I believe that the meanings of Mathematics that been suggested and summarized into 6 points are precise. This is because it conveys the application and implementation of Mathematics I daily life. For example, the study of patterns and relationship are often been applied in real life situation where as it  also represent the way of thinking and how to solve problems that we encounter into. Besides, in Mathematics not merely about numbers, but also patterns that can lead to art i.e, Escher Pattern. Next, it act as the tool of communication as it can conveys the ideas clearly and precisely by using data, statistics and many more. 

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In my opinion, Mathematics is important in my life as a trainee because it train me to instil the moral value. For example, the rationality that we gain under the practice in solving Matheatics problemscan be applied in daily life. We need to be able to form judgement in the classroom based on the evidence, able to reason out out decision and have to be open minded and able to think logically. Besides, it promotes the self- relience in ourselves. A teacher should be responsible, independent, industrious and have a good self confidence in order to disperse the knowledge to the student.

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Pythagoras was an  Greek philosopher, mathematician, and putative founder of the Pythagoreanism movement. He is often revered as agreat mathematician and scientist and is best known for the Pythagorean theorem which bears his name.

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2. Nilai "systematic working" yang menggunakan prosedur dan langkah-langkah pembuktian yg sistematik dari awal sehingga akhir proses pembuktian.

Soalan 2

 https://www.youtube.com/watch?v=Wy1KgKUQTlE

(b) In pairs, determine the mathematical value(s) in the teaching. Provide the evidence(s) that support your argument.

In Graham's progression video also, he talks about what the fraction understanding looks like, comparison of fractions, and the equivalency of fractions. He used concretizing and symbolizing methods to help students to build their fraction understanding among the connection with the whole number.

2. Sociological Value (Openness)
Evidence: 
In the video, Graham realized that rules of teaching fractions were too many and he also uncovered some of these rules. So, he is trying hard to avoid and unpack the rules that we teach with fractions and share the rules that he has uncovered and unveiled to the others.

For Control and Progress, from the example given the teacher is arranging the mathematical situation associated with daily life, which shows the control value in the example. Otherwise, another teacher is showing that how it is applicable in daily life also shows control mathematical value. Another example is given to show Progress value, the teacher is trying to use a small activity to let students understand the relationship between fractional number and rational number. An activity is carried out to relate the mathematical knowledge and let the student do generalizing. 

For Openness, the example given shows that the teacher uses question-answer methods to have students express their own opinion at the explanation stage. In this condition, the teacher gives the opportunity to explain the answers and defend, which match the openness values that need to improve expressive skills in a clear and open way, and the teacher is justifying the answer. For  Mystery value, the example given is a student trying to give their opinion, and the teacher raises awareness and admiration through the idea given, like a student is giving an opinion stating that the number can be divided by 4 and 2, also can divide by 8. But one surprise is given by the teacher, 60 can be divided by 4 and 2 both, but cannot divide by 8 and get a whole number. Thus this example also matches the mystery value. In the nutshell, all examples given in the research paper match to the mathematical values. 

Evidences: Graham Fletcher had used subitizing fraction cards as teaching aids to build student’s fraction understanding. Besides, he also used symbolizing methods as support to help students more easier build their fraction understanding. For example,  the meaning of a fraction, comparison of fractions and equivalency of fractions that he shared in the video.

Evidences: Graham Fletcher mentioned that teaching fractions need to introduce rule after rule to help pushing students' understandings. For example, he leverage the things that students know about the identity property and apply them to fractions which is a x 1 = a to help students come up with the understanding the meaning of equivalency of fractions by themself. He tried to make the connection to whole number which is 2/3 x 2/2 = 4/6 is same value as 2/3. Through this rule, students will have well understanding on equivalency of fractions by understanding the rules with the way rule after rule.

Evidences: Graham Fletcher realised that are teachers teach a lot of rules in teaching fractions. So, his goal is to kind of unwrap and unveil or demystify the rules that used when teaching fractions. And he has uncovered some of the rules when teaching fractions in his progression videos and share his teaching method that created by him to everyone.

In the video, Graham Fletcher emphasizes argument, reasoning, logical analysis and explanation. He develops proof and proving skills in the video. He observes from the tasks of the students and tries to make some analysis by proving the theory. He also uses many explanations to help people gradually build their understanding of fraction from the six stages. Among all, it concerns theory identity property (a x 1 = a) in the equivalent fraction. For example, 2/3 multiply the top by two, multiple the bottom by two and we get 4/6. By using the identity property, people will know 2/3 = 4/6 is not only multiple two separate numbers, but it is leveraging our understanding of the theory. Actually, we are multiplying 2/3 times one whole in the form of two over two. So, 2/3 and 4/6 are equivalent fractions where 2/3 x 2/2 (1) = 4/6.