Category: Mathematics

Do Screencasts Have a Place in the Math Classroom?

Last November, I wrote a short blog post titled, Screencasts of Student Math Thinking. In this post, I also included a link to a glog I created containing four screencasts that were created by grade 6 students explaining their group’s multiplication strategies after an initial multiplication lesson. Since that post, there has been a lot of attention around the world on Kahn Academy where students learn from concepts and strategies from videos (screencasts).

I love the idea of screencasting and I think what Kahn Academy is attempting to do is pretty cool. However, I love screencasts even more when they are created by students. When students create math screencasts it enhances their metacognition. It forces them to think about their math thinking not once but multiple times since they can play back their video, watch and listen to themselves explain their strategy or solution. They can edit and record multiple times until they feel that their screencast is appropriate for their classmates to view. The rest of the class can also benefit from screencasts because they can be exposed to different solutions and strategies to the same problem. In addition, the screencasts are more engaging by virtue of them being created by students and using student language. Also, with websites like Screencast.com screencasts are not limited to the hard drive of a single classroom computer but can be accessed via web link from any computer with an internet connection. This would allow students and their parents/guardians to view them from home.

I truly believe in the benefits of screencasting for students in the math classroom. For the past year, I have been religiously promoting it in my school board as a great tool to enhance student metacognition and math communication. Many teachers seem to like the idea of it but I haven’t really seen it fully implemented in classroom. I’ve mainly seen teachers create their own screencasts similar to Kahn Academy and no disrespect to Khan Academy or to teachers but I don’t find teacher/adult generated screencasts very interesting or engaging. I would argue that students prefer to create the screencasts themselves and watch other student created screencasts. So I ask the question Why? Why isn’t screencasting being implemented in the math classroom? Is it too difficult? Too time-consuming?

I’ve embedded the glog that I created mentioned earlier of student created screencasts of their multiplication strategies below.

I would love to know your thoughts on screencasting and how you would implement it in your classroom.

Posted on by Admin 5

Investigating Arrays Using Bitstrips

I created another interactive comic using Bitstrips that would allow students to investigate arrays and multiplication. In this activity, students help Mr. Ro arrange desks into rows and columns for the first day of school (I know, I know, very teacher-directed seating arrangement) by clicking and dragging desks and into their desired position. The comic problem is open-ended to allow students to create arrays with 12 desks all the way up to 24 desks and to create a variety of arrays for the same number of desks. I have shared this activity in http://bitstripsforschools.com and I would love to get feedback on how this activity goes if you try it with your class.

Posted on by Admin 12

Using Bitstrips To Create Interactive Comic Math Problems

Recently, I have been working with staff on integrating Bitstrips For Schools into their classrooms. Most of the ideas that were discussed involved using Bitstrips for literacy, social studies, history, health, and even science. However, math never really entered the discussion. So I searched the shared math activities that were posted by other teachers on the Bitstrips For Schools website and only found a total of eight activities. All of these activities consisted of instructions for students to create their own comics. Here is an example of a shared activity that was posted:

Grades: 6-8, Subjects: Mathematics

Create a 3 panel strip to explain how to calculate the area of a triangle and/or parallelogram.

This particular application of Bitstrips is more of an assignment which focuses on students creating a product based on a set task. However, I thought that this program could also be used to create comics based on math word problems. In addition, these comics could be interactive as well due to a feature in Bitstrips that allows students to take an existing comic, ‘re-mix’ it, and save it as a separate comic.

The following is a primary algebra word problem:

Mr. Ro gave a handful of jelly beans to Jonah and to Sam. When they counted them Jonah had 3 red and 2 green and Sam had 5 red and 4 green. They realized one of them had more than the other. What could they do to make sure each had the same number of jelly beans? Justify your answer.

I took this word problem and turned it into the comic below:

In bitstripsforschools.com, I could share this comic as an activity and assign it to my class so that when they log in it would show up in their ‘Activities’ section. The students would then ‘re-mix’ the comic and rearrange the jelly beans, type their justifications in the caption boxes, and save it as their own comic. Teachers and students could also provide feedback on each others comics via the commenting feature. I could see students creating their own interactive math comic problems as well and sharing with the rest of the class to solve.

I would love to hear your thoughts on this idea. I have shared this activity in the Bitstrips For School shared activity section for teachers. Please try it out and let me know how it works out for you.

Posted on by Admin 0

Opening up Questions in the Math Classroom

As a math teacher, I often became frustrated when I gave a math problem to my students only to have a small percentage of the whole class be able to answer the question correctly. Naturally, many of my students became frustrated too. Consider the following problem:

The word problem above is a very specific problem that only has one answer. The fact that there is only one answer is not a serious issue for me or for the students that need to solve it. The issue with a problem like this is the fact that there is only one way to answer it.

This is a very specific problem that requires a very specific solution. If I gave this problem at the beginning of a grade 6 transformational geometry unit or TLCP cycle (Teaching Learning Critical Pathway) there is a good chance that only a handful of students would be able to solve it correctly. This is the kind of problem that I would give at the end of the unit or TLCP cycle since it is a really good assessment OF learning type of problem. Then it should come as no surprise that this problem was taken from the 2008 EQAO math assessment.

But what if I wanted to use this problem to begin my transformational geometry unit? Well, maybe not in its current form but what if I could “open up” the problem so that it wasn’t so narrow and specific and that a lot more students could solve it. Consider the same problem but with some modifications:

This is an example of opening up a very specific math problem. This open problem has more entry points for students than the previous problem since students have a choice in how they can move and manipulate the mat. Open questions are questions that have more than one answer and are great for differentiating instruction in the math classroom. Open questions allow students to solve problems based on where they are at in their math development.

I actually used the “open” gym mat question last year when I helped a grade 6 teacher introduce her transformational geometry unit. At first, the teacher was hesitant. This approach was drastically different from how she usually introduced the unit. In years past, she would introduce each transformation in isolation. First, a note on translations. Second, examples and demonstrations of translations. Third, practice problems involving translations. The three-step process would be repeated for rotations and reflections (This is also how I used to teach math). Therefore, the notion of giving an open problem to her students that allowed them the opportunity to investigate and use any transformation without defining, modeling or practicing them was pretty daunting. However, the results were very eye-opening and informed the teacher’s next steps for the next few lessons. Here are a couple of the student solutions:

All of the students in the class participated and solved the problem in small groups. As you can see from the gallery, there was a range of solutions from the class that brought up some really good discussion during the reflect and connect portion of the lesson where groups were able to explain their solutions to the class and answer any questions about their transformations. Some topics/questions that were discussed were:

  • efficiency in transformations.
  • What is the most efficient/fastest way to get the mat to the desired position?
  • What’s the purpose of the dotted line AB?
  • points of rotation.
  • Can an object/shape have more than one point of rotation?

This rich discussion was able to occur because of the openness of the question and the fact that students had the freedom to investigate and use their own math thinking to come up with a solution. It was also very powerful for the students to see that none of the groups came up with the same solution to the problem. The range of students’ solutions also allowed the me and teacher to determine appropriate action for the next couple of lessons.

For more information on differentiating math content using open and parallel questions please read the following article. (A very good read!)

Posted on by Admin 3

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.

Posted on by Admin 3

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:

This slideshow requires JavaScript.

Posted on by Admin 7

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 thinkers— the 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.

Posted on by Admin 0

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:

Posted on by Admin 0

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.

Posted on by Admin 8

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

Posted on by Admin 0