The Perfect Teacher

I’ve just graded the midterm for the first class I’ve taught as a university instructor. Unfortunately , a disconnect is developing, one that becomes more apparent everyday I teach and I’m stumped with how to clean up the pieces. I’m not a perfect teacher. I’m not a perfect educator, instructor, or lecturer. But, I’m more than eager to keep trying. I’m more than willing to attempt to create an exceptional learning environment for my university-level students. If that’s true, then what happened? On the first day of class, I told my students that I didn’t just want them to stay because the class was a requirement or a prerequisite. I wanted them to appreciate math in (m)any of the ways that I did. I wanted them to see its practicality and beauty. But, it seems that is not happening.

A failing average on a midterm, a drop-out rate of 33%, anxiety-ridden students- I’m missing the point. Yet, my supervisor is not surprised. Instead, he tells me that this pre-calculus class I’m teaching in university has the highest drop-out rate of any first-year mathematics course. That includes business/applied calculus, pure calculus, and linear algebra (read: new content). A 33% drop-out rate isn’t even at the average yet. I assume I’ll hit the coveted 40% by the end of the term. At first, I thought it was all my fault. I also thought that it being my fault is the worst possible scenario, but that’s not true. The worst possible scenario is that it isn’t my fault. I can fix myself. I can try harder, research longer, study further- be better. Attempt. Reflect. Repeat.

However, if the problem isn’t me, then it’s the class and the theoretical construct of this class. The scope, then, is much wider. It isn’t just 40% of my class that will leave this mathematics lecture hall more frustrated than when they got here. It’s 40% of every class. It’s 40% of all the students who have taken this class since it began, three years ago. So, what’s the next step forward?

I intend on carefully reviewing the material for the next half of the course. I want to continue to inspire them- at least try to. I’ll provide an extra credit assignment because I care less about grades and more about them understanding. I’ll do everything I can to help them succeed, with the recognition that they, too, need to work hard. I’ll work as hard and sometimes harder than them at creating an atmosphere that is conducive of learning. My biggest concern is that I don’t have enough time. I don’t have enough time to teach both thoroughly and quickly. I’ll continue to model the class how I’m expected to. After it’s all over though, I want to reexamine it again. I want to look back at these questions and find answers. I want to know if other instructors here have the same issues as I do. If they don’t, why? I want to know if other universities have the same troubling statistics. Is there a difference between students who attended schools in Canada and those who haven’t? Is there a difference in test scores since the WNCP took over secondary mathematics education in Canada? What are the trends? How do we begin to change it? A thousand tiny ripples.

A Thousand Tiny Ripples

“One of the features that made this lesson typical of teaching in the United States is just this: stating rules, rather than developing procedures, and thereby turning mathematics into a matter of following rules and practicing procedures.”

– James W. Stigler and James Hiebert (The Teaching Gap)

I can open any mathematics textbook and find the same thing: rules and procedures. The WNCP got a few things right. It was time to change for purely-procedural mathematics to something that had a better shot at igniting curiosity in children: discovery mathematics. Unfortunately, save a few individual teachers, no one knew how to begin integrating inquiry into the mathematics classroom. No one was taught how to teach for inquiry.

Today, the problem is not the same as it was ten or fifteen years ago. Today, there are a number of education programs across Canada that are devoting their time and energy to creating effective teaching programs that teach for the development of the child and growth of curiosity. Today, there are countless academics devoting their time, energy, and money to creating cost-effective, time-effective, and simply effective methods of teaching for discovery in the mathematics classroom. Today, the problem isn’t finding the practical, hands-on teaching methods to design and facilitate inquiry-based mathematics classrooms. Even mathematics textbooks, such as the Math Makes Sense textbooks by Pearson, include a number of different hands-on approaches that teachers can use to teach concepts in the classroom. In fact, Math Makes Sense is so popular, that the “new mathematics curriculum” (i.e. WNCP) is known by parents and other stakeholders as “Math Makes Sense math”.  So, if the issue isn’t in the resources, then where is it?

