“Crack-the-Code Contest” – Reshaping Culture with Contests

Background

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Since the very beginnings of my career, education has always been more than refined lessons and enhanced learning, it has been about attitudes. When you teach a subject like mathematics you realize quickly that you have the cards stacked against. Students walk into your classroom with a negative attitude towards the subject that you have grown to love. We stand there bewildered that someone could “hate” math. Students and grown adults freely admit to me, no reservations, there disdain towards the subject. “No offense”, they would say, “I like you, I just hate math”. It is true? Could this many people actually hate math? I started to believe that.

That is until I started to look at where math surfaced outside the classroom. I would see movies like, Good Will Hunting, 21, and Pi, television shows like Numbers, and books like Permat’s Enigma, that would remind me of how intriguing the mathematical mind could be. On a daily basis we see countless examples of people voluntarily gaming, puzzling, programming, reasoning, conjecturing, and proving and made me think, maybe it’s not mathematics that people have a problem with but school mathematics. In her book, Elephant in the Classroom Jo Boaler explains, School maths is widely hated, but the mathematics of life, work and leisure is intriguing and much more enjoyable. There are two versions of maths in the lives of many people: the strange and boring subject that they encountered in classrooms and an interesting set of ideas that is the maths of the world, and is curiously different and surprisingly engaging”. However, we seldom slap the label of “mathematics” on anything outside the classroom. Go ahead, ask someone you know, child or adult, to name something “mathy” and see what they say. If they are anything like my sampling they might suggest, “adding and subtracting”, “fractions”, “algebra”, or “geometry”, all of which are explicit aspects of the k-12 curriculum.

The Inspiration

My goal in math education has always been to reshape the culture and connotations attached to mathematics. I have worked hard to see that happen in my classroom but I realized if we want students to change their attitude towards mathematics, we might need to help them change their definition of what mathematics is. We need to help students label things like games, programming, and puzzles as being mathematical. I wanted to create a trend around mathematical activities. Then, 3 years ago, a new game began creeping into North American news.

Escape rooms were popping up everywhere and no matter how many of them there were, they were always booked up. If you haven’t had the opportunity to try one, the premise is simple. You are locked in a room and using the clues and puzzles around you, you must figure out a way to escape in less than an hour. If you think I am exaggerating about the popularity of these places, you need only to Google map search “escape room” in your area. I managed to find 13 of them within 30 minutes of the downtown Vancouver area. Not only are these businesses on the rise, they have began to diversify their rooms to include various themes which are often supported by additional actors. The Smartypantz company in Gas Town, Vancouver offered rooms like “Thirst for Murder”, where you are trapped in “a nightmarish basement and about to fall victim to a killer’s psychotic and cannibalistic quest for the fountain of youth. There is a glimmer of hope though; you’ve been left alone and there may be a way out”. Or perhaps you would prefer, “Dreamscape” where you are trapped in “a dream, surrounded by a bizarre and artistic alternate reality. If you don’t find a way to understand the clues and solve the puzzles, you will never wake up”. Other cities offer plenty of alternatives, this one my personal favourite:

As I read more and more about escape rooms I grew in excitement. However, unlike most people my excitement wasn’t to go and solve puzzles (I do plenty of that already), I wanted to be a gamemaster. I wanted to be the one that brought mathematical thinking to thousands of people. These individuals were able to promote logical and lateral thinking, problem-solving, collaboration, and communication, all of the things I considered to be the absolute essence of mathematics.

The Structure

So, I went straight to work, creating my version of the escape room. There was only one problem, of course: parents and administrators tend to frown upon locking students in a room for an hour. Not to mention with escape rooms there are only so many people who could participate at a time, I wanted to involve the entire school. My thinking switched from locking people in an enclosure to locking them out of an enclosure. This gave rise to quick garage project the beginning of the “Crack-the-Code” contest. Of course, I needed to give so additional incentive to get into the box so I loaded it with candy and gift cards.

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The premise for this contest was that the students would need to figure out the combination for the locks and obtain the master key by working through a series of puzzles and problems that promoted mathematical thinking. It was imperative for me that this contest be a public spectacle. Not only did I want students to know that the contest was going on, I wanted students to know that their peers were actively involved in the contest. Too often there is a stigma attached to participating in extra-curricular math activities and students are often shamed for their participation. I wanted the students to see that it wasn’t only the academically elite that were involved. Thus, I made the instructions printed on bright orange paper and on the back was the first clue.

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A key element in this contest was the QR code. QR codes can be created use a number of different websites but I used http://www.qrstuff.com/, mostly because it didn’t require me to download any additional software and I could copy and paste directly from the website. QR codes can be scanned with almost any smartphone and will pull up the website that it is linked to. These could also be used along with Google URL shortener which will keep track of number of times each QR code is accessed. Students quickly discovered that not only do you need to solve the puzzle, you also need to determine where the solution is leading you. For example, students would have come across this clue.

coin-puzzle

The solution to this clue was $162 which may have prompted a lot of different locations. Some students went to room 162, others went to parking space 162. Eventually students would find the clue at locker 162 and behind the lock hid another QR.

Students would scan this and up popped another puzzle. This time they were given a puzzle with esther-lim

This puzzle gave a solution of Esther Lim and 3:00. Completely perplexed, many students were stumped by this solution. I could hear groups of students shouting “Who is Esther Lim?” After many Google and yearbook searches students began to notice the brief quotes written at the bottom of each page like the puzzle above which gave them some direction on their next location found in the fine arts building.

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As students solved each of the clues and moved on to the next location they were slowly revealed the numbers to crack the code with the final puzzle leading them to the master key which was kept by the keymaster (myself). This was done so that I would know when the last group was about to finish the contest. I also ensured that I could give another puzzle if I found that the initial puzzles were done too quickly. This way the contest length was always controlled by myself.

The Art of Puzzle-Making

For me the fun part was making the puzzles and writing problems that were simple enough for the grade nines to engage with but difficult enough to cause the grade 12’s to deliberate over (our school is grade 9-12).

Don’t reinvent the wheel

For many teachers this is the frightening part of organization the event, the component that seems to take the most amount of time. Let me reassure any potential organizers, you don’t need to reinvent the wheel. There are lots of great problems out there. Martin Gardener and Sam Loyd have written numerous books packed full of great problems (some of them found here: www.puzzles.com  and www.mathsisfun.com/puzzles ).

Rename the wheel

While you might not need to reinvent the wheel, you do need to rename it. Too many puzzles have posted solutions found somewhere on the internet. If you do not change the problem it does not take much effort on behalf of the student to copy and paste the clue and find a solution in lesson than a minute. For example, there is a great problem on divisibility that asks you to “Create a 10 digit number where the first n digits are divisible by n”. I copied and pasted that exact phrasing into Google and here are search results.

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On top of the countless websites that offer a solution there is a youtube channel that gives solutions as well. Creating a story that goes with the puzzle allows you to bring the problem to life and change it in such a way that it is no longer recognizable. For example, I adapted the above in the following way.

kings-fortune

Write with the end in mind

One essential piece of the contest is that the gamemaster must be able to create a puzzle that leads students to a particular location. This, of course, begs the skills and knowledge of a math teacher. Someone who not only knows how to solve the puzzle but also knows how to change the conditions of the puzzle to create a particular solution. This could be accomplished by using a cipher to change numerical responses into alphabetical or you could simply use a bit of arithmetic to change the final solution. For example, in the problem it was said that the king gave an additional reward to the servant boy who solve the puzzle. That “award” that was given in addition was enough to change the solution to a phone number that belonged to the keymaster.

The Impact

At the BCAMT conference I was able to share a few stories on the impact the contest had on individual students. While I think there is an incredible testimony that comes in the those individual cases, I wish to share a bit of how it reshaped our community.

In particular the participation of the school body was a bit surprising to me. Initially I had put out 100 orange pieces of paper with the first clue on it. By the end of day one every one of those papers was gone. Although I am certain that some of those ended up in the trash, I was happy to see even a 10% participation rate (our school has approximately 500 students). On top of that I was able to use a URL shortener (https://goo.gl/) for the first clue which tracked the number of times students accessed the first clue. This total was near 90 students which meant that, conservatively, 20% of the students had seen the first clue (I wished I had kept the using shorteners for the remainder of the contest). In the end there were 5-6 groups who had made it to the final few stages. This included 2 grade 12 groups, 1 grade 11 group, and 2 grade 10 groups.

