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)

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.

Figure 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.

Figure 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.

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.

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. 

In his book, Visible Learning for Teachers, John Hattie explores some of the issues affecting the maximization of student learning in the classroom. Over thousands of research articles and millions of students Hattie derives his “d” value to assess the effect that various influences have on student achievement (ex. The effect assigning homework may have on student achievement). Perhaps contrary to popular belief, he never suggests that a teacher’s effort cause a negative effect on student achievement. In fact, what is the most striking about his results is that 95% of the influences examined have a positive effect on student achievement. As he explains it, “All that is needed to enhance achievement is a pulse”. No wonder we are so convinced that latest education trend is revolutionizing student impact. We spend countless hours digging into bold well-intended initiatives entangled in a web of education jargon. As dubious as the research is, there is no doubt that the latest education trend will have some positive impact on achievement but then again, so does giving them a healthy snack in the morning. So why are we convinced that are everything we are doing is working?

The truth is that mostly everything we do in the classroom works. Where we are missing the mark is focusing on the methods that show minimal impact instead of investing in highly effective and consistent learning influence. In many cases we are more focused on teaching than we are on learning and when we do redirect our attention to learning, we don’t know really know what to look for. Over the years “learning” has taken on a unique façade to the extent that we frequently confuse learning with “proxies” for learning, the things that are quite visible to an observer but do not necessarily reflect anything learned. Picture walking into a classroom where the students are enthusiastically working their way through a worksheet as the teacher paces up and down the aisles. Students appear to be engaged and upon inspection look to be answering the questions correctly. The classroom is quiet, calm and under control. Certainly these students are learning. Can you be certain?

These proxies for learning give the perception of a learning environment but speak nothing more than a well-managed classroom. We have accepted that learning looks a particular way with very little evidence to support the façade. As a result we adapt our teaching to target these proxies for learning and thus, we are not truly teaching at all. These have become our proxies for teaching: powerpoint notes, virtual classes, extrinsic reward systems, modelling, worksheets, homework checks, and countless other methods that would give us the desired appearance that learning is occurring. It comes to stand that teaching and learning are often unrelated; both are likely occurring in a classroom but are frequently independent of one another.

Truly effective teachers must begin to view themselves as evaluators of their effect of student learning. It begins by being focused on true learning, substantiated by valid evidence, instead of poor proxies.

The depth of Hattie’s research allows us to find numerous teaching strategies that we have all likely encountered. Within each of those he has summarized the effect size they have on student learning. Perhaps more interesting than the effect size of each influence is that so many influences were found to have an effect. This means that everything we do has some impact, perhaps only minimally, but an impact nonetheless. Every decision you make matters. There should be nothing arbitrary in our teaching.

We can summarize his results by stating that any influence above d=0.40 falls into the “zone of desired effects”. This, of course, is filled with many diverse influences but we could summarize many of them by suggesting that the most important thing we can do in a classroom is cause cognitive change, communicate and generate intentions in learning, understand prior knowledge and have enough content knowledge to create meaningful challenges. To further summarize, we need to make students think and herein lays the challenge.

If you have taught for more than a few years you have likely attempted and failed many methods of teaching. In fact, it is in the first few years of teaching that you would have likely come to “find yourself” as a teacher. Certainly the evidence shows that teaching quality is a significant determinant in learning (ex. “d=.90 for teacher credibility) and further that teacher performance typically improves in the first few years. What may surprise you is that measures of performance begin to plateau between 3-5 years of teaching (Rockoff, 2004; Kukla-Acevedo,2009). Again, it is in the first few years when a teacher finds themselves and apparently does not let go for the remainder of their career. Presumably they have encountered many novel ideas for their classroom and likely have experimented with many of them. They have probably uttered the words “This isn’t working; I need to try something new”. Now, it is possible that many ideas do in fact work while other may have no effect at all, while still there may be many that may have a negative effect. Sadly, in most cases, we have no evidence one way or another. So perhaps we are asking the wrong question. Perhaps instead of asking “how do I make my students think?” we should be asking, “how do I know what I am doing is making my students think?” We must begin to emphasize the evaluation rather than move on to a new method. It is time we look to an evidence-based approach as the grounds for change and effective methods.

