The Magic School Bus (1994, September). Make Mistakes [film]. Retrieved from http://www.scholastic.com/magicschoolbus/index.htm

When I was in elementary school, my favourite television show was The Magic School Bus. The show featured a class of 8 young students, a sentient bus with transformative powers, an anthropomorphic lizard and a theme song by Little Richard. Little Richard.

Surprisingly, the best part about this show wasn't any of the aforementioned features, nor was it that awesome Halloween special when the class got trapped in a haunted mansion (a close second, to be sure). The real magic of The Magic School Bus lay in the hands of Ms. Frizzle, the class' fearless teacher and bus driver. Every kid who grew up watching The Magic School Bus wished Ms. Frizzle could be their teacher. Ms. Frizzle made an entire half hour of scientific education the most fun and exciting thing on TV; quite often I found myself "ahead of the game" in my actual science classes at school, because "Ms. Frizzle already taught me that".

The Friz (voiced by none other than Lily Tomlin) certainly did teach myself, and many other "90's kids" about everything from biological symbiosis to the inner workings of the human immune system. But one of the less obvious lessons provided by Ms. Frizzle was that of the value of a Growth Mindset. Ms. Frizzle never explicitly preached to her audience class about fostering a Growth Mindset - in fact, The Magic School Bus aired over 10 years before Carol Dweck would go on to publish Mindset: The New Psychology of Success. However, Ms. Frizzle could often be heard reciting one of her trademark catchphrases, telling students not to be afraid to "take chances, make mistakes and get messy!".

I've long since graduated from wanting to be in Ms. Frizzle's class to wanting to be Ms. Frizzle herself. I've always liked her thematic outfits, outlandish earrings, and absolute gusto for learning. I want my students to be just as engaged as Ms. Frizzle's class (and audience). I want my enthusiasm for education to be just as infectious as The Friz's was to her students. And most of all, I want to be just as encouraging to my students as Ms. Frizzle was to thousands of young children across the world.

I think Ms. Frizzle's challenge to "take chances" and "make mistakes" in education should be applied to the ways in which modern educators approach teaching Mathematics in the classroom. We know that many students are intimidated by Mathematics. Too often they feel pressured to only produce speedy and correct answers. The work that is put into finding an answer is often dismissed; only the final product, and only when produced with accuracy, is believed to be of value. Students may shy away from exploring Mathematics because they feel they are "too stupid" or "too slow" to truly apply themselves.

Take Laurent-Moïse Schwartz for example. Schwartz won the Fields Medal* in 1950, but as a student he was not considered strong in Mathematics because his approach to Math was more slowly paced than that of his classmates. Schwartz believed himself to be "stupid" because he could not solve Math problems as quickly as his peers.  It was not until much later in his educational career that Schwartz realized that rapidity had no correlation to intelligence.

As teachers, we have to remind our students that it is far more useful to think deeply and to fully understand how things relate to one another than it is to simply speed through Mathematical problems. Although producing a quick and accurate answer can be gratifying, there is nothing to be ashamed of when taking one's time to solve a problem. And if that problem cannot be solved correctly, it is no matter of disgrace. Making mistakes and taking educational risks is one of the best ways to learn - as a teacher, I would much rather see my students attempting to find the right answer, and making mistakes while trying, than to see them not try at all.

Students who feel that they cannot do well in Math class because they are "not math people" must be reminded to slow down, and to embrace the fact that things may "get messy" at first. Developing a skill of any sort - artistry, musicianship, or even Mathematics - involves putting in time and practice. Making mistakes shows students where to focus deliberate practice and where they need to take the time to slow down. Teachers ought to encourage students not to shy away from mistakes, but rather to feel comfortable enough to embrace the potential for mistakes and to view them as a natural part of learning.

There's a phrase in Latin that can be applied to this Ms. Frizzle / Carol Dweck way of viewing mistakes. Known as the "felix culpa", the phrase can be roughly translated to a that of a "lucky fall" or "happy fault".  In literature, it is used to describe how a series of mistakes or unfavourable events can eventually lead to a happier outcome. In my future math classes, it will be used to describe the wonderful opportunity to improve that only derives from making happy little mistakes.



*In our Math Class modules for this week, I was amused to hear the Fields Medal likened to that of "winning an Oscar in Math". The Fields Medal actually originates from my hometown; its namesake, John Charles Fields is a Hamiltonian :)

I Love Hamilton Ontario (2017, September 22). An Absolutely Real Photo of John Charles Fields [not photoshop]. Retrieved from John Charles' Fields' diary, I think.

