MTH121E 2024 Mathematics Education (IP) Study Guide PDF

SCHOOL FOR GENERAL AND CONTINUING EDUCATION {SGCE} FACULTY OF EDUCATION EAST LONDON CAMPUS MTH121E 2024 MATHEMATICS EDUCATION (IP) Study guide Prepared by: Ms. Z.F. Qwesha 1 UNIT 1 CAPS AND FIVE LEARNING...

SCHOOL FOR GENERAL AND CONTINUING EDUCATION {SGCE} FACULTY OF EDUCATION EAST LONDON CAMPUS MTH121E 2024 MATHEMATICS EDUCATION (IP) Study guide Prepared by: Ms. Z.F. Qwesha 1 UNIT 1 CAPS AND FIVE LEARNING STRANDS 1.1 Introduction In Section 2, the Intermediate Phase Mathematics Curriculum and Assessment Policy Statement (CAPS) provides: teachers with a definition of mathematics, specific aims, specific skills, focus of content areas, weighting of content areas and content specification. 1.2 What is Mathematics as defined by CAPS? Mathematics is a language that makes use of symbols and notations to describe numerical, geometric, and graphical relationships. It is the human activity that involves observing, representing, and investigating patterns and quantitative relationships in physical and social phenomena and between mathematical objects themselves. It helps to develop mental processes that enhance logical and critical thinking, accuracy, and problem- solving that will contribute to decision making. Key elements of the definition of Mathematics Mathematics as a language – symbols and notations Mathematics as human activity i.e. we learn mathematics by doing mathematics; mathematics is dynamic i.e. evolves over time. Development of mental processes – logical and critical thinking, accuracy and problem solving, as well as procedural fluency. Practical Implications of the Definition of Mathematics Teaching:  Teachers should use correct mathematical language including symbols and notation  Teachers should teach mathematical language.  Teachers should use appropriate teaching approaches that are mainly learner centred. (e.g., use of jigsaw, cooperative teaching, and learning). 2 Learning:  Mastery of correct mathematical language by learners is key as it enhances conceptual understanding, procedural fluency and strategic competence.  Active participation of learners enhances mental processes required for logical and critical thinking.  Development of learner s who are problem solvers. 1.3 Content Areas in Mathematics Mathematics in the Intermediate phase covers five content Areas: Numbers, Operations and Relationships Patterns, Functions and Algebra Space and Shape (Geometry) Measurement Data Handling 1.4 Specific Aims The teaching and learning of Mathematics aim to develop: a critical awareness of how mathematical relationships are used in social, environmental, cultural, and economic relations. confidence and competence to deal with any mathematical situation without being hindered by a fear of Mathematics a spirit of curiosity and a love for Mathematics an appreciation for the beauty and elegance of Mathematics recognition that Mathematics is a creative part of human activity deep conceptual understanding in order to make sense of Mathematics Acquisition of specific knowledge and skills necessary for: -- the application of Mathematics to physical, social, and mathematical problems -- the study of related subject matter (e.g. other subjects) -- further study in Mathematics. 1.5 Specific Skills To develop essential mathematical skills the learner should develop the correct use of the language of Mathematics 3  develop number vocabulary, number concept and calculation and application skills learn to listen, communicate, think, reason logically and apply the mathematical knowledge gained learn to investigate, analyze, represent, and interpret information learn to pose and solve problems build an awareness of the important role Mathematics plays in real life situations including the personal development of the learner. 1.6 Instructional Time in Intermediate Phase The instructional time in the Intermediate Phase is as follows: SUBJECT HOURS Home Language- 6hrs First Additional Language- 5hrs Mathematics- 6hrs Natural Sciences and Technology- 3,5hrs Social Sciences- 3hrs Life Skills- 4hrs Creative Arts- (1.5) Physical Education- (1) Personal and Social Well-being (1.5) TOTAL = 27.5 hrs. The allocated time per week may be utilized only for the minimum required NCS subjects as specified above and may not be used for any additional subjects added to the list of minimum subjects. Should a learner wish to offer additional subjects, additional time must be allocated for the offering of these subjects. 1.7 Weighting of content areas The weighting of Mathematics content areas serves two primary purposes: guidance regarding the time needed to adequately address the content within each content area guidance on the spread of content in the examination (especially end- of-the year summative assessment). The weighting 4 of the content areas is the same for each grade in this phase. WEIGHTING OF CONTENT AREAS as per CAPS DOCUMENT Content Area: Grade 4, Grade 5, and Grade 6 Numbers, Operations and Relationships- Grade 4: 50%, Grade 5: 50%, Grade 6: 50% Patterns, Functions and Algebra -1 Grade 4:10%, Grade 5:10%, Grade 6:10% Space and Shape (Geometry)-Grade 4:15%, Grade 5:15%, Grade 6:15% Measurement- Grade 4:15%, Grade 5:15%, Grade 6:15% Data handling- Grade 4:10%, Grade 5:10%, Grade 6:10% CONTENT AREAS IN GR 4,5,6 10 15 50 15 NOR SPACE and SHAPE MEASUREMENT DATA HANDLING * The weighting of Number, Operations and Relationships has been increased to 50% for all three grades. This is an attempt to ensure that learners are sufficiently numerate when they enter the Senior Phase. 