Today, the issue is getting pre-service and in-service teachers to “buy into” nontraditional approaches to teaching mathematics. Every teacher was a student (either traditionally or nontraditionally), and in Canada, the large majority of them attended public school with traditional mathematics teachers. What has become typical in the classroom has been typical for an incredibly long time and the teachers who continue to teach typically are the ones who were taught typically. Consider the habit of biting your nails. At some point, typically when a person is an adolescent, they become aware that they are biting their nails. Some stop and some don’t. Those who do often have a difficult time quitting. I know I did and I was fourteen when I decided I didn’t want to do it anymore, and I was much younger than that when I was told that I shouldn’t. Now, imagine if you bit your nails until you were twenty-five, unaware you were biting them and only then, did someone tell you that you shouldn’t. How much resistance would the person who just told you to stop get from you? Perhaps you’d go through denial, anger, frustration- eventually acceptance. Those feelings rang true for me as a pre-service teacher, trying to navigate my way through a Bachelor’s of Education. Those feelings also rang true for many of my future colleagues as well. Being told that your understanding of teaching and learning is faulted (broken, even) is a tough pill to swallow. In fact, it’s a whole bottle of pills because it starts with one situation (e.g. calculators in the classroom), where you are headstrong for or against something, and the world comes crashing down in one simple counterexample- a situation or scenario that was never part of your lived experience.

It’s not fair or even reasonable, however, to expect teachers to have enough experiences that cover the basis of teaching and learning. For the record, most mathematics teachers are teachers because they excelled in traditionally-taught mathematics. They didn’t need a calculator for their multiplication tables in ninth grade. Numbers were easy. Algebra was easy. Calculus was, often, easy. So, if we can’t go back in time and change the experiences pre-service teachers had as students, what is the solution?

In The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom, Stigler and Hiebert (2009) discuss the implications of their study completed in 1999 that detailed the drastic differences in American, Japanese, and German mathematics teaching. Their final conclusion was that American teaching needed a makeover, one that would require time, money, and support. But, they wouldn’t stop there. They would include several key changes that need to be made in the American education system in order to dramatically improve education in the United States:

  1. Build consensus for continuous improvement- This section highlights the importance of designating (a) person(s) for the job of continuing education for teachers. It is not only the responsibility of the teacher, but also of the school board and government. Furthermore, aiding all stakeholders in recognizing the importance of small, gradual shifts as opposed to dramatic ones is key to constructing a solid foundation of improvement. No as-seen-on-TV-products. No gimmicks. Currently in Western Canada,  stakeholders (in particular parents and politicians) have been attempting to uproot the WNCP on the notion that it has done nothing to help young students better understand math; on the contrary, it is furthering misunderstanding of mathematicsAlberta and British Columbia have seen the biggest backlash, but Saskatchewan and Manitoba are not far behind (check out WISE math, and initiative developed by professors for the University of Regina and Manitoba). Unfortunately, according to many extremely successful programs across the globe (e.g. Finland), inquiry-based mathematics may not be a fad, but will take time to grow with the right support and resources. In Canada, there seems to be no consensus for continuous improvement. Instead, mathematics education, like so many things, has taken the political approach: either discovery or traditional mathematics. The middle ground is compromise and no one wants to do that. Until the middle ground is not seen as a compromise, but rather the best solution, a consensus will never be found.
  2. Set clear learning goals for students and align assessment with these goals- In Saskatchewan, a set of learning goals exist since the revision of the mathematics curriculum began rolling out over ten years ago known as outcomes. Saskatchewan outcomes, as well as the outcomes of all other Western provinces, emphasize conceptual understanding, while the indicators of each outcome offer the content knowledge indicative of said outcome. Whether or not clear assessment exists, however, remains to be seen. Most prairie provinces, as well as BC, still utilize some sort of standardized testing method, although many provinces are decreasing the weighting of these exams, such as Alberta from 50% to 30%. British Columbia is currently looking at assessment models that do not include formal grades or even report cards. Continual, formative assessment is the solution given by Stigler and Hiebert (2009), but with rising student numbers and falling educational assistant (EA) numbers in classrooms, how are teachers going to create environments of continuous assessment?
  3. Restructure schools as places where teachers can learn- Unlike the first two points, creating an environment of learning for teachers is less obvious in schools. Professional Learning Networks (PLNs) are one way where teachers are invited to collaborate and cooperate together. In high schools, this is most often done in subject-specific areas. Unfortunately, from my experience, PLNs have the most success on paper and, unless supported properly by administration, are not successful. The biggest reasons for this, I’d argue, is because PLNs are not given enough time to grow, are forced to grow, and are not seen as being important for the overall development of the student. By meeting only once per month, PLNs are not provided with the opportunity for collaboration. By forcing PLNs, something evident in a school district that I work for, PLNs are not encouraged naturally, and many foster negative feeling towards them. The biggest deterrent, however, is lack of recognition of the effects of cooperation and collaboration on student learning. All teachers want their students to learn and to learn in the best possible way. If teachers knew PLNs could do that, perhaps they would focus more on them. The only solution to this is education, where teachers are continuously provided with the opportunities to develop and utilize proper researching techniques, even whilst teaching. If teaching is a profession, then it ought to be treated as such, where teachers hone researching skills to better develop their teaching and pedagogy. Unfortunately, in Canada, this is no the case.

