What is effective professional development? Why this topic of inquiry? I, like other classmates, had some difficulty settling on a topic. I made a list of the issues in mathematics teaching that were critical to me. As I looked at the list trying to prioritize and choose what was most important I realized that what I was really trying to get at was how can we help teachers change, to engage, to use new methods and approaches?
To teach mathematics for understanding is not an easy undertaking if one wants to be truly effective. There are many hurdles to clear when you are changing your teaching approach. Teachers who continue to rely on direct instruction may feel they are teaching the curriculum outcomes but they are not focusing on how students will make sense of what they are trying to teach.
Some of the barriers that come between a teacher and their willingness to change include ( but are not limited too: time, understanding, beliefs, knowledge, dispositions and work load.
-teachers already spend countless unpaid overtime hours, planning, correcting, assessing, committee work etc.
-Teachers themselves are struggling with understanding the concepts they are to teach.
- the changes to assessment standards are huge
In fact, they are being bombarded with so many new issues that have identified as necessary components of teaching mathematics for understanding they become overwhelmed and shut down.
Experience and research has led me to believe that mathematics must be taught using inquiry-based methods. I have learned that teachers who are interested in making the change from traditional teaching approaches to the reform are required to obtain new pedagogical, mathematical, and professional knowledge.
At its root change is based in belief, a teacher's beliefs about math are what will influence her/his instructional decisions. To change beliefs requires a lot of work and most teachers are working hard as it is. For me its as author Maya Angelou said "When you know better you do better". For teachers to know better they must be engaged as learners. They must be helped and supported and if the support for change is not given, we will continue to see a stubborn resistance to real change in the mathematics instruction currently used in many Newfoundland and Labrador classrooms.
This is where professional development is needed. However, the usual professional development leaves something to be desired in terms of affecting change. According to Ball and Cohen, (1999) "research indicates that professional development sessions are often "intellectually superficial, disconnected from deep issues of curriculum and learning, fragmented and non-cumulative". Ball and Cohen also say that PD sessions provide little opportunity for teachers to develop deep, flexible, conceptual understanding of mathematics. So I am off on my journey to find out what makes effective professional development and the surrounding issues that make this so difficult to receive.
Saturday, October 24, 2009
Math Musings
As I read through articles I hope will be helpful with my inquiry project I am often struck by what I find. Recently I read "Improving Mathematics Instruction through Classroom-Based Inquiry", Ebby,Ottinger and Silver, (Teaching Children Mathematics, October 2007) Some of what I found as I read, made me think about Phoenix Park and the concerns I have already expressed in previous discussions (blog and class). To understand what I mean I think I should explain the basics of what I read.
The article describes "a university mathematics educator's efforts to support teachers in adopting a stance of critique and inquiry by developing a teacher research community" Ebby designed a course that would give teachers the opportunity to work together, research ideas and make their classrooms served as the site for inquiry into their own teaching practices.In one of the examples discussed the teacher involved discovered she was thinking wrongly about equity. She thought equity was all about allowing students total choice in all things. She based her thinking on what she had learned from Making Sense: Teaching and Learning Mathematics with Understanding ( Hiebert et al. 1997)which said that 'equity entails the assumption that all children can learn mathematics, as well as the assumption that each student must have the opportunity to learn mathematics with understanding" The teacher took this to mean that she must allow her students the choice of working together or alone and so took a hands off approach in her first cycle of inquiry. Over time she found that the students needed more 'explicit guidance' about how to work collaboratively and communicate with one another.
This is a point I raised in the last class, we expect students to work together in partners or groups and explore a problem without having shown them how. Some students may exhibit these skills naturally as part of their innate inquisitive nature, for others, the quality of learning could only be enhanced by knowledge of how communicate thinking with one another. It is true that I may not yet know everything about the preparation process the Phoenix Park teachers went through in developing the curriculum and tasks for students. I wonder if they too thought incorrectly about equity. Their hands off approach to the on or off task behaviours of students may indicate they thought students needed to be given total power of choice. Would direct guidance on how to communicate and work with these open-ended projects have benefited those students who didn't engage in this type of learning. Would their collaboration skills improved to the point where the students themselves came to value the usefulness of working this way? It will be interesting to find out.
