Rhetorical or Meaningful; Are We Just Talking to be Heard?

We have all had that experience of knowing someone, whether it be in our personal or professional lives, who seems to talk and ask questions just to be heard. As educators, we should strive to make the direct questions we ask of our students effective and meaningful. There is great value in the use of open-ended questions where students are asked to dig deep and make their own connections to the topic at hand, however now I would like to consider strategies used to as effective and direct questions of your class. In their Literacy and Numeracy initiative, the Ontario Ministry of Education, published the Asking Effective Questions in Mathematics which can be accessed through the EduGAINS site.  A number of helpful guides to higher level question-asking for the Math classroom are featured and one example of such is pictured below:

Posing such questions allows for the students to make connections and allows them to make inferences and draw conclusions based on their observations and their reasoning skills. The use of prompts provides the opportunity for the teacher to give the student the floor to use their own voices to work out a problem verbally, justify and support their thought process, and essentially, to think out loud. If we want to welcome literacy into the Math classroom a great start would be to keep encouraging our students by leading them in a way that they become comfortable thinking out loud. 

Connections with Terry Fox

Making Connections

Making connections to personal life can help in math. Terry Fox lessons for any subject always are great when the school is participating in his fundraiser or walk. Terry Fox was a true hero. In this lesson, students will take a closer look at these numbers and use their prior knowledge to create math equations using actual numbers from this amazing story.

http://www.terryfox.org/SchoolRun/_Library/2012_Intermediate_Lesson_Plans_NATIONAL/N_-_Mathematics_-_Patterning_and_Algebra.pdf

Number Sense and Numeration: Solve multi-step problems arising from real-life contexts and involving whole numbers and decimals, using a variety of tools (e.g., graphs, calculators) and strategies (e.g., estimation, algorithms)

Patterning/Algebra:  Represent linear patterns using a variety of tools  Determine a term, given its term number, in a linear pattern that is represented by a graph or an algebraic equation  Describe different ways in which algebra can be used in real-life situations

Scratch

My son is very into Scratch Jr. right now. I think coding is very important skill for all students to learn. ( I am new and learning just like the students) Scratch Jr. is a great start and even students in Grade 7 and 8 can use their integers, transformations, and spatial reasoning skills while having fun. Scratch is amazing and students can use so many math skills while they learn to code. 

https://www.scratchjr.org/

Here is a gallery of scratch projects that teach math concepts. Check it out!

https://scratch.mit.edu/studios/6423/

Big Ideas

Over the past couple of weeks, some colleagues and myself put together a chart outlining Mariam Small's "Big Ideas" for the Ontario Math curriculum, and matched these with the specific expectations from it as well; specifically for Grades 6-8. If you are unfamiliar with Small's "Big Ideas", here is a link to a PDF outlining them, with examples. Our document can be found here.

Feel free to use the chart for your own long range plans, and please comment on this post if you notice any need for changes that we should/will make looking forward!

Bansho- A Mathematics Instructional Method

A colleague of mine the other day mentioned Bansho, and not having a strong math background from my teacher's college days, I had never heard of it before then. Having said that, after doing a bit of research and discussing Bansho's concept with them, I realized that more and more we are seeing this method of mathematical instruction make its way into our Ontario classrooms. I suppose then it is no surprise that the Ontario Ministry of Education has made many publications regarding this concept, one of which you can view here for more information regarding Bansho. ***UPDATE October 24th, 2015: here is also a lesson plan template/exemplar explaining how to structure a Bansho lesson.

My last blog mentioned Bump It Up (BIU) walls in the classroom, and Bansho provides the teacher as well as their students with the opportunity to create and assess student work in real time in an inclusive and cooperative educational environment. It focuses more on what Dan Meyer calls "patient problem solving" (see video below for more information on this concept), which is having students use their knowledge of math and how it works to problems in order to clarify students' understanding of how math works, and why it is meaningful.

Anyways, I wanted to post a blog about this to provide anyone who is interested with these resources to see how they can implement Bansho into their mathematics programs, and why it is a great tool for improving not only student success/understanding, but to help reshape/redefine the aging textbook/linear mathematic programs from the past. As Dan Meyer states, "Math needs a makeover", and here in Ontario we are doing this one classroom at a time!