As seen above, the recommendations made by Stigler and Hiebert (2009) are not without their direct implications in the Canadian classroom. Prior to reading The Teaching Gap, I believed completely in the overhaul of Western education and I believed that it would take a revolution to change it. However, although the overhaul is still necessary, instead it may come in the way of silent, subtle changes (e.g. consensus, clear learning goals, schools as environments for teachers to learn)- a thousand tiny ripples to create the tsunami.



Stigler, J. W., & Hiebert, J. (2009). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. Simon and Schuster.

Developing a Profound Understanding of Mathematics

“If a teacher’s own knowledge of the mathematics taught in elementary school is limited to procedures, how could we expect his or her classroom to have a tradition of inquiry mathematics?”

– Liping Ma (1999)

After reading Knowing and Teaching Elementary Mathematics, a study conducted by Liping Ma, I began to consider three things. First, how does the notion of stereotype play into our understanding of Chinese mathematics and American mathematics? Second, how do Canadian elementary teachers fair in regards to their understanding of elementary mathematics? Third, if Canada is down there with the States, how do we change? And fourth, is it possible to be both “limited by procedures” and limited by inquiry mathematics? This blog post will address the first three questions, and a future blog post will address the latter. Before getting into any of these questions, however, a brief summary of the study is necessary.

In her study, Ma details the apparent deficiencies of American teachers in elementary mathematics, compared to their Chinese counterparts. She does so through a series of well thought-out interviews with a number of participants, both who teach in the United States and China. Overall, she determined that many more Chinese teachers have a “profound understanding of fundamental mathematics” [PUFM] (p.120). PUFM is the combination of four principles: connectedness (the ability to make connections among and between mathematical concepts), multiple perspectives (the ability to be flexible towards mathematical approaches), basic ideas (the ability to be aware of principles of mathematics), and longitudinal coherence (the ability to recognize the interconnectedness of the entire mathematics curriculum and its applications). Although both groups had similar procedural knowledge in understanding many elementary topics, the Chinese group had a deeper PUFM, enabling them to encourage thought-provoking discussion amongst students to facilitate mathematical growth. Interestingly enough, when comparing the PUFM of the Chinese teachers and pre-service teachers, it became evident that the pre-service teachers had little PUFM. Rather, it was the current Chinese teachers who seemingly developed it while teaching, through colleagues, students, and self. Chinese teachers were “stimulated by a concern for what to teach and how to teach it, inspired and supported by their colleagues and teaching materials” (p. 143).

When I first began reading this study, I was initially struck by the obviousness of the entire study. Of course Chinese teachers are better at teaching mathematics than American teachers. It was the moment that I thought this that I began to look at both the study and my prejudice more critically. Why do I believe this to be true, before reading the entire paper? We (western society) have a general tendency to think of Asians as being good at math and their schools to be extremely rigid and rote in practice. Did this study support those notions? Partly. The Chinese mathematics teachers had a more in-depth understanding of mathematics and could, therefore, pass that understanding off to their students through facilitation. It seems, however, that the stereotype that the classrooms were more rigid that American classrooms is false. In fact, in the chapter about subtraction with regrouping, it seemed that the Chinese teachers were utilizing inquiry with manipulatives to teach it effectively, which included a large group discussion following the activity- a pivotal part to the discovery learning model. This instructional method was evident in all areas studied by Ma. So, although Chinese students are continually outperforming American students on international examinations, the reasonings do not lie in the rigidity of the mathematics curriculum in China. Rather, it seems they are a result of the PUFM of the teachers, which enables teachers to effectively utilize an inquiry model.

