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This article investigates the interplay of time words with how children position hands on an analog clock. Using a mathematics discourse framework (Sfard, 2008), we analyzed how students interpreted precise (e.g., 2:30) and relative (e.g., half past 11) times, finding that particular words are dynamically interwoven with activity. Interviews with students in Grades 2 and 4 revealed that different prompts led to different narrative descriptions about time on the clock, with precise times leading to whole-number descriptions and relative times to part-whole descriptions consistent with fractions. Subsequent analysis of assessment performance for students across Grades 2–5 corroborated that specific time prompts led to particular clock interpretations. Implications for theory and the K–12 treatment of time measure are discussed.

Teacher preparation programs face increasing demands to demonstrate the competencies of prospective teachers in their programs, while maintaining a focus on developing high-leverage instructional practices in their methods coursework. This study used a mixed methods approach to examine how the implementation of a simulated edTPA elementary mathematics task influenced prospective teachers' experiences in an elementary mathematics methods course. The course curriculum featured cognitively guided instruction (CGI) as the exemplar for understanding and implementing problembased, cognitively oriented pedagogy. Our findings indicate that elements of both CGI and the simulated edTPA mathematics task worked synergistically to enhance opportunities for CGI-type lesson enactment, support productive changes in beliefs, and contribute to the prospective elementary teachers' (PTs) preparation.

This study examines how collective activity related to multiplication evolved over several class sessions in an elementary mathematics content course that was designed to foster prospective elementary teachers' number-sense development. We document how the class drew on as-if-shared ideas to make sense of multidigit multiplication in terms of partial products and to reason flexibly about products. We document how the class overcame the challenge of accounting for partial products in multidigit multiplication, including particular activities and ways of reasoning that facilitated progress. The results provide insights into how prospective elementary teachers' understanding of multidigit multiplication can develop during a content course and how a sequence of instructional activities and practices can productively leverage the resources that they bring in support of that development.

This article describes how teachers in a professional development course responded to what they noticed about children’s participation in elementary mathematics classrooms and how what they noticed was connected to the teachers’ positionality toward equitable mathematics pedagogy. The study connects scholarship in teacher noticing about children’s mathematical thinking to scholarship in equity and mathematics teacher education. Data from teacher reflections on classroom video were analyzed to identify how teachers attended to, interpreted, and responded to what they noticed about children’s participation. Findings suggest that a lens of participation supported teachers as they considered how to provide more equitable mathematics instruction. Further, the depth to which teachers noticed children’s participation was connected to their positionality as equitable mathematics educators.

www.nctm.org I never even considered what I was getting into. I love teaching mathematics, and I was good at it. However, my principal found a way to support a position for a math support/specialist teacher for our building and asked me to consider applying for the position. Well, I applied, got the job, and was so excited! And then—wow. Nobody could have prepared me for some of the “stuff” that has now become routine regarding the leadership-related demands of my math support/specialist teacher job. Who knew? Who knew that it was seemingly impossible to get into some classrooms to offer my support? Who knew that teachers who used to socialize with me outside of our workplace would “forget” to invite me to our school’s happy hour? Who knew that I would have to literally customize coaching/mentoring strategies for each teacher with whom I worked? Who knew my job was as much about leadership and relationships as it was about mathematics teaching and learning? Elementary Mathematics Leaders

“T o whom do you turn in this school for advice or information about mathematics instruction?” (Spillane, Healey, and Parise 2009, p. 413). When teachers in forty-four schools were asked this question, they were more likely to indicate a teacher leader in their school, rather than the school’s principal or any other administrator. But, who are these teacher leaders for mathematics, and what do they do? In many schools, such leaders are knowledgeable, on-site teachers who support their colleagues’ efforts to interact about all facets of mathematics teaching—curriculum, lesson planning, student work, assessments, and school improvement. Because these teachers are facilitating on-site, jobembedded, professional learning, many school districts are formalizing Influencing Student Achievement

This study investigated how elementary teachers in a mathematics lesson study made sense of student learning, teaching, and content, as related to using representations in teaching multidigit subtraction, and how changes occurred over time in their talk and practice. The lesson-study process paved a group talk path along which teacher talk shifted from superficial to deeper consideration of student learning. By providing a context in which interactions of diverse ideas drove teacher learning, lesson study facilitated teachers making connections between the craft knowledge of teaching and scholarly knowledge. Individual teacher talk paths varied within the group path, and one teacher's learning path and the interaction of different learning paths is discussed.