The Math Club

Math club has always been a bit of difficult club to get off the ground. I have always had a consistent group that participates every year but it was seldom more than 10-12 students. Not only that, the students involved were the same students that did exceptionally well in the classroom. That tells me that students still think that grades translate to mathematical thinking and while most of our students who do well on contests are also strong students, I have had plenty of students who are not straight A students do well on contests. I wanted to entice more students to join the math club, even if they only tried one contest. Instead of asking students to show up to meetings I simply asked them to join a Google classroom group created for the math club. I did this for two reasons. One, I wanted them to at least receive information and decide whether or not they wanted to participate. Two, I wanted lots of people to be a part of the math club. I wanted students to tell their friends they had joined the math club, even if they didn’t write a single contest. By doing this I could slowly remove the stigma attached to math club being for “math geeks”.

So how does math club tie in with “Crack-the-code”? Well, I also decided to use the math club Google classroom to provide extra details on the contest along with some hints. Students were told that they would receive details about future contests that they may wish to participate in. On day one of the contest I had 7 students signed up for math club. By lunch… by day 2… by the end of day 2.

Launching your own contest

If you feel a little overwhelmed you are in good company. The first contest I launched had plenty of issues that I had to work around. That together with the fact that I was attempting to run the entire contest by myself made it challenging to think about anything else the entire week. However, each year I learn a few lessons and the contest runs smoother than the previous year. Here are a few lessons I learned.

  1. Don’t put soda in the box – During a staff meeting I was called to the office after the secretary had informed me that my box was dripping. Apparently the student thought they could shake the contents loose.
  2. Don’t make your window out of glass – I had a student this past year shake the locks on the box and ended up putting the lock right through the glass. It has since been replaced with plastic.
  3. Get a team to help – One of the biggest difficulties I had was getting someone to look at the puzzles and determine a solution before I launched the contest. At one stage I had the most bizarre solutions from students only to discover my question had been written incorrectly. Because the clues were written on Google docs, I was able to update them instantly to avoid any other groups having difficulties.
  4. Control the pace – I intentionally put in some clues that had solutions that were only accessible during certain times of the day or week. One clue, in particular, was found in our auditorium, a location that students could only access once a week. This allowed students who were falling behind to catch up. Other clues were found on license plates that were driven away at the end of the school day.
  5. Support those lagging – As long as students were interested in participating I wanted to make sure they knew they were still in it. In some cases I did a little mini lesson on solving systems of equations for students who had written out algebraic statements. Other times I prompted them to stick to the high school building when looking for a clue. Once the majority of students were on stage 2 or 3, I also started giving hints on stage 1. This ensured they were always trailing close behind.
  6. Keep in grade-neutral – You want to maximize participation and while a more developed thinker in grade 12 will have an advantage over a grade 9, you want to ensure everyone is able to participate. Giving puzzles involving quadratic equations or calculus concepts may inhibit students from getting involved.

You want to get started?

My goal as a math educator has always been to change the culture and connotations surrounding mathematics in North America. I do this with my friends, my family, my colleagues, and my students. However, I have come to a point in my career where I realize I do not operate on an island. If we want to see change in Vancouver, British Columbia, or even Canada we, as educators, need to work together; to share, to support, to inspire. If this contest is something you are interested in launching at your school I would be happy to come and share my experience and for practical ways you can get started. Please don’t hesitate to email me or hit me up on Twitter for any tips or resources. If you have something similar at your school I would love to hear your story.

danielwoelders@gmail.com | Twitter: @woelders

 

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Is “Cancel” the Right Word? How Students Understand Reducing Rational Expressions

Research Investigation – How Students Understand “Cancelling” When Reducing Rational Expressions

Throughout their academic journey students accumulate mathematical ideas, attitudes, strategies and vocabulary. As we introduce new curriculum much of our time is spent trying to understand what students know. This semester was no different as a group of grade 10’s funneled into my classroom and we began a series of mathematical conversations that would continue for the remainder of the course. Amidst these conversations was this notion of “cancelling”. This word was carelessly thrown into explanations often to describe the process of getting rid of a term. For example,  would be described as “cancelling ”, thus one might speculate that cancelling results in zero. However, this term was also used when describing, when the 2 gets “cancelled”. This term is so frequently used in algebra that it often creates confusion with students and this confusion became very evident in our classroom when students were required to work with linear equations. In one particular lesson students were asked to rearrange the equation  into slope-intercept form (). Final answers from students took the form of . When asked how the students came to these conclusions, the term “cancel” came up over and over. It seemed that the way students understood the process of “cancelling” in rational expressions plays a big role in the errors that they produce in reducing algebraic expressions. This experience fueled a desire to shed light on student beliefs and provided an excellent opportunity to investigate how students understand “cancellation” (or “reduction”) of rational expressions.

The subjects in this investigation were 50 grade 10 students half of which had completed a unit on factoring (a skill that plays a significant role in reducing rational expressions). Given the diversity of perspectives on this topic, it seemed appropriate to gather data from every student in the form of a survey (see Appendix). This survey not only addressed what students understood about “cancellation” and the rationale for it, it also picked at common misconceptions in reducing rational expressions when working with variables, radicals and integers. Additionally, part of the intention of the survey was aimed towards self-discovery for students as a way to test their methods and verify reductions.

After compiling surveys, there appeared to be a pattern to how students perceived “cancellation” leading to three general lines of thought. The general groups include, cancelling when the numerator and the denominator have a common factor, cancelling when a term in the numerator and denominator have a common factor,  and cancelling when each/all numbers in the numerator and the denominator have a common factor.

  1. Cancellation occurs when the numerator and the denominator have a common factor

This method, of course, lends itself well to reducing rational expressions. In fact, it describes the reason rational expressions are reducible. Ideally, this would be how all of our students understand reducing. Unfortunately, even amongst those students who have been exposed to factoring, this method is seldom employed by students, and makes up only 20% of those surveyed. Students who used this method were identified as explicitly factoring before the division (“cancellation”) occurred (fig. 1).

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Students who understood the reduction in this way were often found using language such as “factor”, “take out” or “remove” to explain why the cancellation is allowed. In other words, it is essential that a common factor first be identified before the reduction occurs. Understood in this way it becomes irrelevant whether the expression included variables or radicals as long as a common factor could be found. In general, students using this technique demonstrated a stronger vocabulary, were better able to justify why the reduction occurred, and were able to verify that the reduction was correct.

The problem with this strategy, of course, is that it requires that students are able to factor various types of polynomial expressions. Given students limited exposure to any polynomials aside of linear expressions, it is not surprising that only a select number of students are able to factor when given a question outside the context of the factoring unit. So how are students who are unable to factor understanding the reduction? The second group of students reflects a common notion held by grade 10 students.

2. Cancellation occurs when a number in the numerator and the denominator have a common factor

Emphasis should be placed on “a number” and may often be understood as “a term in the numerator is divisible be the denominator”. Students who reduced expressions in this way, identified only a single term need be divided by the denominator. Of course, no error would be detected when the numerator has a single term, as was the case with Question 4C (Q4C) (fig. 2).

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This train of thought was likely established early for mathematics students, with the reduction of fractions. Student are often encouraged to reduce fractions by considering the greatest common factor in the numerator and denominator and then “cancelling” it out. This harmless strategy helps students reduce fractions but seems to cause a stumbling block when binomial expressions are introduced (fig. 3). Students who understand reductions in this way often look for a single term in the numerator that can be reduced the same way a fraction can be reduced. Expressions like  are seen as  and a simple fractional reduction is employed. These students often explained their reduction using words like “divide into”, “fraction”, and “like term”. One student in particular explained “I believe cancellation is allowed where there are equivalent values of the same multiple of the same term are in both the numerator and the denominator”. The interpretation simply put is that the cancellation is allowed if you have common factors and they belong to like terms. This explains why the student chose to divide into the 8 instead of the 6x in Q4A (fig. 3). The student sees the 6x as being an unlike term and thus indivisible by 2.

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An additional element of these students’ understanding was the close association of “elimination” with the word “cancellation”. These students identified a common factor (such as a 2 in Q4A) and the crossing out resulted in the “elimination” of the denominator. In this particular question it is difficult to determine whether the student perceived the cancellation to mean that nothing remains (eliminated) or that a one remains. However, in Q4D, a number of students stated their final expression as  (fig. 4). Neglecting “+1” suggests that the students see the cancellation as resulting in zero or “eliminating” the term.

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Students who understood reductions in this way were unable to explain their rationale for why cancellation was allowed and seldom had any method to verify their reductions. However, almost all students identified the problem with the understanding they held when it came to reducing integer expressions. Students who held this belief were still able to correctly answer Q10 and Q11. These results differed from our last group of students and the beliefs they held about reducing rational expressions.