Another though that came up during this study, almost as immediate as my initial prejudice, is how well Canadian teachers would do in a study like this. Do Canadian teachers have PUFM? If not, why? Are we able to place American and Canadian teachers in the same deficit category? In the United States, in order to be a secondary mathematics teacher, you must obtain a mathematics degree and do a teacher-preparation course, which varies in length. To be a elementary teacher, you need a four-year degree in elementary education. The process to become a teacher is similar in Canada. However, during school, students are subjected to many more standardized tests in the US than in Canada, including SATs and NCLB. Unfortunately, rote memorization is often synonymous with studying for standardized tests, while critical thinking gets left on the curb. Since Canada is using fewer standardized tests (and none at a national level in high schools), where do Canadian preservice and in-service teachers fall? Unfortunately, I do not have answers to these questions, as the literature is quite limited on the topic. More limited is the information on how pre-service teachers, who have gone through the latest remodelling of the provincial mathematics curriculum in Saskatchewan understand mathematics and whether or not this is different than those who experienced the former version of the curriculum. These are important questions that need to be discussed because of the cyclic tendencies of teaching and learning. As Ma puts it,

It seems that low-quality school mathematics education and low-quality teacher knowledge of school mathematics reinforce each other. Teachers who do not acquire mathematical competence during schooling are unlikely to have another opportunity to acquire it (p. 145).

In order to effectively dismantle the cycle, Ma makes several suggestions, one of which is teacher preparation. Ma writes: “[Teacher preparation] may serve to break the circle”(p. 149). Currently, there is an understanding of elementary mathematics as low-level and basic. It is when we start to unpack what these words actually mean, and recognize that they are incredibly limiting with our preservice teachers and the students they will teach, can we begin to chip away at the succession of misunderstanding on mathematics. Like Ma, I believe that there are a number of different steps that need to be taken in order to ensure the success of a new tradition and cycle in mathematics education. However, possibly unlike Ma, I believe that this solution begins with teacher education. A teacher education program that focuses not only on mathematics content, but also attitudes of and abilities in mathematics is important. Because elementary education in Canada is not subject-specific (like the secondary programs), one cannot assume that pre-service students will appreciate or enjoy mathematics. Some may even have math anxiety, as many people do. This needs to be combatted during the four years of university, or it can be passed on to the students of future teachers.

Many questions arose as I read through Ma’s study. The questions raised are important to furthering the success of our elementary education programs in Canada. Currently, public opinion is quite dichotomous in regards to discovery-based mathematics. There are many academics, parents, and teachers who support inquiry-based learning. On the other end though, there are academics, parents, and teachers who are looking to “go back to the basics” in an effort to reintroduce fundamental mathematics through algorithms and formulae. I argue for a less binary examination of the problem and and more spectral look. It is not a matter of teaching one way or the other; it is a matter of teaching in ways the effectively reach the maximum number of students in a classroom. As Ma puts it:

…the change of a classroom mathematics tradition may not be a ‘revolution’ that simply throws out the old and adopts the new… the two traditions may not be absolutely agnostic to each other. Rather, the new tradition embraces the old” (p. 153).

Cultural Bias Meets Mathematics

“We were amazed at how much teaching varied across cultures and how little it varied within cultures.”

– Stigler and Hiebert (1999)

Over the past year, I have had the opportunity to wander through countless mathematics classrooms. Some were designed strictly for high school mathematics: motivational posters plastered on the walls, desks or paired tables in neat rows, a Smart board at the front of the room, and a hand-in bin for any assignments that needed to be corrected. Others were middle school classrooms: posters were brighter, occasionally desks weren’t in rows, but groups of four, a Smart board was still at the front of the room, and their wasn’t a hand-in bin. Others still were elementary classrooms: often carpeted, with one or two bulletin boards designated for learning math, geometric shapes smiling through neon colours, tables of four with rocking chairs or medicine balls for the students who needed to move, and a large 12 x 12 multiplication table at the front of the classroom, only half filled in because the third graders hadn’t yet learned 9 x 7. These rooms all had subtle differences: colours, desk orientation, places to hand in assignments. Yet, the larger similarities were more telling: whiteboards or Smart boards at the “front” of a rectangular classroom, pencils and paper found throughout, desks and chairs. And students’ faces. There faces tell the most important part of the story. When I would tell students to put away their books and take out their math textbooks, eyes would roll, voices would sigh, and bodies would close right up. Not every student behaves this way, but there are at least several in every class. Their body language alone tells me that they do not wish to be there in that classroom learning mathematics. Why?