1.8 Specification of Content and Progression The Specification of Content shows progression in terms of concepts and skills from Grade 4 to Grade 6 for each Content Area. However, in certain topics the concepts and skills are similar in two or three successive grades. The Clarification of Content in Section 3 provides guidelines on how progression should be addressed in these cases. The Specification of Content should therefore be read in conjunction with the Clarification of Content. See 5 The main progression in Numbers, Operations and Relationships happens in three ways: - the number range increases - different kinds of numbers are introduced - the calculation techniques change. The number range for doing calculations is different from the number range for ordering numbers and for finding multiples and factors. As the number range for doing calculations increases up to Grade 6, learners should develop more efficient techniques for calculations, including using columns and learning how to use the calculator. These techniques however should only be introduced and encouraged once learners have an adequate sense of place value and understanding of the properties of numbers and operations. Contextual problems should consider the number range for the grade as well as the calculation competencies of learners. Contexts for solving problems should build awareness of other subject and content areas, as well as social, economic, and environmental issues. 6 7 UNIT 2 Mathematical Language AND Conventions in Primary Schools What is Mathematics? CAPS Mathematics is a language that makes use of symbols and notations for describing numerical, geometric, and graphical relationships. It is a human activity that involves observing, representing, and investigating patterns and qualitative relationships in physical and social phenomena and between mathematical objects themselves. It helps to develop mental processes that enhance logical and critical thinking, accuracy and problem-solving that will contribute to decision-making. Mathematics symbols Mathematics has its own language, much of which we are already familiar with. For example, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are part of our everyday lives. Whether we refer to 0 as ‘zero’, ‘nothing’, ‘nought’, or ‘O’ as in a telephone number, we understand its meaning. There are many symbols in mathematics, and most are used as a precise form of shorthand. We need to be confident when using these symbols, and to gain that confidence we need to understand their meaning. To understand their meaning there are two things to help us – context - this is the context in which we are working, or the topics being studied, and convention - where mathematicians and scientists have decided that symbols will have meaning. Some common mathematical symbols The symbol (+) Words associated with this symbol are ‘plus’, ‘add’, ‘increase’ and ‘positive’. As it stands, ‘+’ clearly has some sort of meaning, but we really need to understand it - within a context. for example, if we see the + symbol written in the sum 2 + 3 We understand that the context is one of adding the two numbers, 2 and 3, to give 5. So here, the symbol + is an instruction to add two numbers together. 8 Let us look at another context in which we see the + symbol. If you study telephone numbers on business cards you will often see them given, for example, as +44 191 123 4567 In this context, the + symbol means that, in addition to the usual telephone number, a person dialing that number from overseas will need to include the country code (in this case 44). So, we see that the + symbol can have completely different meanings in different contexts, and it is important to be clear about the context. The symbol (−) Words associated with this symbol are ‘minus’, ‘subtract’, ‘take away’, ‘negative’ and ‘decrease’. So, if we see the − symbol written in the sum 6 – 4 we know this means 6 subtract 4, and we know the answer is 2. (6-4=2) Let us look at another context in which we see the - symbol. In a different context, we might see −5◦C, meaning a temperature of minus five degrees Celsius, that is five degrees below zero. (Convention) The symbol (×) Words associated with this symbol are ‘multiply’, ‘lots of’, and ‘times. This is just shorthand for adding. For example, if we see 6 + 6 + 6 + 6 + 6 we have five lots of six, or five sixes, and in our shorthand, we can write this as 5 × 6. Let us look at another context in which we see the (x)symbol. In a laptop keypad the word ‘x’ is written as a multiplication sign which has a completely different meaning as multiplication (Convention). 9 The symbol (÷) We can use the division symbol (÷) for dividing. ÷ means 'divide' which is the opposite of multiplication. It asks us to find how many of one number can fit into another. Division. The division slash ⟨ ⁄ ⟩, equivalent to the division sign ⟨ ÷ ⟩, may be used between two numbers to indicate division. For example, 23 ÷ 43 can also be written as 23 ⁄ 43. This use developed from the fraction slash in the late 18th or early 19th century. Let us look at another context in which we see the (÷) symbol. These are examples of division signs in another context (Convention). 10 HIGHER ORDER AND LOWER ORDER QUESTIONS Questions to stimulate mathematical thinking. Good questioning techniques have long been regarded as a fundamental tool of effective teachers and research has found that “differences in students’ thinking and reasoning could be attributed to the type of questions that teachers asked” (Wood, 2002, p. 64). Past research shows that 93% of teacher questions were “lower order” knowledge-based questions focusing on recall of facts (Daines, 1986). Clearly this is not the right type of questioning to stimulate the mathematical thinking that can arise from engagement in problem solving and investigations. Unfortunately, research continues to show that teachers ask few questions that encourage children to use higher order thinking skills in mathematics (Sullivan & Clarke, 1990). Many primary teachers have already developed considerable skill in good questioning techniques in curriculum areas such as literacy and social studies, but do not transfer these skills to mathematics. Teachers’ instincts often tell them that they should use investigational mathematics more often in their teaching, but they are sometimes disappointed with the outcomes when they try it. There are two common reasons for this. One is that the children are inexperienced in this approach and find it difficult to accept responsibility for the decision-making required and need a lot of practice to develop organized or systematic approaches. The other reason is that teachers have yet to develop a questioning