3. Cancellation occurs when each/all of the numbers in the numerator and the denominator have a common factor

Students who view reductions in this way often described the strategy as dividing each number in the numerator by the denominator. How these students drew their lines for cancelled/reduced terms often gave valuable insight about the way they understood the division. For example, figure 5 shows the crossing out relating to both terms in the numerator and only the single term in the denominator.

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This strategy for reducing expressions appears rather harmless and has likely been taught by many teachers, including myself. The problem however exists when the numbers in the numerator represent a single term such as the case in Q4B (fig. 6). If the students’ strategy is to simply divide each denominator into each numerator, then it is likely they will divide both 12 and 6x by 2, as was the case with almost every student who held this belief (fig. 6). These students also incorrectly reduced Q4F and Q9. Even more surprising is that these students, without thinking twice, employed the strategy in Q10 instead of simplifying the numerator. While this did not cause any problems in Q10, it certainly caused issues with Q11 which was often incorrectly reduced.

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Students were often found explaining that cancelling was allowed when “the numbers on top can be divided by the number on the bottom” or that you can simply “divide everything on top by the bottom” instead of each “term” in the numerator. These students were frequently unable to recognize a multiplication as a single term, meaning that even if they factor they may still not be able to correctly reduce. For example, an expression like, , may be interpreted as   and  , perhaps even and . This belief about reducing polynomial expressions is the mostly frequently held belief amongst all grade 10 students. This shouldn’t be surprising when you consider that students are stuck trying to reduce a binomial expression without a strong foundation in factoring.

Implications of these three understanding of reduction extend beyond the grade 10 curriculum and will continually resurface as students repeat common errors year after year. At the root of these beliefs is the students’ first experience with reducing fractions. Those who employ a strategy of looking for the greatest common factor to “cancel”, instead of factoring it out, will likely develop into the same students found in group II. This, of course, highlights the importance of early factoring with younger students. This should be done both in the context of a fraction reduction and a polynomial division, even if it was simply done with integers. For example, reducing  could be seen as  and factored as  and then reduced to

Perhaps underlying these misconceptions is the problem with the term, “cancellation”. By giving it a name based on what it looks like rather that what it is causes students to frequently “eliminate” terms rather than “reduce”. While shortcuts given by teachers often offer a temporary method of simplifying expressions, they may create a list of issues when the student is called to extend their understanding in more complex mathematics.

Investigating student understandings to seemingly simple ideas, such as the one in this study, provides valuable insight for teachers to interpret student beliefs and redirect accordingly. However, even more valuable than information regarding redirection is the information provided about misdirection. That is, the way we introduce students to ideas like “cancellation” often misdirects them and prompts them to extend incorrect notions to other areas of mathematics. If nothing else, perhaps we will think twice when we tell students to simply “cancel” terms.

The Arbitrary and Necessary – Dividing the Curriculum and Addressing Real Mathematics

Every year we are subjected to the pervasive conversation that precedes provincial exams and other standardized testing. “I haven’t memorized the special triangles”, “I need to remember SOHCAHTOA”, “What’s the formula for surface area of a prism?” and other similar phrases are uttered as students seek to prepare themselves to “do math”. Inevitably many students arrive at the conclusion that math is a collection of facts, terms, algorithms to be memorized. As math teachers we are bothered that students could reduce something so obviously abundant in abstraction to a mere list of items to be memorized. Yet, we continue to propagate that perception, undeniably teaching algorithms, definitions, and mnemonic devices simply because there are some things they just need to know. We know this is not true of all things but recognize that language, notation, and symbols need to be memorized in order to participate in the discussion. At the same time we also understand that conjecturing, problem-solving, and patterning are to be developed through experiences. It appears that the curriculum is divided into what Hewitt (Hewitt, 1999) describes as the “arbitrary” and the “necessary”. These descriptions not only carry a framework for viewing the math curriculum, they also demand an awareness of teaching practices as each of the components is addressed.

The arbitrary can be thought of as the collection of labels, symbols, terms and conventions that have been agreed upon and established at some point in the past and would have the perception of being rather arbitrary to the student in the present. These are the elements of the curriculum that could have seemingly been named, drawn, or executed differently; there is no reason they must be done or named so. The arbitrary cannot be derived or worked out through one’s own awareness and, as a result, students only come to know these elements by some external source such as a teacher, textbook, or other authority (Hewitt, 1999). While the term arbitrary carries with it the connotation of being unnecessary, this is not the intention of the author. Far from it, the arbitrary is imperative if a student wishes to participate in the mathematical community and engage in the conversation. If a learner does not commit these items to memory, they are forced to invent names and conventions, isolating themselves from the community. Therefore, the teacher is left with the duty to not only introduce the arbitrary by, often, explicit means but also to assist students in the process of memorization. While the learning of the arbitrary is not mathematics it does play a significant role is how student engage with and communicate mathematics, that is, how they engage in the necessary.

The necessary curriculum is where the mathematics exists. It does not require an external source, as they are mathematical properties and relationships rather than conventions. These items can be discovered, worked out, and verified based on what students already know to be true. Students come to know the necessary through an awareness instead of memorization and thus, the teacher’s role is to provide appropriate activities that develop a student’s mathematical awareness. The implications for both student and teacher are important as it demands the student attend to different cognitive processes and that teachers attend to different practices. This processes are quite different than working with the arbitrary.

If the arbitrary requires students to memorize, then it is expected that the teacher’s role is to assist in that process. The term memorization here, is used by Hewitt to mean the conscious practice of a learner to obtain and hold on to information so that it can be accessed in the future (Hewitt, 2001a). Of course, memorizing a term, symbol, or convention apart from the properties is mathematically immaterial. Thus there exists two realms of the curriculum that must coexist, the arbitrary conventions and the necessary properties. The role of the teacher is to decide the most appropriate pedagogy that unites these realms. We must be sensitive to the order with which the student engages in these realms, as it requires a different cognition. Take, for example, a teacher who decides to introduce a term prior to the student becoming aware of the property. This is commonly delivered as a definition, such as “sine is a trigonometric function that is equal to the ratio of the opposite side of a given angle to the hypotenuse”. The result is that the student engages with both term “sine” and the property the same way, as an arbitrary element to be dealt with in the realm of memory. While the definition may be true, it scarcely reflects an understanding or awareness of the properties of sine. Other techniques of initially stressing the arbitrary include visualizing, associating and introducing mnemonic devices. However, the root of the issue remains: once students believe they have established a connection to the properties they have little need to attend to them anymore. Properties become static and students fail to question, restructure or extend the properties.

An alternative approach would be to start by stressing the properties and through time necessitate the term or symbol associated with that set of properties. As an example consider the properties of trigonometric ratios. Most students, as of grade 10, have come to know the properties of slope and often associate it as a ratio between the changing output units to the changing input units (or “rise/run”). Students can likely visualize changing slopes, noting the changing angle with the horizon as the slope increases. Can slope be determined by angle alone? Can angle be determine by the slope ratio? By engaging students in the realm of awareness they begin to draw relationships between the angle and the ratio of opposite to adjacent sides (which they may call “rise” and “run” sides of the triangle). This, of course, is the tangent ratio. However, the term tangent is irrelevant at this stage and the teacher may deliberately choose to avoid using the word until it is critical, perhaps at the point where students are required to communicate the relationship. This task lends itself nicely to considering other ratios of a triangle that include the hypotenuse side and the realization that there are only so many ratios that can develop. While this approach may act as an initial “glue” between term and properties, in carries with it no guarantees that it will be memorized. This requires practice, repetition, and reinforcement of the language, often by subordination[1].

As a third and final approach the teacher may opt to stress both the name and properties at the same time. In this approach the teacher is conscientious of having students become aware of new properties as the term is being introduced. This approach is ideal in the situation where the term is required early in the discussion or perhaps when the teacher suspects the student may already have been introduced to the term but possess a limited understanding of the properties. Take, for example, the term prism. Most students have heard the term but have limited exposure that likely only includes rectangular and triangular prisms. A teacher may opt to uncover more properties by dividing 3-dimensionsal shapes. Holding a rectangular prism and stating, “This is a prism”, then holding a triangular pyramid and stating, “This is not”. This activity could continue until the teacher is satisfied that all students have a working definition of the term, prism. This could be accomplished by asking students to determine which category a cylinder belongs to, followed by asking why they would place it in the category.

