1. Write a half page, single-spaced reflection on the article attached below. What you choose to reflect on from the article is up to you, but I suggest that you reflect on aspects of the reading that resonated with you, that you found most interesting or most illuminating for your own mathematical understanding.

2. Interview a high school student, college student, high school teacher, or college instructor (basically interview one person that you would expect to know a little something about functions). Conduct your interview over zoom. Record the interview to the cloud and change you settings to get an automatically generated transcript. Use the same interview protocol as in the paper. In the paper pay close attention to the dialogue between the interviewer and the interviewee for the kind of probing questions that the interview asks. Strive to do the same in your interview.

Interview protocol

Start by inviting the participant to sketch by hand on the same coordinate system graphs of y=x^2 and y=(x-3)^2 and ask them to explain how they generated their sketch.

Next, invite participant to compare their sketch to that of computer generated sketches (use Desmos or another web based graphing program).

After they compare sketches, ask the participant to comment on the location of the graph y = (x – 3)^2 relative to the graph of y = x^2, relating, where necessary, to the discrepancy between participants sketch and the computer generated sketch, or to the discrepancy between the intuitive expectation and the “known” result.

If the issue did not come up naturally, ask the participant to discuss the graph of y = x^2−3 and compare it to the graph of y = (x – 3)^2.

Write a one page max, single-spaced account of the mathematical reasoning of the person you interviewed. Use exact quotes (like they did in the paper). After summarizing their reasoning do your best to “diagnose” their reasoning and compare their response to those in the paper.1 attachmentsSlide 1 of 1