style that guides, supports, and stimulates the children without removing the responsibility for problem-solving from the children. Higher-order Questions (HOQ) Higher-order questions are those that the students cannot answer just by simple recollection or by reading the information “verbatim” from the text. Higher-order questions put advanced cognitive demand on students. They encourage students to think beyond literal questions. Higher-order questions promote critical thinking skills because these types of questions expect students to apply, analyze, synthesize, and evaluate information instead of simply recalling facts. For instance, application questions require students to transfer knowledge learned in one context to another; analysis questions expect students to break the whole into component parts such as analyze mood, setting, characters, express opinions, make inferences, and draw conclusions; synthesis questions have students use old ideas to create new ones using information from a variety of sources; and evaluation questions require students to make judgments, explain reasons for judgments, compare and contrast information, and develop reasoning using evidence from the text. 11 Lower order questions Lower order questions are those which require students to remember and recall, such as ‘What was the date of Russian Revolution?’ or ‘What is the chemical composition of hydrochloric acid?’ These types of questions tend to be convergent in that they lead to a fixed or already known answer. They have value since they can be used to check memory and basic knowledge. However, if low level questions are over-used students may feel unchallenged or even bored. Levels of mathematical thinking Another way to categorize questions is according to the level of thinking they are likely to stimulate, using a hierarchy such as Bloom’s taxonomy (Bloom, 1956). Bloom classified thinking into six levels: Memory, Comprehension, Application, Analysis, Synthesis and Evaluation (requiring the highest level of thinking). Sanders (1966) separated the Comprehension level into two categories, Translation, and Interpretation, to create a seven-level taxonomy, which is quite useful in mathematics. As you will see as you read through the summary below, this hierarchy is compatible with the four categories of questions already discussed. 1 Memory: The student recalls or memorizes information. 2 Translation: The student changes information into a different symbolic form or language. 3 Interpretation: The student discovers relationships among facts, generalizations, definitions, values, and skills. 4 Application: The student solves a life-like problem that requires identification of the issue and selection and use of appropriate generalizations and skills. 5 Analysis: The student solves a problem in the light of conscious knowledge of the parts of the form of thinking. 6 Synthesis: The student solves a problem that requires original, creative thinking. 7 Evaluation: The student makes a judgement of good or bad, right, or wrong, according to the standards he or she values. 12 Table 1. Levels of thinking and guide questions LEVELS OF THINKING GUIDE QUESTIONS Memory: What have we been working on that Recalls or memorizes information might help with this problem? Have you seen something like this before? Translation: How could you write/draw/model what Changes information into another form you are doing? Is there a way to record what you have found that might help us see more patterns? Interpretation: What is the same? What is different? Discovers relationships. Can you group these in some other way? Could there be a connection between these…? Can you see a pattern? Application: What do you think comes next? Why? Works towards solving a problem-use of What could you do to explore this appropriate generalizations and skills connection further? Are there any rules that can be followed? Analysis: What have you discovered? Solves a problem- conscious knowledge How did you find that out? of the thinking Why do you think that? What made you decide to think that way? Synthesis: Who has a different solution? Solves a problem that requires original, Are everybody’s results the same? Why? creative thinking /Why not? What would happen if this changed? Evaluation: Have we found all the possibilities? How Makes a value judgement do we know? Have you thought of another way this could be done? Do you think we have found the best solution? 13 What are open and closed questions? Closed questions are questions that can be answered with one word or a short phrase, while open questions are questions that will most probably be answered with a lot of words. Examples of closed questions are questions with yes/no answers, or where answers can be selected from a list. Open questions often start with words like how, what, and why, and give people an opportunity to elaborate when answering them. What are closed questions used for? Closed questions can be used for testing a person's understanding of a subject by giving them the choice between 'yes' and 'no'. They are also good for opening conversations, as people can answer them without feeling that they have to say or explain too much. Closed questions can be used when trying to close a deal and getting someone to say yes. What are types of closed questions? There are three main types of closed questions used in math: True or false questions, which have one correct and one false answer. Multiple-choice questions, which can have two or more correct answers. Multiple-answer questions, which only have one correct answer. What are 2 examples of Closed-ended questions? Closed-ended question are questions that can be answered with one word or a short phrase, examples can be: Do you like ice cream? What is your favorite color? 