Nov 26, 2012


LinkedIn Groups

Group: Math, Math Education, Math Culture
 Marsigit Dr MAOutside perception is sometime not good to the students, because they tend to judge rather than to facilitate. Further read my web
And also my works on, click FMIPA, click Pendidikan Matematika, click APPLY and chose Marsigit. Thanks
William Galinaitis"Scientific American Mind" (july or august issue?) has summary articles on how various factors affect the mind of early learners. Brain activity related to behavior and learning seems to be composed of genetic and environment ( cultural, family, chemical...) factors (You already know this.) How ever I was suprised how much stress that the children experience may affect their ability to control their classroom beahvior, and therefore their ability to focus on a single activity. Just a thought outside the Idea that it is only the system and not the student.

Frances WintersIs it not perhaps that math is like walking. Thousands of years ago our ancestors began to walk on two legs, some better than others. Took ages before everyone was good at it and took walking for granted.

Ginetta NistoranI've noticed more and more these days that math is taught using memorization and mechanical repetition, rather than an understanding of mechanics and logic. Very often the students are able to solve a problem very similar to the one taught in the classroom, but as soon as the layout changes, they are not able to see a pattern, or the fact that they need to apply the same concepts in a different form. For me, that means a superficial learning, based more on memorization than on logic.

Henry Schaffer@Ginetta - in some fields we call this "plug and chug". One remembers the method, plugs in the new numbers, turns the crank and out pops the answer.

I agree that this superficial - and should *not* be more than a small portion of math.

Marsigit Dr MAThe architectonic of Kant teach us that mathematics is the business of the students' architectonic. So, for me, whatever the external criteria to measure students mathematics is always mislead. The genesis competence of mathematics is really their architectonic. So, the problem for the teachers is how to uncover mathematics depth inside of the students. Until then you get the criteria of the success of your teaching mathematics after you hear the students' claim that mathematics really belong to them.
Further read my web
And also my works on, click FMIPA, click Pendidikan Matematika, click APPLY and chose Marsigit. Thanks
Marsigit Dr MAOur further perceptions on how to educate (mathematics) can be read at the following:

Henry Schaffer@Marsigit - you seem to be saying it isn't possible to measure a student's progress in math. Am I interpreting you post correctly?

William GalinaitisAgree with Marsigit: An educator is important in the moment when a learner is trying place new knowledge in to the context of their own understanding of the world. Sometimes I can "read" the student (ask them questions, have them try to explain a concept to others...) and provide the appropriate stepping stones for them to integrate the new material into their framework or modify it.

Plug and chug has its place. You have "memorized" a significant amount of material about the world. This allows you to quickly recall disjoint pieces of information and formulate them in to a sentence for communication. If you had to look up the definition of each word in the formation of a sentence, you would probably forget what you were trying to communicate.

Susan NorthridgeI agree with many of the previous comments. Practice is key and I find that my hardest working students (not necessarily the most brilliant ones) are the most successful. I also agree that there is something lacking in the way basic math is being taught in the lower grades. I teach calculus and I am always amazed at how many of my students still struggle with basic fractions.

Anne PattersonI agree that fractions are an ongoing issue for students at all levels. It sure makes a case for switching over to the metric system for ALL measurement!

Judy Dobles, MBAMy observation is that if the desire is there, people then show the dedication to learn. In American culture it is socially unacceptable to be illiterate but socially acceptable to be innumerate. Our first step is to begin a culture change to show people that math is the underpinning of everything we do so that they want to learn math.

Marsigit Dr MA@ Henry Schaffer and others: By nature it is impossible to measure a student's progress of math using a certain approach or criteria. Objective test is very bad because it is gambling. I have been trying to promote new paradigm that LEARNING is constructing activity anywhere and anytime, not much depend on teacher. Consequently, MEASURING the students competent of math is also in the means of anywhere and anytime, i.e. continuously and using various approach (tools), e.g. portfolio. The criterion test is really dangerous to the students because it is the act of REDUCTION or simplifying of students' characteristics. This will produce partially psychological character of students and ultimately produce problematic students. So then I think there is no choice for the teachers to acknowledge, trust and empower the students in term of facilitating their needs in learning math as their effort to construct their own life (math). Thank's

Henry Schaffer@Khaled - I think that this thread has mostly been about counting - arithmetic - algebra and geometry. Not very theoretical math areas.

@Marsigit - I disagree that the teacher, and teachers' practices - are mostly irrelevant.

