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South African Journal of Education
versión On-line ISSN 2076-3433
versión impresa ISSN 0256-0100
S. Afr. j. educ. vol.28 no.2 Pretoria may. 2008
Mathematics for teaching: observations from two case studies
Mercy KazimaI; Vasen PillayII; Jill AdlerIII
ISenior Lecturer in mathematics education at the University of Malawi. Her research interests include mathematics for teaching, teaching and learning mathematics in multilingual classrooms, and mathematical modelling. E-mail: email@example.com
IIHead of the District Management Information Systems Unit for the Gauteng Department of Education in the Ekurhuleni North District and has taught mathematics for 11 years. His research interest is mathematics for teaching. E-mail: firstname.lastname@example.com
IIIDirector of the Marang Centre for Mathematics and Science Education and Chair of Mathematics Education at the University of the Witwatersrand and at King's College, London. Her international research interests include mathematics teacher education and mathematics for teaching. E-mail: firstname.lastname@example.com
We report on two case studies in which we investigated mathematics for teaching. We were interested in the mathematical knowledge teachers need to know, and know how to use, in order to teach mathematics well. The two case studies focused on the teaching of probability in Grade 8 and the teaching of functions in Grade 10. We discuss the mathematics for teaching probability and functions in terms of the mathematical 'problem solving' or 'mathematical work' demanded of the teachers as they taught the two topics. Among the findings are the interesting differences between the demands on the teaching of functions and the demands on the teaching of probability in these two cases. We argue that mathematics for teaching needs to be understood as shaped by the particular topic being taught, as well as by how teachers select to introduce and approach the ideas and concepts they are teaching. We conclude with a discussion of questions raised for mathematics teacher education, together with implications for practice.
Keywords: mathematical knowledge; mathematics for teaching; mathematical work; problem solving; teacher education
The teaching of mathematics does not only require the teacher to be knowledgeable about the topic that is to be taught in the sense that the teacher is proficient in solving any problem within the topic. The teacher needs to know and be able to do more than doing the mathematics for him or herself. The teacher needs mathematical knowledge that is useful and usable for teaching. For example, a teacher must be able to select and clarify appropriate mathematical goals for any lesson taught, and link these with the approach used to teach an idea or concept; a teacher must be able to sequence mathematical tasks into appropriate lesson designs; the teacher must be able to 'evaluate' on the spot the mathematical worth of a learner's explanation or argument; and a teacher needs to be capable of interpreting curriculum materials and also be in a position to explain these to learners and parents if necessary; a teacher must be able to listen to what learners are saying and also be able to pose questions that enable a learner to advance their mathematical thinking. This is by no means an exhaustive list but suffices to show a specificity to the mathematical work of teaching, work that is different from, for example, the mathematical work of a mathematician.
Ball, Bass and Hill (2004) refer to this work as mathematical problem-solving done in and for teaching. They argue that it is productive to think about the kind of mathematical work that teachers do as a special kind of mathematical problem-solving enacted in the practice of teaching. More recently, Hill, Rowan and Ball (2005) have referred to this as the mathematical work of teaching. In this article we report on two similar studies that explored the mathematical work of the teaching two teachers engaged with as they went about their work of teaching mathematics in secondary classrooms in two different Gauteng schools in South Africa. As explained in Adler and Pillay (2007) the notion 'mathematical work' is used to describe the mathematical entailments of the work the teacher does to provide learners with opportunities for mathematical reflection with a focus on the mathematics the teacher draws on to accomplish these tasks. The two studies in question formed part of the QUANTUM1 research project, specifically its interest in mathematics for teaching (MfT).
As will be seen, the studies reflect two different approaches to teaching mathematics one that is well recognised, especially across the further education and training band (FET)2 in present day South Africa, whilst the other reflects an attempt to embrace new curriculum goals in South Africa, particularly activity-based learning.3 These contrasting approaches were not part of the design of the study. Rather they emerged through the study. Together, however, they provided an important window into the mathematical work of teaching. By observing these two teachers in practice we saw that how they chose to introduce concepts, ideas or procedures to their learners, which was in turn a function of their teaching approach, led them to have to deal with different mathematical work. At face value it is obvious that different approaches will demand different mathematical work. We will argue that the specificity of these differences and how they manifest in classroom practice raises significant questions about descriptions of mathematics for teaching on the one hand, with implications for mathematics teacher education on the other.
Mathematics for Teaching
We are using the term 'mathematics for teaching' (MfT) to refer to 'specialised mathematical knowledge that teachers (need to) know and know how to use in their teaching'. Various studies have delved into the dichotomy of 'content versus pedagogy' in an attempt to define the professional knowledge base of teaching. Some of the foundational work that informs and enlightens this includes, inter alia: Shulman's (1986; 1987) notion of Pedagogic Content Knowledge; Ma's (1999) notion of Profound Understanding of Fundamental Mathematics and the notion of Specialised Knowledge for Teaching Mathematics (Ball & Bass, 2000).
Various researchers (Ball and Bass, 2000; Ball, Lubienski & Mewborn, 2001; Ball et al., 2004; Even, 1990; McNamara, 1991) allude to the notion that knowing mathematics for teaching requires knowing in detail the topics and ideas that are fundamental to the school curriculum and beyond. Many seem to support the ideas of Ball and her co-researchers that in mathematics, knowing in detail entails being able to unpack or decompress mathematical ideas so that they can be accessible to learners. In other words, teachers need not only know how to do mathematics but should also know how to use mathematics in the practice of teaching (Adler, 2004). The unpacking that Ball et al. (2004) are referring to is unpacking from mathematics in relation to learner thinking. Unpacking requires understanding the mathematics of a concept, how this might develop in learning, and then the relationship between these i.e. between what others have referred to as the epistemic and the cognitive. Tall, for example, argues that, in teaching, it is important to start from cognitive roots, rather than mathematical foundations. He defines a cognitive root as 'an anchoring concept which the learner finds easy to comprehend, yet forms a basis on which a theory may be built' (Tall, 1989:40). He offers the function machine, for example, as a cognitive root for the development of the function concept; and argues that this connects more effectively with, and lays foundations for, learners' conceptual thinking, than, for example, beginning with a version of the formal definition of a function.
Others argue similarly that the way mathematical concepts or ideas are introduced to learners is critical. Researchers working with the French theory of didactic situations, for example, emphasise the importance of what they call the 'first encounter' the first moment of the didactic process or process of study. They argue that in any pedagogy practice, the first encounter with a mathematical idea or concept needs to be purposefully designed (see Barbé, Bosch, Espinoza, & Gascón, 2005).
Being able to 'unpack' mathematical ideas, design first mathematical encounters, discern appropriate cognitive roots all elaborate Shulman's notion of pedagogic content knowledge (PCK). Shulman described PCK as a 'blending' of subject matter knowledge and pedagogic knowledge' and he distinguishes this knowledge from what he calls 'subject matter knowledge'. Elsewhere, we have argued that the boundary between these categories is not clear in the practice of teaching, and hence refer to the more inclusive notion of mathematics for teaching. We have argued further that knowledge-in-use is always towards some purpose and so never outside of social activity (Adler & Huillet, forthcoming). In relation then to mathematics used in teaching, be it unpacking or introducing concepts, it is inevitably institutionalised,4 constrained by the context of schooling, i.e. particular curricular, particular social practices and so forth (Pillay, 2006a).
The question for us is what is it that teachers need to know and be able to do as they unpack mathematics, design first encounters, or navigate between the cognitive and the epistemic and their inter-relation? Does it matter, and research suggests it does, how mathematical ideas are 'rooted'? Does this differ across mathematical topics? And if these are important, then where and how are teachers to learn these mathematical dimensions of their work? Our studies with teachers on their classroom practices began to illuminate these questions. We focused on two specific content areas in mathematics: Functions and Probability. Following Shulman, there have been a number of topic-focused studies in mathematics (e.g. Marks, 1992 fractions; Sanchez & Llinares, 2003 functions; Even, 1990 functions; Stacey, Helme, Steinle, Baturo, Irwin & Bana, 2001 decimals). Our case studies therefore also add to this growing literature and research.
The case studies
Teaching linear functions in Grade 10 and teaching probability in Grade 8 formed the backdrop against which the two studies were conducted.5 One of the critical questions asked in both studies was what mathematical work of teaching does a teacher engage in as he/she goes about his/her work (i.e. how do they use mathematics in order to teach linear functions or probability)? The two studies were set up in similar ways. The identification of an appropriate case began with finding a reputably successful, qualified and experienced secondary school mathematics teacher who was willing to participate in the research, and who worked in a well-functioning school. These criteria were to ensure that, on the one hand, the teacher had a reasonable foundation of mathematical knowledge and, on the other, that we would capture teaching in situations where not only was the teacher qualified and the school functional in the technical sense, but also where teaching was professional: the teacher was recognized by the school hierarchy and or/the wider staff and field as competent and committed. The second step was to negotiate with the teacher the observation and discussion of a particular topic and set of lessons over at least one week. In both studies the identification of these teachers was also opportunistic in nature since in the case of the functions study, one of us (VP) knew the principal of the school and this gave him access to the school. Whilst in the case of the probability study, he knew the teacher and the teacher in this instance was the principal of the school. Both teachers observed taught at public schools in Gauteng. Data were collected in 2005, prior to the introduction of the new FET curriculum in Grade 10.
With the functions study, Nash6 had had previous experience teaching the section of linear functions to Grade 10 learners and observation revealed his teaching strategy as quite typical of secondary mathematics teaching. He presented his learners with mathematical ideas or concepts, provided examples of how to carry out procedures to solve related tasks, whereafter learners practised these through textbook set exercises. This kind of pedagogic practice is often referred to as traditional.7 With the probability study, although Vuyo8 was an experienced mathematics teacher, he was teaching the topic for the first time. He used an activity-based approach which was in line with the principles governing the new curriculum currently being phased in to South African schools and already implemented in Grade 8.
It is important to take cognisance of the fact that in both studies the intention was not to evaluate the teachers. Neither was our intention in contrasting the work of these two teachers to judge one or the other as 'better' in any way. Our purpose was to learn from them the mathematical demands of teaching these different topics, viz. linear functions and probability, in the ways that they had chosen, and through this to further our understanding of the mathematical work of teaching.
Methodology, data collection and analysis
To engage in this type of investigation of the mathematical work of teaching that Nash and Vuyo grappled with as they went about their work, we had to observe the teachers in practice. Structured observations were done while the teachers were teaching and we were non-participant observers in the class. An observation schedule was designed based on the categories of teaching discussed below. We also video-recorded all lessons so as to keep a more 'permanent' track of the lessons. The video data made it possible for us to focus in more closely and in greater depth, and also to elaborate categories and expand our analytic frame where needed. All video recordings of the lessons were transcribed. Interviews with the teacher augmented the data-collection strategy. However, it is the video data that we focus here.
In order to describe the teachers' work, we needed to analyse each and all of the lessons in full. We therefore needed to divide up the video transcripts in a way that enabled us to analyse the teachers' work over time. In the wider studies, from which this study was drawn, we describe the methodology and data analysis in more detail (Adler & Pillay, 2007), in particular how and why our unit of analysis was what we call an 'evaluative event'. In Bernstein's (1990; 1996) terms any pedagogy transmits evaluation rules, this is to say, in any pedagogic practice, teachers transmit criteria to learners of what it is they are to come to know. In other words, at various points in time the teacher needs to legitimate aspects of the pedagogic discourse (in relation to what it is he wants learners to know), and in order for the teacher to do this he or she must exercise some form of judgement. We were able to chunk all transcripts into a series of evaluative events, the beginnings of which were marked by the teacher introducing an idea or concept, and the ends of which were marked by a move to legitimate or to fix meaning.
In each of these episodes, we were able to focus on what and how the teacher introduced the concepts or ideas he or she was teaching, the mathematical work of teaching the teachers were faced with and enacted, as well as the resources the teacher called on in this work. Our focus was on the tasks of teaching, elaborated in the next section. We also discuss how the two teachers introduced new ideas, concepts or procedures in their class. We do not focus on the resources called on, what we have described elsewhere as legitimating appeals. The extracts that follow in the Findings section each illustrate an event, as well as present our analysis of how mathematical ideas were introduced, and the teachers' mathematical work that ensued.
The framework used to analyse the mathematical work of teaching in the two case studies was drawn, in part, from Ball et al. (2004). Ball and her co-researchers suggest eight aspects of mathematical work of teaching that teachers could engage with as they go about their work which they describe as:
Design mathematically accurate explanations that are comprehensible and useful for learners;
Use mathematically appropriate and comprehensible definitions;
Represent ideas carefully, mapping between a physical or graphical model, the symbolic notation, and the operation or process;
Interpret and make mathematical and pedagogical judgements about learners' questions, solutions, problems, and insights (both predictable and unusual);
Be able to respond productively to learners' mathematical questions and curiosities;
Make judgements about the mathematical quality of instructional materials and modify as necessary;
Be able to pose good mathematical questions and problems that are productive for learners' learning; and
Assess learners' mathematics learning and take the next steps
(Ball et al., 2004:59)
Ball and her co-researchers developed these through a detailed study of teachers' classroom practices, arguing that these tasks of teaching repeatedly appear in teachers' work. This is a practice-based notion of MfT which we found useful in our study of classroom teaching of functions and probability. However, we condensed Ball et al.'s eight categories into six because we found some of the eight aspects overlapping. Specifically, responding to learners questions, interpreting and making pedagogical judgements about learners' productions, and assessing learners' mathematical learning and taking next steps, are all about working with students' ideas. Our six categories were as follows:
Defining attempts to provide a definition
Explanations teachers explain an idea or procedure
Representations teachers represent ideas and in various ways
Working with learners' ideas teachers engage with both expected and unexpected learners' mathematical ideas
Restructuring tasks teachers change set tasks by scaling them either up or down
Questioning teacher asks questions to move the lesson on
There is a shift in how we describe these categories. Our interest was in capturing what the teachers did irrespective of whether this was correct, appropriate, or productive. In the extract below, which illustrates an event, we point to how we recognised these teaching tasks in our data. Across all lessons observed, we recorded all the instances where the teacher responded to a demand for, or judged the need for, a definition, an explanation, a representation, or any of the other aspects above, as the lesson progressed.
For example, in Extract 1, taken from Nash's lessons on functions, a learner asks a question to which he responds with an explanation. Therefore our analysis recorded this as an explanation.
Extract 2 is from Vuyo's lessons on probability. It also illustrates the teacher providing an explanation. This explanation did not follow a question from a learner as above, but rather followed interactions with learners on possible outcomes from a head/tail coin-spinning activity, where it was evident that learners did not distinguish between HT and TH as two different outcomes. This extract is thus at the same time an illustration of the teacher working with learner ideas. The full event (in the methodological sense) from which this extract is drawn is elaborated later and includes discussion of how the activity was introduced, the interactions that followed, and all the teaching tasks that were evident