The article describes "a university mathematics educator's efforts to support teachers in adopting a stance of critique and inquiry by developing a teacher research community" Ebby designed a course that would give teachers the opportunity to work together, research ideas and make their classrooms served as the site for inquiry into their own teaching practices.In one of the examples discussed the teacher involved discovered she was thinking wrongly about equity. She thought equity was all about allowing students total choice in all things. She based her thinking on what she had learned from Making Sense: Teaching and Learning Mathematics with Understanding ( Hiebert et al. 1997)which said that 'equity entails the assumption that all children can learn mathematics, as well as the assumption that each student must have the opportunity to learn mathematics with understanding" The teacher took this to mean that she must allow her students the choice of working together or alone and so took a hands off approach in her first cycle of inquiry. Over time she found that the students needed more 'explicit guidance' about how to work collaboratively and communicate with one another.
This is a point I raised in the last class, we expect students to work together in partners or groups and explore a problem without having shown them how. Some students may exhibit these skills naturally as part of their innate inquisitive nature, for others, the quality of learning could only be enhanced by knowledge of how communicate thinking with one another. It is true that I may not yet know everything about the preparation process the Phoenix Park teachers went through in developing the curriculum and tasks for students. I wonder if they too thought incorrectly about equity. Their hands off approach to the on or off task behaviours of students may indicate they thought students needed to be given total power of choice. Would direct guidance on how to communicate and work with these open-ended projects have benefited those students who didn't engage in this type of learning. Would their collaboration skills improved to the point where the students themselves came to value the usefulness of working this way? It will be interesting to find out.
Chapter 6 - Finding Out What They Could Do.
Scott( sorry for name mix up), you certainly had an interesting chapter to deal with. The findings that you presented helped bring together issues raised by the previous two chapters. You did a good job of highlighting the statistics in this chapter and helping me make sense of what Boaler found. Great Job, enjoyed it a lot!
As I said above, I found this chapter so very interesting. The fundamental differences between AH and PP that were exposed by Boaler certainly helped delineate what is good and what is not in mathematics education. It was helpful to me as a questioner of " what does this looks like?"(inquiry based learning, as was raised by Terri-Lynn last Thursday. Boaler's activities were designed to "require students to combine and use different areas of mathematics together", and the activities did. Students were not always successful in completing these activities and consequently showed they couldn't combine math areas because the understanding of those areas and their connections to one another had not been made/learned.
When students made 'nonsensical answers' such as a roof's angle being 200 degrees. They demonstrated what I call 'non-attending thinking'. It is obvious in this book and my own experience that students usually do not stop to ask ... does my answer make sense? More than that though they don't have that inner circuitry, that instinctual sense about math that would even cause them to pause or to think they need ask questions. Questions don't pop up because in their minds, math is not about thinking it is about doing.
Another point of interest for me were the differences in the results for year 9 students compared to year 8. There has to be a connection between the improvement and the length of time the students at PP had been involved in this kind of learning. Prior to year 8 their experiences in mathematics classrooms were essentially the same as the students in Amber Hill. More proof, to my mind at least, that inquiry or project-based learning works!
As I stated in class, although perhaps not very coherently, I have some concerns about Phoenix Parks approach. I want to make it clear that I think Phoenix Parks methods to be far superior to Amber Hills. My concern is not about the curriculum and teaching methods. It is regarding the loopholes I see in the programs structure or framework. Specifically, I am speaking about the lack of teacher redirection when students are completely off task and with the lack of organization of student written work. I am questioning whether or it would be beneficial to have some standard for student attending to tasks in order to promote their involvement with the mathematics at hand. Further to that thought, I wonder if the PP teachers can even determine if there are areas of content that have not been covered at all. I do understand the need to build an atmosphere of openness and trust so that students will continue to explore, inquire, question, and contend with the solution to a problem. As for organization of written record of work, its as Dr. Stordy pointed the other night,assessment is not just about what the students put down on paper. Still I wonder how one could sort through the mess of papers to find out what the students recorded and what if anything the written record says about that students needs. I'm sure the answers to some of this will become clearer as we progress through the book. Obviously something informs the project formation by teachers, it is possible my questions come from my limited experience with these methods and that these concerns are really not valid at all. We'll see!