I also disagree with what I think you are saying "By nature it is impossible to measure a student's progress of math using a certain approach or criteria. Objective test is very bad because it is gambling." Asking a student 6 x 7 = ? or "Prove that the sum of the angles ..." are object tests. They are "gambling" in the sense that they are a partial sample of the entire subject area. But a (validly chosen) sample does give valid information about the universe sampled.

Henry Schaffer@Anne - I use the metric system now and then - and when I want to divide 2 liters of orange juice equally among 7 people, can I describe this without using fractions? :-)

larens imanyuel@Susan.
To teach fractions effectively one needs to teach division as the inverse of multiplication. For multiplication one aggregates unit squares into rectangles. For division one may need to divide unit squares into smaller rectangles. One also needs to teach that one is the multiplicative identity. Rational arithmetic and its rules naturally follows from this, so there is no need for students to have a hard time with fractions. Teachers generally switch representations when going to fractions, so leave students confused.

larens imanyuel@Anne.
Scientists use different systems of measurement to naturally fit the physical system with which they are working. To omit this fact by only teaching decimal arithmetic and the metric system is to do students a disservice by alienating them from real science.

Marsigit Dr MA@ Henry Schaffer: What do you expect by questioning the student 6 x 7 =?. Do you expect that the student will deliver his answer 42 ?. What really my concern as a problem is that if you just expect that the students just think about 42 ? Why should, at the first stage of their learning, we do not tolerance to look at other possibilities of answers? In my perception, 42 is just a very static ideot answer. The better and more brilliant answer is his STRUGLING to find out the answer 42. Why? Because it reflect his nature of life (math). It is very good that at the early step the students my get wrong answer. While this aspect will not emerge and not ever emerge when you use OBJECTTIVE test or CRITERION test. That is really my struggling how we implement mathematics education that in line with the student NEED; no just accord with the teacher's expectation (or system outcome expectation).

Henry Schaffer@Marsigit - "What do you expect by questioning the student 6 x 7 =?. Do you expect that the student will deliver his answer 42 ?." Actually I do. If not, then the student has a deficiency in arithmetic and attention should be paid to remediation.

I don't expect or want the student to "just think about 42" - but I do want them to be able to do arithmetic correctly. As far as "tolerance to look at other possibilities of answers" - well, other answers (e.g. 5 or 112 or 29 or 77) aren't correct.

"42" is the correct answer to this arithmetic - it isn't idiotic, and it shouldn't be a "STRUGGLE". If it is, then it's premature to ask the question and instead the student should review more basic arithmetic.

If we never ask such "objective" questions, and don't build one simple arithmetic as the grounding for more advanced math - we will usually fail to have our students be able to understand, let alone master, the more advanced math.

Victor Guskov@Henry, I agree with you completely.

Victor Guskov@Marsigit, your opinion is unacceptable for me.

Gail MillsYes Henry! Learning takes many forms. There is a time to experiment and ponder and a time to master the givens of accepted fundamentals and knowledge. Operating comfortably with numbers does not destroy the thinking and creative abilities. Mastery provides a foundation and vocabulary to discuss abstractions.

Marsigit Dr MA@ Henry, Victor and Gail: Thank a lot for the responses. Ultimately, I think there is no adult or older people, including me, not to expect that younger people should have a correct answer of 42 for math problem 6 x 7 =? Implicitly, it was stated in my previous statement. But this is not the case that I perceived as a problem of the nature of teaching math. I in purpose have striven to provoke you that, pedagogically, the real problem of teaching 6 x 7 is not just guessing to get or memorizing 42, but the process of getting or producing 42. Then, the next problem is about what kinds of the PROCESS, who and how it to be promoted amid the balance between the role of teacher and his/her students. In many places, many teachers teach the students just to memorize 6 x 7 = 42. That's I called it as idiotic math i.e. learn math without understanding and processes. Some points I agree with you and I understand the worry of Victor and Gail.

However, I still don't understand about your point of STRUGGLING in math. For me, struggling is something ontologically an aspect of our life (math). As an adult or older people we need to promote to younger people the means of struggling of every aspect of life (including math).

@ Gail Mills: No certain pedagogy action means to destroy thinking and creativity unless it is partial, not complete, and under certain motive of adult. So, there is no the best way to educate people. However, the best way to educate people is if it is without PREJUDICE.

@ Victor Guskov: Having my explanation you may change your position. I expect your elaboration.


