The IPFW General **Education** Area I **quantitative** **reasoning** requirement for the College of Arts and Sciences is met by completing, with a grade of C or better, any course listed in the **quantitative** **reasoning** section of Area I (except MA 101),or any **Mathematics** course numbered 153 or higher. Note that for some degree programs, only a subset of the courses listed in the **quantitative**

The data analysed in this research were both **quantitative** and qualitative. The **quantitative** data was mathematical **reasoning** ability test (KPM) score after the teaching and learning process finished. While the qualitative data analysed comprises of various types of errors made by students, the **quantitative** analysis was carried out to draw conclusion based on the proposed hypothesis. It was used to determine students’ achievement, while the qualitative data was used to analyze errors and the learning strategy using the PMRI. Before carrying out the hypothesis test, the normality test of Shapiro-Wilk was applied. The achievements were analyzed using the t-tet. The data obtained is homogenic on Mann- Whitney or heterogenic when normally or not-normally distributed. Indicators for the **mathematics** **reasoning** ability are as follows (1) using or interpreting mathematical model such as formula, graph, table, scheme, and drawing conclusions from them; (2) solving problems using appropriate method such as arithmetical, geometrical, or analytical; (3) communicating information effectively using symbols, visual, numerical, or oral representation; and (4) assessing the accuracy level of the conclusion based on the quantity of information. It also comprises of nonroutine problems on fraction.

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This paper explores teachers ’ perceptions of their learners ’ pro ﬁ ciency in statistical literacy, **reasoning** and thinking. Research in Statistics **education** has prompted a move away from the teaching of statistical skills, towards focusing on the development of statistical literacy, **reasoning** and thinking. The recent South African Grade 10 – 12 **Mathematics** curriculum change re ﬂ ects this move. A speci ﬁ c challenge for South Africa is that teachers should understand the new intended outcomes of statistics when assessing learners. The participants (n = 66) included Grade 12 **Mathematics** teachers (females = 40%) from a district in the Free State, South Africa, selected through convenience sampling. A **quantitative** research approach was used by administering a 13-item Likert scale questionnaire with the Grade 12 **Mathematics** teachers. The responses were summarised descriptively as frequencies and percentages. The results indicated that two in three teachers perceived their learners to obtain acceptable pro ﬁ ciency in statistical literacy as de ﬁ ned by the literature. In contrast, only one in three teachers perceived their learners usually or almost always to be pro ﬁ cient in statistical **reasoning** and statistical thinking as de ﬁ ned by the literature. The ﬁ ndings of this study showed that about half of the **Mathematics** teachers do not see the connection between the action words in the curriculum, and the aspects of statistical **reasoning** and statistical thinking to be assessed. The large percentage of teachers uncertain about the pro ﬁ ciency of their learners in statistical **reasoning** and statistical thinking leads to the conclusion that teachers need to be provided with pre-service or in-service training strengthening their Subject Matter Knowledge and Pedagogical Content Knowledge related to the key intended outcomes of statistics assessment, that is, pro ﬁ ciency in statistical literacy, **reasoning** and thinking.

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Our every day is different from the earlier days and this is largely due to the contribution of science in our life. **Mathematics** is a science whose subject matter has special forms and **quantitative** relationships of the real world. **Mathematics** is a branch of science, which deals with numbers and their operations. It involves calculation, computation, solving of problems etc. Its dictionary meaning states that, ‘**Mathematics** is the science of numbers and space’ or ‘**Mathematics** is the science of measurement, quantity and magnitude’. It is exact, precise, systematic and a logical subject. **Mathematics** reveals hidden patterns that help us to understand the world around us. Now, much more than arithmetic and geometry, **mathematics** today is a diverse discipline that deals with data, measurements and observations from science, with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems. It may also be defined as, ‘**Mathematics** is the study of quantity, structure, space and change; it has historically developed, through the use of abstraction and logical **reasoning**, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. We can’t live happy life without science. The science has become integral part of our life. Science has also influenced educational enterprise and hence it is also the integral part of our educational system. Learning of science has become unavoidable part of general **education**. The modern civilization is a scientific civilization. In this age the modern society is completely drawn into the scientific environment. Today science has become an integral part of our life and living. Now we cannot think of a world without science. I think teaching is an art and there are born teachers. But there are majority of teachers, who can improve upon by experience of practice and utilization of various methods of teaching science. The basic aim of teaching any subject is to bring about desired change in behaviour. The change in behaviour of student will be indicated through students' capacity to learn effectively. This is only possible by adopting various methods of teaching. The teacher cannot utilize any method to any type of students in any type of environment. One has to choose and adopt