Once again, these tasks ensure that the student has been exposed to both term and properties and that the glue is placed, but they do not, however, guarantee that glue will keep them fixed. Committing something to memory requires repetition, and ongoing use. If the term is not integrated into the dialogue amongst students it moves completely to the realm of memory, where not only is the term stored but an associated property as well. I say an associated property because often there is only one property that the student associates with the word and frequently it is the property required of them in traditional examinations and thus emphasized or even explicitly stated by the teacher. Herein lies another problem for teachers who wish to devote their time to the practice of awareness, most standard test questions do not ask students to reflect the necessary. The teacher is not only persuaded to frontload the arbitrary, they also mislabel and inform the necessary, leaving the student to either accept it as truth and commit it to memory or work it out for themselves as to why it might be true. Our students rarely spend time in the realm of awareness simply because too often teachers do not identify which elements of the curriculum are the necessary.

The necessary, where mathematics is truly found, does not require students to be informed. Students come to know the necessary by being educated in awareness, informing their decisions and guiding their approach to novel situations. This is perhaps the issue that we have with teaching problem-solving. There is no algorithm or formula to be memorized for solving all math problems. It is only by developing a student’s awareness that they become “problem-solvers”. The question then is two-fold: what aspects of the curriculum belong in the realm of awareness? And how do we educate awareness?

The problem with educating awareness is that it is not something that can be explicitly taught. The good news is that every student walks in to the classroom with some level of awareness. Thus, instead of starting from scratch, the teacher may choose to work with the student’s awareness and allow it to guide their pedagogic decisions. To continue with the example of trigonometry, consider a lesson in which the teacher uses dynamic geometry software and displays the angle between an adjacent side and the hypotenuse increasing. After noting the ratio between the opposite and adjacent side is increasing as the angle increases, the teacher may ask students to estimate the ratio of the opposite and hypotenuse side when the angle is 60 degrees. One student guesses ½ and the other guesses 1. Which answer is reflective of the properties of sine? Perhaps the guess of ½ is quickly dismissed since it is so far off. Instead the teacher applauds the guess of 1, despite the student falsely drawing a relationship that all trigonometric ratios increase as angle increases. Too often teachers seem only interested whether or not students are attending to the right thing rather than what they are actually attending to. The truth is that the teacher will only know where their awareness lies if the focus is shifted from the answer to the process. One of the simplest and most revealing questions you we can ask our students is “Why?” The first step to developing awareness is to reveal awareness so that it may be worked with.

Once awareness is revealed the teacher’s response may vary but one thing remains, the necessary will only remain in the realm of awareness so long as the teacher refrains from informing the student how they ought to understand it. Students must work out their awareness. This may be accomplished by helping students see the consequences of their decisions or directing their attention to a particular item. Drawing attention to properties and relationships is different than revealing them. When drawing attention the teacher is questioning rather than stating and the student takes responsibility for the explanation, not the teacher. By simple asking, “What do you notice when the angle increases?” we are calling students to attend to invariance amongst all that varies. Here, students come to know what is necessary through their own awareness.

Mathematics could be described as an awareness of relationships and properties. It is through this awareness that students come to know what is absolute and necessary in mathematics. Although only a student can come to know these items, it is the responsibility of the teacher to inform only that which is arbitrary and as Hewitt suggests, “create a culture in the classroom where awareness of mathematics is the currency of communication” (Hewitt, 2001b).

The division Hewitt makes between the arbitrary is a valuable tool for providing teachers with a framework to address learning outcomes and develop lesson plans. The division allows teachers the freedom to lecture, practice, and repeat when addressing the arbitrary, and problem-pose, explore, and question when addressing the necessary. While the framework is an excellent resource for teachers, it does not necessarily answer all elements of the curriculum. Mathematics courses can often be about learning the tools, elements that may fit into either category or perhaps none at all. Take, for example, a Venn diagram. It is unlikely that a student will come to know how a Venn diagram works without external instruction, leading the teacher to believe it falls into the realm of memorization. However, what is there to memorize about a Venn diagram? Even the name itself is irrelevant (even amongst the items which are arbitrary). While one could argue that mathematical tools are arbitrary due to explicit instruction, it is difficult to deny the fact that the tools are derived from the properties themselves. Take for example the quadratic formula or any generalization for that matter. The teacher may introduce the tool by directing student’s awareness but it almost always evolves into an equation being memorized. In such a circumstance perhaps it is the discretion of the teacher to determine whether reconstructing a generalization or memorizing an equation is more cognitively demanding. Even when the teacher opts to have students recreate generalizations by denying access to the equation, who is to say they will not memorize the steps to construct the generalization. What realm might generalizing fall into?

The division between the arbitrary and necessary also fails to consider course content and grade level. The assumption Hewitt makes is that all grades and courses are abundant in both the arbitrary and necessary. Consider the primary years when students are introduced to a high volume of semantics and syntax. While there may be opportunities to develop student’s awareness, the majority of a teacher’s time is spent in chants, repetitive exercises, and daily practice to ensure that all that is arbitrary stays ‘glued’ with the appropriate associated properties. Is it possible that not all courses are created the same, that in fact some are heavy in memorization while others could be almost exclusively understood through student’s awareness?

Finally, I question whether simply placing your attention on something constitutes an awareness. Hewitt (Hewitt, 2001b) states “where I place my attention affects what I become aware of, and what I am already aware of affects where I place my attention”. Is it possible that students could place their attention on something and not become aware of it? It is my opinion that this could certainly be the case. In fact, our students are often guilty of it. So then, perhaps it is a matter of time invested in attending to it. Perhaps, given enough time, students could become aware. It is unclear to me whether that is a shared opinion of the author. Is it possible that students will never come to an awareness even after investing a considerable amount of time attending to it?

While these questions may indicate uncertainty or fallacy in the ideas presented by Hewitt, it was certainly not intended. Hewitt provides a useful tool for teachers to assess how they reveal their curriculum and what guides their teaching practices and interactions with their students. The framework used here exposes the ugly truth that despite mathematics being about the realm of awareness, we constantly inform students of what they need to know. The nature of mathematics is distorted and students receive the rote memorization that suppresses mathematical thought and is seldom maintained throughout adulthood. Perhaps by attending to the division of the curriculum we too could come to an awareness of what truly constitutes mathematics and present it in its unblemished form.

 

References

Hewitt, D. 1999. Arbitrary and Necessary Part 1: A Way of Viewing the Mathematics Curriculum. For the Learning of Mathematics, 19(3): 2-9

Hewitt, D. 2001a. Arbitrary and necessary: Part 2 assisting memory. In For the Learning of Mathematics 21(1), 44-51.

Hewitt, D. 2001b. Arbitrary and necessary: Part 3 educating awareness. In For the Learning of Mathematics 21(2), 37-49.

[1] Subordination, as Hewitt describes, is “designing the rules of a task so that students are ‘forced’ into having to go through something (call it A) in order to work on the main task (call it B)” (Hewitt, 2001a)

Making Trigonometry More Than SOHCAHTOA

Every year that I have taught high school mathematics I have had the opportunity to introduce students to trigonometry. This is an opportunity show that trigonometry is not only a tool to solve triangles but also a different way of describing the slope of a line. Despite my best efforts, students seem to complete the course and carry forth a limited understanding about trigonometric ratios. If asked what they understand about trigonometry they consistently answer, “SOHCAHTOA” and when prompted to explain what that means they reply, “Sine is opposite over adjacent”. No doubt this is a description of the sine ratio at a very basic level but in no means does it touch at the very heart of its purpose. Rarely do you hear a student describe sine as a relationship between an angle and the ratio of its opposite side to the adjacent side of any right triangle. While the argument could be made that they knew it but couldn’t explain it, this hardly seems likely given my follow up with students following the unit. Some of the following questions have been posed to students previously and were intended to get at their conceptual understanding of trigonometric ratios.

Could there be more than three ratios? Explain your reasoning

Some of the more comedic responses include “because there is only three buttons on our calculator” or “yes, there are more than three ratios, sin­­­­-1 , cos-1, and tan-1”. However, the most common response was there was only three ratios because there are only three sides or perhaps that there were only three angles. These are true statements in a roundabout way but they miss the idea that three sides can only be paired in three ways. If they saw the ratios in this way they might have some insight as to why there can actually be six ratios if you were to flip all three ratios upside down. Students might even argue why the sine ratio needs be opposite/hypotenuse instead of hypotenuse/opposite. This may lead to the discovery of the graph of each of these functions.

Does the sine ratio have a maximum value? If so, what is it? Explain your reasoning

Responses to this question varied in the past. Often students responded incorrectly that the sine ratio maxed out at 90 degrees because you can’t have a triangle with two 90 degree angles. Part of this, of course, is true but students misunderstood the nature of the increasing sine ratio and often mistook it as a positive linear correlation. They assumed that the ratio increases as the angle increases. This becomes more evident in the next question.