Have you ever watched a movie that employed the use of the "Worthless Treasure" trope? Usually, it goes something like this:

1. Hero fixates on achieving some goal or finding some highly valuable treasure
2. Hero goes on adventure to find treasure
3. Hero finds treasure
4. Treasure is not what is expected; it ends up being some kind of MacGuffin or worthless item 5. Hero comes to realization they didn't really need the treasure to begin with; the "real" treasure was in the adventure of finding such an item
6. Wise Old Man stock character confirms this by saying some variation of "you didn't need the magic treasure to prove you're a hero, the real magic was inside you all along!"



While most film critics would argue that we need to get away from the Worthless Treasure trope, I'd say as educators, we could use a lot more of this particular experience, especially when it comes to Mathematics.

A lot of students unwittingly view themselves as "academic heroes". They've got a treasure to find, and if they don't find it, no one will ever see their academic value.The treasure, of course, is the "right answer". Too often, the value of Mathematic experience is placed on finding that right answer and solving the Math problem correctly. If the right answer can't be found, students believe that all is lost - they've failed their academic quest.

As teachers, we have to remind students that it is the journey, the effort and process of solving the Math problem, that is truly rewarding and valuable. Finding right answers can certainly be gratifying, but if everything is viewed as either right or wrong, much of the value of Mathematics is lost. The hard work that goes into solving an equation, even if the final answer is not quite right, is still valuable. It is this effort, that allows students to find the most valuable treasure of all - that of a Growth Mindset.

Teachers must be careful to avoid praising students who produce correct answers as "smart". When students are told they are "smart", they believe that they have been perceived as clever, and wish to maintain this image. In order to ensure that they continue to produce correct answers, and thus cultivate their image as a clever student, they will shy away from academic challenges. These students are left with the impression that they have a fixed intelligence, and thus feel the pressure to continue to be seen as "smart". They learn to value only the final product, rather than the process of hard work, because they do not see effort as a means to expand one's intelligence.

However, when teachers praise students for their effort and diligence, rather than for some innate acuity, students can see that their hard work is valued. They understand that even if there is a chance they do not find the correct answer, the effort put into finding it is rewarding in itself; practice will allow them to build upon their skills and eventually they will be able to tackle harder problems with greater proficiency. A Growth Mindset is fostered by valuing the endeavour itself and the persistence necessary to expand one's abilities in Mathematics.


Teachers have the ability to play into the "Wise Old Man" trope in this situation. When students are discouraged because they are working hard, but not achieving the results they had hoped to see, we can remind them, "you didn't need the right answer, the math was inside of you all along!". Or something like that.

Welcome back, teacher candidates. Year Two Mathematics is officially underway!
Last year, I related the theme of my blog, the cartoon television show Adventure Time, to my goal of wanting to promote Math as a positive experience for my students. I love how the show puts a positive spin on Mathematics, using it as an expression of awesomeness and a favourite past time of Finn, the young hero of the series. 

I think it is critical for students to become familiarized with the fun and enjoyable side of Math, as Math is often stereotyped quite negatively.

As we have learned in Week One, the general Math Mindset that pervades academic experiences is a negative one. Poor attitudes toward Math, Math Myths, problematic brain growth, and harmful stereotypes have adversely impacted the general public's experience in the field of Mathematics.


The image is actually from a Spanish telenovela, Senhora do Destino. Across the internet, very few people will describe it as an image of Nazaré Tedesco. Instead, it is colloquially referred to as "Confused Math Lady". 

Confused Math Lady literally embodies the two most prominent mindsets that deter students from enjoying Mathematics. It illustrates the belief that Math is confusing and difficult; something that many people are supposedly just "not good at". It also brings to focus the popular, albeit erroneous, myth that "girls can't do math". 

As a young student, I was often exposed to the unfortunate belief that girls cannot participate in domains such as athletics, science or mathematics with the same competence of their male counterparts. In my youth, I assumed that one day the boys in my class would outgrow this mindset and stop teasing girls about what we could and could not do. 

Unfortunately, some myths manage to pervade all ages and genders. In my adult life, I have still witnessed examples of women being discouraged or questioned when operating in the field of mathematics. And both male and female students avoid math as much as possible, because they find it confusing and cannot understand how it will be useful in the "real world". 

So how do we avoid promoting the Confused Math Lady stereotype?

In the video Stanford Math Myths and The Brain 2, Montse says that people have been turned off from Math primarily because of how it is taught. She states that people often describe Math classes as "monotonous, too abstract and not linked to life". To combat this, Montse suggests creating classrooms that treat Math as a "living, breathing subject with lots of ways to access it". Providing students with fun exposure to Mathematics can help to encourage students to practice their Math skills. Many people who find Math "confusing" feel like they are "not Math people" and therefore will never be good at it. These people buy into the myth that Math is only for certain people, when in reality, anyone with a Growth Mindset can become a good Mathematician! When we avoid Mathematics, we do not give the prefrontal cortex in our brains the change to flex its muscles and become stronger when solving Mathematical problems. Just as athletes and musicians must practice to develop their skills, Mathematicians must practice as well! 