Bump It Up Walls in the Mathematics Classroom

Ok, so Bump It Up Walls (or Interactive Performance Boards) have been around in many school boards for a few years and we are seeing them in almost every classroom. Personally, I really like the idea of what a BIU wall can provide my students, as each level clearly shows students what a particular level of achievement looks like. Besides this, students actively engage in the classroom with this wall by developing and deconstructing an assignment's success criteria and learning goals in order to gain a better understanding of what each levelled exemplar has and/or does not have. Thus, at any given point while constructing an assignment, they can look back at the wall, success criteria, rubric, etc. to help figure out where their work currently is and formulate steps of how to "bump it up". Similarly, this serves as a great tool to help students assess their peers work to provide meaningful and transparent feedback. If you'd like more information about what a BIU wall is, or what it looks like, you can check out Stephanie Kennedy's video explaining her BUI wall and/or read Raine6's blog about them. 

The purpose for my blog post today is to shed light on a BIU issue that I have struggled with to date, and that is how to formulate a successful BIU wall in the Mathematics classroom. My background has typically provided me with opportunities to engage with BIU walls in Language, Art and Social Studies courses, as these disciplines encompass tasks that include formulated writing via reading comprehension, research skills, creativity, etc. Thus, providing levelled exemplars is as easy as having students assess success criteria (SC) and mark sample questions; gauging which parts of the SC are evident, and which are not. Math is a little more tricky with regard to this type of marking, as often students understand math answers as either "correct" or "incorrect", so how can we assign levels when the perception of math as a discipline is so black and white?

After some research on the internet, and viewing countless amounts of Math BIU walls online, I'm starting to get the impression that the focus for a levelled approach in Math is on the process and representation of student answers, much like the Social Sciences/Arts BIU walls, however, there is a stronger emphasis of how the answer is represented. For example, depending on the success criteria, students are often encouraged to solve a problem however they see fit and, therefore, the levelled approach is based on what how student demonstrates how/what they know as opposed to an answer being "correct or incorrect". Here is a neat example video from Pat Johnson, showing his grade 1's Math BIU wall.

For interests sake, here is a link to some Pintrest BIU wall exemplars!

Daniel Tammet - The Boy With The Incredible Brain

What practical uses can be garnered from studying a genius such as Daniel Tammet? The answer may be "very little", or perhaps we just don't know yet. What I do know is that Daniel Tammet is absolutely fascinating. Imagine being able to recite Pi to 22,514 digits! This is what Daniel did in 2004, sitting in front of three invigilators for over five hours reciting digit after digit without error. His story is told in a documentary entitled "The Boy With The Incredible Brain". Daniel is doubly unique in that not only is he an autistic savant, but he also is a very-high functioning autistic individual. Being high-functioning allows Daniel to share how he thinks and how his brain processes numbers. It is the way in which he processes numbers which I found so fascinating in the documentary. For example, he can perform very large number multiplications without using any traditional mathematical technique. He visualizes them as shapes and when one number interacts (i.e. multiplies) with another number, he visualizes two shapes and the gap in between forms a new shape and a new number. Perhaps further studies of Daniel's brain functioning will allow us unlock greater potential in ourselves and students.

One thing I have taken away from Daniel is his view on intelligence. In an interview with Scientific American, he says "I know from my own experience that there is much more to “intelligence” than an IQ number. In fact, I hesitate to believe that any system could really reflect the complexity and uniqueness of one person’s mind, or meaningfully describe the nature of his or her potential". He goes on to say "Even if we cannot measure and assign precise values to it in any “scientific” way, I do very much think that “intelligence” exists and that it varies in the actions of each person. The concept is a useful and important one, for scientists and educators alike. My objection is to thinking that any ‘test’ of a person’s intelligence is up to the task. Rather we should focus on ensuring that the fundamentals (literacy, etc.) are well taught, and that each child’s diverse talents are encouraged and nourished". It is this last sentence that struck a chord with me. As a math teachers, we surely want to ensure the math fundamentals are learned by our students, but as educators we should also strive to develop each students' own individual talents.