As I said above, I found this chapter so very interesting. The fundamental differences between AH and PP that were exposed by Boaler certainly helped delineate what is good and what is not in mathematics education. It was helpful to me as a questioner of " what does this looks like?"(inquiry based learning, as was raised by Terri-Lynn last Thursday. Boaler's activities were designed to "require students to combine and use different areas of mathematics together", and the activities did. Students were not always successful in completing these activities and consequently showed they couldn't combine math areas because the understanding of those areas and their connections to one another had not been made/learned.
When students made 'nonsensical answers' such as a roof's angle being 200 degrees. They demonstrated what I call 'non-attending thinking'. It is obvious in this book and my own experience that students usually do not stop to ask ... does my answer make sense? More than that though they don't have that inner circuitry, that instinctual sense about math that would even cause them to pause or to think they need ask questions. Questions don't pop up because in their minds, math is not about thinking it is about doing.
Another point of interest for me were the differences in the results for year 9 students compared to year 8. There has to be a connection between the improvement and the length of time the students at PP had been involved in this kind of learning. Prior to year 8 their experiences in mathematics classrooms were essentially the same as the students in Amber Hill. More proof, to my mind at least, that inquiry or project-based learning works!
As I stated in class, although perhaps not very coherently, I have some concerns about Phoenix Parks approach. I want to make it clear that I think Phoenix Parks methods to be far superior to Amber Hills. My concern is not about the curriculum and teaching methods. It is regarding the loopholes I see in the programs structure or framework. Specifically, I am speaking about the lack of teacher redirection when students are completely off task and with the lack of organization of student written work. I am questioning whether or it would be beneficial to have some standard for student attending to tasks in order to promote their involvement with the mathematics at hand. Further to that thought, I wonder if the PP teachers can even determine if there are areas of content that have not been covered at all. I do understand the need to build an atmosphere of openness and trust so that students will continue to explore, inquire, question, and contend with the solution to a problem. As for organization of written record of work, its as Dr. Stordy pointed the other night,assessment is not just about what the students put down on paper. Still I wonder how one could sort through the mess of papers to find out what the students recorded and what if anything the written record says about that students needs. I'm sure the answers to some of this will become clearer as we progress through the book. Obviously something informs the project formation by teachers, it is possible my questions come from my limited experience with these methods and that these concerns are really not valid at all. We'll see!
Michelle 's Presentation
I just realized that in my blog on chapter 5 I hadn't extended congratulations to Michelle on a job well done! Your presentation was wonderful, it was clear and concise it helped us track and make sense of what was said in the chapter. Good on you... as the British would say! ... or is that Australian? , never mind, still applies. Sorry for being tardy with comments!
Sunday, October 18, 2009
Phoenix Park
Standard 1 of NCTM's 'Professional Standards',(1991)is entitled "Worthwhile Mathematics Tasks"
The teacher of mathematics should pose tasks that are based on-sound and significant mathematics:
-knowledge of students' understandings, interests, and experiences;
-knowledge of the range of ways that diverse students learn mathematics
and that:
-engage students' intellect;
-develop students' mathematical understandings and skills;
-stimulate students to make connections and develop a coherent framework for
mathematical ideas;
-call for problem formulation, problem solving, and mathematical reasoning;
- promote communication about mathematics;
- represent mathematics as an ongoing human activity;
-display sensitivity to, and draw on, students' diverse background experiences
and dispositions;
and
-promote the development of all students' dispositions to do mathematics.
Phoenix Park's mathematics program certainly met this standard. The curriculum at Phoenix Park was teacher designed, it required the teachers to "know a lot about the students - what they knew what would be most helpful for them to work on" in this they meet the descriptors from the NCTM standards. In fact the mathematics tasks that the students of Phoenix Park were given to work on allowed for openness and creativity. The teachers supported this creativity by making "deliberate efforts not to structure the work for students", they did not give closed answers to student questions instead they would reform the question in a way that invited the students to explain what they knew and to identify what they needed to find out, something I am currently working on in my own instructional style. The students were guided to make connections and to reason and communicate their thinking and in doing so developed a mathematics disposition that was based on the belief that it was more important to think in mathematics than to remember rules. I was very impressed with the learning opportunities for students at Phoenix Park.