Teachers must be sure to provide students with a positive, Growth Mindset-inspired view of Mathematics. They must provide welcoming exposure to Math in the classroom, and seek to illustrate how Math pervades many aspects of daily life. It is also important for teachers to be especially welcoming to female students, who may still be impacted by societal norms and stereotypes that assert that they are better suited to subjects that are more "left-brain" oriented. It is important that young girls see themselves as capable Mathematicians and feel just as welcomed in academics domains as their male counterparts.

When a young woman asks her teachers what limitations girls face in terms of Mathematics, there should only be one answer:

Like the vast majority of my classmates, I was not looking forward to starting EDBE 8P29 at Brock University. I had previously believed that the last math class I would ever have to take in school was Grade 12 Data Management & Probability, and I was less than ecstatic to discover that this was not the case. Math had never been my strong suit, and although I was never "bad" at it, I certainly did not feel as though I would ever be able to teach it confidently. I quickly decided that my Mathematics class would be my least favourite and most difficult of the semester.

As you may have guessed, I couldn't have been more wrong. And for that, I am very glad. As it turns out, learning to teach math is actually one of the best ways to increase your own confidence in your mathematical abilities! Over the past few months, I have come to view myself as relatively well-informed individual on the subject of Whole Number Operations, Fractions, Integers, Proportional Thinking, Patterning and Algebra, Geometry and Spatial Sense, Measurement, Data Management and Probability.

The most enlightening and comforting concept illustrated in EDBE 8P29 is that everyone approaches math in their own unique way. Just as we are all unique individuals, we all bring our own unique perspectives to mathematical problems. When we solve problems, we often do so in ways that are reflective of our personal creativity. We learn differently, and as a result we discover ways to solve math problems that are different from the ways others may approach the same problem. Teachers must be aware of this when instructing students. Differentiated Instruction is crucial when teaching mathematics. Teachers who focus more on questions rather than correct answers enable their students to learn in an environment that supports their creativity and natural sense of inquiry.

As a teacher, I want to fill my math lessons with open-ended questions that allow for students to establish a sense of engagement and wonder in the realm of mathematics. I do not want my students to feel anxious about right answers or following strict formulas. Instead, I will teach them to look for alternative methods and original solutions to subjects of inquiry. I will provide my students with as many real-world examples of how mathematics can function in the world around them. I will also give students opportunity to participate in the Gamification of math, so that they can experience the fun side of mathematical challenges.

I want my students to feel excited about the possibilities for inquiry that math class provides. When they think of math, adjectives like "boring" or "difficult" will be far from their mind. Instead, they will think of math as an expression of conquest and positivity, much like Finn, lead character of Adventure Time.  Just as I mentioned in my very first post, Adventure Time does a great job of equating mathematics with fun and success. I can only hope that one day I'll hear students using mathematical terms in the same positive way that Finn does. Who know? Maybe one day when I tell my students that I'll be taking them on a field trip to Tim Hortons I'll be lucky enough to hear one of them shout "that's so rhombus!" or "totally mathematical!" or "algebraic!" amongst the usually "yippies!" and "yays!".

Cartoon Network. (2012, January) Algebraic! T-Shirt [photograph]. Retrieved from www.cartoonnetwork.com

• Assessment FOR Learning is the process of seeking and interpreting evidence that can be used to decide where the learners are in their learning, where they need to go and how best to get there. The most common example of this is Formative Assessments that teachers use to determine what students already know.
• Assessment AS Learning is the use of a task/activity to allow students the opportunity to use assessment to further their learning. This kind of assessment usually manifests through self or peer evaluation. 
• Assessment OF Learning describes assessment that becomes public and results in statements or symbols about how well students are learning. It is usually represented in the form of reports, judgements about the quality of learning, and assigned grades.

As a student, my only concerns with assessment centred on Assessment Of Learning. I hardly took notice of formative assessment or peer evaluation. My interest was divided only between scores on test and assignments, and grades on report cards.  This meant that whenever Report Card season crept up, an undeniable increase in anxiety seem to arise. As a high achiever and a confirmed perfectionist, Report Cards represented the totality of my need to be successful. Although I performed very well academically, I always managed to feel nervous about my grades. What if I got a B instead of an A? Did it mean I was losing my intelligence, slipping from my high standards in the previous grade?