In the two extracts above we have provided illustrations of how we recognised one of the tasks of teaching (particularly providing explanations), as these appeared in both sets of data. In each example, meaning is legitimated through an explanation by the teacher. The nature of the explanations are clearly different and while this aspect of the two studies (the nature of legitimating appeals) is not in focus in any detail in this article, we will discuss this further later. Our recognition of the other tasks of teaching followed a similar process.
In order to gain some insight into the mathematical work of teaching in each of the two studies, it was necessary for us to exhaust all the data as this provided us with a comprehensive grasp of the teachers' work. This enabled us to see the mathematics for teaching as it was demonstrated by each of the teachers. This therefore led to tallying occurrences so that we could obtain a picture of presence or absence and frequency. In this sense quantification was used to structure an overview of the data analysis. Tables 1 and 2 succinctly capture the mathematical work of teaching that each of the teachers engaged with.
From the two tables we see the different emphases in the mathematical work of teaching that the two teachers enacted. Firstly, we can see that while Vuyo was mostly working with learners' ideas and restructuring tasks, Nash mostly engaged in explanations and representations. It is also intriguing to note that, while Vuyo worked across all these tasks of teaching (though in greater and lesser degrees), we recorded no occurrences when Nash was either engaged with learners' ideas or restructured tasks.
In the next section we provide extracts from each of the teacher's lessons. We focus first on the most predominant work of teaching that each of the two teachers engaged in: thus working with learners' ideas for Vuyo and explanations for Nash. The extracts we selected illustrate these. They also illuminate the way in which each teacher introduced ideas in their lessons. It was in the inter-relationship between these that we were able to interpret and explain the different mathematical work done by these two teachers.
In one of the lessons observed, Vuyo gave his class an activity that required the learners to investigate all possible outcomes from tossing two coins. This marked the beginning of an evaluative event for analysis. The learners were asked to practically toss the coins and record all the possible outcomes. The introduction to this aspect of probability was then categorised as through an activity. Learners worked in groups and recorded their work on charts. Later groups presented their solutions to the rest of the class. Extract 3a shows part of the discussion that followed the activity where one group was presenting to the class. In the extract, H represents 'heads' and T represents 'tails' on a coin.
Whole-class discussion followed where learners demonstrated that they considered HT and TH the same. Order did not seem to matter to them. The discussion made reference to a coin game called kapukapu which the learners were familiar with. Kapukapu is a game where two players chose either heads or tails, then a number of coins are thrown in the air and allowed to land on the ground. The number of heads and the number of tails are counted and the winner is one with the highest number. In the kapukapu game, order does not matter, only the total number of heads and tails. Similarly, learners thought that in the class activity what mattered was the total number of heads and tails and not the order. Some learners did not even seem to understand that there was an order. This is understandable because in a kapukapu game that the learners were familiar with, HT or TH would be reported as one head and one tail, order is not part of the game. At this moment during the lesson, Vuyo had to work with the learners' ideas (that order did not matter) to effectively get the learners to understand the concept of 'all possible outcomes' where order does matter. Consequently, the mathematical work of teaching that was demanded of Vuyo was to work with learners' ideas, in particularly here their intuitive and everyday notions of head/tail options.
Of course as Vuyo dealt with the learners' ideas, he also engaged in other mathematical tasks of teaching such as explaining or questioning. Across the lessons, there were many instances where there were additional demands on the teacher while he was working with learners' ideas. Hence the spread of mathematical work evident in Table 1.
As an illustration, further on during the discussion in Vuyo's class, learners still seemed not to understand that order mattered; they still thought that HT and TH were the same thing. In an attempt to demonstrate to his learners that HT and TH were different outcomes, Vuyo explained as shown in Extract 3b:
We see here that Vuyo offers an example of animals to explain the difference between HT and TH in a coin toss. It is important to note that Vuyo's explanation as shown in the above example followed the discussion with learners which revealed their ideas. In observing the probability lessons, there were many of such instances where learners' ideas were evident. This could be because probabilities are visibly experienced in everyday life. Learners thus bring their everyday knowledge and experiences to the classroom. In particular, the activity here of coin tossing brings into the mathematics class objects used in everyday life, be these as money or within a game as discussed.
More significant than the fact of the coins was the nature of the activity. Vuyo's activity-based tasks provided opportunities for the learners to talk about their everyday experiences. Consequently, learners' ideas derived from their everyday experiences were exposed, and the teacher needed to work with such ideas in order to move the lesson on. One of the demands in the teaching of probability then is the conflict between learners' experiences and the counter-intuitive concepts or mathematical notions related to probability.9 The unpacking Vuyo needed to do in relation to a probability event and all possible outcomes required that he work between learners' intuitions and mathematical foundations. This opens the question as to whether coin tossing activity (seen in many textbooks) is appropriate for establishing, in Tall's terms, a cognitive root for the notion of an event and all possible outcomes in probability theory. It is our contention that this question points to one of the key aspects of mathematics for teaching, and illuminates that this kind of knowledge for teaching has topic specificity.
During lesson 3 on functions, Nash demonstrated, on the board, how to draw the graph of the function: 3x - 2y = 6. He used the dual intercept method of substituting 0 for x and y in the function to get the y-intercept and the x-intercept. For this example he got the points (0,-3) and (2,0), respectively. Extract 4 presents the discussion that followed.
Extract 4 is an example of how Nash conducted his lessons. His explanations dominated the lessons. Learners participated mostly by responding to some of his questions, asking questions of clarification on what to do, and doing some class exercises. As can be seen from the extract, learners were asking questions to which Nash responded with explanations. We did not classify these as working with learners' ideas, as the questions were about what to do. Looking at Nash's explanations, we see that the most common explanations were procedural, following questions from learners about what to do (for example, Learner 1's question of whether they didn't need other points to draw the straight line graph). Procedural explanations were also observed when Nash was commenting on learners' written work. This work of explaining mathematics stands in contrast to the dominant work Vuyo did in his class.10
So, how are these differences observed in the mathematical work done by Vuyo and Nash to be interpreted and explained? What work did they (need to) do? What insights into the notion of mathematics for teaching follow?
In order to provide an answer to these questions we first consider the manner in which the two teachers chose to introduce the concepts in question. In the case of probability, Vuyo introduced concepts through activity- based tasks which brought out learner productions either anticipated or un- anticipated. The teacher therefore needed to work with the learners' ideas in order to proceed with the lesson and move learners from their intuitive notions to mathematical notions. Vuyo's learners had their first encounter with aspects of probability in a form of activities, which often brought out ideas from learners' everyday experiences. Vuyo then had to work from the learners' everyday knowledge or prior experiences to the mathematical idea he wanted them to grasp.
In comparison, in Nash's lessons on linear functions, first encounters were with a range of representations of a linear function. In the extract above, Nash (re)presents a linear equation in a particular form, from which the line is best drawn using the dual intercept method. Nash's work was then on exemplifying the important components of this procedure. This followed an initial representation of a linear function as a set of ordered pairs in a table of values, and then as an equation in standard form where a line could be drawn using the gradient-intercept method. In extract 4, Nash's students asked questions. However, these questions did not lead Nash to engage with their thinking. He was able to move his lessons on with explanations of what to do. The spread of explanations prevalent across Nash's lessons were either solicited by students through questions or errors, or a function of Nash's own judgement that an explanation was needed.
So what did we learn from these two teachers, and their work as they taught probability and functions in their respective classrooms? In the first instance, we suggest that their work differed. Both teachers responded to learners and engaged in explanations. The emphases, however, differed considerably. For Vuyo to move from the activity he set to the probability ideas he wanted his learners to understand, he was confronted with intuitive reasoning on the one hand, and everyday knowledge on the other. We have illustrated that these demands were a function both of the nature of probability and the way in which he chose to introduce learners to these ideas. Probability invokes everyday knowledges and these are often counter to mathematical orientations to the same 'notion'. Moreover, task-based activity provides the space for these intuitions to become visible in the lessons.
Of course, probability lessons do not need to proceed in this way. A teacher could present formal definitions of probability concepts, together with a set of rules or formulae to solve typically problems, and some examples of how to do such calculations (e.g. the number of possible outcomes in an event). In this case, it is less likely that the kinds of demands Vuyo faced would materialise. He would be able to proceed with explanations, similar to how Nash proceeded in his class. And it is possible, that as in Nash's class, many learners would be successful in executing these.
In the same way, lessons on linear functions could proceed through task- based activity. Nash would likely then have to work with how learners interpret the tasks, the meanings they bring, and those he wishes them to grasp. Depending on the kinds of activities used to introduce functions, learners' everyday knowledge and experience might too enter the class and require engagement by the teacher.
Our point is that the mathematical work of teaching is complex, and not adequately captured in categories of tasks of teaching in and of themselves. The work of these teachers reveals that the topic being taught matters, as does they way in which teachers approach the topic and how they introduce mathematical ideas in class. Activity-based teaching is likely to illicit learners' ideas. Mathematics for teaching with activities thus entails working with learners' ideas. In the case of probability, there are significant demands on the teacher here as intuitive notions, structured as they are by everyday experience, run counter to related mathematical notions. They cannot be easily ignored or passed over. Elsewhere (Kazima & Adler, 2006) we have shown that Vuyo indeed faces significant challenges in moving between learners' intuitions, his visual examples that attempt to move these notions towards mathematical ones, and then the formal mathematical notions themselves. In contrast, if linear functions are taught through tightly sequenced representations coupled with ongoing explanations, then learners' thinking does not need to be engaged. This is evident in Nash's lesson when he responds procedurally to a question of whether "you don't need all the other parts". This was an opportunity to probe the learner's thinking, and to have the class puzzle over whether and why two points define a line, so illuminating his assertion of the efficiency of the dual intercept method for drawing a line. Hence our argument that MfT is a function of the topic being taught together with the teachers' approach to teaching, which includes the way ideas are introduced.
It is in this sense then that the categories of teaching tasks (the mathematical work that teachers do) offered by Ball et al. (2004) are useful only to a point. Each needs to be understood more specifically in relation to particular topics in mathematics, and to particular approaches to teaching. In addition, one task not identified, yet suggested by our study, is what others have called 'first encounters'.
Conclusion and implications
There are a number of important questions that arise particularly in relation to mathematics teacher education? Our studies confirmed that the tasks identified by Ball et al. (2004) are not only mathematical, but they take on specific meanings across topics, and across different approaches to teaching. And so the question we posed at the beginning of the article: where do prospective teachers get to know about and be able to engage with, for example, learner thinking; with the range and rationales of possible first encounters; with ways of unpacking notions like the linear functions? What opportunities are there for practising teachers to come to understand that a different approach to teaching requires that they engage in new ways with mathematical thinking and mathematical tasks. And where might these be learned? The main rationale for mathematics teacher education is to prepare and support teachers to teach mathematics well. How then do mathematics teacher education programmes include these mathematical foci that are specific to teaching? How do these programmes provide sufficient engagement with all the relevant topics in the school curriculum? Typically, teacher education includes courses in the subjects that do not extend to include related tasks of teaching. These are dealt with at a general level in methodology courses. Topic-specific tasks of teaching and their relation to approaches to teaching mathematics are rarely addressed. This is a significant challenge for mathematics teacher education, which needs to be embraced if our goals for improving the preparation and support of mathematics teachers, and so the quality of mathematical teaching, are to be realised.
This work formed part of the QUANTUM research project on Mathematics for Teaching, directed by one of us (JA) at the University of the Witwatersrand. The material is based upon work supported by the National Research Foundation under Grant number FA2006031800003.
1. QUANTUM is the name given to an R & D project on quality mathematical education for teachers in South Africa. See Adler and Davis (2006a) for elaboration of the first phases of the project.
2. The further education and training band is the term used to describe the grouping of Grades 10, 11 and 12 in secondary schools across South Africa.
3. Reform in education in South Africa led to the implementation and phasing in of a new curriculum which is still in the process of being phased in.
4. Barbé et al. (2005:240) describe the notion of institutionalised mathematics as "the teacher has some 'given data', such as curricular documentation, textbooks, assessment tasks, national tests, etc. where some components of a mathematical organisation, as well as some pedagogic elements and indications on how to conduct the study, can be found. This is how the educational institution 'informs' the teacher about what mathematics to teach and how to do so". We would add that classroom interactions didactic practice also shape what comes to be mathematics in school.
5. For detailed reports on these studies see Pillay (2006a ; 2006b) and Kazima and Adler (2006), respectively.
6. Pseudonym used to refer to the teacher in the functions study.
7. 'Traditional' is typically used to capture and refer to an approach to mathematics that relies on a textbook where the pattern of mathematical presentation is a description of a rule, convention or procedure, followed by a few examples and then an exercise. Investigative tasks, following a more inductive approach to a concept, thus building on learner activity is typically absent from these kinds of textbooks. However, these textbooks have served the curriculum demands of the day more than adequately.
8. Pseudonym used to refer to the teacher in the probability study.
9. See Kazima and Adler (2006) for further discussion of this aspect of teaching probability.
10. There is much that is interesting in this extract, both in relation to the learners' question and Nash's responses. The way in which these were handled focuses on what to do. The questions about other points can be interpreted as profoundly mathematical how many points define a line? The way this was dealt with reflected the resources drawn on by the teacher in his explanations, and these repeatedly turned on procedures and what to do. This aspect of MfT lies in legitimating appeals in use, and is therefore beyond the scope of this paper.
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Kazima M & Adler J 2006. Mathematical knowledge for teaching: Adding to the description through a study of probability in practice. Pythagoras, 63:46-59. [ Links ]
Marks R 1992. Pedagogical content knowledge: from a mathematical case to a modified conception. Journal of Teacher Education, 41:3-11. [ Links ]
Ma L 1999. Knowing and teaching elementary mathematics: teachers' understanding of fundamental mathematics in China and the United States. New Jersey: Lawrence Erlbaum. [ Links ]
McNamara D 1991. Subject knowledge and its application: Problems and possibilities for teacher educators. Journal of Education for Teaching, 17:113-128. [ Links ]
Pillay V 2006a. An investigation into mathematics for teaching; the kind of mathematical problem-solving a teacher does as he/she goes about his/her work. Research Report. Johannesburg: University of the Witwatersrand. [ Links ]
Pillay V 2006. Mathematical knowledge for teaching functions. Paper presented at the 14th Annual Meeting of the Southern African Association for Research in Mathematics, Science and Technology Education, University of Pretoria, South Africa. [ Links ]
Sanchez V & Llinares S 2003. Four learner teachers' pedagogical reasoning on functions. Journal of Mathematics Teacher Education, 6:5-25. [ Links ]
Shulman LS 1986. Those who understand knowledge growth in teaching. Educational Researcher, 15:4-14. [ Links ]
Shulman LS 1987. Knowledge and teaching: foundations of a new reform. Harvard Educational Review, 57:1-22. [ Links ]
Skemp R 1987. Relational Understanding and Instrumental Understanding. Mathematics Teaching, 77:20-26. [ Links ]
Stacey K, Helme S, Steinle V, Baturo A, Irwin K & Bana J 2001. Preservice teachers' knowledge of difficulties in decimal numeration. Journal of Mathematics Teacher Education, 4:205-225. [ Links ]
Tall D 1989. Concept Images, Generic Organizers, Computers & Curriculum Change. For the Learning of Mathematics, 9:37-42. [ Links ]


