Topics, Tasks and Activities in Mathematics
Provoking and promoting students to learn mathematics have been crucial and vital for math educators. A number of students do not like the subject Math. Most of them tend to avoid activities that involve mathematics. Students usually do not utilize their mathematical thinking in order to analyse the situation or the problem given. In this paper an examination of tasks and the utilisation of social constructs that could promote students and learners the initiative in learning and engaging in mathematical thinking.
Activity
The mathematical task that has been used recently is called Bricks for Books. The problem states the following:
The library in your town is building a wing. The public can buy bricks that will become a part of the new walkway connecting the buildings. Companies can buy bricks that look like this for 0 each. Families can buy bricks that look like this for 0 each. Individuals can buy bricks that look like this for each.
1. a. How many “family” bricks would it take to cover the same area as one “company” brick?
b. How many “individual” bricks would it take to cover the same area as one “company” brick?
2. Here is a sketch of what the brick walkway will look like when it is completed. It is in a pattern, but some of the bricks are missing. Use either “company” bricks, “family” bricks, or “individual” bricks to complete the pattern.
3. How much money will the library collect from the brick donations found in the pattern above?
4. Can you design a different pattern of bricks to cover the same walkway space?
5. Does it make a difference in the total donation figure if you use a different pattern of bricks? Justify your answer.
The activity is not easy and it needs deep mathematical thinking. And when students are subjected to activities that require mathematical thinking they are most likely to feel disinterested. In order to engage the students in the activity or the problem the educators must do the following tasks as pointed out by Mason and Wilder (2006) in their book Designing and Using Mathematical Tasks.
Block 2
Task 1: Encountering constructs- In the book of Mason and Wilder the first construct that educators will encounter is Accepting-Asserting. The perception in the first task is that mathematical activities become a collection of experiences that has a strong association with letting students accept the idea of mathematical thinking in different activities and then asserting its importance not just in answering the questions but in everyday lives.
The construct label is an important part in the construct of designing and utilising mathematical tasks wherein the activities or experiences in the past could be helpful in illustrating mathematical thinking in activities. In the case of the activity which is called Bricks for Books, while the students are busy in answering and understanding the activity or the problem, the educator could utilise the idea of Accepting-Asserting by asking them to stop what they are doing and develop a strategy wherein the activity will become more fun and interesting for them. The educator could do that by asking the students to do the activities in groups or by partners and turning the activity into a contest wherein the first group that will give the correct answer will receive prices. Through this activity the students will not just accept the idea of using mathematical thinking because they are required to but because they want to use it.
Providing students with other forms of learning and harnessing their mathematical thinking and skills will enable them to take more active interest on the subject.
Task 2: Influences on thinking- The view regarding precept was developed by Gray and Tall. Precept refers to the process in which the perception of the learner or the student changes as the actions that the students do become objects without their awareness. The processes become ideas, at the same time, facts or theorems implied within the actions emerge as properties that must be discussed and justified. In the case of mathematical thinking, educators may influence the thinking and perception of students regarding mathematics. Just like in the first task, the educators must let the students turn the idea of acceptance to assertiveness. The educators have the biggest influence over the learners through activities that will make the learning more enjoyable and changing the perception of students regarding mathematical thinking.
Task 3: Structure-of-a-Topic framework- In this framework, it is composed of the collection of constructs regarding thinking and ways in which the activity or the topic might be psychologised. In applying the structure-of-a-Topic framework, in the Bricks for Books, the educator could begin by asking oneself on the things that comes into mind when he or she hears the words pattern and shapes. Follow up questions could also be made such as; what problems regarding patterns and shapes that will arouse the interest of the students? When does the topic occur?
The framework Structure-of-a-Topic is a useful tool for updating the topics and it also makes it easier for the educators to keep in mind the important parts of the topic. As mentioned earlier, the idea of Accepting-Asserting is a useful construct in developing mathematical thinking among the students. In the case of the activity, Bricks for Books the educator could modify the task by doing it by group or partners and making the activity a contest wherein the first group that will answer correctly will have an incentive in class. And in order to assess if the task is effective or not the activity will be tested and evaluated in just one group while another group or class will still utilise the regular form of mathematical thinking.
Block 3
In block 3, educators evaluate the methods of teaching and learning mathematics through research. In this block, the educators focus on issues associated with assessment of the tasks and methods in promoting mathematical thinking to students in relation to the utilisation of technology such as computers.
Sustaining and concluding activity- after implementing the modified activity in math class and assessing the effects. Sustaining it if the impacts are positive is another task that the educator must do. However, maintaining it is not an easy task to do, especially if the teacher is handling more than one class with different types of students. One way in which educators could sustain the positive impact is to stimulate the students to actively make choices in class, by giving them options on how to do the activity the students could then modify the tasks in their own understanding.
