What is the purpose of the Mathematics curriculum? How does Mathematics relate to the other curriculum areas?
Mathematics is always underlined as one of the basic constituencies of classical education. Professionals believe that it helps students develop a sense of wonder that is continuously developing, which allows them to deepen their knowledge in different spheres and learn more critical information. In addition to that, it makes learners perceive the world in connection with order, pattern, and relation. It is emphasized that the study of mathematics reveals the beautiful order of things so that its main aim is to describe it in the framework of the natural world. Today, the representatives of the general public mainly see it as a language that is used only by scientists. However, the mathematics curriculum is expected to develop reasoning skills and logical thought that are advantageous in everyday life. All in all, it tends to focus on the development of a general understanding of order and harmony that is taken from a concrete foundation (Treloar, 2012). Even though mathematics is focused on arithmetic and geometry, it is often tightly connected with other curriculum areas. For example, during the classes of physical education, students can be asked to gather in groups of four or count the equipment that is required for sports.
Within the mathematics curriculum, name the three strands that need to be developed and explain their function and the sequence for each of the three mathematics strands.
The mathematics curriculum should be discussed regarding each of its standards that reveal the content that should be learned. In this way, it will be advantageous to consider the number and algebra first of all, as they are developed simultaneously. Students resort to both number sense and a range of strategies for the calculation to get an understanding of different operations. They learn the concepts of variable and function, as well as work with equivalences. All in all, they obtain a chance to maintain investigations, find solutions and provide reasoning. The second standard deals with measurement and geometry that are discussed together due to their practical relevance. In this framework, learners improve their understanding of size, shape, position, and movement. They focus on the properties of different objects and figures to reach geometric arguments. Finally, it is significant to consider statistics and probability. Students learn to identify critical data, analyze it and summarize its interpretation. They assess the likelihood and define probabilities due to the developed ability of critical thinking and reasoned judgment. This standard presupposes that students utilize both experimental and theoretical approaches when evaluating statistical information. What is more, they are expected to provide personal intuitions about particular data (ACARA, 2012).
Describe three mathematics materials and explain how you move from the concrete to the abstract.
Different mathematics materials can be used to move from concrete to abstract. The most advantage can be obtained from those tools that can be moved around by students but not just demonstrated by the teacher. In addition to that, these objects should belong to the students’ world so that they are already familiar with them. For example, cubes can be used in the framework of mathematics instruction. They can be gathered in units, flats, and longs. Cubes are helpful when dealing with various procedures, including counting, ratios, and addition, etc. Two-sided counters in a shape of a circle can also be advantageous when focusing on percentages, integers, and division, etc. Finally, beads on a string can be used for basic mathematical operations and counting.
From the very beginning, students should be introduced to an object, cubes, for instance. They learn particular skills with their help. They can count cubes, holding them in their hands, and getting familiar with their attributes. Then, learners are expected to move to the representational stage. They can draw cubes with simple representations so that they do not use the very objects anymore. Such replacement leads to an abstract understanding of the skill. During the last stage, the abstract is reached. Students are encouraged to resort to their language to share information and ensure proper solutions (Degarcia, 2008).
Define how control of error functions in selected mathematics materials. Give at least 3 examples.
Control of error is significant to point in education because it allows children to correct themselves in the case of any mistakes. In this way, they can avoid teacher’s corrections and remain motivated to continue working (NAMC, 2013). For example, when working with cubes, children can be asked to count them. It will be advantageous if a teacher writes the numbers of the objects and puts them in proper order so that students can check whether their answers are correct. When teaching division with two-side counters, teachers can use some boxes for answers. It will be beneficial to put the results under them. Focusing on the beads on a string, a teacher should prepare beads of different colors so that when multiplying all yellow items students obtained the same number as when multiplying all green ones.
Clarify the role of language in the mathematics curriculum and give examples of materials.
Language plays a great role in mathematics because it allows teachers to introduce a specialist mathematical vocabulary. It is known as mathematical discourse and is often used in class. It includes technical terms, more general terms, and everyday words that obtain a new meaning. The first group of words is the most critical because teachers need to provide students with the vocabulary that have never faced before. It may be the most challenging group of words in this way. For instance, a teacher can resort to such words as equilateral or quotient. The second group of words is easier because it consists of those words some children already know. But there is a possibility that learners find these words challenging to use as they are needed only in particular situations. For example, children maybe not sure regarding the meaning of factor or frequency, which may lead to confusion. Finally, the last group of words includes the simplest terms. Even though they are used to describe unrelated ideas, these elements of mathematical vocabulary can be easily memorized. These words include such ones as difference and area. In addition to that, it is critical to mention that even the usage of those words that help to develop syntax (such as: and, then, a) may affect the sense of the sentence (Barwell, 2011).
Describe the direct aims and the indirect aims of two mathematics materials.
Different mathematic materials can be used when working with students. They all have both direct and indirect aims within a class. The first ones are connected with purely mathematical considerations, while the second ones may refer to the wide variety of other spheres (Balsam Montessori Program, 2017). For example, when working with beads on a string, the direct aim which is considered by the educator is likely to be to teach students counting to 20 or something like that. However, at the same time, such a tool can be used to fulfill an indirect aim of the development of fine motor skills and color differentiation. Similarly, two-sided counters can be used with the focus on the direct aim of mathematic activities, such as division. At the same time, it will surely help an educator to develop students’ fine motor skills, personal independence, and intellect. It will assist in color differentiation and strengthen one’s will. In general, a direct aim is developed based on the class topic. It allows to ensures that children obtain the required knowledge and skills. An indirect aim, in its turn, is often connected with practical life and other vital achievements.
