I always feel a little bit guilty when it comes to "trying new things" with students. I think it's partly because the students in practicum aren't "my" students, and instead these are the students for whom my sponsor teacher is ultimately responsible. Because of this, as much as I want to try out different ideas, and no matter how supportive the sponsor teacher is (and he has been lovely, for sure), I always feel as though my experimentation may prove to be unethical, or may detriment my students. So although I understand the optimism with which Heather Robinson writes about "still [having] insights into [her] aspects of pedagogy and instruction" (pp. 62) despite lesson plans not working for her, I still feel as though I am a bit of a cheat, taking "bits and pieces" from other people to create my "own" practice. I suppose that all knowledge is built on previous experiences and does not require a person to "reinvent the wheel", so to speak, but when students' success is at stake, ironically, the students' enjoyment and ability also become the ultimate reason why a teacher would try something out in the first place. Heather Robinson's article sounds like me, though; I have yet to teach with a purpose other than for teaching math, and am still somewhat detached from the lessons I teach. I love math, I really do, but I also know its applications, and that's what it makes it relevant to me. Making mathematics relevant to thirty students, times six blocks, is a huge challenge!
Although I would like to say that my detachment from math teaching is because I am not writing lesson plans for my own class, but for someone else's, I ultimately know that I need to write my lesson plans, and find a reason to teach math. I think this is largely due to an overlap of my personal opinion and the classroom mixing together. I like the idea of provoking students and discussion. I don't know how much liberty I can take to doing that in a classroom that isn't my own, though, and I've been losing quite a bit of sleep over it. IB is supposed to be interdisciplinary, yet every classroom seems like it isn't; it's just a bunch of stuff to learn. I know that there are students who will just learn the math because they're expected to, but I want them to enjoy it, as well. Where can we get questions like the "Alternative Assessments" and "Enrichment Masters" which Heather Robinson mentions in the article? What about the ones we've used in our EDCP 342A class? I'd love to expand my resources to include such materials. There's also a social justice in mathematics textbook in the Education Library which I'd like to pick up.
I love the idea of Jigsaw activities; one concern in math, however, is whether or not students will teach each other incorrectly. However, this is something that seems to work better in my math 9 class than when I taught them, so I am considering having students teach each other in that sense, because they aren't benefiting from lecture.
The other issue I have is that the evaluation of mathematics remains the same even though the methods of instruction are all different. Mathematics remains a written form. I would be interested in having an interview setting, instead of tests, for one test out of the year, to see how well students can explain their ideas. No prompting, and correct answer = top mark, slight prompting and correct answer = A- or B+, etc. That would certainly help for students who have problems with written output, but perhaps are very strong verbally. Some countries certainly had interviews as part of their evaluation. These would have to be short, however, likely spread across an entire week; perhaps they could be an option for students to complete instead of a test, if they were struggling a great deal. This sort of interview type testing is used informally in various Skills/Learning Assistance periods when students write tests under the supervision or with aid of a support worker.
Thursday, 14 November 2013
Sunday, 10 November 2013
Practicum Experience Response
I didn't find anything easy about teaching math at my sponsor school. It's not that I found the subject difficult, but teaching math in a way that was still exciting to students was a challenge. When it comes to relative difficulty, however, I realized that I preferred to use the whiteboard over the tablet in a classroom, which is a huge revelation for me, because I always thought I did really well with the tablet. There's still something about moving around when you teach that, according to one student, is more appealing about teaching on a whiteboard, whereas I found that the tablet keeps you locked in one place, for the most part. In general, tablets are big and bulky and can't really be held up for walking around the room. I found it easy to talk to students, though; that was nice. I think I'm still young enough to be able to talk to them (I realized I use a slight touch of slang, like "s'all good" or "no worries"; though, is that slang, or just an expression used by my expression? I definitely called a large math expression a "holy mother of a cow". This seemed confusing to some, but I had their attention, which was nice. It was pretty easy for me to connect with the students, which was nice. They weren't shy when I would ask them questions, which was nice, and I asked them for their honest feedback. Although there was slight hesitation, I pointed out that I was a student learning to be a teacher. Although that's a dangerous move when it comes to keeping authority, they were still able to tell me what they liked and what they didn't like.