Solving arithmetic and algebraic word problems is a key component of the Singapore elementary mathematics curriculum. One heuristic taught, the model method, involves drawing a diagram to represent key information in the problem. We describe the model method and a three-phase theoretical framework supporting its use. We conducted 2 studies to examine teachers' perceptions and children's application of the model method. The subjects were 14 primary teachers from 4 schools and 151 Primary 5children. The model method affords higher ability children without access to lettersymbolic algebra a means to represent and solve algebraic word problems. Partly correct solutions suggest that representation is not an all-or-nothing process in which model drawing is either completely correct or completely incorrect. Instead, an incorrect solution could be the consequence of misrepresentation of a single piece of information. Our findings offer avenues of support in word problem solving to children of average ability. Solving arithmetic and algebraic word problems is a key component at every level of the Singapore primary mathematics curriculum (Curriculum Planning Division, 1990; Curriculum Planning & Development Division (CPDD), 2000a). However, the solution of word problems is one of the most problematic areas in the mathematics curriculum. Why are children challenged by word problems? What sort of instructional strategies can be provided to support children in such tasks? Success in solving elementary word problems requires more than sound conceptual knowledge (Riley, Greeno, & Heller, 1983). Findings from the solution of physics word problems show how skilled physics problem solvers solved problems first by constructing elaborate representations of the problem, rather than solving directly from the text-based problem description. These representations often include diagrams that make certain relationships and constraints conspicuous

Secondary mathematics preparation pro grams throughout the United States focus on higher-level mathematics topics such as calculus, discrete mathematics, and abstract algebra. Mathematics educators agree that secondary mathematics teachers should have the deep understanding of the structure of math ematics that is inherent in these higher-level courses. Mathematics methods courses required in these pro grams generally focus on the mathematics traditionally taught in middle school and high school classrooms, as identified by the individual states' standards. By focusing on higher-level mathematics, how ever, we are not preparing prospective mathematics teachers to cope with the reality that exists, especially in inclusion classrooms: namely, that some secondary students do not have the skills they were expected to develop in elementary school. One of the general recommendations made by the authors of The Math ematical Education of Teachers is that teacher prepa ration "courses on fundamental ideas of school math ematics should focus on a thorough development of mathematical ideas" (CBMS 2001). Research has found that teachers of mathematics are more effec tive when they have a profound understanding of the content that is being taught and the content to which students have previously been exposed (Ma 1999). It is clear that professional programs must enable pre service teachers to develop the knowledge and skills they need to teach elementary mathematics skills and concepts when necessary. As early as 1904, David Eugene Smith identified this need in his program descriptions of secondary mathematics teacher prepa

Van De Walle (2007) describes dividing one fraction by another in this way: “Invert the divisor and multiply is probably one of the most mysterious rules in elementary mathematics” (p. 326). Tirosh (2000) concurs and cites research suggesting that “division of fractions is often considered the most mechanical and least understood topic in elementary school” (p. 6) and that students' performance on tasks involving division of fractions is typically very poor. These claims are reflected in the difficulties that college students experience in courses for mathematics for elementary teachers when they try to explain why the invert-andmultiply algorithm works. See the following problem.

When teaching mathematics content courses designed for preservice elementary teachers, we often meet resistance when we ask our students to learn mathematics conceptually; they consider us out of touch with what works in the classroom. Preservice elementary teachers question the need for conceptual understanding, feeling certain that elementary students cannot and will not understand mathematics in this way.

The process that teachers used to increase communication during problem-solving investigations. It demonstrates examples of students’ struggles and successes.