14 UNIT 3 Learning Theories: Constructivism and Multiple Intelligences Constructivism What is constructivism? It is one of several theories of learning, building upon what cognitive theorists discovered about how mental structures are formed. John Dewey was one of the founders of this theory. He believed that for learning to occur, education needed to move away from behavioral methods and instead create models of teaching and learning where students were actively involved in the learning process. What underlying principles is constructivism based upon? Each learner constructs his or her own knowledge based upon his/her unique experiences with the world and the meaning he or she gives to those experiences. Knowledge is constructed; it cannot be "delivered" or "imprinted" on others. Learners build their own knowledge by exploring environments, manipulating objects, testing hypotheses, and by drawing conclusions. In other words, learners learn by seeing, doing, and connecting. Notice the active verbs here. Behavioral objectives are written specifically with observable, measurable events in mind which are typically fact-based. Constructivist-based objectives are more open- ended and do not usually culminate in a test or quiz. The teacher plays an important role in a constructivist classroom because it is his/her role to create the activities and environment in which students can learn. The teacher must develop problem-oriented learning activities that have a meaning, are purposeful, and are based on student interests. The teacher must provide a multitude of resources from which learners can construct their understanding (this means text, pictures, sounds, manipulatives, community members, web resources, and more). The demonstration of learning is often a product of some type with a relevant outcome (instead of a test or even a paper). The teacher must construct a classroom atmosphere where students respect each other and where each person can contribute; this atmosphere must also be one in which there is room for more than one correct answer. Collaborative learning becomes a part of the learning environment, as does the opportunity for interdisciplinary learning (after all, isolated knowledge is meaningless). The teacher helps students understand what they need to know to solve problems. Technology can help to foster this type of learning environment in several ways. Computers have had a powerful influence in decentralizing the learning process. A vast array of resources can be on every desktop, accessed by only a few keystrokes. As a result, the teacher now no longer holds the only set of keys to learning. In addition, the limited access to computers in the classroom forces teachers to figure out how to best 15 utilize the resources they have–and often student-centered, collaborative groups work well. There is now an abundance of rich educational software developed with a constructivist approach in mind. Technology tools such as word processors, spreadsheets, CD databases, and multimedia programs can be used by students to show their understanding of concepts and principles. Because many of these programs now incorporate more than just text, it means student understanding can be reflected in many ways that text alone does not provide. The use of the Web also facilitates a constructivist environment because of the wealth of resources that can be accessed by using it. Multiple Intelligences Howard Gardner gives us an alternative glimpse at how learning occurs. His theory stems from brain research. Gardner proposes that there are eight intelligences (possibly nine, though the ninth has not yet been totally validated). His theory purports that various areas of the brain are responsible for different functions, and if one area of the brain is damaged, certain abilities are affected. His belief is that we have intelligence in all eight areas (maybe nine), but that we are stronger in some than in others and that what makes each of us unique is how those intelligences interact. There are many implications of this theory on teaching. First, we must value all of the intelligences, not just the verbal/linguistic and logical/mathematical as we have in the past. Gardner's theory tells us that it is not reasonable to expect everyone to function alike because of physiological differences within the brain. It means that teachers need to provide multiple opportunities for students to learn which touch on several of the intelligences. Stronger intelligences can be used to nurture weaker ones. As a result, we can help students who are not strong in math or writing to develop their skills in these areas through their strengths. It's not how smart you are, but HOW ARE you smart! Verbal/Linguistic intelligence involves the successful manipulation of words and phrases to convey ideas, feelings, and moods. Excellence in this intelligence is best seen in poets like e. e. cummings and writers like Ernest Hemingway. A biological link can be demonstrated by the phenomenon known as Broca's aphasia in which damage to a specific area of the brain called Broca's area, results in the loss of a person's ability to form simple sentences, even though the ability to understand words and sentences is unimpaired. The ability to communicate and articulate thoughts in a linguistic mode is highly valued in most cultures. People with strengths in this area think in words and can use words effectively both in writing and in speaking. 16 Technology can help to address student needs in this area. Word processors and desktop publishing programs, especially those with voice annotations or speech output work into this strength well. Multimedia authoring tools, such as Hyper Studio, fit in well here because of students' ability to incorporate text into their products. Telecommunications tools, which are predominantly text-based, also fit into this intelligence well. Logical/Mathematical intelligence, which has been also termed "scientific thinking," employs the use of observation, induction, and deduction to solve problems. Evidence that supports this ability as an intelligence includes the existence of child prodigies such as Euler and idiot savants who exhibit highly focused intellectual abilities similar to those portrayed in the movie Rain Man. Also, logical-mathematical abilities are highly prized in Western culture. The syntax for communication in logical and mathematical forms has been highly developed because of to the Western emphasis on these abilities. People with strengths in