When students realize assessment is criterion-referenced and properly spread between Assessment For, As and Of Learning, the pressures surrounding assessment can be reduced, and students can avoid some of the anxiety that it entails.

In terms of Mathematics, teachers sometimes need a reminder that there is more than one way to solve a problem mathematical equation, and therefore more than one way to assess students. Students and teachers can sometimes find themselves focusing too much on the “right way” or traditional way to solve a problem, when in reality, there are almost always multiple ways to arrive at an answer. Teachers should especially keep this in mind when assessing students’ work. If a student’s method for solving a problem differs from their own, it is not necessarily wrong. It is important to study the entire work involved in a student’s solution, rather than just marking the answer as “right” or “wrong”. Often, small errors are made that effect the student’s final answer, even though the student understood the processes and concepts necessary to solving the problem. In cases like these, students still demonstrate an understanding and/or application of Specific Curriculum Expectations, and thus it is important that they assessed with more than just respect to Assessment of Learning.

This motivates me to actively avoid ever giving a student a "zero" when they solve problems in math class. As long as the student shows their work, I believe there is opportunity to discover what is right about the student's wrongs.

One of the highlights of this topic was the in-class mini lesson provided by two of my peers this week. May and Kaytlyn created a very helpful interactive lesson about Data Management using an app called Nearpod. I loved this app, because it allows teachers to use tech in the classroom, in a way that creates a collaborative learning environment for students. Not only did May and Kaytlyn teach us about the Measures of Central Tendency, such as mean, median, mode and range, but they also taught us how to teach it. Nearpod is not necessary for teaching these concepts, but as my two peers demonstrated, it can be very helpful when doing so. I would love to create my own Nearpod lesson, similar to theirs, when I teach Data Management to my own students.

Our discussion of Probability also provided me with inspiration for my own teaching. My instructor, Sue, shared the brilliant idea of having students directly interact with theoretical and experimental probability at Tim Hortons! During Roll Up The Rim season, teachers can take students on a field trip to a local Tim Hortons where they can compare the results of the theoretical and experimental probability of winning a prize in Roll Up The Rim. As a teacher, I would buy every student in my class a hot chocolate, and then have them roll up their rim. The actual number of students with winning cups could be compared to an estimated number of winning cups determined by the class, based on the contest statistics provided by Tim Hortons. After comparing the class' winning number of cups to what Tim Hortons asserted would be the average number of winning cups, students would be able to recognize that theoretical probability represents what we think will happen based on statistics and that experimental probably represents what actually happens when we carry out the event. If my previous experiences with Roll Up The Rim serve as any indication, chances are we will have two very different results!

Nystrand, Shawn. (2011, March, 26). Canada's New Motto [photograph]. Retrieved from http://bit.ly/2gQsUmb

Despite the fact that some students may be left disappointed when they don't win a Napoleon Gourmet Grill (oh wait, that's me), I think they will still really enjoy the outing to Tim Hortons and the chance (ha ha, probability is a great unit because there's probably going to be many opportunities to sneak in a pun or two) to test out experimental probability. And the added bonus of a tasty hot chocolate doesn't hurt either!

How many tools can you think of that help us measure?

To reinforce this concept, we played the "Mini-Metric Olympics". This involved the enjoyable experience of playing "sports" such as Paper Plate Discus, Paper Straw Javelin, Cotton Ball Shot Put and The Big Foot Contest. These "sports" made measuring distance and size into a fun and competitive activity. Yet again, the Gamification of mathematics proves to make simple concepts, like measurement, not only engaging to students, but it also makes these concepts applicable to real-life situations.  I can't wait to host my own Mini-Metric Olympics with my own students, as I love the interactive and collaborative learning environment in creates.

AIMS Education Foundation. (1987,  January). Mini-Metric Olympics [worksheet]. Retrieved from http://www.nclark.net/mini-metrics.pdf

Measurement can be made fun through more than just the Mini-Metric Olympics, however. As a class, we also learned about more measurement-themed games such as Area Dice and Measurement Jeopardy. We also discussed the different ways in which measurement can be interpreted. Remember when I asked you to picture tools that help us measure? We discussed many tools - everything from the humble 30 cm ruler to a high-tech galvanometer. But what if we move beyond these tools and think outside the box?




As Drew Brees and Elmo point out in the video above, you can use anything to measure ... even a potato! When young students are first introduced to measurement, this concept of open-ended measurement can be extremely helpful. Students can learn to use everyday objects to estimate the length, width or height of any object, including themselves! Measurement should never be introduced as a restrictive concept with limiting rules. Instead, using open-ended and relative measurement, as well as gamification can help students to view measurement as a site of wonder and inquiry.