The most difficult thing was probably to make math exciting. I did a lot of activities which were very upbeat and didn't allow for practice of material as much as I think I should have. I tried the "Japanese model" of teaching we discussed in one of the articles (Hoffier?), but I quickly found that students have come to expect being fed information and practicing it, so when I gave them a problem, there was such a large amount of complaining! My goodness! And with the small number of students in the school, by the time the next grade 8 class had come in, the previous class had already informed them that the class involved "a very difficult activity that nobody got", even though the students had learned all the information they needed to solve the problem. Although I don't expect all students to be problem-solvers, I realized that for the second class, I needed to explain a little more about how to solve the problem I had assigned, and from there, some people pounced on it immediately. Also, time management! Tiiiiime management, I'm still struggling with this so much! I've finally got control of my students, which is thanks to my sponsor teacher, who suggested that I have clear transitions between activities, and that worked. Also, I realized how not to be boring. That was huge. I realized as I was boring myself to tears, that other people were probably also pretty unimpressed at the moment, and so I became a little more theatrical. It helped!
I think the most interesting thing about teaching math is how helpful it is to have a problem worked out ahead of time. A problem in surface area, for instance, is much easier to teach if you have information prepared ahead of time and pre-organized. Another lovely thing; I divided up my boards into three columns to organize my work, and that helped tremendously. Finally, no matter how you try to turn cards into a teachable moment in probability and statistics, the junior level will always want to play poker, so don't use playing cards to organize people into groups....
The most difficult thing was probably to make math exciting. I did a lot of activities which were very upbeat and didn't allow for practice of material as much as I think I should have. I tried the "Japanese model" of teaching we discussed in one of the articles (Hoffier?), but I quickly found that students have come to expect being fed information and practicing it, so when I gave them a problem, there was such a large amount of complaining! My goodness! And with the small number of students in the school, by the time the next grade 8 class had come in, the previous class had already informed them that the class involved "a very difficult activity that nobody got", even though the students had learned all the information they needed to solve the problem. Although I don't expect all students to be problem-solvers, I realized that for the second class, I needed to explain a little more about how to solve the problem I had assigned, and from there, some people pounced on it immediately. Also, time management! Tiiiiime management, I'm still struggling with this so much! I've finally got control of my students, which is thanks to my sponsor teacher, who suggested that I have clear transitions between activities, and that worked. Also, I realized how not to be boring. That was huge. I realized as I was boring myself to tears, that other people were probably also pretty unimpressed at the moment, and so I became a little more theatrical. It helped!
I think the most interesting thing about teaching math is how helpful it is to have a problem worked out ahead of time. A problem in surface area, for instance, is much easier to teach if you have information prepared ahead of time and pre-organized. Another lovely thing; I divided up my boards into three columns to organize my work, and that helped tremendously. Finally, no matter how you try to turn cards into a teachable moment in probability and statistics, the junior level will always want to play poker, so don't use playing cards to organize people into groups....
Thursday, 17 October 2013
Reading in Mathematics - response
The article mentions four main aspects as to why students may struggle with mathematics text reading:
1) Density
2) Mixed translation
3) Changing variables, symbols, and graphics
4) Concepts are interconnected
1) In LLED 361, we explored the idea of literacy and multiple modes of expression. One activity which was suggested for "translating" mathematical language is to create two columns, where one contains mathematical text, and the other contains the "translation" of that text. It's a little bit funny, because both are a form of English, but mathematical text is used very infrequently. This sort of activity, I would argue, is only to be used with students who have not gotten into the habit of translating mathematical text or asking themselves questions about the content or question being asked. Much like students may need to be taught how to complete a multiple choice exam efficiently, so to do they need to be taught how to "translate" mathematical language, hopefully without writing down the interpretation in the interest of time.
2) "Or", or "or". Oh, how I've missed you. "The mathematical 'or' is different from the categorical 'or'," my proofs professor used to say. For context, in the "every day world", one would say "please bring me coffee or tea", and this would be the exclusive use of "or", since you would not wish to drink a concoction of coffee and tea, call it cofftea. However, in mathematics, this would be exactly the case; asking a mathematician to bring you coffee or tea would mean that the mathematician would bring coffee, tea, or both.