this intelligence like to work with numbers. They are very analytical and tend to think sequentially; they seek to find patterns and relationships. Database and spreadsheet programs, because of their structure and use of numbers and mathematical formulas, are good technology tools for students with this strength. Calculators are an obvious match as well. Computer programming languages interest students with this strength because of their problem solving and linear nature. Strategy and problem-solving software often appeal to learners with this strength because of the logical way these people think. Multimedia tools also are useful because students can structure their projects as they see fit. Intrapersonal intelligence involves the ability to form an accurate model of one's own self and be able to use that model to operate efficiently in life. It is "knowledge of the internal aspects of a person: access to one's own feeling life, one's range of emotions, the capacity to effect discriminations among these emotions and eventually to label them and to draw upon them as a means of understanding and guiding one's own behavior." (Gardner, 1993) Because this intelligence is mostly private, expression through one of the other intelligences may be required to detect the activity of this intelligence. According to Gardner, the personality changes exhibited in frontal-lobe injuries can also be used as evidence for a biological link to this intelligence. Writers and thinkers in Western culture have emphasized the importance of knowing oneself. People with a strength in this intelligence are typically very introspective. They are self- motivated and disciplined. They will shy away from team activities when given a choice. Independent, self-paced instruction often works well for learners exhibiting this strength. 17 Computer software that is developed for individual learners appeals to this type of student. Competitions against the computer are looked upon more favorably than are contests against other students. In some ways, email and telecommunications tools are good options for learners with this strength because the connection they make with others is still rather removed. Visual/Spatial intelligence is the ability to form a mental model of a spatial world and to maneuver and operate using that model. Damage to the right posterior of the brain has been shown to cause impairment in a person's ability to physically navigate, to recognize faces or scenery, and to notice details in a given environment. Spatial intelligence is not limited to the visual mode–Gardner suggests that the ability of a blind person to size objects through touch is an example of spatial intelligence exhibited through a tactile mode. People with a strength in this area tend to think in images. They know the location of everything and enjoy designing things. Technology can address the needs of these learners in several ways. Drawing and graphics programs are obvious tools. Computer-Assisted Drafting (CAD) programs are enjoyed by these learners, as are programs that allow learners to see information pictorially. Multimedia and web authoring tools allow learners to bring in the images and visual organization they desire. At the primary levels, reading software that contains visual cues can help learners make the connection between text and the meaning of each word. In addition, science probe ware that connects to computers allows students to graphically see the output of the data collected by the instruments. Video production is another technology tool that can target strengths in this area. Musical intelligence is the ability to recognize pitch and rhythm and use this ability to create a musical composition that is culturally acceptable and pleasing. Although musical intelligence is not typically considered an intellectual skill, Gardner suggests that it passes the requirements for being an intelligence. First, the existence of musical prodigies supports the claims of a biological link to the intelligence. Mozart, for example, could compose music before he had received any sort of formal musical training. Second, neurobiological research has shown that certain parts of the brain play a pivotal role in the perception and production of music, and brain damage can lead to a loss of musical ability. In addition, music has served as an important unifying role in several societies and cultures. Finally, music has its own systematic and accessible notation system. This strength is exhibited in ways that go beyond writing and performing music. People tend to think in notes, rhythms, or beats. They notice non- verbal background noises in the environment that others overlook. They need music in the background to study, and if there is none, they might hum or whistle. 18 Software programs that allow students to develop their own songs and music is a match for these learners, as are MIDI (Musical Instrument Digital Interface) devices. Programs that incorporate music and sounds into them will help these learners understand the concepts. Audio CDs can help the learner focus. Multimedia software allows these students to incorporate meaningful sounds into their projects. Bodily-kinesthetic intelligence is defined as "the ability to solve problems or to fashion products using one's whole body, or parts of the body." (Gardner, 1983) Motor cortex control of bodily movement and the evolution of specialized body movements such as those necessary to correctly use a tool reveal an obvious link to the biological nature of this intelligence. Good physical articulation skills are prized in many different cultures, demonstrating a cross-cultural importance of the intelligence to different societies. Finally, using the body to express an emotion in a dance, to participate a group sport, or to precision machine a part illustrates some cognitive features of this intelligence. Learners with this strength need to be physically involved in the learning process. They would rather participate than observe and they respond well to non-verbal cues. These people need to move around; they are sometimes labeled erroneously as hyperactive because they have a