Teaching Kids Real Math With Computers

How often have you heard someone say something like, "kids these days can can't do anything without a calculator"? The thinking behind such a statement assumes that if you can't do math with pen and paper or better yet, in your head, then you aren't really doing math. The flawed logic that drives this assumption is that math = calculations. In the TED talk below, Conrad Wolfram correctly asserts that math is not equivalent to making calculations, but something much greater. So often in our math classrooms students spend a great deal of time performing calculations. Why spend so much time performing calculations when we have machines/computers that can do it much quicker and accurately than ourselves. One of the great takeaways from Conrad's TED talk is that math can be used to solve so many interesting real world problems. Formulating a question, identifying a problem and being able to solve the problem using math is interesting and exciting. What is not interesting and exciting (to many) is the calculations. Yet we have students spend hours and hours doing calculations. If computers were integrated more heavily in the math classroom then students could use math to solve real problems instead of just doing calculations. For example, in the video there is a sample exam question which could be posed (on computer, not pen and paper) which is "which is the best life insurance policy?". Students would use the computer to input variables to determine the answer. Let the computer do the calculation and let the students solve the real problem. Conrad Wolfram suggests that math curriculum should be reformed to emphasize computer based math. I believe this would be far more relevant to students lives, more practical, and definitely more interesting. It should be pointed out that this doesn't mean there isn't a place for hand calculations or that they should be forgotten completely. But if it computer based reforms can make students more more interested in math and perhaps help them in everyday life, then I am all for it.

TED: Teaching kids real math with computers

Rethinking What We Teach as Math Teachers and How We Teach it

When I read Jessica Lahey's article about Steve Strogatz, a Professor of Applied Mathematics at Cornell University  teaching an introductory math course for non-math majors who hate math, I am saddened.  Strogatz has his students submit a math biography outlining their math encounters throughout their life.  He shares that his Liberal Arts students have had unpleasant math experiences and they blame themselves for not understanding math, they feel they are not intelligent enough or talented enough to do math.  Strogatz (and others) teach an inquiry based math program at Cornell University to Liberal Arts students to help them see math in a different light and feel good about themselves and math. He states with the right approach he has been able to turn students views about math around.  He feels this turn around is related to how the math is delivered to these students.  The program he delivers is called Discovering the Art of Mathematics: Mathematical Inquiry in the Liberal Arts.  If you follow the previous link there are student testimonials, quotes and videos describing the experiences they have had in their Cornell math class in comparison to their prior math experiences in high school. 

I believe these messages are important messages for us as secondary teachers to hear. Sometimes I think teachers don't always stop and realized just how much they can affect their students.  Rethinking how we teach math and ensuring we do our best to reach every student using multiple means is important. Taking the extra time can put students on one future path or another, all because of the experience they have had in a classroom. Here is a video of Strogatz at Cornell with students who enjoy math and are studying Applied Math to use in pursuit of future careers.

Wonder Shelf In Math

 

I am so inspired when I read Rafranz Davis' article about providing her math students with a shelf of manipulatives and tools that afford them the opportunity to express creative freedom and ask wonder questions.  She calls this shelf a Wonder Shelf and it has been many years in the making. She includes simple household items from the kitchen, toys, blocks and lego pieces, arts and craft items as well as a variety of forms of technology. Read about at,

http://www.edutopia.org/blog/embracing-student-creativity-wonder-shelf-rafranz-davis

Her students are able to access this shelf during class and before/after school not only to extend their learning in areas they are studying, but also to reach out and explore other topics. I can only imagine her surprise when she turned on an ipad to find stop motion clay figures demonstrating changes in volume of a cylinder, when this was not even assigned.  A student took it upon themself to create the clay figures and produce the stop motion clip possibly as a way to help them understand the concept being taught in class.  How exciting for a teacher to make this discovery!  Students can surprise us in so many ways when they are given the freedom and space to be creative.  I'm certain this particular student will remember changes in volume of a cylinder for many years to come because the memory of creating the stop motion clip in Ms Davis' class will stick with them forever!