It was very surprising to me that time on task for both Amber Hill and Phoenix Park was about equal. I would say that I had a very strong reaction to the descriptions of students being permitted to wander at will and to be noticeably off task for long periods. There was no attempt by the teacher to refocus or encourage them to return to the work at hand and I found this disturbing. I couldn't help but think that a little structure in this area would have benefited the students. I can only suppose that the teachers felt it more important to create an atmosphere in the classroom where students were unafraid of being wrong and willing to explore mathematics concepts then it was to impose any useful level of discipline. I'm not 100% sure that they aren't right in this but I'm a long way from this in my teaching.
The teacher of mathematics should pose tasks that are based on-sound and significant mathematics:
-knowledge of students' understandings, interests, and experiences;
-knowledge of the range of ways that diverse students learn mathematics
and that:
-engage students' intellect;
-develop students' mathematical understandings and skills;
-stimulate students to make connections and develop a coherent framework for
mathematical ideas;
-call for problem formulation, problem solving, and mathematical reasoning;
- promote communication about mathematics;
- represent mathematics as an ongoing human activity;
-display sensitivity to, and draw on, students' diverse background experiences
and dispositions;
and
-promote the development of all students' dispositions to do mathematics.
Phoenix Park's mathematics program certainly met this standard. The curriculum at Phoenix Park was teacher designed, it required the teachers to "know a lot about the students - what they knew what would be most helpful for them to work on" in this they meet the descriptors from the NCTM standards. In fact the mathematics tasks that the students of Phoenix Park were given to work on allowed for openness and creativity. The teachers supported this creativity by making "deliberate efforts not to structure the work for students", they did not give closed answers to student questions instead they would reform the question in a way that invited the students to explain what they knew and to identify what they needed to find out, something I am currently working on in my own instructional style. The students were guided to make connections and to reason and communicate their thinking and in doing so developed a mathematics disposition that was based on the belief that it was more important to think in mathematics than to remember rules. I was very impressed with the learning opportunities for students at Phoenix Park.
It was very surprising to me that time on task for both Amber Hill and Phoenix Park was about equal. I would say that I had a very strong reaction to the descriptions of students being permitted to wander at will and to be noticeably off task for long periods. There was no attempt by the teacher to refocus or encourage them to return to the work at hand and I found this disturbing. I couldn't help but think that a little structure in this area would have benefited the students. I can only suppose that the teachers felt it more important to create an atmosphere in the classroom where students were unafraid of being wrong and willing to explore mathematics concepts then it was to impose any useful level of discipline. I'm not 100% sure that they aren't right in this but I'm a long way from this in my teaching.
Wednesday, October 14, 2009
Chapter 4 follow up
As I read through chapter 5, my thoughts returned time and again to Chapter 4. I felt like I had a little more to say I guess and wanted to make this post before we discuss the next chapter. The more I read the more I realize how much there is to do to make a classroom an inviting, exciting arena for mathematical learning/understanding. The teachers at Amber Hill, however good intentioned they may be, were not being effective, shouldn't they have been able to see that? I turn the mirror on myself and ask if I have not also been guilty of this very thing. I pay much closer attention now to the what and why when planning and get a great sense of satisfaction from knowing that the small changes I've made can be built upon and its all good for the students!
As long as I'm looking in that mirror I must admit that I have been guilty of 'excessive prompting' in the past. Allowing students to think, that break between question/problem and answer is excruciating. I have noted that the students have noticed the changes too, at times they have such quizzical looks on their faces as in " When is Miss going to step in?" It is a bit of a tight-rope walk and a lot of getting rid of bad habits to know when is the right time to step in. I struggle on! What has inspired me to be aware is the idea that Boaler presented from her observations at Amber Hill " The teachers thought that students would not or could not think" this " learned helplessness" is something I do not want to be a part of continuing. I know there will be times when I will 'fall of the wagon' so to speak but it has become an important issue for me.
Finally, when I think about all that I read and learned in chapter 4, differences between teacher beliefs and actions, the development of negative student attitudes toward mathematics, the lack of true mathematical understanding have all been eye opening for me. However, it is the idea that the students from Amber Hill were considered to have inadequacies based on the social group they were identified as having come from, that bothered me the most. The studies Anyon, ( 1981) cites that find that schools in lower socio-economic areas "discouraged personal assertiveness and intellectual inquisitiveness in students and assigned work that most often involved substantial amounts of rote activity" is perhaps, shameful! ( maybe that's too strong a word? Hmm?) As I comb through research for this course I have come across the following quote, "All students regardless of their personal characteristics, backgrounds, or physical characteristics must have the opportunity to study - and support to learn - mathematics (NCTM, 2000, p.12) hallelujah brother, say it again! Equity is certainly one of the critical issues facing mathematics education today, Amber Hill certainly demonstrates that.