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Message Board
Anonymous Message Board
Talk about the novels, new and used books that Anonymous has written!
Author Anonymous's Book Reviews
Fallen: Confessions of a Disbarred Lawyer
A man sets a fantastic goal for himself, to become a lawyer. It is a very high goal for him because to achieve it he had to overcome earlier failures, including poor performance in undergraduate study and a checkered past. The first might keep him out of law school, the second out of the bar. He confronts both, however, as well as his debilitating self doubt, and does become a lawyer, gaining admission to law school, completing it and graduating from it,...
Go Ask Alice
This is the fictitious diary of a young teenage girl, who somehow finds herself lost in the world of drugs during the 1960's. She is introduced to LSD during a party where it was slipped into her drink. She loved how the drug made her feel, and was delighted to learn that there were more drugs out there. Every party she went to, she remembers in bits and pieces, but often only remembers the beautiful colours that the drugs brought to her, and sometimes t...
Go Ask Alice
A fairly normal girl, who is given the name "Alice", gets wrapped up in the world of drugs and revolves in a downward spiral. She is accidentally introduced to drugs when LSD is put in her drink at a party. She traveled away from home with a "friend" who also did drugs. They traveled from place to place, never staying in one place for too long. Alice tells of her life as a druggie, including her new sexual experiences and new drug trips. She eventually t...
It Happened to Nancy
Nancy,a good fourteen year old girl, meets an older college guy named Collin. Collin is supposed to be her best friend and the man of her dreams, but shatters that illusion by raping her, taking her virginity. After Collin leaves Nancy feeling used and broken she finds out she is infectected with the HIV virus, destoying her already frail health. Nancy learns to cope with HIV by the support of her friends and most of all her family. ...
Anonymous booklist
The Bride Stripped Bare
The Bride Stripped Bare plays off the premise that the anonymous diarist's mother found the text after her daughter and grandson disappeared under mysterious circumstances, leaving only their car at the top of a cliff. Their bodies were never found. From there follows the inner secrets of the ostensibly perfect housewife � from her Marrakech honeymoon to her illicit Sevillian affair and her descent into a sexual awakening at the hands of strangers. Her ...