SMART Exchange

For everyone who has a SMART board in their room, I think that it is essential that they know about SMART Exchange. SMART Exchange is a collection of SMART board files made by various people that are free to download and use in your own classroom.

The website is:
http://exchange.smarttech.com/

This site allows you to specify which country you are from in order to choose files made for your curriculum. You can then narrow down your search by subject, grade, and file type. You are also able to search key words to find the topics you are looking for.

I find this site great because I don't have to start from scratch when creating SMART board files if I don't want to. It is nice to be able to use bits of already created files, add your own content and quickly have a SMART board file prepared for class. This also allows you to see the different features of SMART notebook that others have utilized or used differently than you have thought of.

JUMP Math

This program is currently being used in Canada for classes 1-8 and uses a balanced approach called “guided discovery”.  Essentially, students explore on their own (with guidance), and go through a series of challenges increasing in difficulty, receiving immediate feedback.  This program allows students to work at their own pace and breaks down lessons into manageable chunks.  This helps everyone learn better, because they move at the speed they feel comfortable and never get overwhelmed with information.  This means LD kids can learn the material just as well as academic students.  Each small concept is immediately practiced and assessed.  This keeps students more engaged.  By learning a small concept, immediately practicing it, then immediately getting feedback, it resembles more of a video game format of a reward system – immediate feedback for accomplishing a task.  Even students with short attention spans can get through the challenges and learn effectively.  In addition, this format of learning ensures that no child gets left behind and greatly reduces the chances of students developing gaps.  Kids get a more solid foundation on which to build in the future.  The JUMP program also provides training and resources for teachers to make implementing the program easier.  For more information, visit jumpmath.org.

DAN MEYER TED TALKS: MATH CLASS NEEDS A MAKEOVER

In this talk, Dan Meyer explains what's wrong with math education and why it should focus less on teaching kids to solve problems and more on how to formulate them of their own to better prepare students for problem-solving and for life.

He explains the five symptoms that you are doing math reasoning wrong. They are: 1. lack of initiative, 2. lack of perseverance, 3. lack of retention, 4. aversion to word problems, 5. eagerness for formula.

Then he suggests five techniques to engage the students. They are:  1. use multimedia in the classroom, 2. encourages student intuition, 3. ask the shortest question you can, 4. let students build the problem, and 5. be less helpful. 

His instructional technique gets students involved in conversations about mathematics and promotes patient problem solving.

In my opinion I believe that all teachers should encourage patient problem solving in math classroom and guide students to solve problems in ways that makes sense to them.

DAN MEYER TED TALKS: MATH CLASS NEEDS A MAKEOVER

Math in the Flipped Classroom

Math classes seem to be the ideal place to implement a flipped classroom.  Currently, students listen to a lesson then have only enough time to answer the easiest homework questions.  So at home, they’re stuck trying the harder questions on their own, often doing them wrong and getting frustrated.  As a result, we make after school programs like Homework Club which is 95% students who need help with Math homework, so it might as well be called Math Club.  The problem is, everyone’s time is precious, so Homework Club is normally only a 2 day a week thing.  So students can’t benefit from this homework help every night.

Enter the implementation of the flipped classroom.  Students watch videos at night on the lesson, watching it at their own pace and stopping it to take notes or write down questions as necessary.  After that, there is a short quiz that students must answer on the video either online that night, or at the start of class the next day.  Because these are evaluated, the students will typically watch the videos (there are other solutions to ensure students watch the videos, but I won’t get into that here).  The remainder of the class time is devoted to homework, so students have far more time to not only answer the easy questions, but the hardest ones as well.  The benefit here is that now the teacher is available to help.  The student also has peers for assistance as well.  The last few minutes of class can be used doing a little review, having students answer a few questions on personal whiteboards/clickers/etc., or even a Q&A.  Students learn best by doing, so why not use your class time wisely and give it to the students to work through the problems?  But of course, no system is perfect…

I know most people become teachers because the want to teach.  Removing the lesson from the classroom poses several problems.  Teachers no longer get feedback from the students as the lesson progresses, making it more difficult to know what students aren’t getting and removing the option of explaining a concept in a different way.  Having students do a mini quiz before they start the homework helps, but not entirely.  There’s also the time delay between students learning a new concept and putting it into practice.  Some information is bound to be lost.  In addition, unless you’re using somebody else’s videos (and stats have shown that students are not impressed with Khan Academy), teachers now have to make their own.  The amount of time spent learning how to make a video and actually making the videos can be a real turn-off.  For non-tech savvy teachers it can also be a real challenge. 