We have probably all seen children who are reluctant to continue to work with a mathematics problem that is not solved easily. In this article, we describe the work of a second-grade class on two challenging, but engaging problems. Although the problems both took several days to solve, the atmosphere of persistence and perseverance that the teacher had created in the classroom was such that the students relished the challenge and were delighted with what they found.

The Missouri Council of Teachers of Mathematics (MCTM) has conducted an annual elementary mathematics contest for students in grades 4, 5, and 6 since 1985. Twenty-five sites throughout Missouri host regional contests in which school-selected participants (three to five per grade level, depending on the number of students) compete in grade-specific tests that measure mathematical estimation and computation, conceptual understanding, and problem-solving ability. The top three winners from each test event and at each grade level are then invited to compete against the winners from other regional contests at a state finals competition.

The Curriculum and Evaluation Standards for School Mathematics “expresses the consensus of professionals in the mathematical sciences for the direction of school mathematics in the next decade” (NCTM 1989, vi). It represents a response to the call for reform in the teaching and learning of mathematics. As one familiar with the preparation of elementary school teachers examines the Curriculum and Evaluation Standards, a sense of doom pervades the otherwise enthusiastic attitude toward the reform represented by the document. Many practicing and prospective teachers are not adequately prepared to meet the challenge of implementing the curriculum standards.

In 1989 the National Council of Teachers of Mathematics published the Curriculum and Evaluation Standards for School Mathematics. A basic assumption shaping the K-4 standards is that “K-4 classrooms need to be equipped with a wide variety of physical materials and supplies” (p. 17). The 5-8 standards also make the assumption that “every classroom will be equipped with ample sets of manipulative materials and supplies …” (p. 67). In using materials in an elementary mathematics classroom, each student can be placed in a cooperative-task group, with each member of the group performing a specific function. This structure establishes a positive interdependence in which each student contributes significantly to the achievement of a learning goal. The following activity describes how this management system can be implemented in a mathematics classroom.

What is being overlooked by those that hold this view is the relative difficulty of explaining to students the meanings of addition and multiplication of fractions. Though the words and symbols for addition and multiplication are given many meanings in mathematics, the meaning for addition most used in elementary mathematics textbooks is combining groups. See figure 1. The meanings given for multiplication in elementary school textbooks are more diverse and often seem unrelated. For example, meanings given to multiplication include repeated addition, arrays (area, etc.), comparative size, rates, and Cartesian product. Though they are often treated as distinct meanings, they can easily be interpreted as different manifestations of the repeated-addition meaning of multiplication. Figure 2 offers examples of different meanings given to multiplication according to application.

Four preservice elementary teachers were studied during a field-based mathematics methods course. The purpose of the study was to investigate the elements of mathematics teaching and learning the preservice teachers found problematic and how they resolved those problems. Data were collected in the form of individual interviews, group discussions, and individual journals. The preservice teachers exhibited concerns about the classroom context, pedagogy of mathematics, children's mathematical thinking, and, to a lesser extent, the mathematics content. The data indicate a relationship between the preservice teachers' locus of authority and the reflective quality of their thinking.

Twenty fourth-grade children were matched according to performance on a whole-number calculation and concepts pretest and assigned at random to one of two groups: wooden base-ten blocks and computerized microworld. Instruction in each group was designed to orient students toward relationships between notation and meaning. Instruction given the two groups was based upon a single script that extended whole number numeration to decimal numeration, and emphasized solving problems in concrete settings while inventing notational schemes to represent steps in solutions. Neither group changed in regard to whole-number notational methods. Blocks children understood decimal numerals as if they were funny whole numbers; Microworld children attempted to give meaning to decimal notational methods, but were largely in a state of disequilibrium at the end of the study.

Unquestionably, the key to success in learning elementary mathematics is through meaningful experiences and practical applications. The skills of computation, however, need to be nurtured by a variety of systematic practice and drill, which build accuracy, efficiency, and retention. An abundance of mathematical games is available commercially and games are relatively easy to devise by the classroom teacher. The use of other forms of practice has also been found to be advantageous.