Henry Schaffer@Marsigit - While I agree that the student should learn how 6 x 7 = 42 (by rectangle, etc.), they still do have to learn that 6 x 7 =42. Yes, that is memorization - and I see nothing wrong with it. In fact, if a student needs to go back to the derivation each time a multiplication is done, it will take much, much too long. As far as guessing - if a student can consistently "guess" the answers to arithmetic problems - then perhaps it isn't really "guessing"??

As to "struggling" - perhaps we are using the word differently. To me it means need to use unusual effort with a constant stress of failure. I don't consider it is the same as, perhaps, "study diligently".

Victor Guskov@Marsigit, I take 5th graders and should teach them fractions, decimals, etc. Regrettably, too many of them don't possess simple arithmetic skills - addition and subtraction within the limits of 20, multiplication and division within the limits of 100. I suspect that elementary school teachers waste too much time on “the process of getting or producing” instead of practice and memorization.

Gail MillsWith standards-driven education teachers don't have a lot of choice. My 4th grade grandson had 204 concepts to "learn". All the teacher could do, if she stuck with the district's plan, was expose the students, not teach the students. Teaching does not occur if learning does not occur.

Marsigit Dr MA@Henry Schaffer: I am interested with your notions: guessing that it isn't really guessing; struggling that it isn't really struggling; and memorizing that it isn't really memorizing. Really that's all my points. The problem is then how to realize them. Thank's

@Victor Guskov: Again, in my opinion, you too much stressing on your own expectations about what the students do or their capabilities. This is really the main problems of education i.e. predominantly employing by teachers' expectation but lack of developing and employing students' expectation. You also seem in a hurry and not be passion to wait the students to develop math and produce their own concept of math. This is also the biggest problem of math educ. because it is related to the curriculum and the educ.system.

@Gail Mills: How sad the fate of EDUCATION due to the standard-driven educ system. Yes I am aware that amongst the global interaction many countries stressing much on how to compete with others. Consequently, in educ also the system means of competing between everything. Why do we not to promote education based on COLLABORATION/COOPERATION rather than competing. You know that in every scene of competition, there will always be the LOOSER. Who will take responsibility to the looser students? No other people except that of teachers. No other people except that of the people who really think intensively and extensively about the nature of education.

Bradford Hansen-SmithThe best way to overcome struggling is to begin to have some curiosity about what we are doing to the degree that self-discipline and responsibility becomes the result of effortless attention in learning to love through the experience of doing.

Where does this leave math when teachers, as a generalization, do not simulate curiosity about their subject, themselves having little curiosity. No matter how many good teachers, few students will find real interest in the mathematical form. There are other ways to acquire understanding of pattern and thinking abstractly in "logical" systematic ways.

Competition kills curiosity and love for what we do by focusing on self over others. Maybe by opening beyond individual self-importance we can find curiosity enough to love and cooperate, the first step towards collaboration. Learning is grounded in curiosity, yet we still favor competition, going to war with each other to prove ourselves. It is difficult for students to learn when they are in an educational war zone.

Art DiVitoArgh. I'm sorry, but I really don't like "competition" getting bad mouthed like this (I say as I am about to head to a five state Regional US Tennis Assoc. playoff!). "Competition" exists only because activities must have a "limit" (typically the limit is time; volleyball goes with points, tennis with sets, and baseball with outs). If you don't like it, try cricket, ... but even those games eventually end. : )

When folks, especially teachers, speak in terms of competition "killing curiosity," "focusing on self over others," and "creating losers," then it is time to reassess our understanding of "competition." Life itself is not fair. Get used to it. Courses are still passed or failed. Shall we drop that? Shall we just say, "It's okay, Johnny, you don't know dog manure (about fractions, or signed numbers, or whatever), but you're going to the next course, the next teacher, the next grade, the next school"? Real competition is about having fun, learning to cooperate with teammates, respecting your opponents, accepting defeat, being gracious in winning, and advancing the "game." Perhaps above all else, competition is about "getting it done." What students need to do today, more than ever, is to "get it done." The next time you attend a concert, do you want to listen to someone who is accomplished or someone who is not?

Education has its problems today. Competition is not one of them.