hard time being still for long time periods. Software programs using various input devices (mouse, joystick, touchscreen) work well for these learners. Often the fact that they are using the computer and manipulating things by itself is appropriate. Keyboarding and word processing programs involve much physical contact with the keyboard. Programs where learners animate objects or move them around on the screen appeal to them, and the science probe ware described above under visual/spatial learners also is a good match for bodily/kinesthetic learners. Interpersonal intelligence is the ability to understand other people: what motivates them, how they work, how to work with them. Sometimes characterized as charm or charisma, this intelligence is based on one's ability to notice subtle distinctions in mood, temperament, motivation, and intention to sense hidden desires and intentions and to act on this information. Brain research shows that the frontal lobes play a role in this intelligence; damage to this area can results in extreme personality changes but somehow spares other mental factors. An evolutionary link may be the development of this intelligence through the necessity of ancestral species to work and hunt together. Strengths in this intelligence are exhibited by abilities to organize others, usually choosing to take on a leadership role. People with this strength not only enjoy working with others, but actually learn better and more by doing so. Interpersonal learners enjoy working with software that involves groups, especially titles dealing with social issues. For example, many of the Tom Snyder titles are geared for 19 group problem solving and decision making. Group presentations using PowerPoint work well here, as do activities such as television production. Naturalist Intelligence refers to the ability to recognize and classify plants, minerals, and animals, including rocks and grass and all variety of flora and fauna. The ability to recognize cultural artifacts like cars or sneakers may also depend on the naturalist intelligence. This intelligence includes observing, understanding and organizing patterns in the natural environment, including plants and animals. People with this strength enjoy collecting objects from the natural world, observing nature, noticing changes in the environment, caring for pets, and related activities. This is categorized as an intelligence because it's an ability we need to survive as human beings. We need, for example, to know which animals to hunt and which to run away from. Second, this ability isn't restricted to human beings. Other animals need to have a naturalist intelligence to survive. Brain evidence supports the existence of the naturalist intelligence. There are certain parts of the brain particularly dedicated to the recognition and the naming of what are called "natural" things. Technology might address this intelligence in several ways. The use of video and television broadcasts dealing with natural phenomena and objects can help these people learn more about things that they otherwise would not encounter. The same is true with CD databases. The World Wide Web can put them in touch with additional resources as well that can help learners categorize and sort the items. 20 21 UNIT 4 South African mathematics performance in International Arena INTRODUCTION The teaching of mathematics in South African schools is amongst the worst on the world. In 2011, the Trends in International Mathematics and Science Study (TIMSS) showed that South African learners have the lowest performance among all 21 middle- income countries that participated. A recent CDE report further underlines the issue as it found rapid increases in enrolments in private extra mathematics classes, which was partly in response to poor teaching in public schools.1 Such supplementary efforts fail to address the wider deficiencies in mathematics education. Vast improvements in this area of the public schooling system are vital to South Africa’s future socioeconomic prospects: for the learners as well as the development of the country. South African Mathematics Schooling in an International Context International studies often show that South Africa has the worst educational outcomes of all middle-income countries that participate in cross-national assessments of educational achievement, especially in mathematics. We also do worse than many low- income African countries. The Trends in International Mathematics and Science Study (TIMSS) is a recognized international assessment study designed to measure the effectiveness of an education system within a country in relation to mathematics and science. TIMSS 2019 was the 7th cycle of the International Association for the Evaluation of Educational Achievement (IEA) series of large-scale assessments. It is conducted every four years at the Grade 4/5 and Grade 8/9 levels. In addition to assessing learner knowledge in mathematics and science, TIMSS also gathers information in relation to the learner’s context for learning (websitehttps://www.iea.nl/studies/iea/timss/2019). TIMSS Assessment framework The curriculum used by TIMSS is broadly defined as the major organizing concept in considering how educational opportunities are provided to learners and the factors associated with how learners use these opportunities. The TIMSS Curriculum Model has three aspects: the intended curriculum, the implemented curriculum, and the attained curriculum (see Mullis & Martin, 2017, for details. These represent the mathematics and science knowledge that learners are expected to learn as defined by countries’ curricula policies and publications, how the educational system should be 22 organized to facilitate this learning, what is taught in classrooms, the characteristics of those teaching it, how it is taught, and, finally, what it is that learners have learned. TIMSS at the Grade 5 level in South Africa In 2015, South Africa extended the TIMSS program and conducted the TIMSS 2015 Numeracy assessment at the fifth Grade, to gain an insight into education and achievement in primary schools. TIMSS Numeracy offered a ‘less difficult’ mathematics