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Journal of Mathematical Behavior 22 (2003) 437–450 Conceptions of function translation: obstacles, intuitions, and rerouting Rina Zazkis a,∗ , Peter Liljedahl a , Karen Gadowsky b a Faculty of Education, Simon Fraser University, 8888 University Drive, Burnaby V5A 1S6, BC, Canada b Delta Secondary School, BC, Canada Abstract A horizontal translation of a function is the focus of this study. We examine the explanations provided by secondary school students and secondary school teachers to a translation of a function, focusing on the example of the parabola y = (x−3)2 and its relationship to y = x2 . The participants’ explanations focused on attending to patterns, locating the zero of the function, and the point-wise calculation of function values. The results confirm that the horizontal shift of the parabola is, at least initially, inconsistent with expectations and counterintuitive to most participants. We articulate possible sources of this perceived inconsistency and describe a pedagogical approach aimed at resolving it. © 2003 Elsevier Inc. All rights reserved. Keywords: Function; Transformation; Translation; Obstacle; Transformation of function; Horizontal translation 1. Prologue (motivation and questions) Imagine the graph of (a) y = x2 . Now imagine the graph of (b) y = (x − 3)2 . Check your imagination with the graphing device. If you are not surprised, it is probably because you have already explored the relationship between the two parabolas in the past. Most people conjecture that graph (b) will appear three units to the left of graph (a). The surprising result is that (b) is actually three units to the right of (a). As students, we memorized this result as one of the counterintuitive facts of mathematics. As teachers, we attempted to explain this phenomenon to our students, often creating a conflict between their intuition and our explanations. ∗ Corresponding author. Tel.: +1-604-291-3662; fax: +1-604-291-3203. E-mail address: zazkis@sfu.ca (R. Zazkis). 0732-3123/$ – see front matter © 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jmathb.2003.09.003 438 R. Zazkis et al. / Journal of Mathematical Behavior 22 (2003) 437–450 As researchers, we became interested in how teachers and students deal with the phenomenon and what explanations they provide. Moreover, we wished to understand possible sources of students’ “wrong” intuition and sought pedagogical solutions. 2. Background Teaching, learning, or understanding of functions has been an important focus in mathematics education research in the past decades. However, within a variety of research reports focusing on functions, little attention has been given to the transformations of functions. A common treatment of transformations of functions in pre-calculus courses involves a consideration of a graph of a function f(x) on a Cartesian plane. Functions f(ax), f(−x), f(x) + k, and f(x + k) correspond to a dilation, reflection, vertical translation and horizontal translation of f(x), respectively. Discussion of transformations of functions traditionally begins with a consideration of a parabola and then proceeds to graphs of other quadratic relations and other functions. Eisenberg and Dreyfus (1994) conducted an extensive exploration on students’ understanding of function transformations, focusing on visualization of transformations. They acknowledged the difficulty in visualizing a horizontal translation in comparison to a vertical one, suggesting that “there is much more involved in visually processing the transformation of f(x) to f(x + k) than in visually processing the transformation of f to f(x) + k” (p. 58). Baker, Hemenway, and Trigueros (2000) have investigated the understanding of transformations of various functions from a perspective of Action–Process–Object–Schema (APOS) theory (for a description of APOS, see for example, Asiala et al., 1996 or Dubinsky & MacDonald, 2001). They confirmed the observation that vertical transformations appear easier for the students than horizontal transformations. They explained this based on their theoretical perspective, claiming that “vertical transformations are actions performed directly on the basic functions, while horizontal transformations consist of actions that are performed on the independent variable of the function and further action is needed on the object resulting from the first action to get the result of the transformation” (Baker et al., 2000, p. 47). Furthermore, students’ difficulty with function transformation was attributed, at least in part, to their incomplete understanding of the concept of function. Baker et al. agree with Eisenberg and Dreyfus in their observation that an object conception of function may be a prerequisite to the effective understanding of transformations of functions. Borba and Confrey (1996) have presented a detailed case study of a 16-year-old student, Ron, working on transformations of functions in a computer based multi-representational environment. Their study intended to investigate vertical and horizontal translations, reflections around vertical and horizontal lines, and vertical and horizontal stretches of functions. Ron’s attempts to interpret the horizontal translation of a parabola are presented as “problematic.” This is followed by his investigation to coordinate visual actions with changes in other representations. Though it is not explicitly stated in this research report, we infer from the focus of their discussion that researchers perceived other transformations as less “problematic.” In this study, we focus on the difficulty — acknowledged in the aforementioned prior research — presented by a horizontal translation of a function. Our goal is to investigate how students and teachers cope with this difficulty and to explore how it is possible to overcome or at least to reduce it. We examine the understanding of horizontal translation of functions in general, focusing on parabolas in particular. We explore how participants explain the perceived inconsistency presented by a horizontal translation R. Zazkis et al. / Journal of Mathematical Behavior 22 (2003) 437–450 439 and suggest a pedagogical approach to transformations of functions that addresses the mystery of graphs “moving in unexpected directions.” 3. Methodology Participants in this study were preservice secondary teachers (n = 15), practicing secondary teachers (n = 16), and Grade 11/12 students (n = 10). The teachers were volunteers attending courses at the University in which this study was conducted. The students were volunteers referred by the participating teachers, all classified as having “above-average” ability. In a clinical interview setting, all the participants were asked to predict, check and explain the relationship between the graph of y = x2 and the graph of y = (x − 3)2 . The interview protocol started by inviting the participants to sketch both graphs on the same coordinate system and to explain how they generated their sketch. The participants were then invited to compare their sketch to that of a graphing calculator. After the sketch was either confirmed or disconfirmed, the participants were asked to comment on the location of the graph y = (x − 3)2 relative to the graph of y = x2 , relating, where necessary, to the discrepancy between the sketch and the display of the graphing calculator, or to the discrepancy between the intuitive expectation and the “known” result. In addition, if the issue did not come up naturally, the participants were asked to discuss the graph of y = x2 − 3 and compare it to y = (x − 3)2 . Their responses were audio recorded, transcribed, and analyzed. The analysis attended to (1) common trends in explanations, (2) attitudes towards perceived inconsistencies, and (3) differences between the groups of participants, that is, students versus teachers, and preservice teachers versus practicing teachers. 4. Results The fact that the shape of y = (x − 3)2 is a parabola that is congruent to the canonical parabola y = x2 was taken for granted by teachers and students alike. The fact that the graph of y = (x−3)2 is a horizontal translation of y = x2 was also mentioned as requiring no further explanation. Therefore, we turn our focus now to the direction of this horizontal translation. 4.1. Students’ responses The participating students had learned about transformations of quadratics (conic sections) before this study was conducted. Nevertheless, in the very beginning of the interview half of the students (5 out of 10) predicted the spatial location of y = (x − 3)2 incorrectly. There was no sincere attempt on students’ part to explain the phenomenon, though all admitted that the observed location of the parabola was counterintuitive. We chose to present in detail the following excerpt from the interview with Mitch, a Grade 11 student, because it captures most of the themes that appeared in students’ responses. Mitch learned in class about transformations of functions and their corresponding graphs about two months prior to the interview. The interview excerpt presented below takes place after Mitch incorrectly sketched the graph of y = (x − 3)2 . 440 Interviewer: Mitch: Interviewer: Mitch: Interviewer: Mitch: Interviewer: Mitch: Interviewer: Mitch: Interviewer: Mitch: Interviewer: Mitch: Interviewer: Mitch: Interviewer: Mitch: R. Zazkis et al. / Journal of Mathematical Behavior 22 (2003) 437–450 Would you please check this. No problem. (pause) Yes problem. I should have remembered this. It moves in the wrong direction. What do you mean, wrong direction? It’s minus three, so you sort of expect it to move left. But it moves the other way. It moves right. Oh, and why is that? That’s the way it is. It’s always like that. I really knew that. Just remember to do the opposite. But not always. Just when . . . only if your number is in the brackets. Please explain. OK. When you do something to x in the brackets, like x minus three or x plus two your graph moves the opposite way. For x plus two it moves left, two times. For x minus three, the one you wanted, it moves right, three times. And once you remember that, it works for all the graphs. You mentioned something about brackets. Yes, it moves the opposite way only when there are brackets. Without brackets, say for y equals x squared plus three, written this way, [y = x2 + 3], it moves the way you want it to move. This one will move up, and with negative three it will move down. This one is doing exactly what it should. So the other one is doing something it shouldn’t. Not really, just not what one would guess, but the other way around. Does it bother you? Not really. You just see on the calculator what it’s doing and you know it will do it all the time. I wonder why the graph [of y = (x − 3)2 ] moves the way it moves. It’s not what I expected, it’s not what you expected, at least initially. Can you help me understand this? (pause). Maybe try small steps. Try y equals x minus one squared [y = (x − 1)2 ]. You see, it goes right one step. And try y equals x minus two squared [y = (x − 2)2 ]. It will go right two steps, you see them together here [demonstrates with graphing calculator]. So after seeing this you do not expect x minus three [(x − 3)] to do something different, do you? Is this how your teacher explained it to you? I’m not sure he explained this at all. But graph it once and you know how it works. In examining Mitch’s response, we note the initial confusion, which is rapidly corrected based on the feedback from the graphing calculator. The claims “I should have remembered,” “I knew that” demonstrate that the phenomenon is not new to Mitch: he has encountered this behavior of

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