subset: commoncrawl


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The $500 million Superfund cleanup of the canal is being paid for by 20 different parties and individuals responsible for polluting the waterway. The two largest contributors for the tab will be New York City and National Grid.
The EPA has split the canal into three sections and will use a method of dredging and sealing contaminated areas.
At a Community Board 6 Land Use Committee meeting, representatives from National Grid called for changes to the EPA's proposal.
"All of the responsible parties should be involved," said Andrew Prophete, Environmental Project Manager for National Grid.
Prophete was willing to admit that National Grid should be held responsible for helping with the cleanup, but he suggested some changes to the proposal.
He said that the EPA should cap the bottom of the entire canal rather than clean some portions first. One community board members said she was confused how that would be sufficient.
"It looks to me like most of the test sites were very high in metals and various PCBs," she said "I can't really imagine what you're thinking when you say you're only going to clean up hotspots."
"In regions one and two we know that we have significant impacts," replied Prophete. "But what we're saying is that there are systems that you can put in place over these contaminated areas that have absorbing materials."
He went on to say that the EPA's plan to dredge the canal would cause toxins and other chemicals to flow through the water, essentially making the capping process less successful.
Prophete also argued the bulkheads lining the canal won't withstand the EPA's plan.
"The problem with the existing plan is that if you dredge down below you're going to cause these bulkheads to fail," he said. "We want to make sure that there is enough flexibility in the design. Don't pigeonhole the solution."


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A real estate professional who is a member of the National Association of Realtors, a professional association. Realtors include agents that work as residential and commercial real estate brokers, salespeople, property managers, appraisers, counselors and other real estate professionals. More than 1 million real estate agents are realtors, and the term is a registered trademark. Realtors must belong to both a local association or board and a state association.
CA Realty Group Los Angeles Real Estate
CA Association of Realtors
Beverly Hills Real Estate
Hilton & Hyland
Beverly Hills/Greater LA Association of REALTORS
Silver Lake Real Estate
The MLS
Teles Properties
Coldwell Banker
Coldwell Banker
Jade Mills Estates Worldwide
Las Casas Realty, Inc.
Gibson International Real Estate
Prudential California Realty
The Bizzy Blondes
Partners Trust Real Estate Brokerage & Acquisitions
Edgestone Real Estate Inc
Coldwell Banker Residential Brokerage
Dilbeck Real Estate South Pasadena
George Chung Realtors
West Los Angeles Real Estate Group
Nourmand & Associates - Hollywood Real Estate
Wish Sotheby's International Realty
Charles Dunn Company
Coldwell Banker Residential Brokerage
Brock Real Estate
The Noel Team | Sherri Noel
Pardee Properties
Sotheby's International Realty
Keller Williams Realty Los Feliz
Sunset Strip Realty
Silverwood Properties Inc
A Team Realty Inc
Century 21 Beverlywood Realty
CA Realty Training
Westside Rentals
Sky Minor Real Estate
Premier Real Estate Brokerage, Inc.
Real Estate Consultants
The Radcliffe Group
AIR Commercial Real Estate Association
Sunset Group Realty
Ramsey-Shilling Associates
Derrick Ruiz Real Estate Broker
Campbell Wellman
The Rental Girl
CP Financial and CP Realty Inc.
Nanci Edwards Realtor
Clint Lukens Realty
Coldwell Banker Residential Brokerage
Deasy/Penner & Partners
Coldwell Banker Residential Brokerage
The Valerie Fitzgerald Group
Berkshire Hathaway HomeServices California Properties: Los Feliz Office
Coldwell Banker Spectrum
Heyler Realty
Extraordinary Real Estate
ACME Real Estate
For more Realtors results,click here :http://us.wowcity.com/losangeles/?what=realtors
Realtors are expected to be experts in their field and must follow the NAR's code of ethics, which requires agents to uphold specific duties to clients and customers, to the public and to other realtors. Among its many requirements, the code of ethics says that realtors "shall avoid exaggeration, misrepresentation, or concealment of pertinent facts relating to the property or the transaction;" "shall be honest and truthful in their real estate communications and shall present a true picture in their advertising, marketing and other representations;" and shall "pledge themselves to protect and promote the interests of their client" while treating all parties to the transaction honestly.