assessment at the fourth Grade than the mathematics assessment normally administered. TIMSS Numeracy asked learners to answer questions like those posed in TIMSS, except that easier numbers and more straightforward procedures were used. TIMSS Numeracy was reported on the same TIMSS Grade 4 scale. In TIMSS 2019, both mathematics and science assessments were included at the Grade 5 level. South Africa again opted for the less difficult mathematics assessment items; however, the science assessment was the same as all participating countries. The 2019 TIMSS cycle provides the first opportunity to set up a trend line for mathematics achievement and a baseline measure for science achievement. The TIMSS 2019 Grade 5 Sample The Department of Basic Education (DBE) and the Human Sciences Research Council (HSRC) collaborated in conducting the TIMSS 2019. The TIMSS 2019 sample of schools was selected from the 2018 DBE List of Schools that offered Grade 5 classes (school population: 16 254 public and 1031 independent). Statistics Canada selected the national sample based on province and school type (public or independent) as explicit stratification variables and school poverty ranking as the implicit stratification variable. The realized sample consisted of 297 schools, 294 mathematics educators, 295 science educators, 11 903 learners and 11 720 parents. We explain the methodology in detail in the forthcoming TIMSS 2019 Grade 5 National Report. Structure of the Highlights of Grade 5 results report: A total of 64 countries (including the six benchmarking participants) took part in the study. Table 1 provides mathematics average scale scores, together with the standard errors (SE) of the countries who participated in TIMSS 2019. The scores in the tables are in rank order to allow for cross country comparisons. The TIMSS achievement scale is set to a Centrepoint (point of reference which remains constant from assessment to assessment) of 500 and a standard deviation of 100. In mathematics, the top three countries were Singapore (with an average scale score of 625), Hong Kong (602) and Republic of Korea (600) – all from East Asia. The three countries with the lowest 23 achievements were South Africa (374), Pakis¬tan (328) and the Philippines (297), with the South African achievement score significantly higher than Pakistan and Philippines. Country Mathematics Mean (SE) Singapore 625 (3.9) Hong Kong SAR 602 (3.3) Korea, Rep. of 600 (2.2) Chinese Taipei 599 (1.9) Japan 593 (1.8) Russian Federation 567 (3.3) Northern Ireland 566 (2.7) England 556 (3.0) Ireland 548 (2.5) Latvia 546 (2.6) Norway (5) 543 (2.2) Lithuania 542 (2.8) Austria 539 (2.0) Netherlands 538 (2.2) United States 535 (2.5) Czech Republic 533 (2.5) Belgium (Flemish) 532 (1.9) 24 Cyprus 532 (2.9) Finland 532 (2.3) Portugal 525 (2.6) Denmark 525 (1.9) Hungary 523 (2.6) Turkey (5) 523 (4.4) Sweden 521 (2.8) Germany 521 (2.3) Poland 520 (2.7) Australia 516 (2.8) Azerbaijan 515 (2.7) Bulgaria 515 (4.3) Italy 515 (2.4) Kazakhstan 512 (2.5) Canada 512 (1.9) Slovak Republic 510 (3.5) Croatia 509 (2.2) Malta 509 (1.4) Serbia 508 (3.2) Spain 502 (2.1) TIMSS Scale Centrepoint 500 25 Armenia 498 (2.5) Albania 494 (3.4) New Zealand 487 (2.6) France 485 (3.0) Georgia 482 (3.7) United Arab Emirates 481 (1.7) Bahrain 480 (2.6) North Macedonia 472 (5.3) Montenegro 453 (2.0) Bosnia and Herzegovina 452 (2.4) Country Mathematics Mean (SE) Qatar 449 (3.4) Kosovo 444 (3.0) Iran, Islamic Rep. of 443 (3.9) Chile 441 (2.7) Oman 431 (3.7) Saudi Arabia 398 (3.6) Morocco 383 (4.3) Kuwait 383 (4.7) South Africa 374 (3.6) Pakistan 328 (12.0) 26 Philippines 297 (6.4) Achievement trends The TIMSS 2015 and 2019 cycles provided the first trend measure for mathematics achievement. The difference of average mathematics score of 376 (3.5) in TIMSS 2015 and 374 (4.7) in TIMSS 2019 is not statistically significant. This means that there was no change in the achievement performance over these two periods. To meet the country’s developmental objectives, the Medium-Term Strategic Framework (2019–2024) (DPME, 2014)4 set the target for the TIMSS average mathematics score, in Grade 5, of 426 in 2023. Currently, this target does not look attainable. From the TIMSS data, we are unable to explain why achievement scores did not increase. The Grade 5 results are particularly perplexing as the TIMSS mathematics and science achievements increased at Grade 9. We recommend an investigation into the primary school education sector, including issues such as the support provided to primary schools in relation to secondary schools, the nature of teaching, learning and assessments in primary schools. 27 UNIT 5 Numbers, operations, and Relationships Numbers, operations, and relationships focuses on the: development of number sense for learners. This includes the meaning of different kinds of numbers, the relationships between different kinds of numbers, the relative size of different numbers, and representation of numbers in lots of different ways. Place value of the digits By place value we mean the value represented by a digit in a number based on its position in the number. For example, the place value of 7 in 517 389 is 7 thousand or 7 000. However, the place value of 7 in 715 389 is 7 hundred thousand or 700 000. Expanded notation Digits in a number can be represented using expanded notation (breaking down the number). For example, the number 517 389 can be represented using expanded notation, e.g. 500 000+ 10 000+ 7000+ 300+ 80 Counting in ascending order (arranging from the smallest to the largest) and descending order(from the largest to the smallest) Arrange these numbers in ascending order: 34 289. 34 288, 34 287, 34 286, 34 285, 34 284 Firstly, what do you notice in these numbers? The Ten thousand, Thousands, Hundreds and Tens are the same. Then that means only the ones/units are changing. Then arrange them starting from the smallest to the largest: 4,5,6,7,8 and 9. 