Complaints are often heard concerning the difficulty of making mathematics rele vant to students who want to see immediate applications. In elementary mathematics courses, applied examples usually have to be contrived or so simplified as to be un believable. An article on toxemia in News week magazine (see Appendix) prompted this attempt to create applied mathematics problems suitable for the upper secondary or lower tertiary level. In the following sec tions, problems dealing with conditional probability, Bayes's formula, the derivative and curve sketching, and elementary statis tical inference will be considered?all be cause of the implications of two sentences in the mentioned article.

search data indicate sex stereotyping is one of the most important factors in the governance of behavioral development, motivation, and self-concept (Brody). Recent reports from the National Assessment of Educational Progress (NAEP) show that in most other learning areas tested "females tended to do better than males. However, the results for science and consumer mathematics show females at a definite disadvantage. These data seem to reinforce the old stereotypes about female inability to cope with 'technical' or 'logical' subjects" (NAEP 1975 [a], p. 35). Extensive sex stereotyping has been well documented in earlier textbooks (Milnar 1973, Rogers 1975). But has sexism been eliminated from the more recently published elementary mathematics series (1971 through 1975)? To answer this question, the authors examined texts from eight major publishers, chosen on the basis of recent publication dates and wide use of the series in schools. The first-, fourth-, and seventhgrade texts were examined in each series to determine whether sex stereotyping had persisted in activities and occupations portrayed in illustrations and in written problems. This sampling of grade levels was selected to identify differences that might appear as the format changed from pictorial mathematics to symbolic mathematics.

The awareness of sexism in the curriculum (Frazier 1973) and the numerous articles on sex-role stereotyping in elementary textbooks prompted me to take a close look at some elementary mathematics series myself. Since I had immediate access to them, I examined two sets of elementary mathematics texts, kindergarten through grade six, both with 1974 copyrights. I am pleased to report that my examination of these books indicates that apparently an all-out effort has been made to eliminate the sexist bias and offensive sexrole stereotyping.

Many elementary school students have difficulty in developing a fraction concept and in learning to compute with fractional numbers (Latino, 1955; Hartung, 1958; Rappaport, 1962; Riese, 1964; Swenson, 1964; Anderson, 1969). The extensive work of Piaget suggests that concepts which in the past were considered to be simple are in reality extremely complex and develop slowly over a number of years (Lovell, 1966). The problem addressed in this report focuses on the complexity that is inherent in the fraction concept, and on the need for its analysis in order that all necessary defining characteristics that can be taught to elementary school students be identified. This focus is in agreement with Engelmann (1969), who lists concept analysis as the first step in developing teaching strategies for purposes of teaching concepts. The fraction concept was thus analyzed into a number of less complex related subconcepts. Based on Gagn6's (1965b) procedure for developing a hierarchy, selected subconcepts were arranged into a hypothesized hierarchy of dependent subconcepts. This hierarchy, entitled "A Hierarchy of Selected Subconcepts of the Fraction Concept (HSSFC)," is shown in Table 1; the behavior associated with each subconcept is described in Table 2. Each subconcept was depicted by a model that had as its unit one of the following: a geometric region (part-whole model), a set (part-group model), or a unit segment of a number line (a specific form of part-whole model). The rationale for each dependency in the HSSFC was based on (a) mathematical development, (b) sequence in elementary mathematics textbooks, and (c) the limited research that has been done on fraction concepts (Polkinghorne, 1935; Gunderson & Gunderson, 1957, 1959; Gunderson, 1958; Novillis, 1969).

L eading teacher educators such as Bloom (1968), Glaser (1968), and Lindval (1961) have questioned the adequacy of the traditional means-referenced conception of instruction. They maintain the means-referenced orientation is inadequate for most instructional decision making. In its place, they suggest that teachers adopt a goal-referenced instructional model. A form of this model as adapted by Trueblood (1971) for individualizing mathematics instruction is presented in Figure 1. The adoption of a goal-referenced instructional model by mathematics curriculum specialists poses new problems for the training of elementary teachers. Specifically, how should teacher education programs be designed to prepare prospective elementary teachers to perform the many professional tasks demanded by the emerging types of individualized instruction? Few, if any, current teacher education programs are specifically designed to train prospective elementary mathematics teachers how to use a goalreferenced instructional model to implement the emerging types of individualized instruction. To begin to meet this need at Pennsylvania State University, a field-based pilot program was organized to study what impact elementary education majors trained to use two instructional procedures would have on the achievement and attitude of elementary children and to design teacher training modules that prepare elementary teachers to perform the professional teaching tasks required to use one of four types of individualized instruction.