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You cannot learn if you are focused on not feeling well.
Looking for Registered Nurses to Help Sub
Are you a registered nurse? Do you have some free time during the week when you are available to work? MSAD 75 is looking for substitute nurses. Learn more by calling Karen Kilmer in Human Resources, at 729-9961 x271.
Suicide Awareness and Prevention Training
Any employee of a Maine school system must receive Suicide Awareness and Prevention Training. Please view the training materials and when you are finished, meet with a school counselor for certification approval.
Wellness Tip from Lisa West, Member of the District Health and Wellness Team
Are your kids getting tired of the same old boring sandwiches (and parents , are you growing tired of finding uneaten sandwiches in the lunchbox!)? Try these fun ideas and tips to make packed lunches exciting!
Spice up that PB&J by trading out the jelly for sliced apples, pears or bananas. This is a great way to include fruit in the meal while cutting down on the sugar(especially high fructose corn syrup).
Put a sticker on it! A recent study showed that kids will choose fruit over cookies when it has a sticker of a cartoon character on it. Kids notice packaging, so use this to your advantage and put a sticker of their favorite character on a bag of carrots or other healthy food.
Entrovirus 68 - Learn the Facts
Enterovirus D68 (EV-D68) is one of many non-polio enteroviruses. It can cause mild to severe respiratory illness. Mild symptoms may include fever, runny nose, sneezing, cough, and body and muscle aches. The virus likely spreads from person to person when an infected person coughs, sneezes, or touches contaminated surfaces.
Safe Schools/Healthy Students Action Center
This site provides assistance to Safe Schools/Healthy Students and School Action Grantees. The focus of the project is to enhance the safety of the school environment; to reduce self-reported drug, alcohol, and tobacco product use and abuse; to provide early, effective, and affordable interventions for troubled children; and to foster growth in resiliency and the reduction of risk factors in early childhood through family outreach programs.
Seizure Care


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We are one of the most experienced teams on ground with over 150+ years of collective experience in capital markets. Over the past 15 years we have helped entrepreneurs in India build leading businesses across diverse sectors such as Technology, Ecommerce, Healthcare, Financial Services, Consumer Brands, Infrastructure, etc.
Capital is only a small part of what we bring to the table. Our passion for our entrepreneurs' businesses, and ability to add value beyond the clichés is the foundation of our success. Our marquee investors, who include long term institutional investors like pension funds, foundations, endowments. fund of funds and large corporations have backed us across economic cycles – a vote of confidence in our ability to excel in private equity asset class.


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Spider Loc (Feat. Yorz True, T Bo & Colombian Blacc) - Talk About It
new video from sipder loc Feat Yorz True, T Bo and Columbo Blacc. Shout out to Lou
Please click the "Report" button below if the video on this page is not working properly.
Comments will only be displayed by verified emails. No Spamming or Racism.


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January 21, 2014 09:25 ET
CPA Ontario Student Competition for Money, Prizes and Campus Pride
TORONTO, ONTARIO--(Marketwired - Jan. 21, 2014) - Students from across the province - including six from McMaster University - are competing for pride and prizes in the Battle for CA$H in Toronto on January 25. The Chartered Professional Accountants of Ontario holds this annual competition to give university students considering a career as a Chartered Professional Accountant the opportunity to compete, have fun and earn some cash prizes.
A total of 37 three-person teams are traveling to the CPA Ontario Office in Toronto for the competition. Team members are undergraduate business students selected by the CPA Ontario's Board of Ambassadors in partnership with accounting/business society executives from 20 university programs across the province.
McMaster University will be represented by:
The CA$H Competition pits the teams against each other in a friendly competition stressing the use of CPA skills such as communications, teamwork, leadership and time management.
First prize winners get the prestigious CA$H Cup, plus $3,000 for first place and an additional $1,500 for the university's accounting/business society. Additional prizes include $1,500 for second, $750 for third, and $300 each for fourth and fifth place. All teams will also be aiming to succeed last year's winner - a team from Queen's University - as the CA$H champions.
CPA Ontario can put reporters in touch with participating students in most Ontario communities. The top five university teams will be announced at the event on January 25, 2014 with a news release to follow on January 27, 2014. Professional and student media are invited to attend the event.
The university programs taking part are:
About the Chartered Professional Accountants of Ontario
The Chartered Professional Accountants of Ontario (registered name of The Institute of Chartered Accountants of Ontario) is the qualifying and regulatory body of Ontario's almost 37,000 CPAs and 6,000 CPA Students. With a rich history dating back to 1879, the Chartered Professional Accountants of Ontario serves the public interest by developing, monitoring and enforcing the highest standards of professional competence and ethical conduct. Chartered Professional Accountants are valued by organizations of all types for their financial expertise, strategic thinking, business insight, management skills and leadership. The Chartered Professional Accountants of Ontario's website is: www.cpaontario.ca. Information for students can be found under "High School & University".
The Chartered Professional Accountants of OntarioPerry Jensen416-969-4271 or 1-800-387-0735, ext. firstname.lastname@example.com
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CLEARWATER, Fla. — It's still hurricane season, so make sure to properly safeguard chemicals in your home and storage spaces in the event of a storm. Families usually store chemicals such as fertilizers, paints, automotive fluids, pesticides, and cleaners in garages, sheds, kitchens, and bathrooms. If a hurricane's damaging rain were to flood your home and break these containers, dangerous chemicals would be introduced into the environment, endangering public health, wildlife, and our ecosystem.
Now is a great time — before the storm — to rid your home of chemicals you don't use anymore. Take them to the Household Electronics and Chemical Collection Center (HEC3), operated by Pinellas County Utilities. Simply drop off your hazardous chemicals for free at 2855 109th Ave N., St. Petersburg. To learn more, visit www.myclearwater.com or call (727) 464-7500.
If you bring chemicals to the HEC3 center, remember:
– Don't mix chemicals together.
– Keep products in original labeled containers, if possible.
– Place leaky containers in clear plastic bags and transport in boxes with newspaper.
– Put boxes in the trunk or in the back of vehicle away from passenger.
The HEC3 Center cannot accept automotive/marine batteries, biological/infectious waste, empty paint cans, explosives, fire extinguishers, propane tanks, radioactive waste or smoke detectors. Call (727) 464-7500 for more information about the HEC3 facility, or call the city's Environmental Division at (727) 562-4750.
# # #
City of Clearwater Web site – http://www.myclearwater.com


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Big name developers like Bungie, Ubisoft, and Treyarch have budgets reaching into the millions, as well as a staff of designers and programmers working around the clock to bring the next blockbuster to your doorstep. It's an incredibly bustling market, one that frequently brings in more revenue than movies, and propels video games into the mainstream consciousness without ever batting an eye. Of course, that's just one face of the video game industry.
We've seen a virtual barrage of indie games for consoles and mobile platforms in recent years, as small studios — sometimes with just one or two people — create inventive, emotional projects that push the boundaries of what a "game" can be. Journey, the stunning adventure game for the PlayStation 3 from developer thatgamecompany, took home six out of 10 awards at the 2013 Game Developers Choice Awards in San Francisco. Other hit titles, like Bastion and Minecraft, continue to showcase just how powerful indie gaming has become in recent years.
However, this is article isn't here to tout the success of others; it's here to help you create a success of your own to tout. Making an innovative and groundbreaking game is just as difficult as creating a hit song, if not more. It requires a combination of hard work and innovation, a perhaps a dash of genius to boot. We are in no way suggesting that just anyone can sit down and make a game like Super Meat Boy or Limbo off the bat. However, making a playable game is not as insane as you might think; it just takes a little bit of time and patience. Here's our quick guide on how to make a (very simple) video game. No experience necessary.
Note: I designed a little 8-bit game of my own, titled "The DT Express Quest," on behalf of Digital Trends to go along with the article. I had no previous experience crafting video games, nor did I invest any money into making the project, but it's a working video game I can write home about after a mere 10 hours of trial-and-error tinkering. Windows users can give it a whirl and help boost my self-esteem. There may be other ports down the line.
Give it try by clicking the download link below:
Conceptualize the video game
First things first, think about what you're doing before you plow full-steam ahead with your game. It may not be a necessary step if you are just tinkering, but it's one we highly recommend if you want your game to contain at least an ounce or so of substance. It's easy to fall prey to the curse of second guessing your ideas and wanting to go back and continually replace what you've done. You can easily get stuck in this pattern forever. Visualize the kind of game you want to make, but make sure it is within your limitations as an amateur game designer. Making a completely immersive 3D world on par with the likes of Skyrim and Bioshock is out of the question, but that doesn't mean you shouldn't take the time to flesh out your game's beginnings. Below are a few suggestions of things to think about from the get-go. Remember you can always expand later and change things later, but having the basics locked down will help.
- Know the type of game you want to make (i.e. a platformer, shooter, RPG).
- Know the budget. There are both free and premium options ripe for the taking.
- Know the length. Keep it short or attempt something on the sprawling side?
- Know the basic plot. It doesn't have to be elaborate, just have a general idea of the game's goal.
- Know your skill level. Start with something simple and build from there.
* The DT Express Quest Conceptualized: I knew I wanted to create an 8-bit, side-scrolling game incorporating our 404 page. And since we've recently become completely obsessed with coffee here at DT (seriously, our new coffee machine is so sophisticated it may become sentient and try to take over the world), I thought it fitting to throw that in the mix as well. I chose to keep the game short for the sake of the article, offering only one level and few obstacles as part of my video game endeavor – but I also knew that I wanted to expand it down the line. The plot fluctuated from time to time, but it always revolved around our CEO Ian Bell reclaiming the captured espresso machine at the end of the level and garnering points through coffee mugs. The cliche spikes and flame-throwing baddies were a must, as was that over-the-top 8-bit rendition of the Requiem for Dream theme (credit of Joel Pöllänen).