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Many curriculum reforms have focused on the use of contexts and applications of science and **mathematics**. Although the intensity and the role of context use vary across implementations, one aim of all context-based curricula is that students will experience the relevance and applicability of the science content in society and in their personal life worlds (Gilbert 2006). Many studies on context-based interventions report gains in students' attitudes to science and technology, with learning gains similar to those of conventional approaches. Typically, teachers use links to contexts to motivate students and support the learning of **mathematics** content, rather than to develop the ability to explore real-world contexts through the use of **mathematics** (Gainsburg, 2008). An important pillar in constructivist pedagogy is contextualising learning using an authentic environment and real- world examples. A majority of students have difficulties in connecting **mathematics** to real world applications and this could be a reason for failure in **mathematics**. Making **Mathematics** relevant via real world examples 4.2 INQUIRY-BASED LEARNING

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number system consisting of terminating decimals will be in- troduced as a mathematical space or system based on axioms 1 – 3. The role and nature of the axioms in **mathematics** and in constructing a mathematical space will be discussed and given emphasis. The same should be done with respect to the com- plex vector plane where the axioms are: 1) the existence of the unit vector 1 and 2) vector rotation. Vector addition, translation and dot product as well as inversion are introduced here. At all times, both systems will be related to experience. Fractions will be revisited as quotients where the divisors do not contain prime factors other than 2 or 5. Then the student will have the unique experience of extending division to cases where the quotient is ill-defined, e.g., when the divisor has a prime factor other than 2 or 5. In fact, this would be an excellent opportunity to show how a nonterminating decimal arises and how to define and compute with them (Escultura, 2009b) by giving examples. At this point the student will be introduced to some special nonterminating decimals important to science such as and the exponential base e how they arise in **mathematics**.

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Edwards’ (1999) results indicate that students don’t see any need to justify a statement until they are convinced that the statement is true. However, they may think that providing empirical examples is a sufficient justification. Some students go beyond empirical examples and try to create a justification, which may be mathematically sound. These findings are similar to those found in a separate study by Edwards (1997). In the 1997 study, two groups of students were examined, neither of which had any experience with transformational geometry. Their task involved combining transformations. Students were given a combination of transformations, and then asked if they could get the same effect with only one transformation, or “rule”. Edwards found that students tend to over-generalize. For example, many students thought that a double reflection would still result in a reflection, because a double translation still resulted in a translation (this was also the case for a double rotation). When students were asked to explain why certain conjectures were true, they were able to provide “formulas” based on their activities with reflections, translations, and rotations. However, students could still not provide any type of formal proof on their own. When the investigator offered one, the students were able to accept and understand this explanation. Still, students were satisfied with inductive **reasoning**. They tested their conjectures on one or two cases and did not feel that any general proof was necessary. Again, this mirrors the results of Fischbein and Kedem’s (1982) study. A very common problem among high school students seems to be that students are not convinced by a proof. To them, it is too general. They believe its truth only after empirical examples are given. Students do not understand what constitutes a proof, and seem more satisfied giving empirical examples.

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Applicants for admission to the doctoral program in mathe- matics and science **education** must meet the general requirements for admission to both universities with classified graduate standing as outlined in the respective current catalogs. Applicants must also meet the special requirements of this program. These include: (a) an acceptable baccalaureate degree in **mathematics** or science (or a related discipline); b) a master’s degree, or its equivalent, in biology, chemistry, physics, or **mathematics**; (c) a GPA of at least 3.25 in the last 30 semester (or 45 quarter) units of upper division work and at least a 3.5 in the graduate work attempted; (d) good standing in the last institution attended; (e) suitable scores in **quantitative**, verbal, and analytic sections of the Graduate Record Examinations.