Some aspects of competency-based teacher education (CBTE), as they apply to the preparation of teachers of mathematics, are explored in this article. Essential and distinguishing features of CBTE are examined and related to the University of Georgia experience in CBTE, and some nonessential features of CBTE are identified. A “competency” and how competencies may be identified are discussed, along with the effects of a competency orientation on methods courses, student teaching, and internship experiences. Some research trends related to CBTE are also touched on. And finally, the development of a competency is described.

Federal legislation has mandated that a ll children with special education needs should have those specific needs met by the school. Included in this special education group along with the learning disabled, physically handicapped, and mentally reta rded are the gifted and ta lented children. This public mandate and reawakened interest in the early identi fica tion and education of gifted and talented children has brought into sharp focus some serious curriculum problems and interesting challenges to teachers.

T he development of logical reasoning has long been accepted as an important goal of mathematics instruction. The importance of logical reasoning, not only as a long-term goal, but also as a tool for learning and doing elementary mathematics appears to be increasing with the current emphasis on problem solving. The nature of the reasoning processes used by elementary teachers is, therefore, of interest and importance. The use of class and conditional logic among various groups in the general population has been the subject of many studies in recent years; much of this research is reviewed by Wason and Johnson-Laird (1972) and by Roberge (1972). Two studies (Eisenberg & McGinty, 1974; Jansson, 1975) have investigated logical inference among preservice elementary teachers. Overall performance of subjects in these studies was low; Eisenberg and McGinty's population performed somewhat better (an average of 54.5% correct over five problem types) than Jansson's (an average of 36.8% correct over four problem types). Jansson (1975) and Eisenberg and McGinty (1974) revealed that preservice elementary teachers are insufficiently trained in the area of conditional reasoning and advised that development and implementation of instruction in logic for preservice teachers is overdue. Jansson suggested that efficient instruction can be accomplished only after further understanding of the problem is acquired. The current study was undertaken to determine how preservice teachers perform on two types of tasks related to the inference tasks studied by the researchers cited above. One of these tasks is similar to those used by Paris (1973) in a study of logical usage by subjects at five grade levels. The other type of task does not appear to have been studied before. The relationship of the tasks used in the current study to the inference tasks used by Eisenberg and McGinty and by Jansson is clarified by considering truth value as a function (TV) of statements. In traditional inference tasks, subjects are told the values of TV(p C q)# and one of TV(p) and TV(q); they are then asked to determine the missing value TV(p) or TV(q). This type of task is schematized in the first column of Figure 1. Figure 1 also provides schematic descriptions of the two types of tasks

MATHEMATICS educators are once again engaged in a vigorous debate. This time the subject is behavioral objectives. Are they good or are they bad? The arguments pro and con have been rather thoroughly presented. See, for example, the articles of Walbesser (1972), Nichols (1972), Allendoerfer (1971), and Forbes (1971). Before proceeding, however, I would like briefly to remind the reader of the events and debates leading up to the current debate. Soviet space achievements in the 1950s signaled the end of the era of rote learning of elementary mathematics?the "old math." This brought mathematicians and teachers of college-level mathematics into school mathematics (for better or worse). Addition facts, multiplication tables, and time-honored jargon like "invert and multiply" and "minus times minus equals plus" gave way to the concepts and the new jargon of sets, additive inverses, the distributive property, and the empty set. There was much debate in teaching circles (and also among mathematicians) over the pros and cons of the new mathe matics. Nonetheless, the new mathematics greatly changed the curriculum as well as the methodology for all of us. In the late 1960s, the pendulum began to swing again?not back to the old