Some teachers also find the flipped classroom causes a real disconnect between the teacher and students.  The teacher’s role now becomes that of a facilitator; your entire class time is essentially spent wandering around and helping.  The students no longer really see you as a teacher (unless maybe if you make your own videos) but more as an assistant.  There’s seems to be a loss of respect if teacher’s don’t use their own videos; it leads to the mentality of “Well, if they can’t even put in the effort to make their own lessons, why should I put any effort into watching it?”.  It is up to the teacher to really make sure to connect with every student during homework time and make a positive learning environment, which some simply don’t know how to do with a flipped classroom.

Ultimately, teaching isn’t easy and it’s up to each teacher to figure out what works best for them and their students.  Good luck to all the teachers out there; be flexible and never stop learning!

Questioning in the Math Classroom

Questioning the students within a classroom is by far one of the most important aspects of teaching to engage the students in the subject matter and keep them thinking. Not too often do you hear about discussions and questioning being done in a Math classroom. Usually Math classrooms consist of students busying themselves with problems, but rarely ever does anyone ever take them to the next level by asking them open questions and creating discussions. Or maybe that was just my math experience?

Now that I am on the other side of the desk within a math classroom, I can now understand better why I didn't have any discussions or questioning being done. It's not all that easy in a subject that is so concept based that there is much further thinking that can be done. Or so I thought.

The Ontario Government has put together a fantastic resource on Effective Questioning in a Math classroom. This is a great way to open your Math classroom up to engage the students and get them using logical reasoning and critical thinking skills. Reasoning, proving, problem solving, and communicating are all mathematical processes that are required in Mathematics, so creating discussions and questioning it allows for these processes to be used and engages the students.

Even a questions as simple as "How else could you ...?" makes the students look at every problem more logically and use their critical thinking skills to expand on the same problem in multiple ways. This uses their problem solving strategies to bring them to another level of mathematics, rather than just knowing how to solve the problem but understanding it.

Effective questioning is key to an interactive classroom, and no matter the subject there is always a way to incorporate it in the classroom. I know I feel more confident about it now!

Game Shows

I got some inspiration while reading the post on board games, because something that can also be used in the classroom as examples of using mathematic skills are game shows.

Let's see:

The Price is Right - there are many different mini-games in the price is right that involves probability, estimating, pricing items out, and logical reasoning out a problem.
Deal or No Deal - probability at it's finest. You could go through the whole game and simulate how probability changes (increases) as you eliminate each briefcase.

Wheel of Fortune - what is the letter that is most likely to make you money? what are some strategies to approaching each category/words being guessed? Look at the wheel, what is the probability of landing on bankrupt or $1 Million? And, considering the final spin, where they give you the 5 most common letters (RSTLNE)
Who Wants to Be a Millionaire - multiple choice questions - talking about probabilities
Bingo / Lottery - although maybe not age appropriate for younger kids (since they can't gamble) they have all probably played it before
Lingo - Lingo features two teams of two contestants who are given the first letter of a five-letter mystery word and five chances to identify it correctly. The team with the highest number of points earns the chance to correctly identify as many words as they can in two minutes (where in this picture: Red means the letter is correct, a yellow ball means that the letter is beside where they had placed it, and blue means wrong)