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Northern Michigan Home & Cottage Construction: Hydroelectric House
As the managing editor of Northern Home & Cottage magazine, I'm inspired by the rebirth of our Northern Michigan construction industry—after, gulp, those dark days in the wake of the 2008 Great Recession.
What I am seeing delights me: new homes and renovations that are green, efficiently designed and pushing the style envelope in sophisticated directions. Northern Michigan is developing a coterie of visionary architects and the builders who are skilled and experienced enough to execute their plans. Among this group is Michael Fitzhugh, a young architect who cut his professional teeth as the architect of record on downtown Traverse City's State Theatre. Fitzhugh went on to add the posh 101 Park Street project to his resume.
Currently, Fitzhugh is working with Paul Maurer General Contracting on a Leelanau County home that, besides being a paragon of neo-modern style, will boast the first residential use of hydroelectric turbines in the country. Recently I toured the construction site with Maurer, then checked in with Fitzhugh for more information:
This project began with a challenging site—can you give us a word picture? It's at the top of an extremely steep hill with a beautiful view of Lake Michigan at the top. It definitely was challenging, but it also presented a very exciting proposition.
And the plan organically grew from that site? Yes, there was a path through the woods that led up to the ridge where the view was revealed. The path continued along the ridge line. The house really grew out of the natural lines of the site. You enter it from the back of the ridge, and then ascend a stairway located pretty close to where the path in the wood was. And then the home opens up to that view. The windows capture the view. We'll be installing two stacked windows that are 12 feet high and 12 feet wide—that makes a 24-foot tall window. The glass comes from Europe. Nobody in this country makes glass panels that large. They are low-e coated insulated glass. And they are strong enough to withstand the severe winds off the lake.
Maurer's crew will be holding its collective breath as they install those windows, no doubt! But they are only part of the story on this home:
Yes, the homeowner is a partner in a company called Powerdigm. It provides innovations in energy asset management and procurement that allow large energy consumers to get the benefits of energy efficiency without risk or upfront capital by funding projects through private investment and managing the lifecycle of energy equipment. Powerdigm works with energy efficiency firms that have provided energy solutions to large companies like Ford, Goodyear and General Motors. One of the many systems used at industrial plants is the use of waste fluid to generate hydroelectricity as a means to offset a plant's electrical costs.
One of the homeowner's partners in Powerdigm walked the site and came up with the idea of collecting rainwater and groundwater expelled from the home's open-loop geothermal heating and cooling system and using it to generate hydroelectric power to help power the home.
Sounds complicated:
It's known technology that uses small turbines. The difference is that to our knowledge they've never been used on a residential project before. Any complications actually had to do with the site layout—the home's design didn't have to be altered. There needed to be a water collection system and pipes to the turbines.
And could this type of residential hydroelectric system be replicated in other homes—or would you need an especially steep site like this one? They calculated that we would need between a 40- and 50-foot drop on the site. The typical two-story house is 35 tall. So any two-story house on a hill could potentially use this type of system.
How much power will the system generate for the house? The system is still being tested and we hope to have this available in early spring as the house is completed and all of the electrical loads are determined.
This home also utilizes another cutting edge construction system called rain screen siding. Can you explain? This is a new siding method popular on the West Coast—although it is approved for snowy climates. The siding is actually attached so that there is a two-inch space between the house and the siding. That acts as a layer of air insulation. It is also serves as a windbreak because it prevents the wind from actually hitting the home.
And that siding itself is worth commenting on: It's from Europe too. It's a laminate material with that has a wood-grain design and a lot of green qualities such as low VOC and low formaldehyde. Like almost everything else on this house, it's not an off-the-shelf product, so we had to really think through the installation. We didn't want to waste any, so we did shop drawings of the exact dimensions of every panel. Then each panel came with this thin coating of plastic, so here is Maurer's crew on freezing mornings trying to peel it off. They taught themselves little tricks along the way like leaving in the garage so it would warm up enough to make the plastic come off easier.
Okay, now the grand finale. Tell me about those pools on the first floor: Yes, the hot tub and cold plunge pool. Those will also be tied to the geothermal heating and cooling system. Besides being pools, they are integrated into the geothermal heating and cooling system to act as reservoirs for storing hot water and cold water. It's a custom system that we worked on with D & W Mechanical.
Including this spectacular home and beyond, what trends to you see in the Northern Michigan residential construction industry?
Things have really picked up lately, and clients are more aware [than they were before 2008] of how they are spending money. They are making better, more energy-conscious decisions. Before in the industry it was how fast can we build and move on to the next project. Homeowners have learned that if they don't invest in energy efficient systems their house won't hold its value.


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Reviewed by E.K. (age 9)
I thought that this book was intense, surprising and cool. Itĺs like Warriors Darkest hour because that has a giant battle in it. I think that the author Erin Hunterĺs writing style is to make her books very, very, very intense. Even though itĺs intense, I still think you should read this book.
I recommend this book to all readers that like suspenseful books. I like this book because it has 3 mysteries. Also when a journey is going on the book switches off between the journey and what is going on at the place where all the cats are. STILL READ THIS BOOK FOR THE LAST TIME!!