Ask a roomful of elementary mathematics teachers where their students have the most trouble, and the chorus of "problem solving" is overwhelming. Teachers reply this way because it is true. Study after study continues to cite the area of problem solving as the number-one concern in the classrooms of America as was recently well documented (Suydam 1982, 5660). Some students who can perform arithmetical computation quite well often have difficulty solving word problems that require the application of those same computational skills. Why is this? What is so difficult about solving word problems? The answer may lie in the discrepancy that exists between the skills commonly taught

Previous studies have demonstrated that children use oral calculation procedures not taught in school. The present study provided evidence for situational variables that strongly influence the tendency to use such procedures. It also provided a qualitative analysis of the oral mathematics used by Brazilian third graders. Concrete problem situations were powerful elicitors of oral computation procedures, whereas computation exercises tended to elicit school-learned computation algorithms. Oral computation procedures involved the use of two reliably identifiable routines, decomposition and repeated grouping, that revealed the children's solid understanding of the decimal system. In general, the children were far more successful in using oral mathematics than written mathematics. An understanding of children's oral procedures may be useful in developing more successful programs for elementary mathematics instruction.

Much of the emphasis in the teaching of elementary mathematics, however, does not follow this sequence. A concern with standardized techniques for evaluating mathematics ability has led to extensive use of paper-and-pencil tests. Although these tests are well designed for measuring the third stage of concept acquisition ability to use formal mathematics language they pay little attention to the first two stages of concept acquisition, concrete experience, and personal interpretation. This emphasis on tests, in itself, would be harmless, but the influence of achievement testing on curriculum has been greater than the influence of learning psychology on curriculum. As a result, the ability to use formal mathematics language has become the target of instruction and although concept acquisition is still stated as a goal, instructional procedures do not generally reinforce this intention. Ability to recall concept names and ability to match names with concept definitions have become the criteria of both achieve

Appreciation of mathematics is repeatedly and properly touted as one objective of instruction in mathematics. Formally. appreciati on may be interpreted as sensitive awa reness, especially of aesthetic qualities. Those who pursue mathematical studies often obtain aesthetic satisfactions in perceiving relationships, structures. and the “elegant” nature of ce rtain solutions.

During the 1967/68 school year the Stanford project in computer-assisted instruction in elementa ry mathematics was expanded to include schools in Iowa, Kentucky, and Mississippi, in addition to schools in California. As many as 78 students were able to take arithmetic lessons simultaneously on instructional terminals operated by phone line from the computer of Stanford's Institute for Mathematical Studies in the Social Sciences. The instructional terminals (teletype machines with modified keyboards) were located one to a classroom in some schools and grouped in a single room in other schools. Before describing the workshop held at Stanford for Mississippi teachers, a brief description of the drill-and-practice program in arithmetic skills and concepts will be given.

The typical elementary school teacher of 1970 has been prepared as a general elementary school teacher. She may have had no other preparation to teach mathematics than a methods course; at worst, only a few weeks in a general method course; at best, a couple mathematics courses and a three-credit course in methods. Whatever her preparation, the typical teacher follows the textbook with a fair amount of vigor. This is not to be critical of the teacher since she also teaches six to ten other subjects. It can be said, however, that ordinarily she does not have a clear picture nor understanding of what she is trying to do in mathematics.

First, a word about the present reform movement. It is a national educational movement, with great momentum. It is concerned with subject matter, objectives, and the psychology of learning. Some proponents have called it a revolution, in which the new is giving battle with the traditional. Members of profession are asking, “Do you teach the new mathematics or the old? Do you employ the new theory of learning or the old? Are you modem or traditional?" But more about this dichotomy later.

About fifteen years ago our National Government brought suit for the dissolution of the United States Steel Corporation on the ground that this organization was a combination in restraint of trade and therefore was operating in violation of the Sherman Anti-Trust Law. In the year 1919 the final decision in this case was made by the Supreme Court of the United States.

The title of this article is somewhat misleading. If the article has merit at all, the merit is not in the scientifically established validity of its position. If the reader expects to find here a study of the claims of the different theories concerning transfer of function, he will be disappointed, and had better turn at once to the next article. No references are made here to correlation coefficients, although instances of probable error are frequent.