As long as I'm looking in that mirror I must admit that I have been guilty of 'excessive prompting' in the past. Allowing students to think, that break between question/problem and answer is excruciating. I have noted that the students have noticed the changes too, at times they have such quizzical looks on their faces as in " When is Miss going to step in?" It is a bit of a tight-rope walk and a lot of getting rid of bad habits to know when is the right time to step in. I struggle on! What has inspired me to be aware is the idea that Boaler presented from her observations at Amber Hill " The teachers thought that students would not or could not think" this " learned helplessness" is something I do not want to be a part of continuing. I know there will be times when I will 'fall of the wagon' so to speak but it has become an important issue for me.
Finally, when I think about all that I read and learned in chapter 4, differences between teacher beliefs and actions, the development of negative student attitudes toward mathematics, the lack of true mathematical understanding have all been eye opening for me. However, it is the idea that the students from Amber Hill were considered to have inadequacies based on the social group they were identified as having come from, that bothered me the most. The studies Anyon, ( 1981) cites that find that schools in lower socio-economic areas "discouraged personal assertiveness and intellectual inquisitiveness in students and assigned work that most often involved substantial amounts of rote activity" is perhaps, shameful! ( maybe that's too strong a word? Hmm?) As I comb through research for this course I have come across the following quote, "All students regardless of their personal characteristics, backgrounds, or physical characteristics must have the opportunity to study - and support to learn - mathematics (NCTM, 2000, p.12) hallelujah brother, say it again! Equity is certainly one of the critical issues facing mathematics education today, Amber Hill certainly demonstrates that.
Saturday, October 10, 2009
What's Current, What's useful, What's useable?
Inquiry project research has led me down some very different ( and interesting) roads, I'm not the most adept researcher. Just lately I've been considering changing my topic, but haven't reached a decision. One of the reasons is the difficulty I'm having finding information on original topic but mainly its because I've become interested in a couple of other ideas that have cropped up.
New idea maybe: Self -regulated Mathematics Learning??
This area of thought is intriguing to me and seems destined to be connected to differentiated instruction ( althought i've yet to find the connection). Perhaps choosing this will allow me to 'kill two birds with one stone' as I am on the starting end of the learning curve about DI. Mainly I'm interested in any concept or approach that can help inquiry-based learning of mathematics. Students who are good 'problem-solvers' need, I believe, a strong background in mathematics knowledge ( facts,symbols, definitions, algorithms etc) but it is not necessary for this 'knowledge' to be received only through direct-instruction. I truly believe that if we make teachers aware of the 'how to' and not just the 'why' of teaching using an inquiry-based approach we will see it used more and more as an instructional approach. Perhaps my 'inquiry' for this course will lead me to a place that will be helpful in this regard.
I am never sure when an idea, that seems new to me, is actually still current in the field of mathematics. Is self-regulated learning a viable topic or have we moved on .... I guess that's a good question for Thursday!
New idea maybe: Self -regulated Mathematics Learning??
This area of thought is intriguing to me and seems destined to be connected to differentiated instruction ( althought i've yet to find the connection). Perhaps choosing this will allow me to 'kill two birds with one stone' as I am on the starting end of the learning curve about DI. Mainly I'm interested in any concept or approach that can help inquiry-based learning of mathematics. Students who are good 'problem-solvers' need, I believe, a strong background in mathematics knowledge ( facts,symbols, definitions, algorithms etc) but it is not necessary for this 'knowledge' to be received only through direct-instruction. I truly believe that if we make teachers aware of the 'how to' and not just the 'why' of teaching using an inquiry-based approach we will see it used more and more as an instructional approach. Perhaps my 'inquiry' for this course will lead me to a place that will be helpful in this regard.
I am never sure when an idea, that seems new to me, is actually still current in the field of mathematics. Is self-regulated learning a viable topic or have we moved on .... I guess that's a good question for Thursday!
Thursday, October 8, 2009
Critical Reflections - Chapter 4 and My Presentation
I would like to begin by thanking everyone for participating in the discussion surrounding the issues I found in chapter 4, what a relief!