The most common reaction I get from people when I explain to them that I have a math degree is, “Wow, congratulations! You must be so smart. I always hated math.” I typically brush it off the nervous laughter, while inside I’m screaming that the mathematics taught in schools is only the tip of a beautiful iceberg. An article in USA Today shares thoughts on why mathematics is a cause for great stress in people. In particular, it states that the push to get students to understand mathematics at a younger and younger age is problematic when they are not ready. For example, as of the 2015/2016 school year, students as young as eleven were learning algebra in Texas. According to the article, when students are not ready to push the envelope toward abstract concepts, they are more likely to fail. However, I challenge this notion with a program detailed in Out of the Labyrinth: Setting Mathematics Free by Robert and Ellen Kaplan. In the opening chapter, Kaplan and Kaplan describe how they worked with pre-elementary students to discover every number, from zero and one to irrational numbers. What’s more is that they did this without lecturing, instead teaching through the Math Circle. Without getting into specifics in this post, the Math Circle is a series of courses offered to students who wish to expand their ideas about mathematics. If small children have the ability to understand and grasp ideas of irrational numbers, a rather abstract concept, what is stopping them from understanding algebra at a young age as well?

Culture. Our culture has gotten in the way of allowing children the right to explore mathematics on their own time and in their own way. We are too caught up in ensuring that children have all the right answers, but are unwilling to let them discover when a solution they’ve given no longer works and needs adjustment. Save a few fundamental axioms, the mathematics we “teach” to third graders via white board and textbooks is attainable through inquiry, without having the awful side effect of curing curiosity. I am not saying that the memorization of certain mathematical concepts or use of certain formulae is not important. It is time to think beyond the binary of traditional vs discovery mathematics. It is time to embrace a planar view of mathematical knowledge, one that is not even as narrow as a one-dimensional spectrum, but rather a two-dimensional surface. As I read more through Stigler and Hiebert’s book, I realize that only when we decide to break the mathematical barriers that our culture we have created for our children, only then will our children be able to experience the beauty that is mathematics.

Where’s the Teaching Gap?

I’ve just begun reading the Teaching Gap, by James W. Stigler and James Hiebert (1999). There are two passages that resonate with me in the preface, even before I’ve gotten into the meat of the book. The first, discusses on a problem that has plagued modern education since its birth: “… just recruiting more qualified or talented teachers doesn’t lead to improved teaching… In mathematics… teaching now looks pretty much like it did 100 years ago. Why?” (Stigler & Hiebert, 1999, p. xiii) The second passage responds to that question, “Most teachers continue the tradition [emphasis added] of teaching and use the same methods” (p. xiii). It’s safe to say that the most popular teaching methods utilized in classrooms today are considered traditional. We still view a classroom as a rectangular room with rows of desks and a teacher at the whiteboard, pardon me- Smartboard. What words does a classroom like this invoke? Typical. Order. Quiet. Individual. Yet, we can now also imagine classrooms that don’t look like this. Not only can we imagine them, but we can remember them. Whether it’s a completely different way of approaching education than the West, like the Montessori method, or public school teachers redesigning their classrooms, like Kathy Cassidy, a shift towards student-centered facilitating and learning has begun. The first time you step into these other learning spaces, words like “typical,” and “order” are tossed out the window, replaced by imaginativecreative, and (my favourite) chaotic. 