3) I find that students struggle a lot of greater than and less than signs. I know in school, we were always told that the "sign has teeth on it, and the teeth eat the bigger number". But when it comes to reading an inequality, it actually takes a second to recognize which number is bigger, and whether you need to say "greater than" or "less than" as you read from left to right. I would like to suggest that the reason for the discrepancy is because as language users, we are used to reading from a certain point on a page to another point on a page, and in many languages, symbols do not have individual meeting, but must be connected before a meaning is derived. Mathematics symbols all have individual meanings, such as + <=, =/=, etc. Knowing both the meaning of the symbol as well as its collective meaning in a statement means that mathematics reading can be especially challenging.
4) I can't get over the fact that I find myself teaching high school students about fractions more and more often. I don't mind doing so, but to me, it's appalling that they have managed without it. I imagine a calculator has something to do with it. What was the reasoning to change the math curriculum in BC from linear, compounding concepts to more scattered ones? I remember there was an issue with too much repetition; was there any other reason as to why the math curriculum changed so drastically in the last few years?
I will say, however, that I disagree with Marzano et al. in using direct instruction on words. I discussed with my inquiry class, that an inquiry science classroom may have students determine a definition together. I like this idea for mathematics; although mathematics is less fluid than science is; I would say that defining a mathematical concept in a way previously unbeknownst to a student means that the student can learn a great deal from their peers about how to express an idea that they all share but are unable to get across to other students or to the teacher. All they would need at that time would be a label to have a definition.
1) Density
2) Mixed translation
3) Changing variables, symbols, and graphics
4) Concepts are interconnected
1) In LLED 361, we explored the idea of literacy and multiple modes of expression. One activity which was suggested for "translating" mathematical language is to create two columns, where one contains mathematical text, and the other contains the "translation" of that text. It's a little bit funny, because both are a form of English, but mathematical text is used very infrequently. This sort of activity, I would argue, is only to be used with students who have not gotten into the habit of translating mathematical text or asking themselves questions about the content or question being asked. Much like students may need to be taught how to complete a multiple choice exam efficiently, so to do they need to be taught how to "translate" mathematical language, hopefully without writing down the interpretation in the interest of time.
2) "Or", or "or". Oh, how I've missed you. "The mathematical 'or' is different from the categorical 'or'," my proofs professor used to say. For context, in the "every day world", one would say "please bring me coffee or tea", and this would be the exclusive use of "or", since you would not wish to drink a concoction of coffee and tea, call it cofftea. However, in mathematics, this would be exactly the case; asking a mathematician to bring you coffee or tea would mean that the mathematician would bring coffee, tea, or both.
3) I find that students struggle a lot of greater than and less than signs. I know in school, we were always told that the "sign has teeth on it, and the teeth eat the bigger number". But when it comes to reading an inequality, it actually takes a second to recognize which number is bigger, and whether you need to say "greater than" or "less than" as you read from left to right. I would like to suggest that the reason for the discrepancy is because as language users, we are used to reading from a certain point on a page to another point on a page, and in many languages, symbols do not have individual meeting, but must be connected before a meaning is derived. Mathematics symbols all have individual meanings, such as + <=, =/=, etc. Knowing both the meaning of the symbol as well as its collective meaning in a statement means that mathematics reading can be especially challenging.
4) I can't get over the fact that I find myself teaching high school students about fractions more and more often. I don't mind doing so, but to me, it's appalling that they have managed without it. I imagine a calculator has something to do with it. What was the reasoning to change the math curriculum in BC from linear, compounding concepts to more scattered ones? I remember there was an issue with too much repetition; was there any other reason as to why the math curriculum changed so drastically in the last few years?
I will say, however, that I disagree with Marzano et al. in using direct instruction on words. I discussed with my inquiry class, that an inquiry science classroom may have students determine a definition together. I like this idea for mathematics; although mathematics is less fluid than science is; I would say that defining a mathematical concept in a way previously unbeknownst to a student means that the student can learn a great deal from their peers about how to express an idea that they all share but are unable to get across to other students or to the teacher. All they would need at that time would be a label to have a definition.