Shall we go a little bit older?
Let's Make a Deal - the infamous Monty Hall problem (which has already been discussed in this class)
Press Your Luck (also known as Whammy!) - where contestants collect "spins" by answering trivia, and then spin on the electronic board to win prizes or money, or could land on a whammy and lose everything. Three whammy's and you're out of the game (if I remember correctly)
Family Feud (although this is still one of the most popular game shows still around, I put it in the older category because of it's roots with Richard Dawson as its host) - the show surveys 100 people with questions and reports its findings - families battle to answer questions for points, and the final round has two people from one family answering questions and giving the best possible answer within 60 seconds (2nd person cannot repeat).
Card Sharks - using one deck of card, contestants decide if the next card is higher or lower (there's more to it, but that would be the probability aspect of it!)
Match Game - A panel of celebrities would answer a "fill in the blank" statement and a contestant would fill it in, hoping that celebrities might have used the same answer. For each match, one point was earned. In the second round, only those that did not match in the first round would answer (therefore, someone who was behind in the first round could catch up). Winner went on to the final round.


And many more....

So while some of these are more trivia related, you can bring in the concept of mathematics through the chances participants have in actually winning and different stages of the different games. I have always been infatuated with games shows (especially GSN) and whenever I got the chance (cable at the cottage and at my father's house included the channel GSN!!!!), I would be sitting there enjoying the risks people were taking in order to win or get more money (greedy greedy!).

Oh the fun memories I have. Using game shows in the classroom would also increase engagement, because everyone likes to have fun! However, make sure it is not too competitive and that your students know it is just for fun!

You can teach an old dog new tricks!

I was recently looking at my daughter’s math worksheet from her JK class (she is 4½)   The class had to do some reasoning about the number of dogs.

      The question went a little something like this:

           Clifford sees 2 big brown dogs at the pound
              Emilie sees 4 little black dogs at the pound

      Then they had to fill in the drawing of what that represented…   Only Clifford and
      Emilie were provided on the worksheet and my daughter had drawn with the best of
      her 4 year old ability a great representation of the dogs.  And even the numbers (mind
      you they are backwards)

      And then the worksheet asked 2 questions:
           1)   Who saw more dogs?…  To which my daughter had drawn an arrow to Emilie  
               and written her name.
       and
           2)   How many dogs were there in total?..….. and in my daughters handwriting there
               was almost a perfect #7.
  
Seven…?    She got it wrong…  Oh the horror…  What should I do? How could this be?  Think of what all the teachers will say if she will not be good at math.  2 brown dogs + 4 black dogs is 6 dogs… that is the only answer it could be.   The teacher had written Bravo and put stickers on it but maybe she just didn’t want my daughter to feel bad for getting the wrong answer.

All the time I was looking over the worksheet my daughter was staring at me with her big blue eyes and a smile on her face.  How could I tell her that she had done it wrong? How could I crush all the joy and pride she was feeling about her assignment?   So instead of just jumping in and telling her the mistake I took a breath and asked her to explain her worksheet to me.  She was so excited; she read through it and got to the part where she had written seven.  I still couldn’t come out and say that she had got it wrong so I figured maybe if I got her to point them out she might see her own mistake.    I asked her to point to the dogs and show me them while she counted.

Which she did…  
I read: Clifford saw 2 big brown dogs…  she pointed and started counting...  one, two…
             Lisa saw 4 little dogs…   she pointed and counted three, four, five, six….. 

Oh I couldn’t bear it…..  so I asked her why she had written a 7 on the page.  She looked at me and calmly said,  “well mommy” ,as she pointed to Clifford,  “Clifford is number seven” because Clifford is a dog too (A big red one… she had got the colour right).

I found myself reflecting back on this example last month in class when a student got an answer to a problem, but did it in a way completely foreign to me.  As he explained it, I could see where his thought processes were going.  He logically went through the problem.  To me his method seemed wrong, but it was just foreign.  I tried to show the student my method for solving the problem because honestly I thought it would save him time because it was more streamlined, and he just got confused.   In this instance I agreed that he could do the work his way (because it made sense to him) and after he explained his process to me I could see where he was coming from.  He completed the rest of the questions and got prefect. 