Why does order prevail when it’s paired with individual and quiet? One of the most prevalent arguments for learning in a purely traditional sense is to prepare children for the “real world.” In what world today are adults sitting in rows of desks, unable to speak to one another out of fear of being reprimanded? If our humanity has taught us anything, it’s that we work better in groups: cooperation, collaboration, community. We also know that this can get a little chaotic, but if it can lead us towards greatness, then embracing the chaos is the best thing we can do. We often equate “order” and “quiet” with learning, and think of “chaos” as the antithesis of learning. Why? Because somehow our cultural routines became more than routines. They became traditions, not based on anything, but themselves.

Stigler and Hiebert (1999) provide an approach for questioning these habits and the effects they have on the classroom. They call for a:

“shift [in] focus from teachers to teaching… What does it mean to focus on teaching?… becoming aware of the cultural routines that govern classroom life, questioning the assumptions that underlie these routines, and working to improve these routines over time. It means that recognizing that the details of what teachers do… are the things that matter for students’ learning… and that all these details of teaching are choices teachers make” (p. xiii).

To enable this shift demands the help from preservice programs, to encourage up-and-coming teachers to be more reflective on their pedagogies, to question where they have come from, and to grow outwardly.

In general, I consider myself a teacher, although not a typical contracted one, whose future lies in a public school classroom. This summer, as I work to solidify an idea for my thesis, I’ve been given the opportunity to teach a university-level mathematics course. I started with forty-five students. I’m in a lecture-theatre and the only pieces graded from students are quizzes and exams. The final is worth 50%. I feel as though I’ve been forced into tradition. The course syllabus is one page, unlike the hundreds of pages of curricular content for K-12 mathematics. I’m struggling with how to break the tradition, because it doesn’t seem enough to stand in front of 40+ students, many of whom suffer from math and test anxiety, low confidence in mathematics, and a lack of mathematical foundation. The Western ideologies of mathematics education have not worked, thus far, for my students, so why should I, the department, or my Faculty, expect that it should work now? “The most alarming aspect of classroom teaching… is not how we are teaching now but that we have no mechanism for getting better” (Stigler & Hiebert, 1999, p. xix)



Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York: Free Press.


Mathematics is Un-Knowing

What are the expectations we have on mathematics teachers across Canada? Is there a difference in expectation for secondary mathematics teachers and elementary mathematics teachers? If so, why?

Me at the Canadian Undergraduate Mathematics Conference (2013)
Me at the Canadian Undergraduate Mathematics Conference (2013)

What are the expectations we have on our children throughout their academic careers, particularly related to mathematics? What types of skills do we expect our students to develop as they go through mathematics in Canada? Are these skills categorized based on age, grade-level, and/or comprehension, or should they be valued on a continuum, where students develop naturally and without a five-hundred page document telling them what they should know?

Many of these questions may seem obvious. We expect greatness from teachers. In particular, we expect our mathematics teachers to instil mathematical prowess in students, often in the form of mathematical reasoning and problem solving. Our secondary mathematics teachers should  understand the mathematics that they are teaching, similar to our elementary mathematics teachers. Yet, their knowledge banks should look different. A secondary mathematics teacher should have a deep understanding of algebra, fractions, and graphs. Our elementary mathematics teachers should know addition, multiplication, and basic shapes. The reasoning is simple, almost redundant. Teaching demands understanding- history teachers know history, biology teachers know biology, etc.

What this frame of thinking leaves out is the inherent properties of mathematics. What is mathematics but a gigantic structure, one that contains a foundation- pivotal ideas that must be understood before the full value of mathematics can be realized. If mathematics demands a foundation, then perhaps the teaching of mathematics demands more than the understanding of the curriculum. If a math teacher is teaching fractional division of rational numbers, is it not important to comprehend fractions, division, and the ever-elusive idea of ‘the whole’? How does this relate to the division of rational expressions? Does it relate at all?

In terms of the inherent properties of mathematics, who decides them? Regardless of whether or not mathematics is invented or discovered (although this is a fascinating topic to discuss), determining how mathematicians determine what a mathematical foundation looks like is important and charged with cultural bias. For the West, the structure that represents mathematical knowledge is a pyramid, with basic arithmetic skills and algebra at the bottom, and pure math resting on top. For other cultures, however, the power of mathematics lies in its applicability. Who is right? Is someone right? Are we both wrong?