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Hot Topics
A Skeptical Look at the Spooky2 Rife System
Electro Interstitial Scans:
Another Test to Avoid
A Skeptical Look at Anodyne Therapy and Other Low Level Laser Treatment Systems
Dubious Claims for the Molecular Magnetic Energizer
About Device Watch
Federal Regulation
- Overview of FDA's Device Regulations
- What are 510(k) Clearance and Premarket Approval? (updated 12/26/08)
- Check the 510(k) Database
- Search the Premarket Approval (PMA) Database
- FDA Guidance Document for Galvanic Skin Response Measurement Devices
- FDA Policy on Ozone Devices
- FDA Import Alert List (devices with an import ban)
Device Quackery
- Signs of A Quack Device
- Quackery in the Medical Device Field (1963) (posted 4/21/06)
- Big Quack Attack: Medical Devices (FDA, 1980) (posted 12/10/12)
Questionable Devices
- Ab Force
- Activator Adjusting Instrument
- AcuGraph
- Aculife Magnetic Wave Therapist
- Aladdin H TR-1000 and TR-2000
- Alex Chiu's Immortality Device
- Amalgameter
- Aqua Detox / Ionic Cleansing
- Atlas orthogonal percussion instrument
- Bio-Acoustical Utilization Device (BAUD)
Bio-Electric Field Enhancement (BEFE)
- BioEnergy Light Therapy System
- Bio-Ionic System (Evans Rapsomanikis)
- Bio-Lyfe Immune Booster
- BioPhotonic Scanner
- BioPhotrex
- Bioray Light and Sound Generator
- Biotape
- BioTracker
- Body Vibes
- Brain Wave Synchronizers
- Cancer Detector LEC-03
- CardioVision (posted 3/21/10)
- Cavitat ultrasonograph (link to another site)
- Chi Machine
- Copper bracelets for arthritis (posted 9/21/13)
- CTD-Mark I for Carpal Tunnel Syndrome
- CRT 2000 (link to FDA warning letter)
- Crystal Clear Electron Water/Air Machine (John Ellis)
- Dental Metal Activity Meter (DMA-Meter)
- Detox Foot Pads (updated 11/12/10)
- Dynatron (link to another site)
- E-Meter (link to another site)
- E-Z Detox
- Ear Candles
- EAV Devices: Accupath 1000, AcuGraph, Asyra, Avatar, BICOM, Bio-Tron, Biomeridian, Computron, CSA 2001,Dermatron, Diacom, DiagnoMètre, Eclosion, e-Lybra 8, ELAST, Interro, Interactive Query System (IQS), I-Tronic, Kindling, LISTEN System, MORA, Matrix Physique System, Meridian Energy Analysis Device (MEAD, MSAS, Oberon, Omega Acubase, Omega Vision, Orion System, Phazx, Prognos, Prophyle, Punctos III, Quest4, Sensitiv Imago, SpectraVision, Syncrometer, Vantage, Vegatest, Victor-Vitalpunkt Diagnose, Vistron, Vitel 618, and ZYTO.
- EB-305 Energy Balancer
- Electro Interstitial Scanning (EIS System)
- Electromagnetic Field Detectors
- Electronic Muscle Stimulators (some uses) (link to FDA consumer information)
- Electro-Magnetic Induction Treatment System (EMIT System) (posted 2/5/19)
- Ellis Micro-Dynameter
- Endocardiograph (link to another site)
- Energetix Magnetic Jewelry
- F-Scan
- Geoclense Home Harmonizer (link to another site)
- GyroStim (link to another site)
- High-frequency pulsed stimulation devices (link to another site)
- Home Water Treatment Units
- Hothouse FIR Health Builder
- Human Computer Network Interface (HCNI)
- Infratonic 4.3
- InnerQuest Brain Wave Synchronizer
- Interactive Query System (IQS)
- Introspect
- Ionic Breeze
- iTIOVi Nutrition Tracker
- Kirlian Photography
- Laser Facelifts
- Low Level Laser Therapy (LLT) Devices (posted 6/17/09)
- Low-Power Lasers for Smoking Cessation
- Life Force Energy Discs
- Light Beam Generator (LBG)
- Magnagraph Data Collection System
- Magnetic Bracelet
- Magnetic wrist straps for arthritis (posted 9/21/13)
- Magnetic Mug
- Metatron
- Migun Thermal Massage Bed
- Millennium Water Cooler
- Molecular Magnetic Energizer
- Negative Ion Generators (link to another site)
- Neuroinfiniti
- NeuView glasses
- NutriPax
- Oberon (link to another site)
- OligoScan (posted 11/17/13)
- OncoDiagnosticator
- ONDAMED Therapy System
- Orgone accumulators
- Orion Bioscan
- Ozone generators (air cleaners)
- Pap Ion Magnetic Inductor (PAP-IMI)
- Performance Mouthguards (Under Armour, Makkar Pure Power)
- Phiten neckware (link to another site)
- Photon Genie / Photon Genius a/k/a Super Sauna
- Physiospect
- Power Balance Products
- ProAdjuster
- Pyro Energen
- Q-Ray bracelets
- Q2 Energy System
- Qe3Quantum Energy Oscillator
- Quantron Resonance System (QRS)
- Quantec
- Quantum Xrroid (QXCI / SCIO / EPFX)
- QuantumPulse, formerly called Vibrational Integration Bio-photonic Energizer (V.I.B.E.)
- Radionics devices
- ReBuilder
- Reflexology Steering Wheel Cover
- Relax-A-Cizor
- Rife Frequency Generator
- Rita Meter
- ROM Quick-Gym (4-minute exercise machine) (link to another site)
- Sapsheet
- SeaBands acupressure wrist bands (link to another site)
- Scenar 97.4
- Sculptone Vacuum Therapy Machine
- Slim Slippers
- Smarty Whole Body Hyperthermia System
- Sound Amplifiers Are Not Hearing Aids
- Spectro-Chrome
- Spooky2 Rife devices
- Spinal Analysis Machine
- Spinal heat-detecting devices (handheld): Analagraph, Neurocalometer, Nervo-Scope, Nervoscope, Temp-o-Scope
- Stimplus Pro
- Stimulation VII (breast developer)
- The Stimulator
- Syncrometer
- Teslar watches
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- Thyroflex
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- Transfer Enhancement Amplifier
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- VoiceBio
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Legitimate Devices
- FDA Breast Implant Handbook - 2004
- FTC Advice on Hearing Aids (posted 10/20/10)
- Home Healthcare: Medical Devices Checklist (posted 5/16/11)
- Make Sure Devices You Choose Are Designed for You (posted 5/16/11)
- Aquascams (Water-related scams and quackery)
- Bakken Library and Museum: Quack devices
- Misleading Advertisements for Home-Use Devices
- Museum of Questionable Medical Devices
- Dr. Bob's Home Page for Medical Quackery
- Quack Medicine: Imagination Goes Wild
- Quackatorium
- Questionable Devices for Cancer
- Radioactive Quack Cures
- Turn of the Century Electrotherapy Museum
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Thanks all for making the Random League II so enjoyable. Winning or losing just one game in these batles can make a huge diffrence in the final score after 9 rounds. A special tx for benga for organising it. Job well done once again.
I am sure we wil meet most of you again in the RL III.
DYNASTY will be ready to get some revenge.


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Passing the containership SANTA CATALINA while approaching locks from the
Caribbean side of the Panama Canal. From the NOAA ship BELL SHIMADA.
Image ID: ship2170, NOAA's Fleet Then and Now - Sailing for Science Collection
Location: Panama, Colon area
Photographer: Allen Shimada, NOAA/NMFS/OST/AMD
• High Resolution Photo Available
of the U.S. Department of Commerce, National Oceanic & Atmospheric
Adminstration (NOAA),
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Last Updated:
November 10, 2017


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By Clark Groome
New Hope's Bucks County Playhouse was first saved from demolition in 1939. A group of theater artists, including Bucks County resident and famed playwright Moss Hart, turned the doomed gristmill into a theater and, on July 1, opened "Springtime for Harry," starring Edward Everett Horton.
Over the years the Playhouse became one of, if not the, most important summer theaters in the country. Its proximity to New York and Philadelphia, its rural setting and its appealing building lured many American theater greats to its boards early in their careers. Among them were Bea Arthur, Kaye Ballard, Shirley Booth, Kitty Carlisle (who would later marry Moss Hart), Hume Cronyn and Jessica Tandy (in their first appearance together on stage), Colleen Dewhurst, Lillian Gish, Helen Hayes, Kim Hunter, Grace Kelly, Jack Klugman, Bert Lahr, Angela Lansbury, Paul Lynde, Walter Matthau, Liza Minnelli, Leslie Neilson and Robert Redford, to name a few.
Playwrights Kauffman and Hart and Neil Simon premiered shows there. In addition Mary Chase's "Harvey" and James Whitmore's "Give 'em Hell, Harry," started at the Playhouse.
Over the years the Playhouse had its ups and downs. In the 1980s and early 1990s it had something of a revitalization, becoming after years as a semi-professional theater fully professional, meaning it had a contract with Actors Equity, the actors and stage managers union. Over those years I reviewed 24 shows, all in the summertime. The last professional production was 20 years ago in June 1992: A.R. Gurney's "Love Letters."
After that the Playhouse lost its Equity contract, produced less-than-inspiring work and generally fell into financial trouble while also deteriorating physically. It even lost its long-held designation as Pennsylvania's official state theater.
In recent years the building was again threatened with demolition. Broadway producer Jed Bernstein came to the rescue a couple of years ago, determined to save the building — which has been completely renovated — and return the theater to its former glory.
On July 2, almost exactly 83 years after its first show and 20 years after its last Equity production, the Bucks County Playhouse reopened with the Richard Rodgers/Oscar Hammerstein II review, "A Grand Night for Singing."
It's an appropriate choice since Hammerstein, like many other theater artists, called Bucks County home. The show, which recently appeared at the Walnut Street Theatre's Independence Studio on 3, is a delight.
R&H wrote the stage musicals "Oklahoma," "Carousel," "Allegro," "South Pacific," "The King and I," "Me and Juliet," "Pipe Dream." "Flower Drum Song" and "The Sound of Music;" one movie, "State Fair;" and the TV version of "Cinderella." Songs from those productions, often given a new and surprising twist, make the 100-minute production very easy on the ears.
The BCP production is just about flawless. Directed by Broadway veteran Lonny Price, it has an exemplary cast: Courtney Balan, Ron Bohmer, Greg Bosworth, Erin Davie and Kenita R. Miller. They, together with musical director Phil Reno's superb six-piece combo, make this show a stunningly impressive beginning for the Playhouse's new life.
Price's production also features good designers Jim Noone (the simple but versatile set), Nicole V. Moody (the often witty costumes), Kirk Bookman (the evocative lighting), Walter Trarbach (the subtle sound) and choreographer Lorin Latarro.
Throughout most of the show there are pictures of Rodgers and Hammerstein displayed. They seem to be overseeing the proceedings. There's no question that they would be pleased with what they're seeing and hearing. The show and the renewed Bucks County Playhouse are a smash.
Next up for the Playhouse is a production of Neil Simon's "Barefoot in the Park," called "Nobody Loves Me" when it began its life in Bucks County. It will play Aug. 7-26.
For more information about New Hope's Bucks County Playhouse or for tickets to "A Grand Night for Singing," playing through July 29, call 215-315-7788 or visit bcptheater.org