To many times my students have confided in me that their previous math teachers “didn’t get how they did their math”, or marked them wrong just because it wasn’t written in the format that they teacher desired.  I feel sometimes as a math teacher that we are too fast to jump to the solution and not look at how the students get to the answer.  If a student does not respond to a question in the way we were expecting them to, we should not tell them that they are wrong and try to teach them the “right way”.  Who’s to say our way of thinking isn’t wrong?   We should not be penalizing our students for thinking outside the box or coming up with creative solutions.  After all, isn’t the ultimate goal that we want from our students, is for them to be good problem solvers.   Research has taught us that we need to encourage differentiated instruction because we know that students learn from a variety of ways.  So why is it that some teachers still think that there is only one way for a student to solve a problem?   

We need to remind ourselves as math teachers that it is this kind of thinking that we want from our students and it should be encouraged, not corralled, or forced into compliance.  Whenever I come across students who do not follow the norm, I try now to pause and take a breath.  Then I have the student explain how they were able to come up with the answer.  This not only helps them verbalize their ideas (there’s your communication mark), but more times than not, it teaches me a new way of tackling an old problem.   I can then keep that in my arsenal for teaching future students who have problems “getting” my original methods of explanation.

And the next time I see my daughters JK teacher I need to thank her for, not only seeing that my daughter was correct in her thinking process, but most importantly for showing me that I need to pause and reflect a little more over my own teaching.

by Melissa Krausse

Math SAVES the day!

This weekend there was a Criminal Minds marathon on, and I was glued to the television. I did take some time to enjoy the warm weather, but otherwise I was in front of my television with my computer on my lap working away at different tasks I had to complete and needed to work on. All the characters on the show are fascinating, but the most fascinating is Reid who has an eidetic memory and is a great asset to the FBI team who profiles serial killers in order to find them. 

The episode in particular that caught my attention was one that included the reference of the Fibonacci sequence. Back in my undergrad, I worked for a leadership spring camp at Brock University (called Youth University). We did many things with students in grades 5-8 for the 2 ½ days they were visiting with us, including high ropes, rock climbing, leadership games/activities, nature walks, etc. One thing in particular that I recall doing (6-8 weeks in a row, 2 camp sessions per week) was making a necklace on our nature walk (usually on the first or second day) representing the Fibonacci sequence through the colours we chose to put on. (For example, there would be one blue bead, then one red bead, then two yellow beads, then three purple beads, then five green beads, etc. to make up the Fibonacci sequence).

What is the Fibonacci sequence you ask? Well, if you don’t know, the Fibonacci sequence is a set of numbers that starts at 1, with each subsequent number is the sum of the previous two.

So, we start at 1, and the number before it is 0. Creating the sum 0+1=1 to get the next number; so the first two numbers are 1,1. Then you add, 1+1 to get 2; and if we add 2 the sequence you get: 1,1,2. Then you add, 1+2=3 so we add 3 to the sequence to get 1,1,2,3. Add 2+3=5 to get 1,1,2,3,5; and add 3+5=8 to get 1,1,2,3,5,8 …etc.

Here’s the sequence with no words and you might get it a bit better (if you don’t already):

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…(I think you get the point now…)

Anyway, the point of this whole blog, is the fact that something that we teach in middle or high school CAN really be used in real life. Even something as abstract as the Fibonacci sequence and that it can show up in the simplest of things (sitting and watching a television show, for example). Even if your mind isn’t doing the mathematical equations while watching television, you are aware of it and able to connect with the content that much more. (I even posted a picture of Pascal's Hex (or triangle) which has elements of the Fibonacci sequence in it during my last blog post and didn’t even notice it – although recognized the ‘pattern’ as I called it – because I haven’t reviewed the concept in a long time!)

I think students need to see the relevance of what they learn as an incentive to learn. I know that I sometimes have a hard time sitting in a class where I don’t find relevance to it, so I know the importance of providing students with that incentive to learn – and answering that “why are we learning this” question…and not by answering with the simple answer of “because you have to” or “because I’m told to teach this”. Everything has a purpose!

To close this off, I wanted to let you know that because of Reid’s discovery and use of the Fibonacci sequence, the FIB team were able to crack the case and save some people’s lives (as well as their own)!! Yay for math that saves the day!