If the cosine ratio of 60 degrees is 0.5, estimate the cosine ratio of 30 degrees? Describe how you arrived at your answer (this is done without the use of a calculator)

Not surprisingly most students tend to answer that the cosine ratio of 30 degrees would be 0.25. The same reason that students misunderstood maximum values for sine ratio resurfaces. Once again the assumption is that trigonometric ratios and angles form some sort of linear relation. This is potentially hazardous as student begin to interpret the periodic motion of the sine and cosine functions.

Estimate the angle that would have a tangent ratio of 2? Describe how you arrived at your answer.

The response to this question varied drastically, some relatively accurate, others dipped below 45 degrees. The most frequent inaccurate response was 2 degrees, suggesting that either the student had no tools to even venture a guess or they truly believe that ratio and angle were equivalent.

The misunderstandings extend beyond the evidence that I have accumulated in post-unit surveys. It was not uncommon to see solutions to trigonometric statements that make you question whether students really understand anything about the concepts involved in trigonometry. The Sin 60 = x/6 was often answered by stating the value of x in degrees (as opposed to a length unit).

This has bothered me for years but I often dismissed it as the student’s fault. Year after year I taught trigonometry the same way and year after year their grasp of the concepts was dismal. “Students are slow”, I excused. Now I wonder who the slow one is. The approach to the unit was a definition-based approach, filled with equations and algorithms, and the students were able to do exactly that. They could solve a triangle and do the algebra. The following is an example of a common approach I took to introducing trigonometry.

SOHCAHTOA1

Now, this is not to say it is a terrible approach. It necessitates the use of trigonometry and would maybe accompany a real life example really well. However, this introduction was often followed by stating one of the three ratios. Right away the students experience trigonometry from a formula-based approach. It also begins with a foreign term like “cosine” and quickly defines it as opposite/adjacent. To the students cosine is not a ratio, it is a formula for solving triangles. The first lesson consisted of getting them to memorize that tangent was opposite/adjacent, cosine was adjacent/hypotenuse, and sine was opposite/hypotenuse. I would then hand them a worksheet where they spent 30 minutes labelling triangles with the letters O, A, H and theta (another strange adjustment in their understanding of variables that I rarely explained). Historically the second lesson of the trigonometry unit really excited me. It was the lesson where I would get to tell them what that button on their calculator labelled “sin” did. “If you hit ‘sin’ followed by any angle you want, it will tell you the opposite/adjacent”, I would explain. In my attempt to teach trigonometry I deprived students the opportunity to experience the world of mathematics. In the book, “Developing Understanding of Geometry” (Sinclair & Pimm, 2012)[1], the authors identify four big ideas that should drive our approach to geometry. Using these big ideas, I will suggest the problems that emerge in the lesson described above and introduce a new approach that I have recently experimented with and the results of that approach.

Big Idea 1: Working with diagrams is central to geometric thinking.

While students certainly worked on questions that included diagrams, they rarely had to produce one themselves. They were not required to describe triangles or ratios and thus, the language was only necessary because they needed to know which number to plug into the formula.

Big Idea 2: Geometry is about working with variance and invariance, despite appearing to be about theorems.

What I missed in my lessons is the opportunity to see the degree of invariance involved in right angle trigonometry. When a triangle gets bigger, the sides change. When the angle changes, the sides change. There is a lot of change occurring in the triangle. What is special are the things that do not change, the invariance. However, instead of allowing my students to experience variance and search for invariance, I told them what the invariance was. It wasn’t even special to them. This is the danger of a lecture-based approach to the curriculum, students do not experience the treasure found in invariance. To the student everything is invariance. Students learn the invariance, not by discovery but by memorization. It is not surprising that students cannot remember a simple thing like factoring or area of a triangle, they never construct it and therefore rebuilding those ideas becomes impossible. The best they can hope for is a mnemonic device to trigger their memory (examples, “FOIL”, “SOHCAHTOA”, “adds to b, multiplies to c”).

Big Idea 3: Working with and on definitions is central to geometry

Students in the lessons above were never required to work with a definition of sine. They utilized sine ratio, but they didn’t work with or on it. Compare it to being given a hammer, told that it is for driving nails and then asked to drive nails for a few hours. Chances are that you would get really good at driving nails but you miss out on all the other things that hammer could do. This is how students use trigonometric ratios. Show them how it works and then let them do as the teacher does for a few hours. For years I started with a definition, often from a textbook, and left no room for students to contribute to the building of the definition. We were working from a definition, not towards it.

Big Idea 4: A written proof is the endpoint of the process of proving.

This is perhaps one of the most missed opportunities teachers make in the classroom. You would never exempt an English student from writing a conclusion to their essay or a chemistry student from writing their experimental conclusion but rarely will a math student get the opportunity to search, discover, conjecture, test, and then formalize/generalize their conjecture as a written proof. This was rarely occurring in my classroom let alone in my trigonometry lesson. Students believed that the ratios worked for all triangles because I told them that they work in all triangles, end of story, no evidence needed.

So, after teaching trigonometry for years I realized that I was not effectively getting students to grasp the concept of ratio and despite repetitive practice with algorithms they seem to not understand what they were calculating. I had a sense that if any of my unit plans needed to change it would be this one. Thus, I tossed away my lessons and started from scratch.

One new approach I had to teaching trigonometry was piggy-backing it off of a topic that I had felt was rather concrete. The month previous we had dedicated our time to linear functions. More specifically we looked at the arithmetic nature of linear functions and the meaning of slope. We spent days shifting, describing, drawing, and calculating slope. My students experienced slope. If I were to give them a particular ratio, most could picture it in their mind and sketch it out with relative precision. I felt this was an opportune time to begin asking them if they could do the same with angle. And so, I gave them an angle and asked them to visualize it and then draw it as accurately as possible. I realized that they probably had an even greater ability to visualize angles than they did slopes and thus, the first trigonometric ratio, tangent, had already been introduced and understood. I still wanted them to play around with the relationship between angle and slope ratio so I introduced it by diagram. My initial plan was to have them create the slope and angle but without Geometer’s Sketchpad I realized this would be impossible. My options were to have them graph it on paper or have them play with it on Sketchpad Explorer. The decision was really between the benefits of construction versus the benefits of dynamic geometry and variance. Emphasizing Big Idea 2, I chose the latter and constructed this sketch for them to use in Sketchpad Explorer.

SOHCAHTOA2

There were three things I hoped to accomplish by introducing students to this exploration. One, I hoped that students would recognize that steepness could be described in two ways. This related to Big Idea 3 and more specifically addresses Essential Understanding 3a, “Geometric objects can have different definitions. Some are better than others, and their worth depends both on context and values”. Here steepness was could be described in two ways: slope described the steepness relative to the horizontal axis and angle described it relative to any other line. Both valuable concepts and both worthy of discussion in various contexts. Two, I hoped to show them that we have been working with triangles all along. Rise and run create the legs of a right triangle and it does not matter where you place those legs or how big they are, the slope remains constant. In other words, I wanted them to see the invariance amongst the variance. The final thing I wanted to develop was the invariance between angle and slope; no matter where you placed B, slope was unique to that angle. This invariance allows us to use slope as a tool to derive angle and make predictions, as was the case in question 7. Students began to make connections that I had hoped. I overheard students use phrases like, “so 35.15o = 0.70”. I knew once that other ratios were introduced students would need to correct this logic and notation.

The second part of my lesson was targeted to address Big Idea 2 and 3 and to necessitate a change of language. Below is the sketch that students worked through in small groups.

SOHCAHTOA3

The conversation that ensued was encouraging. While there were students who needed a lot of direction on where to look, many dismissed the “slope” measurement that GSP calculated and instead focused on “RISE/RUN”, noting that it stayed the same until the angle changed. Student’s also began to argue over the term “rise” and “run”, suggesting that it was no longer a “rise” or a “run”. While there was no consensus on what the names should be, one student did offer the terms “opposite” and “beside”. I thought this was a good start and did not correct it until I felt the correct language was necessary.

This, of course, gives rise to the third side of the triangle that some students had already began to address. While there were a few students who already knew the name of the third side, none of them had thought about using it in a ratio like we did with opposite and adjacent. For that purpose I created this lesson.

SOHCAHTOA4

Once again I wanted to draw students’ attention to the value of the ratio and how that could help determine the approximate angle. Questions 4 and 5 readdress the predictable nature of the ratio and also helps students narrow their scope to look at one side at time. If the instructions were followed correctly students should be able to determine that when you keep adjacent constant and increase opposite and hypotenuse, the angle increases. Perhaps more surprising for students is that when you keep opposite constant and increase adjacent and hypotenuse, the angle actually decreases. This is something students rarely have the opportunity to see, which is why most students couldn’t tell you what why the cosine ratio increases as the angle decreases. This sketch was followed by more instructions that would explicitly have them determine the ratios.

