet, the standard algorithm is frequently the first (and sometimes the only) strategy that is introduced to students when teaching operational sense. The standard algorithm, like many other division strategies can be a great strategy but only if it is understood. For many students, the standard algorithm for division is difficult to understand which forces them to rely on the memorization of the steps without any conceptual understanding of division. If given the opportunity, students are capable of coming up with their own invented strategies for division. More importantly, these are invented strategies that make sense to them which would lead to a better conceptual understanding. Below are the same solutions to the division problem mentioned above but I have included the students’ final statements that answer the question.

The group of students that used a longer and “less efficient” strategy had a better understanding of division than the group that solved using long division. It is crucial that students be given an opportunity to solve problems and on their own in as many ways as they can and allow them to share and dicuss their strategies with their classmates. By doing so, they may be more likely to make connections between their own strategy with others and move towards a conceptual understanding of a more efficient strategy. When you take a step back as a teacher and let the students loose with math and allow them struggle with guidance, you could be pleasantly surprised with their results and the discussion of their results is where you will find that the most teaching and learning occurs.

What’s Your Gift?

I had the pleasure of attending the Western Regional Computer Advisory Committee (RCAC) Symposium. The day was filled with great keynote presentations by Ian Jukes and Angela Maiers and very informative breakout sessions by presented by passionate educators that fully embrace the implementation of 21st century fluencies in our education system. I truly admire these educators for their drive and passion to share their knowledge and to me are great ambassadors.

As I was riding on the bus back to Hamilton, I had time to reflect and process the “infowhelming” (thanks ian jukes) content that was offered. I kept referring back to Angela Maier’s presentation about the power of children and the fact that we were all born geniuses. She explained that young children have extraordinary imaginations, curiosity, self-awareness, perserverance, courage, and adaptability. However somewhere along the way as children get older and become more educated, they lose these genius-like qualities. Angela brought up a very thought provoking point that really resonated with me. We as educators should not be asking how we can teach 21st century skills to our students. We should be asking how we can keep them. How can we prevent our students from losing their innate genius qualities that they have when they enter our education system? Angela Maiers brought up many great ideas but what I really took away from her presentation was the fact that it should be our goal as educators to help students find their talent, their genius-like quality and guide them so that they can learn how to share their talent and contribute to society.

One month ago, my wife completed a very inspiring pediatric chiropractic course. The instructor for her last session recited a quote from an anonymous person that really stayed with me when I heard it and it came to mind as I listened to Angela’s presentation. “The purpose of life is to find your gift. The meaning of life is to share your gift with others.” It would be a great shame if our students went through their entire education without ever knowing or realizing their innate gift or talent. How great would our students be when they graduate from secondary school if they knew what their gift was at an early age?

As a proud father of two young beautiful daughters (3 year old and  an 7 8 month old), I am consistently amazed by their curiosity, perserverance and courage and I hope that these qualities will still be present when they are 16, 25, 40 and beyond. After participating in today’s conference, I come away with a changed perspective as a father and as an educator.

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.

I choose the red pill…

“You take the blue pill, the story ends, you wake up in your bed and believe whatever you want to believe. You take the red pill, you stay in wonderland, and I show you how deep the rabbit hole goes.” Morpheus, 1999

I can’t help but think of this quote from The Matrix to describe my new perspective on 21st century fluencies in education. I always thought myself to be a knowledgeable individual when it came to using technology in education. I was an I.T. teacher at two of my previous schools, I was a report card administrator, I knew how to use the Microsoft Office programs, I was familiar with a variety of educational software programs, and I also had experiences with Interactive White Boards in my classrooms. In terms of technology in education, I thought I was doing just fine….until I swallowed the red pill…

During a leadership workshop, two colleagues of mine, Zoe Branigan-Pipe (a teaching colleague) and Lisa Neale (now my leadership mentor) passionately spoke about using “web 2.0” in education. By the end of the workshop, I was still hesitant about using web 2.0 in education but was convinced by Zoe and Lisa to at least create a Twitter account and use it for professional purposes. It only took me a couple of tweets about some good math related resources, articles, and some more guidance and encouragement from my meetings with Lisa before I realized the power and the positive impact that social networking can have in education. Sure, I only had 9 followers but those were 9 educators that were possibly benefiting from my shared knowledge through Twitter. 9 people that wouldn’t have had access to my professional knowledge without this simple yet powerful microblogging site.

Twitter was just the tipping point. Soon after I joined Twitter, I created a Wikispace account. As I was searching for Wikis about web 2.0 applications in education, I stumbled upon Classroom 2.0, a social networking group for educators. For a brief moment, I thought my web 2.0 journey had come to an end until I discovered Glogster, Wordle, Jing screencasting, and etherpads. Every site I visited led me to 5 more and the number of websites that I discovered grew exponentially. I soon realized that the rabbit hole was becoming a bottomless abyss!

As a “born again techie” I am overwhelmed yet excited. When I look at the incredible applications of web 2.0 in the classroom and the willingness of so many to share and collaborate, I feel encouraged and proud to be an educator. I look at technology and web 2.0 as the key to student engagement, achievement, and equity. Web 2.0 has also made me realize that my professional learning community is not limited to my teaching partner, my divisional team, my school, or board. My professional learning community is now global with teaching colleagues in places like Johannesburg and California. When I graduated from the Faculty of Education from Brock University in 2002, I never imagined that I would be able to collaborate with educators from around the world and until a few months ago my perception of technology in education was very narrowminded. However, now that I have chosen the red pill, my blindfold has been lifted. I can no longer ignore the power of Web 2.0 and the positive impact it can have on student learning. I want to stay in this techno wonderland and see where this rabbit hole will take me.