 “Knowing that is different from knowing how. But in the model of learning that dominates teacher professional development (as well as most formal education), we assume that if we teach the knowing that, then the knowing how will follow. We assemble teachers in rooms and bring in experts to explain what needs to change—and then we’re disappointed when such events have little or no effect on teachers’ practice. This professional development model assumes that what teachers lack is knowledge. For the most part, this is simply not the case. The last 30 years have shown conclusively that you can change teachers’ thinking about something without changing what those teachers do in classrooms.”-William, D. (“Assessment for learning: why, what, how?”, Lecture, 2009)

Kukla-Acevedo, Sharon. “Do Teacher Characteristics Matter? New Results on the Effects of Teacher Preparation on Student Achievement.” Economics of Education Review 28.1 (2009): 49-57. Web.

Rockoff, Jonah E. “The Impact of Individual Teachers on Student Achievement: Evidence from Panel Data.” American Economic Review 94.2 (2004): 247-52. Web.

You need not look far to sense that our current education system is in a state of upheaval. Proponents of Common Core Standards, Alberta’s Curriculum Redesign and the new BC Ed Plan have all placed mathematics in the limelight of educational reform. Addressing the development of negatives attitudes and confidence in mathematics is likely one of the biggest concerns in education today and there is certainly good reason for such concern. The National Assessment of Educational Progress (NAEP) revealed a decrease in the percentage of students who indicated they like math from 1992 to 2000. Likewise, a survey by Olson (1998) revealed that one-third of students reported that they did not enjoy mathematics. With a rising resentment towards the subject, economists fear the work force will be impacted after research revealed that negative attitudes can decrease the tendency of individuals to consider careers in mathematics-related fields (Haladyna, Shaughnessy, & Shaughnessy, 1983; Maple & Stage, 1991; Trusty, 2002). In fact, further studies have shown that students’ attitudes toward mathematics were the strongest predictor of their participation in advanced mathematics courses (Ercikan, Mccreith, & Lapointe, 2005). In the field of mathematics education, this is nothing new. Educators as early as the 1960’s have held the belief that attitude plays a fundamental role in learning and achievement in mathematics (Neal, 1969) and while recent programs in BC and Alberta have identified the steps needed to generate a shift in attitude there is limited literature that summarizes the research about the causes of negative attitudes.
Before exploring the causes of these beliefs, it is important to define exactly what constitutes an attitude being negative. While some argue that attitudes have a revolving definition in math depending on the instrument used to measure it (Leder, 1985) it is generally accepted that an attitude, either positive or negative, is at the least an emotional disposition in relation to the subject (McLeod, 1994) and at most “an aggregated measure of a liking or disliking of Mathematics, a tendency to engage in or avoid mathematical activities, a belief that one is good or bad at Mathematics, and a belief that Mathematics is useful or useless” (Ma & Kishor, 1997). With this working definition and the current research it could be shown that negative attitudes stem from 1) perceived nature of the subject, 2) academic labels and expectations, 3) parental influence, and 4) teacher beliefs and instructional practices.