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Photo credit:

If we can all agree that there exists a foundation for mathematics that is itself mathematics, when is it appropriate that children should understand a particular concept. Is it appropriate at all? At some point, a line must be drawn. Why? Because if mathematics has a foundation, then it is important to teach it to children before they can fully understand the complexity and beauty of mathematics. What if, however, through learning concepts beyond the foundation, that understanding the foundation is made more obvious? If this is true, then what is a mathematics curriculum but a prescribed way of learning disguised by the theory of multiple intelligences.

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Photo credit:

For what multiple intelligences does not account for is the sequence at which a learner learns, discovers, and understands. If it is true that different students better learn when taught through (a) particular method(s), such as visually or kinesthetically, then the sequence of learning may also matter. Furthermore, the foundation of mathematics could be more complex than initially thought.

What are the consequences of this on teaching? Is it enough that elementary mathematics teachers understand ‘elementary mathematics’ and high school teachers understand a ‘little bit more than elementary mathematics’? Currently, professional teaching programs across Canada reflect the idea that elementary mathematics teachers need to know less math than their secondary counterparts. In addition, as reflected

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Photo credit:

by many programs, it is imperative that secondary mathematics teachers understand calculus. Why? Because through knowing calculus, teachers will better understand high school mathematics. In fact, elementary mathematics teachers are even encouraged to understand a little beyond the mathematics they are teaching. Why?

The questions asked in this post are not rhetorical; rather, they are important questions that have been and need to be researched in order to encourage change in mathematics teaching in school. Whether they have solutions remains to be seen, but as my math philosophy goes: “The solution is the least important step to understanding mathematics. Rather, it is the process that holds significance. Above all, however, is the idea that there are always multiple processes to the same solution.”

Social vs. Anti Social Media

This week was bitter sweet. I particularly enjoy logging on every week for the #greatedtechdebate. It always makes me think. This week the debate was We have become too dependent on technology and what we really need is to unplug.

Up until this debate, I was always about the idea of unplugging. I have always felt this was best for me. For me the idea of unplugging gives me the ability to step away from the hectic life I have chosen to lead and live in the moment.  I particularly liked Prince Ea in this video.

This made me want to unplug almost instantly. Can we not even have conversations anymore? Are we really making ourselves more lonely by relying on social media for self gratification?

But then I listened intently to the disagree side. There main point was in today’s world there is not point to unplugging. They challenged the idea about relationships online not being “real”. How often do we use technology to communicate with loved ones?  and what does unplugging really mean? Does that mean no social media or no electronics at all?

It really got me thinking about how much so many people rely on technology to connect with the people they care about most. If you are long distances away from friends or family we rely on social media to help us connect with the people closest too us. I know when I can have a conversation with a good friend or family it helps me de-stress and decompress. It really got me thinking is it even possible to unplug? How much of technology is ingrained in our lives? So much of technology is used in schools, homes, and work places can we really unplug?

Let me know what you think.



Connect or Disconnect? Why Not Both?

With the final debate for ECI 830, I find myself yet again unable to commit to the agree or disagree side. Have we become too dependent on technology? Do we need to unplug? Yes and no. I think in some ways we have become dependent on technology and we are missing out on certain things. But at the same time our society revolves around technology and to unplug from that would mean essentially unplugging from our lives, which is unrealistic.

Photo Credit: hine via Compfight cc

When out socializing with friends or family, put down your phone. There is no need to check Facebook or text. Live in the moment and participate in the human interaction. But you’re worried about your kids who are home with the babysitter? Leave your phone on, or check in half way through the evening. There is no reason you need to completely unplug. You can be connected, while still interacting and being fully present with the people you are with. This is the balance we need to find and achieve. Be disconnected enough to appreciate the moments you are living, while being connected enough to feel at ease with your responsibilities. Being connected gives us way too many opportunities that we could have only imagined before this age of technology, why give that up? Take advantage of it. But don’t forget to live your life. Tyler admits to the fact that he doesn’t see the need to unplug, nor does he want to. Logan also has similar thoughts and recognizes the benefits of being plugged in. So if you want to unplug- go for it! If you don’t-then don’t! It’s that simple.