In sustaining and making conclusions regarding the tasks or activities keeping the record is very important because the records are the ones being analysed, interpreted and edited by the educator. The following are the different types of records that educators could utilise in making conclusions and sustaining the activities:
Field notes- these are written records which include descriptions and remarks made by the educator regarding the performance of the students after the implementation of the new activity.
Audio recordings- it includes the interviews between the teacher and the students, as well as, recorded group discussions. It could be helpful in the assessment since the audio recordings provide an accurate detail of the results of the intervention.
Video recordings- just like audio recording, it also includes interviews and discussions between the teacher and the students, but it is mostly used when the educator wants records that are visual.
Teacher or learners work- the activities or the works that are produced by the teacher and the learner while in the course of the work such as rough notes from discussions.
After obtaining various data from different records, the educators could now use the data and assessed it. Making a transcript of conversations from the video and audio file will make it easier for educators to determine the important data that may be relevant in his or her research.
After presentation of the task, as an educator I observed that I became more involved with mathematical thinking. The class became more enjoyable because the educator and the students are participating enthusiastically on the task. As an educator I became I observed that the class became more fun during the modification of activities. In the case of the students, I observed that they also became more involved in the task because prior to the modification, the attitude of the students is very unenthusiastic and they just accept the activities and do what was required for them to do. However, when the activity was utilised in the class, the students became very eager to determine the answers to the questions. Aside from the increased enthusiasm, I also observed that the students have became more cooperative and interactive with other their classmates during the activity.
Through out the entire activity that lasted for one week, wherein everyday the students are given new set of problems and groups there are significant changes that took place in the class. First, the students started to look forward in the math subject which rarely happens since most students prefer finishing the class as soon as possible. The students become more excited on the questions and problems that they will encounter. Another surprising incident is that the students have researched and asked more questions regarding math topics. The students became more interested with the subject that is why they are the ones making an initiative to research on various topics. The incidents are very significant to me as an educator because it shows that the steps utilised in order to make the subject become more enjoyable and interesting to students have proven to be effective. And the idea of students anticipating and researching for math topics is usually difficult to attain for an educator. Getting students to become interested in mathematics is a very important and significant event in the life of a math educator since those types of incidents rarely happen.
Ethical Issues
In research in mathematics educators and researchers often times used transcripts, and audio and video recordings in assessing the topics and activities needed to be implemented. The researches on mathematics often times draw observations and conclusions from those three in order to obtain evidences for discussion and justification. And in order to gain materials that are important and useful a high level of trust is important. There is a risk involved as viewed by the learners and teachers, especially if the research is done by a third party rather than if the research is in collaboration with them. In order for the researchers to avoid mistrust from the educators and the students they must view them as a knowledgeable source of information rather than subjects being observed and studied. Recordings, whether audio or video must require consent from the people being interviewed, as well as, guardians or parents if the individuals being interviewed are not yet on his or her legal age. There are various levels of consent that researchers could utilise:
First is the agreement between the researcher and the person or group of persons being interviewed that the file will be only for personal use of the researcher and must not be utilised by any other persons.
Second is that the persons interviewed gave permission to the researcher that he or she may let the transcript or the record be heard by another individual.
The last is the permission from the individuals subjected in the study to use the transcript in public forums.
In any type of study, anonymity is important, particularly in cases that are sensitive, the names of the participant is not required or must be changed as well as the location of the participants.
Mathematics is one of the toughest subjects students encounter in class whether they are in the primary or the post graduate. Often times, most of the students despise the idea of encountering math problems and utilising mathematical thinking in order to answer the problems given to them by the educators. The educators play an important role removing the stigma accompanied by the math subjects among students. They are the individuals that have the power to influence and change the perceptions of students regarding math and encourage them to use mathematical thinking. The teachers must create classes that are interesting and enjoyable and promotes the appreciation for math and mathematical thinking. One of the ways is to develop a class that fosters cooperation and interaction with students, since math is a difficult subject having another individual help in answering the problems will be beneficial for both the students since they could exchange insights and information regarding the topic. Another is by providing incentives to students that do well in class, since it is a hard subject rewarding those students will encourage others to follow. And finally, the educators must be open to communications, a number of students fail to address their concerns on topics on math subjects because of fear that the educators may scold them. Teachers are not the only individuals that must be involved, but the students, as well; the students must also cooperate with the educators and ask questions if they did not understand. Students must engage together with their educators in analysing the topics and activities in the classroom in order to increase the mathematical understanding between the learners and the educators. The students and the educators must work together in order to create a classroom environment that fosters mathematical thinking and understanding.
Credit:ivythesis.typepad.com
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