Belief about the Nature of Mathematics
It is a common belief amongst both parents and some educators that the reason so many students struggle to enjoy mathematics is because the subject itself doesn’t offer much to like. This is rationalized by the fact that mathematics is not like the other subjects. While the humanities and sciences introduce new material in the same language, math appears to present itself with very different terminology. Even foreign languages, despite various pronunciations, will at least use the same notation as other subjects. It could be perceived that the unique language, notation and complicated syntax of mathematics present a stumbling block for young learners and as research suggests it may be the communication, not the process, of mathematics that is problematic (Ashby, 2009). With such a rapid and early emphasis on the language of mathematics, it is easy to understand how students may misunderstand the overall picture of math and associate their beliefs with the language instead. Much of the frustration students have is likely due to a failure to successfully and consistently understand the tasks they are given.
It is also a commonly held belief by the average math student that not only is there only one right answer but that there is also only one correct way to solve a mathematical problem (Schoenfeld, 1992). It is this perceived absolute nature of mathematics that further contributes to negative attitudes and anxiety in mathematics.
More recently a study by Thompson and Denisse (2005) found that 58% of students viewed learning mathematics as a memorization of formulas and rules. Not surprising less than half of those students found mathematics enjoyable. According to the authors, such beliefs likely influence students’ intrinsic motivation to engage in mathematical tasks and interfere with their ability to rehearse and play with the subject. These factors ultimately lead to lower achievement in academic contexts and link themselves to another source of negative attitudes toward mathematics.

Academic Labels and Expectations
It has been suggested by recent research that students’ attitudes and anxiety increase with age (Nicolaidou & Philippou, 2003; Mato & De Le Torre, 2010). These negative attitudes do not develop from the students’ sudden independent awareness that math is an unpleasant subject. In fact, Maria de Lourdes Mata (2012) makes a compelling argument that the relationship between age and attitude stem from increased pressure to perform well and the existence of over demanding tasks. Nicolaidou and Philippou (2003) showed frequent and repeated failures in mathematical tasks create negative attitudes which may become relatively permanent. The authors point out that students’ first interaction with school mathematics is generally positive and as they progress, their attitudes become decreasingly positive and often negative by the time they reach high school. This is to say that attitudes are not the product of the student’s self-actualization of their ability but instead the product of how academia labels their ability with performance standards and standardized tests. Studies of secondary school children and adults have consistently revealed that performance in mathematics (as defined by academia) is significantly correlated to positive attitudes towards mathematics (Lam, Doverspike, Zhao, ,Zhe, & Menzemer, 2000; Nicolaidou & Philippou, 2003; Mato & De Le Torre, 2010). As our schools increase the frequency of standardized arithmetic testing and consistently formalize mathematical content into rigid, unimaginative lessons we label our students and create a dichotomy of mathematical giftedness and weakness before students even have an opportunity to form an opinion about the subject. The reality is that math attitude and performance is highly correlated (Nicolaidou & Philippou, 2003). Students with positive attitudes achieve higher results; students with higher results had more positive attitudes (Sanchez & ZImmerman, 2004). These findings are likely due to the students’ perception of themselves, and while our school’s standardized assessment contributes significantly to this belief, it is not the only contributor to the development of mathematical self-belief.

Parental Influence
In 1956, Thomas Poffenberger and Donald Norton published the results of a survey focused on the factors that determine attitudes toward arithmetic. Though 60 years have passed, it is hard to argue with the application of the results today. The studied concluded that one of the determining factors for children’s initial attitude towards mathematics was the influence of their parents. More than 50% of parents in the survey reported that they expected above average work in all subjects except math, where an average achievement was acceptable. The authors argued that parental expectation of children’s development in math is likely due to parents, themselves, experience with minimal achievement. Despite a lack of research supporting the influence of genetics in mathematical ability, it is typical for students to identify their mathematical ability with the belief held by their parents. Students with a fixed mindset, as Carol Dweck might suggest, come to believe that even their mathematical ability is fixed and likely inherited by their parents. It would appear that the easiest way to determine a student’s initial attitude towards math is to identify the parents’ attitude about the subject. More recent studies corroborate these findings (Furner & Duffy, 2002) and find that math attitudes, despite being largely environmental, are inevitably passed from parent to child.
This development of self-image, of course, can be moderated and overcome in the right environment. While initial attitudes towards mathematics may be the result of parental influence, it is the ongoing exposure in the classroom and thus the beliefs of teachers in early education that may ultimately determine students’ enduring attitude.