This is my 3rd class with Alec and Katia and I find myself having this debate each semester. Even though I don’t agree with her completely, Sherry Turkle has some interesting insights on the debate. Where is the balance between being connected and living in the moment? I don’t think there is a correct answer, I think each and every one of us is responsible to find the balance that works in our lives. Some people’s jobs or families may require them to be connected 24/7 while other people may feel the need to disconnect for periods of time.

Thanks for a great semester ECI 830, have a great summer!


Photo Credit 

Dialled in, plugged in, and loving life.

Look up“.

Great advice for some, but not all. I think in some cases, we need try to reflect on the value of looking down… and what looking down allows us to do – connect, answer, and learn. Yes, we can go to a concert and hear musicians tell us to unplug and live in the moment, and we need to, but what about getting a video of Chris Martin singing “Fix You” for your friend who couldn’t be there because they were sick?

Some moments don’t need a camera and there is value in appreciating things without a digital record… but, like every other debate we’ve had, we need balance, moderation, and an open mind. Is taking a video of a child’s first steps also not living in the moment? Or is it an opportunity to reflect and relive the moment years later? Humanity is evolving, constantly rewiring the hardware of our brains, and with this includes modern connectedness and socialization which occurs by looking down and utilizing our technology and devices. We are comfortable with looking down when it helps us learn with PLNs or to help facilitate learning and friendships, but are quick to antagonize it when people don’t appreciate moments the way we might want them to. There is a challenge to begin to recognize that who we are today involves a link between offline to online life. This is echoed by the concept of augmented reality, and as we learn about what our digital footprints are, and adjust our digital identity to improve this, we improve our IRL identity as a result… we project a better us to live up to. (But this can create pressure to please, so we need to continue to reflect and be fair to ourselves).
Optimism versus the facts against being plugged in.

“Plugging in” has been called a way to avoid dealing with loneliness,


Loneliness via Wikimedia Commons

but perhaps it is rather an opportunity for less boredom, two states of mind that are, at times, difficult to differentiate from one another. I would seek to argue that perhaps we are more engaged and stimulated than ever before, but is there a backlash to this? We are all capable of multi-tasking and some evidence points to the idea that I am, in fact, wrong. Having too much going on at once is imposed by tech and causes higher levels of stress… including how connected we are and the inherent expectations for shorter response times. I would argue that I feel efficient when I get a lot done in a day, and am capable of getting a lot of things done thanks to technology, and have a lot of positive means of coping with the potential stress that occurs as a result. I want to be involved and I feel fulfilled when I am… or am I just afraid of missing out?

Fear of missing out is a reality for some, and some may tell you that technology is making this worse, but there is also learning to be had when struggling with this this fear. Speaking from personal experience through toddler to teenager, I have been completely wrapped up in what others are doing, and over time learned to accept the things I may be missing out on for what is more important, isn’t that what growing up is and has been for some time? Some argue that technology can be an addiction, observing others make trips home to retrieve devices that, without, individuals would feel naked. I have a hard time agreeing that technology is an addiction, we have it to connect and it is something that we feel improves or is needed in our lives. How is this different than applying the argument to being addicted to our cars or other modes of transportation? It is a part of our lives that improves our lives, and the fact that I feel that I “need” it to get to work wouldn’t be considered addiction or “bordering on obsession”, so many things would therefore border on obsession. My love of hockey, teaching, cats, and my family, borders on obsession. However, the points listed above make my life better, no question about it. Does being plugged in actually make my life better?

Does being plugged in legitimately make your life better?

Does being plugged in make your students’ lives better?

If you think it doesn’t, then stay unplugged.

For me?

I am dialled in.

I am plugged in.

I am educate-in.

And I am loving life.


EC&I 830, have a great summer.

Logan Petlak



Logan Petlak 2016-06-22 20:34:46

What has stood out for me this year? (include a referenced picture to do this).


CCI School Pic via Twitter

Some things that stood out for me this year were:

  • Communications Tech 90 – this class was so awesome. I wish Mr. Petlak taught me every single period of the day.

What am I looking forward to next year?        

What have I learned about in this class? Have I progressed with Z Type? Where did I start at in WPM and where am I now?


CLASSIC #uno #cards #hostel #gamenight

A photo posted by Justin Cannon (@2jcannon20) on Jun 22, 2016 at 2:13am PDT


If you liked this blog, wait till you see my other one about education.