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Western Square Dancing — Thursdays from 7:30 p.m. to 9:30 p.m. Sets in Step Dance Club offers this class at the Community Center of La Cañada Flintridge, 4469 Chevy Chase Drive. Ron Durkee is the professional caller. Dress is casual and the price is $6 (cash) per lesson. Refreshments included. A partner is suggested, but singles are also welcome. For more information call Betty Woodhull at (818) 249-2651.
Music Appreciation — Tuesdays beginning Feb. 15, from 2:30 p.m. to 4:30 p.m. Dr. Ted Stern, professor of music at Glendale Community College, conducts this off-campus music appreciation class for adults. This semester will cover "Film Composers and Other Related Stuff." Classes are held at Crescenta Valley United Methodist Church, 2700 Montrose Ave., Montrose. Cost for the 15-week course is $90. For more information call Dianne at (818) 248-8110 or (818) 445-2300.
Pasadena Bead and Design Show — Jan. 13-16, 10 a.m. to 6 p.m. Artisan bead makers featured along with designers showcasing beads, gold and silver, jewelry, art clothing, metalwork, antiquities and more. Hilton Pasadena, 168 S. Los Robles Ave., Pasadena 91101. Admission $10. Valet parking at $6 is available. For exhibitor and workshop lists go online to http://www.pasadenabeadanddesignshow.com.
Heritage Oak Walk — Saturday, Jan. 15, 11 a.m. Wen Wang leads an informative walk among the five heritage Coast live oaks at Descanso Gardens, 1418 Descanso Drive. Meet at the Center Circle. Free with admission: $8 for adults; $6 for seniors and students with I.D.; $3 for children 5-12. For information call (818) 949-4200.
Clock and Watch Convention — Saturday, Feb. 5, 10 a.m. to 2:30 p.m. Chapters 75 and 133 of the National Association of Watch and Clock Collectors host the annual Greater Los Angeles Regional Clock and Watch Convention at the Pasadena Convention Center Exhibit Hall, 300 E. Green St., Pasadena 91101. Open to the public one day, Feb. 5. Thousands of antique and vintage clocks and watches will be for sale or viewing. Admission is $12.
Food of the Gods — Saturday, Jan. 15, 10 a.m. to 1 p.m. The Huntington Library, Art Collections and Botanical Gardens, 1151 Oxford Road, San Marino 91108. Kids will explore stories of the mythical gods, goddesses and heroes depicted in the bronze sculptures in the art galleries, then cook up a feast inspired by Roman mythology. Led by chef and art educator Maite Gomez-Rejón. Ages 7-12. Fee includes one accompanying adult. Huntington members, $55; non-members $65. To register call (626) 405-2128.
Spanish Dance — Saturday, Jan. 29, 3 p.m. The La Cañada Flintridge branch of the county of Los Angeles Public Library hosts Spanish flamenco dancer Susana Elena and Spanish classical guitarist Almer Imamovic as part of an ongoing series of programs aimed at providing free arts and literary events for children and their families. Susana Elena introduces Spanish dance to children as young as 5. At the show's end children are invited on-stage to join in a simple rumba flamenca danced with scarves. A castanet lesson is offered to those interested after the show. Almer Imamovic plays compositions created for the Spanish guitar. The library is located at 4545 N. Oakwood Ave.
Musica Angelica presents "Bach Songs and Organ Works" — Saturday, Jan. 15, 8 p.m. Guest artists are Ida Aldrian, mezzo-soprano and Dmitry Bondarenko, organ. Neighborhood Church, 301 N. Orange Grove Blvd., Pasadena 91103. Tickets are $39 general admission; $15 students with valid I.D. For more information visit http://www.musicaangelica.org or call (310) 458-4504.
Pasadena Master Chorale hosts Cornell Glee Club — Sunday, Jan. 16, 4 p.m. La Crescenta Presbyterian Church, 2902 Montrose Ave., La Crescenta 91214. The concert will open with a performance by the Pasadena Master Chorale of the "Introit" and "Kyrie" from Mozart's Requiem, K. 626. Soloists will include soprano Lauren Buckley and pianist Renee Gilormini. Jeffrey Bernstein will conduct. The Cornell Glee Club will then present all-male choral music from a variety of cultures, ranging from compositions from the European Renaissance to folk music of the Americas. Tickets are available at the door for a suggested donation of $20. For more information about the Pasadena Master Chorale, visit http://www.pasadenamasterchorale.org.
Haydn's "Drum Roll" Symphony — Saturday, Jan. 22, 8 p.m. Performed by the Los Angeles Chamber Orchestra, conducted by Ignat Solzhenitsyn. Alex Theatre, 216 N. Brand Blvd., Glendale 91203. Ticket prices are $100, $85, $50 $18. Tickets available online at laco.org, or by calling (213) 622-7001, ext. 215. Tickets, if still available, will be sold at the Alex box office on the night of the concert.
Dueling Pianos — Sunday, Jan. 23, 4 p.m. Jack Lantz and Kemp Smeal of La Cañada Presbyterian Church will square off at the Steinways for "Three for Two — Duo Piano Concert." The music is grouped by three, with three selections of each style of music, which will range from classical music to hymns to the Great American Songbook and show tunes, all featuring pieces arranged for two pianists working at separate pianos. The concert will be held in the church sanctuary, 626 Foothill Blvd., La Cañada. Tickets are $15 and can be purchased in advance at the church office or by calling Lori MacDonald at (626) 794-1522. They will also be sold at the door. For additional information, visit http://www.lacanadapc.org or call (818) 790-6708.
CPR course — Thursdays, Jan. 13 and Feb. 10, 6 p.m. both dates. The Crescenta-Cañada YMCA Heartsaver AED (automated external defibrillator) course covers adult, child and infant CPR; adult and child AED; adult, child and infant choking; and stroke detection and intervention. The class lasts approximately three hours and the certification is for two years. Fees are: CC-YMCA member, $40; program member, $50; non-member, $60. Class sizes are limited and pre-registration is required. Call (818) 790-0123. Classes are at the Y, 1930 Foothill Blvd., La Cañada.
First-aid course — Thursdays, Jan. 20 and Feb. 17, 6 p.m. both dates. This Crescenta-Cañada YMCA Heartsaver First Aid Course covers first-aid basics; medical, injury and environmental emergencies. This class lasts approximately two hours and is held at the Y, 1930 Foothill Blvd., La Cañada. Class fees are CC-YMCA member, $10; program member, $15; non-member, $20. Pre-registration is required through the Y's membership office. Call (818) 790-0123.
Parkinson's Support Group — Fridays, Jan. 21, Feb. 18 and March 18. The Crescenta Cañada YMCA hosts monthly meetings for the La Cañada Parkinson's Support Group. This group is free to join and will provide research and support for those with Parkinson's or movement disorders. For more information contact Nancy Turney, Senior Services director, (818) 790-0123, ext. 225.
To submit calendar listings, send an e-mail with the pertinent details to email@example.com.


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We are excited about a number of new products that are now available. They have been in development for a while, and we think they were worth the wait!
We really like the SON Delux "Wide-Body" generator hubs. The wider flanges make for a stronger wheel than the standard Delux hub with its very narrow flange spacing, and the wheel looks a nicer as it fills out the fork.
Now the "Wide-Body" hubs are available with Schmidt's "connector-less" SL system, in which the hub transmits energy to a contact plate in the dropout. No longer do you need to unplug any wires as you remove your wheel.
You need a special fork with the insulated contact plate, so you can't just install one on your existing bike, but it's a neat option for a new bike. The standard "Wide-Body" Delux hub with connectors remains available as well. Both are in stock with 32 and 36 holes. Supplies with the SL system are limited for now, but we hope that this will become a standard production model in the future. More information is here.
The Grand Bois fork crown for wide tires now is available in stainless steel, in addition to the cromoly steel version. To me, the fork crown is the focal point of the bike. It's what I first notice when I look at a bike. Classic bikes often had chrome-plated fork crowns. Stainless steel can be polished to a chrome-like finish without the cost and environmental hazards of chrome-plating.
Stainless steel must be silver-brazed, so a good fit is important. (Silver cannot "bridge" gaps like brass can.) The Grand Bois crown is designed specifically for the Kaisei "Toei Special" fork blades, which use the "old Imperial" oval cross-section at the top and have a very slim profile at the bottom for optimum shock absorption. The blades fit into the crown with a perfect fit, ideal for silver-brazing. More information is here.


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The collaborative nature of the Ann Romney Center for Neurologic Diseases allows Brigham and Women's Hospital to provide the highest level of care while promoting discovery and accelerating the translation of scientific findings into clinical practice.
Comprised of a group of multidisciplinary biomedical research laboratories, The Center for Neurologic Diseases has a central objective in the application of a wide range of current biological methods to the elucidation of the pathogenesis and treatment of certain chronic, unsolved neurological diseases: multiple sclerosis, Alzheimer's disease, and Parkinson's Disease.
Leaders in the field, Dennis Selkoe, MD, and Howard L. Weiner, MD, co-direct The Ann Romney Center for Neurologic Diseases. In addition to a wide range of clinically related research, many of the laboratories that make up The Ann Romney Center for Neurologic Diseases have an additional focus directed and understanding basic human and animal immunology.
Phone: 617-525-5200Fax: 617-525-5252US Postal Service or Package Courier:Ann Romney Center for Neurologic DiseasesHarvard Institutes of Medicine77 Avenue Louis PasteurBoston, Massachusetts 02115
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