Teacher’s Beliefs and Instructional Practices
While it has been often reported that negativity toward mathematics is felt more frequently in secondary school (Nicol, 2002), data from more recent studies suggests that negative experiences are most commonly developed in primary and middle school (Uusimaki & Nason, 2004). Researchers propose that these negative experiences are the result of the teacher, with 72% attributing their negative experience with a primary school teacher rather than social factors or mathematical content. These negative experiences may very well be correlated to the academic experience and preparedness of the teachers themselves. A 2000 study by National Survey of Science and Mathematics Education (Weiss et al., 2001) found that only 1% of elementary mathematics teachers held a degree in mathematics. The studied also concluded that only 44% have completed coursework in algebra and 36% in statistics and probability, areas that the National Council of Teachers of Mathematics suggest should be introduced in primary school (NCTM, 2000). It stands to reason that we have either failed to attract math experts to elementary school or we have failed to train elementary teachers as experts in their teachable subjects.
In addition to inadequate competency in mathematics, there appears to be a lack of dynamic and progressive teaching practices in elementary schools. When asked if they place heavy emphasis developing students’ interest in mathematics, 52% said they did. When it came to developing an understanding about the history and the nature of the subject, 3% reported giving special attention to the task. This priority of instruction is also seen in the teachers’ models of instruction, where 83% of teachers report to using textbooks or worksheets at least once or twice per week, while 43% reported to have never used extended mathematics projects and investigations as a source of mathematical development. This approach in the classroom reflects a belief that mathematics is static and non-explorative. It develops an emphasis on the absolute nature of mathematics and robs students of the opportunity to experience the aesthetics, discover the innovation, and contribute to the development of the subject.
Elementary school years have been identified as a pivotal period in developing students’ attitudes. Teachers’ beliefs about mathematics play a crucial role in their instructional practices (Charalambos, Philippou, & Kyriakides, 2002) and in their students’ achievement and development of beliefs and attitudes towards mathematics (Emenaker, 1996).
Though early attitudes may be formed in elementary school, it certainly does not get high school teachers off the hook. These attitudes are often solidified and endure throughout high school as a result of increased formal instruction, standardized testing, and growing connotations attached to math. The truth is that many of the underlying issues that affect students’ attitudes are cyclical and amplified with every new experience our students have in math. Students walk into the math classroom and are introduced to new notation and terminology, one of the most abstract concepts about the subject. When they struggle to grasp it, our assessment is quick to identify them as a weak mathematical thinker which is accentuated when they go home and realize that their parents also struggled in math. Returning to school they are given repetitive, arithmetic worksheets that produce either correct or incorrect solutions, yielding an easy comparison to their peers, further ranking them as adept or inept mathematicians. This lack of confidence leads to anxiety and a withdrawal from mathematical tasks which only further divides the dichotomy of abilities and leads to negative experiences when their mathematical knowledge is called upon. These students grow up and become parents and teachers whose negative attitude towards mathematics is echoed for another generation. The pattern continues; that is unless there is intervention.
By developing an understanding about the roots of negative attitudes we are better able to equip our teachers and provide timely interventions for students. The first of these interventions likely occurred in 1989 by the NCTM with the Curriculum and Evaluation Standards, revised again in 2000 as the Principles and Standards for School Mathematics. In this publication, traditional approaches of teaching algorithms and specific methods were discouraged and replaced with an approach that encourage students to think through the math, establishing connections between mathematical concepts and promoting communication and multiple representations. With these documents acting as a prompt, British Columbia has also begun to develop an education plan that addresses these key ingredients. In the new BC Ed Plan, the Ministry of Education plans to reduce the amount of scripted curriculum allowing teachers to focus on personalized learning experiences and developing a better overall picture about the nature of mathematics. While the effect of these programs has yet to be seen, it is only by timely and intentional interventions that we will influence the beliefs of a generation of young learners immersed in a culture inundated with the negative connotations attached to math.

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