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1.3 Theory and literature Owing to the wealth of literature pertaining to the learning of mathematics at university level in general and the material of Analysis in particular, it was de[r]

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relatively long interaction. The questions can develop during the interview process. To avoid the miss information and to save the validity of data, so it should use recorded during in the research activities whether audio or audiovisual.The procedure of research data collection is a task-based on the interview. The researchers do depth interview to the subject about the completion of the given TPM. The researchers as the primary instrument in the interview process to check the results of each answer in mathematical problem-solving. The data is collected in the micro-teaching laboratory at the University of PGRI Madiun with the time set together among researchers, subjects, and the Head of English Department (Kaprodi). Interviews have conducted twice, and the first interview performed when the subject was working on TPM1. And the second interview had done when the subject was working on TPM2. From the results of the interviews revealed the relational **reasoning** profiles of **mathematics**' teacher candidates in the mathematical problem-solving.The Research's indicator of relational **reasoning** profile which done by the subjects includes: In the stage of understanding the problem 1) the subject shows the existence of a problem from the matter. 2) The subject knows the relationship of similarity/difference information between ideas/concepts with other ideas/concepts by explaining. 3) The subject can show the existing information on the matter so that can understand the relationship of similarity/difference between ideas/concepts with others. 4) The subject can know what is known and ask for the problem. In the planning phase of problem-solving; 1) Subject can show information to answer the questions. 2) Subjects can name the basic information to respond to the questions too. 3) The subject can plan to solve the problem. 4

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programs, institutions, university systems or states is strongly discouraged except when the scores are used as one indicator among several appropriate indicators of educational quality. Use of concorded scores. Concordance tables are available at www.ets.org/gre/concordance to help score users transition from using Verbal **Reasoning** and **Quantitative** **Reasoning** scores on the prior 200–800 score scale to using scores on the current 130–170 score scale, and to facilitate the comparison of scores of individuals who took the General Test prior to August 1, 2011 with those who take the revised General Test. The concordance tables show the relationship between the two score scales.

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As the main issue, students must develop competence to decide when and how it is appropriate to use available ICT tools and to use them. These new requests put demands on those teachers who have not developed corresponding competencies themselves. Various designs should be made to made to integrate technology with **mathematics**. The design of these sessions does not see mastering the software as an end but as a means for the teacher to reach his teaching goals. The Principles and Standards for School **Mathematics** recommends that technology is an “essential tool for teaching, learning, and doing **mathematics**” (NCTM, 2000, p. 24). Geogebra was developed for supporting the learning **mathematics** through free exploration in the less constrained environments. Even though GeoGebra can influence what is taught, teachers need to design the suitable instructions and environment that best support this approach. Well-applied GeoGebra can support requirements of learning outcomes as it helps the children process mathematical concepts through investigation and problem solving. Thus Geogebra can act as one of the most important tool in the hands of the teacher for teaching learning of **mathematics** thereby acquiring the ultimate goal of **education** i.e., all round development of the individual!

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One was to determine the degree of relationships between the GRE General Test scores (GRE Verbal Reasoning [GREV], GRE Quantitative Reasoning [GREQ] Test scores), Undergraduate GPA (UG[r]

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Geometer’s Sketchpad, Grapher, MacCalculator, Word, iCal, and PowerSchool. This elective meets every other day for the entire school year and is designed for our non-AP/Honors students. Clearly, these electives would not have been possible without the one-to-one initiative. What we feel we are bringing to our students is an opportunity for a deeper understanding of and broader appreciation for **mathematics** through the use of technology. Students are making connections using many types of applications and are beginning to use their laptops as tools for problem-solving. The second change is in each classroom. Many of us now use the vast supply of Applets available on the internet for demonstration purposes in classroom instruction. We can now “record” our lessons and make them available as documents or podcasts through our local intranet to students who have missed class. Class documents (handouts, etc.) can now be distributed using the local intranet. We offer extra credit in the form of a Problem of the Week that must be completed electronically using laptop applications. Most teachers make use of iCal and encourage students to subscribe to a class calendar for assignments and important dates. Students now have a Peterson’s ID number for SAT and AP Test Preparation. Teachers can make daily