Briefly discuss the sensorial aspects of the mathematics materials and how this provides a foundation for further understanding of mathematics.
The sensorial aspects of the mathematics materials deal with students’ senses and their development. In general, they can be used to enhance children’s knowledge of colors, shapes, textures, forms, and other characteristics of different objects. It is significant to mention that they resort to controlling error. Facing some chaotic situation, a child receives an opportunity to categorize and classify both the items introduced by the teacher or additional information. In this way, psychological and neurological development is supported. Due to the utilization of sensorial materials, children have an opportunity to prepare for intellectual life. They develop critical thinking, proper judgment, and association. In this way, the further understanding of mathematics is also likely to be enhanced because students will be motivated to work and reach significant achievement. In addition to that, well-developed sensorial aspects can allow children to understand the situation extremely quickly. It is easier for them to remember different forms and associate numbers with mathematics vocabulary. Activities within this area allow dividing into different groups such items as cubes and rods so that they are immediately recollecting those specific characteristics these items have (Boure, 2008).
What principles are followed in presenting math concepts?
When presenting math concepts, teachers should take into consideration several principles. First of all, it is significant to ensure both why something works and then how does it work. It seems to be advantageous if an educator reveals the reason for required actions because such an approach allows ensuring procedural understanding. Then, it is also vital to remember those goals that reveal the necessity to teach and learn this subject. It allows to practice further studies, and ask for advice, etc. In the same framework, it is also critical for me to ensure that learners have an opportunity to practice on the Internet instead of any misunderstanding. The next principle deals with the usage of those tools students resort to when studying. If they include those items children already have at home, it is no critical to look for some additional improvement. Finally, it is significant to ensure that students are willing to get to know more information connected to math. They should be motivated to do their best and to achieve success (Miller, 2015).
A prospective parent is asking when his/her child is ready to count. How would you explain to the parent based on the Montessori philosophy?
If I am approached by a parent who is willing to get to know when her daughter is ready to count, I would like to tell her that it is not an urgent progressive change, which she should wait for. Considering Montessori’s ideas, I need to explain to her that age is not that critical. It is vital to have some support from the outside so that there is a desire to make everything close to one another. Before calculating, it is significant to ensure that the child can cope with easier operations. In this way, it is expected that some improvement will be observed while the educator can attract the child’s interest. I would emphasize that I try to give my learners some freedom so that they do not feel obliged to follow my rules but just oblation some. I would state that my role is to assist a child but not to make one follow some norms (Hendron, 2012).
What mathematics materials help the child gain mastery of the decimal system?
Trying to ensure that students gain mastery of the decimal system, educators should start to teach them counting in tens, using cubes and stairs. Then the number of rods should be given. The names of the numbers can be discussed along with the rods they belong to or considered separately. It is also possible to focus on the decimal system with the help of the spindle box and cards (“Montessori sensorial education”, 2016). All in all, the decimal system will be introduced with its special names from the very beginning so that a child learns the required vocabulary before being occupied in decimal learning activities. It is possible to use fun games later. For instance, life problems can be discussed so that their relevance is considered.
List all the activities which contribute to the child’s ability to count with understanding to 1000.
Child’s ability to count to 1000 can be enhanced with the help of a special application that consists of 6 section that allows corresponding the names of numbers. During the first one, kids count from 1 to 11 and then to 20. Using the Quality game, they receive an opportunity to focus on matching. Thus they match numbers and objects. Unlike the previous game, this one allows revealing that number. With the help of the Hundred hoard, numbers up to 1000 can be learned. It is also possible to let children match the name of the number pronounced by the educator and its equivalent. There is also a possibility to play a game vice-versa, focusing on the sounds of the numbers and finding the way they are written. Finally, it is possible to play a tracing game (Teachers with Apps, 2016). It is possible to play the Math City as well, building a multi-story house, counting the bread and platforms. In addition to that, teachers can introduce other games that do not require any technical materials. For example, children can be asked to count numbers, substituting all “3” with a clap. They can also count in an ordinary way but treat ten as one (Pramono, 2014).
How would you help a child to develop an understanding of mathematical four operations in Montessori math?
If there is a necessity to help a child to develop an understanding of mathematical four operations in Montessori math, I will try to resort to some learning games so that he/she got interested and involved in this process. I will resort to the bead material to explain to a child how to build different numbers. I will try to let a child maintain all activities under my supervision but will emphasize self-correction. Using math with dominoes, I will teach him/her to add numbers. With the help of cubes, I am going to explain subtraction. I will use matching puzzles to explain the principles of division. Finally, I will use tables and pyramids for multiplication. Of course, it is not possible to resort to the same approach all the time that is why it is critical to focus on particular children and their characteristics so that the approaches selected for teaching correlate with one’s needs and abilities.
References
ACARA. (2012). Content structure. Web.
Balsam Montessori Program. (2017). Montessori method. Web.
Barwell, R. (2011). The role of language in mathematics. Web.
Bourne, L. (2008). All about Montessori math. Web.
Degarcia. (2008).Concrete-representational-abstract instructional approach. Web.
Hendron, R. (2012).Introduction to mathematics. Web.
NAMC. (2013). The importance of control of error in the Montessori environment [Web log comment]. Web.
Montessori sensorial education and early math experiences. (2016). Web.
Miller, M. (2015).Four principles of deeply effective math teaching. Web.
Pramono, E. (2014). Montessori Math City offers a solid way for kids to learn numbers up to 1,000. Web.
Teachers with Apps. (2016). New & improved Montessori numbers – Learn to count from 1 to 1000. Web.
Treloar, T. (2012). The purpose of mathematics in a classical education. Web.