Wednesday, 16 October 2013
Micro-teaching Retrospection #1
Here is a table outlining the feedback received for each section:
| Aspects to maintain | Suggestions | Interactivity |
|---|---|---|
| - Walking around to help students | - Raise volume | - Encouraged group interaction |
| - Pace | - More clear definition of slope at beginning or end of lesson | - Asked good guiding questions |
| - Good intro | - Have the app available for students to practice | - Advanced group got a chance to discuss more |
| - Good applet and technology use | - Fun/relevant | |
| - Addressed confusion | - If more time had been available, drawing more lines for practice would have been helpful | |
| - Eye-contact | ||
| - Visuals | ||
| - Group work | ||
| - Positive words/ feedback/ answered questions effectively | ||
| - Relating topic to “real life” |
I felt that the lesson went well, overall. I was really excited by the hook, because there was immediately a connection between slopes and skateboarding/snowboarding/skiing. There were several things which I would like to improve, in particular. Firstly, I would like to improve my teaching for intuition. Although this is a very broad goal, I will narrow it below. It is very difficult to wander from traditional teaching styles of stating definitions, demonstrating problems, then practicing problems of definitions. I think that our group may have chosen a topic which was a bit too ambitious to cover in its entirety in 15 minutes, and given the applications of a slope in calculus, it would not be wise to rush the lesson. I would stick by my choice to leave negative slopes until the end of class, however, because it has been decided, by a combination of the definition of a slope as well as the selection of which axes are positive or negative, that a line falling from the second quadrant to the fourth quadrant has a negative slope. If horizontal axes increased to the left and decreased to the right, the sign on a slope would be reversed. The magnitude of slope is really crucial here, the sign of the slope can be discussed later in the lesson. I think a first start to developing inquiry teaching would be emphasizing what is necessary and what is arbitrary in mathematics, such as the sign on a slope; I feel that this is a good guide as to what needs to be learned and what can be (self-)taught in an activity.
My second point for improvement would be the understanding of when to introduce definitions. After all the articles we've read in class, especially the Hoffman/Brahier article, I feel as though introducing a rote definition is almost a faux pas in mathematics, because if it is used incorrectly, a definition becomes not a summative statement but an introduction of something arbitrary being passed off as necessary. After all, why did we actually choose for slope to be a ratio of the change in vertical to the change in horizontal, rather than the other way around? Rather than a concept of "more steep", we would likely instead use the terminology "less flat". However, everyone who has ever encountered any wheels in their life will have an intuitive sense of what slope means, even as a matter of survival. Finding a mathematical definition is a fine balance between over-complicating an intuitive definition and over-simplifying a (often) physical concept. When is a place for definitions during a math lesson? I have always encountered definitions at the beginning of a math lesson, especially in calculus. However, could calculus also be taught intuitively, without starting from the definition? After all, this obsession with definition (I would argue) originates with the formation of proofs. The definition of a concept being proved is one of the most useful tools for verifying a mathematical statement around that concept. When, if ever, does mathematics become really intuitive in an upper level? What is the evolution of the necessity of a formal mathematical definition as a function of time in a math classroom?
My second point for improvement would be the understanding of when to introduce definitions. After all the articles we've read in class, especially the Hoffman/Brahier article, I feel as though introducing a rote definition is almost a faux pas in mathematics, because if it is used incorrectly, a definition becomes not a summative statement but an introduction of something arbitrary being passed off as necessary. After all, why did we actually choose for slope to be a ratio of the change in vertical to the change in horizontal, rather than the other way around? Rather than a concept of "more steep", we would likely instead use the terminology "less flat". However, everyone who has ever encountered any wheels in their life will have an intuitive sense of what slope means, even as a matter of survival. Finding a mathematical definition is a fine balance between over-complicating an intuitive definition and over-simplifying a (often) physical concept. When is a place for definitions during a math lesson? I have always encountered definitions at the beginning of a math lesson, especially in calculus. However, could calculus also be taught intuitively, without starting from the definition? After all, this obsession with definition (I would argue) originates with the formation of proofs. The definition of a concept being proved is one of the most useful tools for verifying a mathematical statement around that concept. When, if ever, does mathematics become really intuitive in an upper level? What is the evolution of the necessity of a formal mathematical definition as a function of time in a math classroom?