SOHCAHTOA5

The use of Geometer’s Sketchpad throughout these activities gave students access to the nature of the ratios and allowed for students to spend more time getting a feel for how the ratios are associated with the angles. These tasks were followed by a bit more terminology as students felt it necessary to give names to these ratios so that they could remember what each one described. However, before I shared “cosine” and “sine” I had students explain what the ratios told them about the triangle. Students commented on the idea that any ratio could be used if you knew 2 of the sides and employed the Pythagoras theorem. In a way, they suggested, all of the ratios are part of a family. They also commented that the opposite/hypotenuse ratio went from 0 to 1 and depending on the value, you could estimate the angle. Meanwhile, adjacent/hypotenuse ranged from 1 to 0 and could also be used to determine the angle. I suggested that perhaps it would be easier to use a table that told us all these values instead of trying to memorize them. And that is precisely what we did. For 3 days students used nothing but trigonometry ratio tables to determine angle and a calculator was later introduced as a device that could store these tables for us.

As we come to the end of our unit, I have seen some students slip into the routine of “SOHCAHTOA” but there has been a number of difference in the students’ approach to solving trigonometric problems. One difference I observed was that students understood the range of values for each of the ratios. They understand why cosine and sine have a maximum but tangent does not. The second difference was that students began looking for patterns. They manipulate the triangles and look for clues that will help them determine future scenarios (I even saw a student start his own table of ratios). Thirdly, trigonometry became more geometrical and less algebraic. I was not interested in whether students could solve a ‘fill-in-the-blank’ algorithm. My questions were geared to estimation and patterning. Finally, students seem to enjoy a personalized math experience. Trigonometry became something different than the rest of the curriculum. They controlled the triangle, they controlled the values, and they controlled the experience. While there were adaptations I would make, this new approach to trigonometry contributes great value to not only the students’ understanding of trigonometric ratios but more importantly an understanding about the nature of mathematics.

[1] Sinclair, N., & Pimm, D. (2012). Developing essential understanding of geometry for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics.

Avoiding Passivity in a Flipped Lesson

A Flipped Exploration: Using Technology for Student Exploration outside the Classroom

After spending some time observing, discussing and experimenting with flipped classrooms I have become increasingly more concerned about how math teachers are using it. Despite some perceived benefits there were some obvious drawbacks as a teacher, one of which is missing the opportunity to lecture. I can’t imagine there are many teachers out there that despise, from a personal level, lecturing. It is an art and gives teachers an opportunity to inspire students and, despite the fact that I rarely lecture due to its ineffectiveness when compared to alternative strategies, I still enjoy a good old fashioned lecture. However, not being able to lecture isn’t my biggest concern in regards to flipping a classroom.
Having used both a lecture-based approach and a flipped classroom and I can assure you there are effective and ineffective ways to use it. One of the rising problems with using a flipped classroom in a math class is that teachers are simply replacing their lecture and notes. Now you might be saying, “Yeah Woelders, that’s the point” but much of the ineffectiveness of math teachers and negative attitudes from students arises from creating this type of traditional environment. The type of environment that has a student sit in a class and watch someone else solve the problem and then goes home and spends an hour doing worksheet or textbook questions. Now our modern-traditionalists are using a flipped classroom to have the student go home, watch someone solve the problem and then come back to school and spend an hour on a worksheet. By using either approach we fail to minimize the impact on student learning and worse, we maximize the impact on student’s negative attitudes and perceptions about mathematics. To quote Paul Lockhart (2009), “Routinization, memorization, and soul-crushing repetition are not the goals of math education.”
In a subject where “doing” is fundamental, lectures and flipped lessons can destroy a student’s opportunity to engage in the true nature of the subject. Don’t get me wrong, there is content knowledge that needs to be communicated and perhaps that could be the role of flipping but the experiences that you present them in the classroom will have the greatest impact on their learning. The arguments against a lecture-based or flipped classroom in a math class are the same arguments I would present against having students take notes in a math class.

The Problem with Traditionalist Flipping
1. It does not require students to think. Why would they think when a teacher has so nicely organized, defined and synthesized all the information for them?
2. Two, it makes the teacher the source of information and the students are recipients of knowledge instead of creators of knowledge. In most classrooms this is how the teacher establishes a clear hierarchy; however there are some disastrous effects in creating an atmosphere where students expect to receive information. Not only will students cease to explore the material, they will also stop asking questions because it is assumed that the answer to all of their questions is found somewhere on the board or in the video.
3. It does not promote dialogue. This is similar to the previous point but we can add that when students become non-participants in their learning, there is little feedback for the teacher when assessing the learning process.

Again, I’m not saying it should not be done in a math class but it is essential that the teacher make effective use of his time with the students in class. This means addressing of those problems listed above and ensuring that students have the opportunity to inquire, explore and construct their understanding. My concern is that math teachers are using flipped classrooms as a modern façade on a traditional approach. It is apprenticeship through observation and by employing a traditional approach in our math classes, our rapidly paced, closed, procedural lessons develop shallow, procedural knowledge but it’s worse than that; it shapes the identity of the learner and gives a particular perception about mathematics and their learning. It reflects to students that mathematics is all about remembering rules and procedures and thus they develop a reliance on their memory to solve problems. They scour problems for cues, in a similar way their textbooks do, rather than interpret the situation mathematically. This impact extends outside the school walls and into the students real life application of math. Because the real world does not cue thinking, students abandon school mathematics and create their own strategies, the same strategies that could be employed in the classroom.
I am not opposed to flipping. In fact, I spent two days at the CanFlip Conference in Kelowna in June. However, my desire is to see them used more dynamically. I don’t want students to become passive learners, I want them to interact and play with math as they explore the concepts. This Piagetian learning (learning without being taught) is the reason that I have begun exploring various ways we can make a flipped classroom come to life with the use of some highly-effective technological tools that are accessible by all students and teachers.
In particular, I introduced students to two popular programs in math education, Desmos and Geometer’s Sketchpad. The purpose of this “flipped exploration” is that students would familiarize themselves with the general function graphical transformations and translations (ex. the effect of a,b,c, and d in af(b(x-c)) + d) and this would serve as the foundation into the conjectures they would make for sine and cosine functions. The subjects were 16 grade 11 students enrolled in the International Baccalaureate standard level math course, nearing the end of their first year. At this stage in the course, they had been exposed to composite functions, along with the graphical transformations of exponential, log, quadratic and reciprocal functions. Students had also been made aware of a basic f(x) = sin(x) and cos(x) graph. In order to make this a true flipped exploration, students were simply given a link, posted to the class website (Edmodo), and asked to follow the prompts given in the task.
After clicking the link, students were lead to a page created on Desmos (Fig. 1) and asked to create their own function. Due to their familiarity it was expected that students would opt to use a linear, quadratic or exponential function. Once the function was created, students were asked to play with the sliders and determine their impact on the graph. As a means of feedback, I had set up an assignment to submit on Edmodo and had students write a brief description of each slider. As a way to connect the ideas to the current topic of study, the students were asked to predict and the summarize the sliders’ effect on f(x) = asin(b(x-c)) + d.

fig1Figure 1 – Direct link to the assignment constructed in Desmos that students would be led to.

Upon returning to class I had the students attempt a short quiz (Fig. 2) in which they were given the graphs of two functions and asked to estimate the approximate values of a, b, c and d in the transformed graph.
My hopes were lofty; I hoped that, best-case scenario, students would be able to communicate how the values of a,b,c and d alter the function and that they would build a new relationship with transformations that would be applicable to all functions instead of seeing them as individual function transformations. Worst-case scenario, I hoped that they would at least familiarize themselves with Desmos and view it as a useful tool in searching for patterns and analyzing graphs.

fig2Figure 2 – Graphic attached to question posed in class.

Despite ambitious expectations I realized that, like any new task, there would be constraints on what the students would be able to accomplish. , I expected that students would either forget or not bother to spend time doing the task and that this would affect their understanding and participation the following day. Secondly, the nature of the task was novel to students and not presented in the typical format for this class. As Ruthven (2008) suggests, “crafting of lessons around familiar activity formats and their supporting classroom routines helps make them flow smoothly in a focused, predictable, and fluid way”. This certainly was not a task that had been made a part of routine. Thus, I expected that some students would have trouble navigating themselves between two different websites (Desmos and Edmodo) and would likely not bother to complete the reflection piece that would provide valuable information with what the students were able to derive and where their misunderstanding occurred.
As predicted, these problems occurred and the night was spent troubleshooting as students encountered various problems along the way. Often these problems occurred because students would either delete or alter the original instructions, the “f(x)”, or shift the graph in such a way that the function would not be visible. Students were encouraged to simply re-launch the link if any problems occurred and this seemed to extinguish most of the fires.
Upon analyzing student submissions via Edmodo, it was not surprising to see a diversity of responses. Amongst the most interesting feedback are the responses from students recorded below.