28  NB-Write the full number, e.g. 34 284, 34 285…. In descending order, you will start with the bigger number, e.g.34 289, 34 288. Describing numbers in word form Numbers can be represented or written in words. For example, 399 can be stated as Three hundred and ninety-nine. 587 256 in the word form is Five hundred and eighty- seven thousand, two hundred and fifty- six. Comparing numbers Comparing numbers is the process of identifying the number that is greater than, smaller than, or equal to another number. There are signs that we use to compare the numbers: Greater than/ smaller than ()NB- The sign opens on to the side of the greater number) , e.g. 567> 555 Use of basic operations There are four basic operations in mathematics: 1. Addition- We use this operation when we want to find out the total of the two or more numbers, e.g. 5+3= 8(The total of five and three is eight) 5+3+2= 10 (The total of five, three and two is Ten). 2. Subtraction- is used to find the difference between two numbers, e.g. 10-5=5 (The difference between ten and five is five). 29 3. Multiplication helps us to find the total number of items quickly, e.g. 3+3+3+3+3+3= 18 can be written using multiplication (6x3 or 3x6). NB-The order of how you place the numbers does not change the product, e.g., 6x3=18 and 3x6= 18. Division helps us to separate or divide the number of objects into equal size groups. To divide, we need to know the total number of objects or to know the total number of groups (groups of dogs) 5 groups of dogs or 15 total number of dogs 5 groups of learners Total number of dogs is 15 and the total number of children is 5. If we divide the number of dogs equally to each child, each child will get 3. (15÷5=3). 30 UNIT 6 The Use of ICT, synchronous and asynchronous teaching and learning in mathematical settings The use of ICT (Information and communication technology) in Primary mathematics provides great support for the teaching and learning in mathematics. However, it can only really provide effective support where its use is ‘transparent’. This is the aim of developing student ICT capability and in this course, we will investigate and examine how to use ICT in the math classroom where it helps teachers demonstrate and explain mathematical ideas along with helping students develop their mathematical knowledge, skills and understanding. ICT provides teachers with opportunities to capitalize on the idea that it can help students visualize mathematical ideas and concepts. It can provide teachers and students with resources to help them concentrate on the learning objectives within the National Curriculum and not get bogged down with other issues. Finally, even though there are technology in mathematics education that enables students to review and consolidate mathematical skills such as ‘drill and practice’ programs, these are not the focus of this course as they do not fully develop ICT capability. How to best use ICT in Mathematics When using ICT in mathematics primary, there are many opportunities to choose from. However, the best option is to ensure that your students control the technology and that they make the decisions thus developing their higher order thinking skills. Yes. There is a technology that teaches mathematics, and these are known as integrated learning systems or are commonly known as subject-specific software. They defeat the purpose of understanding how to integrate ICT in teaching maths. The advantages of using ICT in mathematics should be: o It enhances student learning of mathematics. o It motivates them to learn mathematics. o It develops student ICT capability. o It promotes higher order thinking skills. 31 The right ICT tools can, therefore, provide a conceptual construction kit that can transform students’ mathematical knowledge and practices in your curriculum. When looking at the Australian Curriculum, for example, there are many opportunities to embed ICT into the curriculum and to develop ICT capability. As a result, we focus on these opportunities in our advanced online professional development for primary teachers that helps them understand how to integrate ICT in teaching mathematics. It encourages a whole school level approach to integrate ICT in primary mathematics. Another key aspect that you need to be aware of when using ICT in mathematics primary is that it also involves a shift in teaching approaches. Integrating ICT into meaningful subject-related learning activities is never straightforward. Various pedagogical approaches need to be adopted for students to not only develop ICT capability transparently alongside mathematics learning but to also understand that when they are using technology in mathematics that they view it as a tool. Great ICT tools for teaching Mathematics o Databases – the handling of data is a very important part of a mathematics lesson. It involves analysing information collected by the students themselves during a hands-on, practical activity. This can be done with real and relevant data making it an authentic learning experience. o Spreadsheets – these are designed to help you work with numbers and students can use to do the same too. They can be set up as several machines that can repeat calculation processes quickly and easily. They can be used to help solve problems where repeating calculations can help find the answer. o Interactive whiteboards with the aid of digital projectors – the combination of these two with the addition of the computer itself will allow us to teach mathematics using whole-class teaching methods. It can also be used to help you to demonstrate to students’ various techniques that they need to use during the maths lesson such as spreadsheet skills. o Programmable toys – robotic toys can be used as a catalyst for problem-solving from early childhood to primary education. 32  o Desktop publishing software – this is a great idea for when investigating and designing objects on the screen. They are very useful for studying 2-dimensional objects as they allow you to create simple shapes quickly and easily. Some have built-in functions that allow students to rotate and reflect these shapes once created. ICT offers powerful support for teaching numeracy and mathematics. It can be where the teacher uses ICT alone or when students use it to gain the benefits from its features such as allowing them to develop ICT capability. 33