Monday, 7 October 2013
Yopp - Response
Oh dear. Conjectures in a math high school classroom? Hmm. It's not that I don't think students are capable of doing it, nor do I feel guilty about temporarily frustrating my students so that they can learn something from the process. However, I don't know that the current sort of abstraction present in curriculum would serve students well, or even at all, when disproving conjectures. I would worry that students would get turned off completely from math. I imagine that the "math" that people say they "hate" is really all instrumental/arbitrary. In fact, math is so much more than that; what the general public thinks is math is just the methodology of processes used (mostly) in arithmetic. It seems to me that it is easier for a teacher to use and teach methodology than it is to effectively teach someone how to solve a problem or disprove a conjecture. And on that note, how do we wish to mold students? Do we wish for students to have a high aptitude of mathematics, logic, and abstraction, for students to be laborers capable of completing conversions and estimating necessary materials, functioning taxpayers (or both)? Given that arbitrary math is so disliked, is it worth using conjectures in a classroom? I remember rolling my eyes at any proof that came my way in a high school, because nobody ever bothered to explain why proofs were useful/necessary. My logic said, "I could see it, why did I need to prove it?". Also, at what age level would this be done? Grade 11/12? Earlier? Would the conjecture methodology be a lens or occasional teaching tool? I would like to learn more about this!
Sunday, 6 October 2013
Hoffman/Brahier response
I enjoy problem solving. It's a useful tool in any classroom. Again, by saying it is a tool, I suppose that I limit it from being a way of teaching. There were two scenarios of which I was reminded in this article. I will name two.
The first scenario was when I worked at a math camp over the summer. One of the students in the class did well with rote learning and practice (e.g. learning how to simplify exponential expressions). However, he acted out often, he would distract other students, and he created a challenging learning environment. I found the "remedy" for this was to bring out University of Waterloo math contest problems for him to do. He devoured each and every one of them. He went through any puzzle books I brought in, as well.
The other side was when I took Math 414 at UBC, where I would travel to certain high schools in Vancouver and conduct math workshops. The layout of the workshop involved students being given a certain list of problems to solve in groups, then they would present the solutions to the class. Most students would work on the set of problems they were given, but they wouldn't really pay attention to those presenting alternate solutions. The students presenting were also very hesitant to come up to the board, and in some cases, students would quietly call each other "stupid" or "slow", which I found absolutely vulgar.
I love any puzzle, it's always a delight for me to complete puzzles, especially ones like Kenken (which, if I have my own classroom, I would love to use in a grade eight classroom on the first day in order to gauge how students understand logical thinking and how their basic math facts are). I feel like I could fill up a whole year with nothing but problems. However, would this not require a readjustment of curriculum? One of the IRPs for grade 10, for example, under "Relations and Functions" is that a student must be able to express an equation in slope-point form, slope-intercept form, an general form. So there would still be a need for "arbitrary learning" as was mentioned in the other article. I was unclear as to how these things are balanced in Japan. How is new material introduced in a Japanese classroom? Do students find additional information elsewhere (e.g. online), or do they still have an instructional portion to their classes?
Thursday, 26 September 2013
Hewitt Response
(I'm all over the place in this response, I apologize. I had a hard time maintaining a continuous stream of thought for this one.)
I appreciate that Hewitt makes a distinction between the arbitrary and the necessary. Sometimes, I have the initial reaction as the author did to say, "Because it just is" when asked about an arbitrary point in mathematics, such as positive rotations being counterclockwise. It's another convention decided upon within mathematics, and I remember one of my professors stating it as such. The one practical issue which arises from this article relates to classroom management. My understanding is that Hewitt prefers an inquiry-style of teaching over a direct-instruction one, as direct-teaching tends to create more overlap of teaching both arbitrary and necessary information, whereas inquiry-style learning tends to lean toward using arbitrary information and allows students to determine necessary information for themselves. Initially I wondered about students who find it sufficient to trust a result without proof, and how to use inquiry to engage them. However, I am learning that there is much more to inquiry than previously expected; I have also had the opportunity to see various types of inquiry work, including some IB (International Baccalaureate) assignments. One of the assignments involves calculating a side length based on sine ratio of an equilateral triangle located in a circle, followed by finding the area of the associated triangle, followed by finding the area of two other regular polygons in the circle, then generalizing to n-sided polygons. I would like to design a few assignments like that, just to get a better understanding of inquiry. I have not yet designed a lesson based on inquiry. A lot of my work tends to be teacher-focused; however, I am working on changing this.