Amber Hill is a school with some critical issues in mathematics that need resolving. The teachers are caring, more than competent, "All the mathematics teachers were well qualified mathematics specialists", and efficient. Their effectiveness, however, is an entirely different matter. I do believe it would be wrong to say that 'no' mathematics learning took place at all ( and Boaler doesn't) but it must also be said that very little true mathematical knowledge or understanding happened.
To read about these classrooms is to recognize the forms of instruction, the ways students felt, the motivations behind teachers' actions. The old adage " people in glass houses shouldn't throw stones" comes to mind. It was heartening to read that "students did not blame their boredom on the intrinsic nature of mathematics" Still we sure have a long way to go folks! See you next Thursday!
Amber Hill is a school with some critical issues in mathematics that need resolving. The teachers are caring, more than competent, "All the mathematics teachers were well qualified mathematics specialists", and efficient. Their effectiveness, however, is an entirely different matter. I do believe it would be wrong to say that 'no' mathematics learning took place at all ( and Boaler doesn't) but it must also be said that very little true mathematical knowledge or understanding happened.
To read about these classrooms is to recognize the forms of instruction, the ways students felt, the motivations behind teachers' actions. The old adage " people in glass houses shouldn't throw stones" comes to mind. It was heartening to read that "students did not blame their boredom on the intrinsic nature of mathematics" Still we sure have a long way to go folks! See you next Thursday!
Critical Reflections - chapters 1-3
I guess I was caught up in preparing for my presentation, I've just realized I hadn't posted a reflection on Chapters 1-3, mea culpa Luckily I'd made a few notes, here they are:
After reading Schoenfeld's introduction, it did in fact "induce me to read on",( of course the fact its required reading played a part too!) This being my first graduate level course I am unfamiliar with the rigors of research and what makes a reliable study, suffice it to say I was suitably impressed by Boaler's explanation of her study and have no doubts that her findings are accurate and founded in truth. I was struck but the inclusion (in both Schoenfeld's and Boaler's introductions) of gender as an issue. I must have been living with my head in the sand, I so thought that had been dealt with ( this was an issue yea those many years ago when I did my Bachelor's degree) . Upon reflection I wonder if it wasn't because I didn't see gender in math. In other words I don't expect the boys to do better than the girls, I just expect ( and find) that there are always a mixture of abilities in any classroom.
I was also struck by the idea of reality versus facade in education. When Boaler described Amber Hill, its Principal, its reception area, classroom/hallway behaviour, I started to get a picture of the school, as was her intention. This was then contrasted with what was happening at Amber Hill,( type of instruction, level of learning etc) and one quickly realized that 'you have bite into the chocolate to find out what flavor is on the inside'. I'm sure as we progress through the book Amber Hill will reveal itself to have some redeeming qualities. I look forward to the journey!
Postscript: Now that I've looked so closely at chapter 4 I realize that as Boaler states (p.47) "the portrayal of mathematics at Amber Hill is quite bleak" More about that in my next entry!
After reading Schoenfeld's introduction, it did in fact "induce me to read on",( of course the fact its required reading played a part too!) This being my first graduate level course I am unfamiliar with the rigors of research and what makes a reliable study, suffice it to say I was suitably impressed by Boaler's explanation of her study and have no doubts that her findings are accurate and founded in truth. I was struck but the inclusion (in both Schoenfeld's and Boaler's introductions) of gender as an issue. I must have been living with my head in the sand, I so thought that had been dealt with ( this was an issue yea those many years ago when I did my Bachelor's degree) . Upon reflection I wonder if it wasn't because I didn't see gender in math. In other words I don't expect the boys to do better than the girls, I just expect ( and find) that there are always a mixture of abilities in any classroom.
I was also struck by the idea of reality versus facade in education. When Boaler described Amber Hill, its Principal, its reception area, classroom/hallway behaviour, I started to get a picture of the school, as was her intention. This was then contrasted with what was happening at Amber Hill,( type of instruction, level of learning etc) and one quickly realized that 'you have bite into the chocolate to find out what flavor is on the inside'. I'm sure as we progress through the book Amber Hill will reveal itself to have some redeeming qualities. I look forward to the journey!
Postscript: Now that I've looked so closely at chapter 4 I realize that as Boaler states (p.47) "the portrayal of mathematics at Amber Hill is quite bleak" More about that in my next entry!
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