How does the value of ‘a’ affect your graph?

Student 1:“It flips the graph upside down”
Student 2: “It makes the graph skinnier”
How does the value of ‘b’ affect your graph?
Student 3: “It makes it go backwards”
Student 4:“It makes the graph wider or skinnier”
Student 5:“a and b do the same things…”
Student 6: “when a and b are zero, the graph is flat”
How does the value of ‘c’ affect your graph?
Student 7: “The graph moves whatever way c moves”
Student 8: “moves it right or left”

How does the value of ‘d’ affect your graph?

Student 9: “d moves the graphing up and down”

It was upon this analysis when I realized many of the gaps in students understanding. For example, it seemed evident although students recognized that “a” and “b” caused the graph to get wider or thinner, they were unable to identify the difference between a vertical and horizontal expansion. This was reflected by the student who expressed “a and b do the same things”. This was certainly not something to be concerned about since the students also failed to make any connection between the “stretch factor” and the value of “a” and “b”. In other words, they could identify that the graph was thinner when “a” was 10 but they could not tell you how thin it would become. Perhaps the most valuable piece of information they derived from controlling “a” and “b” was that “it flips the graph upside down” and “when a and b are zero, the graph is flat”. Of course, predicting a graph would be flat when “b” is zero is rather dependent of the type of function and was likely true for a student who only experimented with the functions suggested (although they wouldn’t have seen anything for the log and reciprocal function).
In determining their understanding of the effect of “c” and “d”, it appeared that most students understood the relationship to the vertical and horizontal shift. However, it became apparent that students were slightly confused that a positive value following x would result in a shift to the left. In other words, many students would predict that a possible equation for a quadratic graph shifted to the right by “c” units would be written as “f(x) = (x+c)2”. The effect of “d” seemed well understood following the task.
Students were also asked to extend their understanding to sine and cosine functions. At this point in the course students already understood terms such as “amplitude”, “period” and “shift” and so the following questions were intended to get some feedback as whether the students had connected the ideas that they learned in the task and extend it to trigonometric functions.

How is amplitude, period and shift of a sine function affected by the values of a,b,c and d?

Student 1: “increasing ‘a’ will increase the amplitude”
Student 2: “making ‘a’ negative will make the amplitude negative”
Student 3: “The amplitude will be the same as ‘a’”
Student 4: “increasing b makes the period shorter”
Student 5: “b controls how many periods you see. When b is 10, there are 10 periods on one side of the y-axis
Student 6:“when b is zero”
Student 7: “c and d shift it side to side and up and down”

Once again there was a strong indication that students understood the relationship between “a” and the amplitude, and “b” and the period (ex. “increasing ‘a’ will increase the amplitude” and “increasing b makes the period shorter”). However, there was still a gap between the actual values of “a” and “b” and the scale factors of the graph. It would appear that students were reluctant to go much further than a general description of the transformation. The concern, of course, is that the students may be forming strong concepts based on conjectures that have not been tested. This appears to be the case for student 5who believed that the value of “b” will determine how many periods are right of the y-axis. This may, in fact, be accurate with a particular window but this student had failed to adjust the window to determine the validity of the conjecture. It seemed evident that students had built some misconceptions which would need to be worked out in class. More than that, it was clear that students were not being pushed to analyse the exact effect of each value and to look for patterns in the numbers and measurements.

While this task gave students control over the values of a,b,c, and d and could observe the effect on the graph, I also wanted students to have control over the graph and see the effect on a,b,c and d. This, to me, was fundamental in the use of technology in the classroom and is what separates the flipped lecture from the flipped exploration. By giving students the opportunity to manipulate both the graph and the function, we allow student to program the computer. In a video lecture the opposite is occurring, the computer is programming the student. To this end I created a task on Geometer’s Sketchpad (fig. 3). In this task students were asked to stretch out the amplitude and the period note the changes to the equation. Although this task was given in class, students had very few instructions aside of the ones given in the task itself. The results this time were much more encouraging with all students identifying exactly how “a” and amplitude were related. After encouraging students to set their period at clear values and then record the value of “b” a few groups were able to derive the numerical relationship between “b” and the period.

fig3

Figure 3 – Window that students viewed after opening the assigned sketch on Geometer’s Sketchpad.

While these exercises were useful as a way to explore and experiment with various tools as a teacher, it is questionable whether they had as much benefit to the students. It is arguable that the initial task of manipulating the values of a,b,c, and d to see their effect on the graph was simply replacing what student could do with pen and paper or perhaps a simpler technology tool, their graphing calculator. In this sense, some might see the task falling into the category of “substitution” on the SAMR model developed by Dr. Ruben Puentedura. This, arguably, is true. However, Goldenberg (2000) challenges teachers to evaluate some criteria when considered the use of technology for a particular task. In reference to these criteria, he suggests “a well-designed lesson has a central idea and focuses students’ attention on it, without distraction by extraneous ideas or procedural details”. One of these procedural details is having student use a t-table and computing various values in order to plot the graph. This lesson is not about evaluating inputs and outputs, thus Desmos reduces the investment made on computation and focuses the students to the actual task of describing changes to the graph. It is true that this task could be replicated without Desmos but the program allows students to interact with it quicker and easier. With this in mind, it seems evident that Desmos “augments” (according to SAMR) the task. Desmos also helps students make conjectures about the relations, using the sliders (Laborde, 2001). This tool is what Pea (1985) considers a “visual amplifier” when observing properties of the graph.
The merit of the use of Desmos for this activity does not boil down to the question as to whether or not the task had a positive impact on student learning, it is a question as to whether it had as much impact as another task would have. The order of Desmos and then Geometer’s Sketchpad was decided in response to student feedback and so Sketchpad was used to compliment the task. However, upon reflection I question whether the order should be reversed in the future. By reversing the order, students would be able to manipulate the graph and construct conjectures that could then be used to and tested on Desmos. This, in addition to directed questions, would steer students down a foreseeable path and perhaps focus their attention a bit more on the stretch factors and “a” and “b”, which appeared to be major stumbling blocks for students. This foreseeable path may also help eliminate one of the biggest constraints I had failed to recognize about doing this activity as a “flipped” lesson: that I could not predict where students would take this task. I could not see what students were seeing and I could not direction them down appropriate tasks.

While this flipped class/technology experiment may have failed on many levels, it is in those failures that any useful information is derived. What surfaces are the criteria in determining what lessons could be accomplished independently and which tasks should be done in the context of the classroom. Prior to this activity it was assumed that a flipped lesson is simply a video lecture or a set of notes to read. This is certainly a common approach teachers take but likely only because it is one way to guarantee that your students receive exactly what you intend on them receiving. However, for teachers who seek to help students develop conjectures, create context for classroom discussion, and are who open to a diversity of observations, a flipped exploration may be an appropriate tool providing that the teacher is anticipating misconceptions that arise. Tasks given in this way lend themselves to readdressing the technology as a way to test conjectures and develop proofs.

Perhaps one of the most important aspects in this whole task is that the culture of student learning. By assigning tasks that students are asked to look for patterns we are creating an environment of active learners. More than that we are allowing students the opportunity to play with programming tools, and thus play the role of “programmer”. Similarly to Papert’s and his reflections with the use of Logo, I would argue that students are acquiring a new image of themselves as mathematicians and as programmers, a role previously viewed as complex and reserved for the intellectually gifted. This not only increases self-confidence but also gives students control over the math. As Papert (1980) asserts, “The child, even at preschool ages, is in control: The child programs the computer. And in teaching the computer how to think, children embark on an exploration about how they themselves think. The experience can be heady: Thinking about thinking turns the child into an epistemologist, an experience not even shared by most adults.” The classroom cultural effect and development of students as learners is valuable enough for teachers to consider integrating explorations in their flipped classrooms.

Sources Cited
Goldenberg, Paul. (2000), Thinking (And Talking) About Technology in Math Classrooms.The K-12 Mathematics
Curriculum Center: Issues in Mathematics Education. Education Development Center, Inc.
Laborde, C. (2001). Integration of Technology in the Design of Geometry Tasks with Cabri-Geometry. International Journal of Computers for Mathematical Learning, 6, 283-317.
Lockhart, P. (2009). A mathematician’s lament. New York, NY: Bellevue Literary Press.
Papert, S. (1980). Mindstorms. Brighton: Harvester Press.
Pea, R. D. (1985). Integrating human and computer intelligence. New Directions for Child and Adolescent Development, 1985(28), 75-96.
Ruthven, K., Hennessy, S., & Deaney, R. (2008). Constructions of dynamic geometry: A study of the interpretative flexibility of educational software in classroom practice. Computers & Education, 51(1), 297-317.