Transition to 9

This past summer, I tutored students who had just graduated from grade 8 and were preparing to go to grade 9 math classes. I tutored in a one-on-one environment, so we could really hone in on their own difficulties (note that at the same time, a colleague of mine was running a "Transition to 9" camp, where she would do the same thing, just in a small group setting).

To prepare myself for this, I grabbed a grade 9 textbook and looked at what topics might be covered. To no surprise, linear functions came up quite frequently and was definitely the "big idea" of grade 9 math. So, I really wanted to focus on linear functions (you know, the good ol' y = mx + b) to make sure my students understood how these functions worked but more importantly, why they were so important.

We started with figure out slope. Really, this is just to figure out if change is constant over time (which it always was, since quadratic functions aren't introduced until grade 10). To make this information relevant, we talked about human growth. We took milestones throughout life (with the help of some research of course) and just graphed them and quickly noticed that this was a non-linear function. Then, we looked at selling chocolates (something they had done as part of a school fundraiser). We graphed the relationship between the number sold and the cost and found that this was a linear relationship. Then we worked with calculating slope (rise/run).  We used geoboards to create linear functions and used this information to figure out slope. They quickly got the hang of it.

 

When we started working on using our slope and a point to create our final equation, they asked "why do we even need to do all this work?" The answer was simple enough: it's easier to give an equation to someone as opposed to giving them a set of points.

Once I knew they understood the concepts, I wanted to ensure they got enough practice working with creating equations given two points or given a graph and then to make a graph based on the equation. We spent weeks just practicing these skills (mind you, they only came in once a week for an hour). We graphed whatever information they found interesting, including comparing revenues of two business models (these kids were really interested in how they could make the most money doing the least amount of work).



By the time September came around, these students felt comfortable with their math skills. When they started learning about linear equations, they were pros and could do it without any difficulties. One of my students had a 60% in grade 8 math, but they ended up with an 86% in grade 9 math. I feel as though I had done my job as a teacher and this student's confidence level was surprising, even to his parents.

The only thing left to worry about, though, was EQAO.

BANSHO

During my grade 5 placement, I was introduced to the concept of BANSHO, but I never really got to experiment with it as I was in an open-concept school and the resources were limited. So, on my grade 2 placement, I got to try it out and I must say, it was a huge success!

For those who don't know, BANSHO is a Japanese teaching technique where students are provided a problem and are allowed to solve it using whatever strategies they chose. After, their answers are grouped according to strategies chosen and are put on display so they can see the various ways of solving the same problem. It's a great teaching tool and I highly recommend trying it out!

I was teaching grade 2 measurement, so students were just introduced to the concept of perimeter being the "total distance around an object." I read "Jim and the Beanstalk" by Raymond Briggs (a continuation of "Jack and the Beanstalk"), where the giant enlists Jim's help to get a new wig, some dentures, and some glasses. The pictures show Jim measuring the giant. Next, I showed students the problem: The giant wants to remodel his garden just like Jim helped "remodel" him. He needed to find the total amount of fencing for his new gardens. I provided pictures of the gardens and measurements of each straight side and had students trying to figure out the total perimeter. They were allowed to use whatever math manipulatives they wanted.

Walking around the class, I was able to make anecdotal notes of what the students were doing -- I made notes of tools being used (snap cubes, hundreds charts, counters, standard algorithm, base ten blocks) and if students were comfortable using that tool (were they using it correctly?).

Consolidation at the carpet

 I gave students about 30 minutes to do whatever they could. Some still didn't finish while others were showing their answer in multiple ways. Next, we did the consolidation at the carpet. I made stand-up signs of each strategy (as the children described what they did). Then, if students solved using, say, base ten blocks, they put their work in that column. We discussed and reviewed each addition strategy. After school, I posted these strategies on our math learning wall, which students referred to throughout our work with perimeter.

Our BANSHO wall

BANSHO is definitely an amazing teaching strategy that can easily be adapted into any classroom, regardless of the grade level. I am hoping that I will get to observe and create BANSHO lessons at the intermediate and senior levels.