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During the course of second level **education** in Ireland, students complete two standardised sets of examinations. The ‘Junior Certificate’ exams mark the end of compulsory **education** at age 14 (see Fig. 3), while the ‘Leaving Cert’ exams are taken at the end of secondary **education** at age 17. The above graph, Fig. 2, shows the percentage of the total student population completing the Higher Level Maths course for the Leaving Certificate. Presumably due to the mathematical aptitude required in the study of engineering, 81% of the degree-standard engineering courses across Ireland require that students achieve a minimum of 55% in their Higher Level Maths final exams. The percentage of students achieving this standard each year is also shown on the graph in Fig. 2. However, due to the presence of this formal barrier, there is a significant portion of the Irish student body is being prevented from choosing engineering – reducing the attraction pool to which engineering courses may market to approximately one tenth of the high school leaving cohort.

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Stream selection is very complicated process. So it should be carrier oriented. It is basically includes a process of curious thoughts, mental conflicts, mental pressure and decision making regarding the choice of proper subject selection. The factors which affect the stream selection are: Aptitude, Academic achievement, locus of control, self-assessment of ability, vocational awareness, aptitude, gender and interest in the subject etc. Thus choosing **mathematics** as a career oriented stream will open the doors of many opportunities in different areas. **Mathematics** **education** trains student to make and use measurement, includes the study of computer program, algebra, statistics, geometry, calculus etc.

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The skills on use of the different fundamental operations, recognizing the value of honesty, integrity and trustworthy as future assessors of funds, solving business related problems, and values (i.e. accountability, honesty, collaboration, competence, communication) that are developed in **mathematics** subjects are widely used in the field of accountancy. The findings of the study concurs with Clark (2009) who stated that accounting use math to forecast future cash flows based on several different probable scenarios (probabilities), and then use the math concept of time value of money to determine the value of those cash flows in today’s dollars. In short, **mathematics** is indeed useful in accounting. Although most accounting educators readily acknowledge that mathematical ability has a significant impact on students’ performance in accounting courses, to date no statistical research has appeared that numerically quantifies the effect. The present research does not only confirm the relativity of **mathematics** skills in accounting but it also provides a result regarding the specific skills developed in each **mathematics** subject.

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Problem-based learning has come to dominate the field, but the stark question is this: If PBL is so good, why are we apparently so deficient in imparting that ‘tacit knowledge which directs higher order reflection and **reasoning**? 1 And more cogently, where is all the revolution McMaster Medical School made half a century ago? 7 Are we now indeed diminished to the status of mere pattern recognisers? If so, PBL may indeed be the fad some fear. 9, 10

While there is no universally agreed-upon definition of what QR is, there is agreement on what Q is not - it is not the same thing as **mathematics**. 4 QR involves mathematical thinking, uses mathematical analysis, is often taught by mathematicians, but it is not identical to **mathematics**. What, then, is QR? QR is a habit of mind; a way of thinking. Whereas **mathematics** deals with abstractions (symbols) and generalization, QR involves data (numbers) and context. The literature yields helpful information about the elements of QR (see Table 1) and typical goals of QR courses (see Table 2).

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A video-recording was made of the first session with each of two classes (Year 9, set 4 out of 5, Year 8 set 5 out of 5). This was viewed afterwards by two separate Heads of **Mathematics** groups in Hampshire as well as by the geometrical **reasoning** group. There was a great deal of very productive activity in each lesson. What surprised Peter was the mathematical vocabulary that was used by the pupils when explaining their results to each other both in pairs and as a larger group. There was much kinaesthetic work taking place – pupils explained with their hands as well as their voices. Pupils enjoyed using overhead projectors and explaining to their peers. In the Year 8 class he was delighted to see two of the lowest attainers, who normally lack confidence, volunteer to present their work to the class. The plenary session involved two groups explaining the **reasoning** to the others. It was noticeable that although the overhead projector transparency did not include much written **reasoning**, the pupils spent most time explaining their **reasoning** to the class. This was pleasing since that was the key objective of the lesson.

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