The Nature of Mathematics: Challenging Our Axioms

For those who have had the privilege of running the International Baccalaureate program at your school, you probably encountered the Theory of Knowledge course. In fact, you have probably been told that your physics course, just like the history course, should have aspects of the theory of knowledge in that field of study. Sadly mathematics is the one course that rarely gets explored. We discuss factoring, derivations, trigonometric functions but rarely do we discuss the true nature of mathematics. Perhaps it is because we haven’t thought enough about the subject itself, perhaps we believe math is just a collection of facts, rules and algorithms to memorize.  How are those rules established in the first place though? The interesting thing about math is that it needs to start with a claim that is generally accepted as being true, an axiom if you will. Students have plenty of axioms, they just don’t know they have them. They don’t realize (nor do we) that many of the “facts” taught to them in school were only possibly if they believed something prior to that. We build off of axioms on a regular basis but every once in a while we are called to question them. Picture the first time you were told that for every counting number you were aware of there was also another negative number. Certainly you would have questioned the reason for such a thing and perhaps even denied its existence until someone proved to you that it has the same qualities as the rest of the numbers. Suddenly you felt better about it, knowing that you don’t have to throw away some foundation axioms that you have built your math on.  What happens when someone presents something that you believe is false but they are able to prove it to you? You have two options: 1) You accept their proof and their claim. 2) You throw away their proof by claiming the axiom in which the argument is built is false. There is danger in the later. If you claim there axiom is false, you can no longer build your math upon it, for it would be absurd to claim something is false in one scenario and true in another. The scenario should not define its veracity.

I had the opportunity to lead students through an axiomatic conundrum that started like this:

From the list of numbers below, how many different values is there?

½, 0.5, 1/3, 0.333…, 0.99…., 1.

Most students who look at this list would claim that there are 4 different values: 0.5, 0.33…,0.99…, and 1. My claim to them is that there are only 3 different values, that 0.9 repeating is the same value as 1. Students vehemently disagree with this and demand some sort of proof for my claim. This in itself lends to the idea that students have established axioms about math and in this case the axiom is that for every sequence of decimal numbers there must exist only one numerical value. This is in one-to-one correspondence.

Proof 1:

1/9 = 0.1 repeating, 2/9=0.2 repeating, 3/9=0.3 repeating….8/9=0.8 repeating, therefore 9/9 = 0.9 repeating. QED?

I then ask “How many of you believe me now?” Inevitably I have convinced one student. For the rest of the students they reject my proof because they are willing to throw away an axiom that I have just created, that is that any n integer divided by 9 will create an infinite repeating n. Why are they so quick to dismiss my proof? Because there is nothing built on that axiom (most of them just realized that 1/9=0.1 repeating).

Proof 2:

Sum of an infinite series = The first term/(1-ratio). Since 0.99999… can be written as 0.9+0.09+0.009+0.0009…we can claim that it is a geometric series, where the first term is 0.9 and the ratio is 0.1. Plugging this into my equation…

Sum = 0.9/(1-0.1) = 0.9/0.9 = 1   but we also said it equals 0.99 repeat. Therefore, 0.9 repeat = 1.

Some students reject this one even faster than the first one, especially if they really didn’t understand the unit of series and sequences. In their minds the proof falls apart once I introduced this fictitious equation that I invented. They have no problem throwing away the axiom of the equation and once they have done that my proof falls apart. Once again the axiom does not carry much weight because of the math that is built upon it.

Proof 3:

Let n = 0.9 repeat, then 10n = 9.9 repeat. Subtracting the two equations we get 9n = 9 and n=1. However, n also equals 0.9 repeat. Therefore, 0.9repeat = 1

This is where some students get converted. They are sceptical but they find it hard to argue with. Everything they know about algebra tells them that this is allowed. You can subtract one equation from another because they know add or subtracting to both sides of the equation maintains equality. The isolation of n also is based on a few additional axioms that they have used for the last 4-5 years. They cannot dispute this proof because it would require them to throw away multiple algebraic axioms and at this stage in their mathematical experience there is a lot built on them. Yet still, some are willing to throw away some of their basic, foundational understandings of algebra and reject the proof.

Proof 4:

If 0.1 is between 0 and 1, and 0.01 is between 0 and 0.1, and 0.001 is between 0 and 0.01, then we can say that there will always exist a value between any two unique values. We can say more. We can say that there exists an infinite amount of values between any two unique values. From this supposable infinite set of numbers, give just one value that exists between 0.9 repeat and 1.

Here is where they are stuck. The proof is built off of one of the most foundational axioms they have, that is, there always exists a value between any two different values. The reason there are no values between 0.5 and ½ is that they are the same value so by extension the only reason you cannot name a value between 0.9 repeating and 1 is that they are the same value.

This is the nature of mathematics. Proofs built upon axioms that are foundational to our most basic understanding of mathematics. Students rarely get the opportunity to question mathematical claims, nor the opportunity to develop a proof.  This activity highlights both of these and introduces the elegance and beauty that can be found in a proof through the reflection of mathematical creativity, observable simplicity and intellectual intricacy.  

The Notes Debate: Note-taking vs. Note-making

There is no shortage of research out there on the value of notes. Some of this research dates back to the 1920’s when C.C. Crawford questioned whether notes actually improve student test performance. Since then there has been hundreds of studies on various aspects of student note-taking.  Bligh (2000) found that students were better able to recall lecture material if they took notes, while Kiewra et al.(1991) observed that there was a positive correlation between the extent of note-taking and the score on delayed assessment. Further studies (Carterm 1975) also suggest that the process of reviewing notes appears to have an even greater impact than the act of writing the notes in the first place. With all this overwhelming evidence about the positive impact on note-taking, why is there still a debate about the use of it in classroom?

What many studies fail to differentiate is the type of notes that the students are writing. Teachers and students alike seem to miss the difference between making notes and taking notes. These are very different processes. Making notes suggests that the student is cognitively engaged with the material as they wrestle for appropriate language in an attempt to synthesize the information presented to them. In this sense the benefits come not from the act of writing but through the act of communicating and as students organize their knowledge, they improve their recall and identify gaps in their own thinking. Note-making requires critical thinking, thus, facilitates retention and applications of the information.

Note-taking, on the other hand, looks very different. Note-taking is when a student copies directly from the board or verbatim from the teacher. This process encourages very little cognitive function and is usually more physically demanding than it is mentally. There is growing research that indicates the difference between these forms of written notes. Bretzing and Kulhary (1979) compared students that copied notes (note-taking) with students that created their own set of notes (note-making) and found that students who took verbatim notes scored lower on comprehension tests than those who were cognitively engaged while they made notes. Einstein et al. (1985) found that successful college students engaged in deeper levels of processing during note-making, and that note-making itself “enhances organizational processing of lecture information.” (p. 522).

What does this mean for teachers in the classroom? Well, it means that copying notes from the board is not getting students cognitively engaged and while the notes may play a small role in developing student understanding, it is minimal and only impacts those students who would otherwise be passively engaged with something else (perhaps doodling). Our goal, of course, is to spend our time in pedagogical practices that produce the biggest impact in student learning. This is just one of the reasons amongst many that I have highlight as to why note-taking should be eliminated from our teaching repertoire. While I may only be able to speak most accurately about the subject of math, I imagine this rationale can be extended cross-curricular.

  1. Copying notes does not require students to think. Why would they think when a teacher has so nicely organized, defined and synthesized all the information for them?
  2. Copying notes makes the teacher the source of information and the students are recipients of knowledge instead of creators of knowledge. In most classrooms this is how the teacher establishes a clear hierarchy; however there are some disastrous effects in creating an atmosphere where students expect to receive information. Not only will students cease to explore the material, they will also stop asking questions because it is assumed that the answer to all of their questions is found somewhere on the board.
  3. Copying notes does not promote dialogue. This is similar to the previous point but we can add that when students become non-participants in their learning, there is little feedback for the teacher when assessing whether the learning outcomes have been met.
  4. Copying notes discourages a collaborative community. This could also be said of “making notes”. Remember, the key ingredient in higher-level processing is that the students organize their thoughts in order to communicate them. This same process can occur when students are in groups and asked to communicate their ideas to one another.
  5. Copying notes is a huge investment of time.