L/iscovery can be an important process in elementary mathematics education. The unit described here encourages self-discovery and allows the student to occasionally make a discovery that is new even to the teacher. All that the unit requires is a pencil, some paper, a creative imagination, and a "Let's see what would happen if . . ." attitude toward the problem. It is based on the concept of the digit sums of numbers, specifically of sets of numbers that have a pattern linking them together. The topic of digit sums (or digital roots) is an old one, sometimes referred to as "casting out nines," that is experiencing a revival in many textbooks. The topic is simple enough for elementary school children to handle with ease (this article is based upon experiences with fourth-grade children), yet it is one that appears in advanced-level number theory. The ideas are more than mere curiosities.

How much emphasis should be given to symbolic logic in secondary-school mathematics classes?

Recently i had occasion to think somewhat about the teaching of elementary arithmetic, algebra, and analysis—the three A's of mathematics. It occurred to me that some mathematics teachers might be interested in the personal reactions of a mathematical physicist, although these may well be commonplace to education experts. Let me begin with the first A—arithmetic.

vJuided discovery lessons may lead to unexpected discoveries as well as to those that were intended by their teachers. The following account is based upon the experiences of a small group of advanced college students in a course designed to fulfill the spirit of the Nuffield Foundation film, / Do and I Understand. The first part of this article describes a lesson brought by a student with a specific goal to enable the learner to predict the height of an unknown right triangle when certain other information is provided. Because of a prevailing spirit of inquiry additional available discoveries were made. The reader will realize that the experiences described here involve not only ratios and proportions but also similarity, primes, composites, and common factors. The lesson began using a large pegboard surface that had been installed to form part of the wall of the classroom. The student in charge, Shirley Harris, directed the class's attention to a triangle outlined in colored yarn on the pegboard. It was a right triangle with a measure of 30 spaces for the base, Bu and 36 units for the height, Hu as seen in figure 1. She invited someone to construct another right triangle with a base, B2, of 10 units, using vertex A of the first triangle. r/

The “Object-A-Screen” consists of a translucent plastic screen and a panel of low-voltage lights which is shielded to form round dots on the screen when the lights are tumed on. There is a total of eighteen lights, three rows of six each, controlled by a remote panel of switches mounted in the same three-by-six pattern. The position of each switch on the control unit directly corresponds to the position of its related light on the screen. To differentiate among the three rows, each row is a different color: white, yellow, and red, respectively, from top to bottom.

Individualized instruction—what's that?” This question was frequently asked five or more years ago. Today, there is not a good educator around who has not heard the two magic words: individualized instruction. However, to have heard about individualized instruction and to know what it means are two different things. There are an infinite number of interpretations of the meaning of individualized instruction; in other words, it means different things to different people. The Winston Senior Dictionary (1957) defines individualize as “to characterize with distinctive qualities” and defines instruction as “the act of teaching; knowledge imparted.” To put these two together would give a feasible definition of individualized instruction—“teaching with distinctive qualities.” However, this definition is very inadequate, since the distinctive qualities are not described; the definition would also fit poor instruction.

Visual aids—films, still pictures, models, bulletin boards, and so on—are among the most useful tools in education, but they do not teach without intelligent planning and use (Glenn O. Blough and Albert J. Hugget, Elementary School Science and How to Teach It [New York: Dryden Press, 1957], pp. 33–34). Bulletin boards can play an important role in today's mathematics program. However, a bulletin board, if it is to be successful in achieving its purpose, must gain and be worthy of the class's attention.

A source of difficulty and confusion in learning algebra and geometry is found in their terms and symbols. Most terms are Latin words and some symbols are hieroglyphic like. It will be seen that a considerable number of the suggestions of the Committee on Terms and Symbols amount to replacing foreign phraseology by English. The work of the Committee whose report has been recently printed is worthy of all praise. Practically every proposal will commend itself to the thoughtful teacher.

This is an attempt to bring the concepts of mathematical logic within the grasp of the high school student and, at the same time, develop a better foundation for indirect proof.

Many mathematicians think that a more modern mathematics must be introduced into the high schools and colleges. This means that some consideration must be given to basic principles.