Back to Hewitt's article; I also think that distinguishing between arbitrary and necessary avoids misuse of memory aids like, "two negatives make a positive". I've seen multiple abuses of the "rule", but if students were able to follow through with a brief proof, I feel that this could be much better understood. For example,
Dr Math provides an excellent explanation on their website:
I appreciate that Hewitt makes a distinction between the arbitrary and the necessary. Sometimes, I have the initial reaction as the author did to say, "Because it just is" when asked about an arbitrary point in mathematics, such as positive rotations being counterclockwise. It's another convention decided upon within mathematics, and I remember one of my professors stating it as such. The one practical issue which arises from this article relates to classroom management. My understanding is that Hewitt prefers an inquiry-style of teaching over a direct-instruction one, as direct-teaching tends to create more overlap of teaching both arbitrary and necessary information, whereas inquiry-style learning tends to lean toward using arbitrary information and allows students to determine necessary information for themselves. Initially I wondered about students who find it sufficient to trust a result without proof, and how to use inquiry to engage them. However, I am learning that there is much more to inquiry than previously expected; I have also had the opportunity to see various types of inquiry work, including some IB (International Baccalaureate) assignments. One of the assignments involves calculating a side length based on sine ratio of an equilateral triangle located in a circle, followed by finding the area of the associated triangle, followed by finding the area of two other regular polygons in the circle, then generalizing to n-sided polygons. I would like to design a few assignments like that, just to get a better understanding of inquiry. I have not yet designed a lesson based on inquiry. A lot of my work tends to be teacher-focused; however, I am working on changing this.
Back to Hewitt's article; I also think that distinguishing between arbitrary and necessary avoids misuse of memory aids like, "two negatives make a positive". I've seen multiple abuses of the "rule", but if students were able to follow through with a brief proof, I feel that this could be much better understood. For example,
Dr Math provides an excellent explanation on their website:
Let a and b be any two real numbers.
Consider the number x defined by
x = ab + (-a)(b) + (-a)(-b).
We can write
x = ab + (-a)[ (b) + (-b) ] (factor out -a)
= ab + (-a)(0)
= ab + 0
= ab.
Also,
x = [ a + (-a) ]b + (-a)(-b) (factor out b)
= 0 * b + (-a)(-b)
= 0 + (-a)(-b)
= (-a)(-b).
So we have
x = ab
and
x = (-a)(-b)
Two things that are equal to the same thing are equal
to each other, so
ab = (-a)(-b)
Also, suppose a teacher is instructing part of the lesson, and a student asks why a certain rule applies, perhaps one which would have been covered years earlier. At what point can a teacher, if ever, say that something is "the rule"? I know as an undergrad, I was often extremely frustrated when a professor would say, "You should know this by now", because they rarely ever continued with, "But if you don't, here's where you can find how to do x, y, and z." Because of such experiences as an undergraduate student (whether in math courses or others), I tend to over-explain things when I teach, and I was flagged for it by my school advisor when I conducted my two-week practicum last year. I remember a sea of blank stares in a classroom when teaching part of the rational expressions unit; I had questions arising over why I needed to multiply both the denominator and numerator of a fraction in order to add it to another one (in a grade 11 precalculus class). Although I explained it on the spot, I also wonder about the remainder of my students; if I had waited to answer the question, I imagine the student would've understood how to simplify the remaining part of the question, except for that one line of finding common denominators. What about one student who looked absolutely bored as I was going through the explanation of how to add fractions? Next time, can I open that question to the class? Would that have been a better thing to do? Perhaps it would've maintained attention a bit better.
Finally, Hewitt states, "A teacher taking a stance of deliberately not informing students of anything which is necessary is aware that developing as a mathematician is about developing awareness rather than collecting and retaining memories. Furthermore, this stance clarifies for the students the way of working which is appropriate for any particular aspect of the curriculum - the arbitrary has to memorized, but what is necessary is about educating their awareness." This makes me wonder: How many students know a single of the (over four hundred) proofs available for the Pythagorean Theorem? Yet how many can use it effectively throughout their high school mathematics career, onwards? Do students need to be able to explain every math result they encounter? I wonder, is there time to do so for every particular topic of study? How early on can you begin with inquiry in mathematics, or to acknowledge conventions? Wouldn't it be confusing to students to say in kindergarten when first learning numbers to say, "We use the Arabic number system because it is the most useful to use in calculations and we can establish place value based on the order of the numbers, but had we been in another country, our numbers would've been based on a placement of horizontal and vertical bars"?
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