Elias Gourtsoyannis@Art. I agree. Perhaps not with the tone. But, you're right.
I once participated in a Mathematics competition in California. At the
time, I did not even know trig identities. I revised until late night from
the textbook. Next day, I took part. Several schools. It was a big
multiple-choice test. Together with some additional questions. The top
prize was an aluminium state of the art log-log type slide rule. In the
test, I did not even make the top 50%. But, I had an idea. The competition
was sponsored by the slide rule makers. The top prize was for sale for $32.
Now, I had some pocket money I had not spent. Given to me by the AFS,
monthly. So, I promptly bought it. It had a beautiful leather case!

Perhaps one of the reasons I eventually studied math?

Marsigit Dr MA@Bradford Hansen-Smith: Your ideas are challenging. I agree with you about curiosity and the concept of educational war zone. However, from your explanation, I found some in-synchronize notions. Curiosity is very difficult to be directly connected with self-discipline, responsibility and understanding of pattern and thinking abstractly in "logical" systematic ways. Why? Because curiosity is something happened in the very early stage of everything and it is original. It should be free, neutral and not have a certain burdened. Even the 7 month age baby has his/her curiosity. It is clear that it cannot be compared with the notions of self-discipline, responsibility and understanding of pattern and thinking abstractly in "logical" systematic ways. While the later is coming from the very powerful authority e.g. adult people (authoritarian teacher or pure mathematician). I agree with you on individual self-importance as the first step towards collaboration.

@Art DiVito: Comparing educational activity with other subjects e.g. sport, art is to some extent not proportionally appropriate. Education is sometime about long term program rather than short term program as you said because of limitation of time. If you put LIMITATION as the main factor of education, I assume that there is not appropriate foundation or theories of education. You feel you have just very limited time because you feel you have everything to teach, while you may perceive that your students have nothing. I prefer to give the small amount of knowledge to my students in which they are in a hurry running to come forward to me; rather than I bring a huge amount of knowledge but they are running leaving me. So, the concept of limitation much depend on our perception. It is you as the winner who said "Real competition is about having fun, learning to cooperate with teammates, respecting your opponents, accepting defeat, being gracious in winning, and advancing the "game."" I prefer to hear it from the looser rather than from the winner.

Marsigit Dr MA@Elias Gourtsoyannis: How wonderful your experienced in the process of learning math. However, it may be difficult for other people to follow you. Your experience is unique and only a few, while the teacher in a certain class should take responsibility for all of his/her students (both the winner and the looser). Your experience is your creativity. Regrettably, teachers can not teach CREATIVITY. They can only facilitate or develop the scheme in order the students are to be creative. As we know that other students are also unique. Of course it is his uniqueness that the most important of his value.

Art DiVito@Marsigit. Marsigit, every winner has lost tons more than they have won. It is in losing that we know we are human; and we generally learn more from losing than from winning. We have to learn to handle losing (gosh, I wish I could do that when I golf!), not shy away from the challenge. ... As for analogies with education, ... education could learn from a few. A wise man once said (I believe it was an ancient Greek, but I cannot find it; I wonder whether Elias would recognize it) the now very politically incorrect, but nonetheless true: "A nation that draws too broad a distinction between its scholars and its warriors will have its lessons taught by cowards and its battles fought by fools."

Marsigit Dr MAArt DiVito: Thanking for the response. Again I wish to say that it is you as the WINNER or you as the POWERFUL SUBJECT/AUTHOR/TEACHER or you as the COORDINATOR/SPONSORSHIP of Playing Game or you as the PEOPLE who always have the AUTHORITY to TEACH/EDUCATE ...who said that "every winner has lost tons more than they have won. It is in losing that we know we are human; and we generally learn more from losing than from winning. We have to learn to handle losing (gosh, I wish I could do that when I golf!), not shy away from the challenge.". And also again I prefer to hear it from the LOOSER or from the STUDENTS or from the OBJECT or from the WEAKER or from THE PEOPLE who have no authority to speak. I do really more trust to the last because they are the MORE. As you know that at every single game there will be always very-very few WINNER (first, second, third). You can imagine how frustrated, sad or even feel devastated most of the looser football team in the UK Olympiad (99 %), compare with just MEXICO who defeat Brasil in the Final (0,001%). For the people who really love football playing (not football game) they also feel like the looser. Imagine please!

By relating the scholar and the warrior in a very short distance, you look a very pragmatics people. In my opinion, because education is about long term program, it still needs idealistic people. As Immanuel Kant said :"Practice without theory is BLIND, and theory without practice is EMPTY". So, the scholar without its warriors is empty, and the warriors without its scholar is blind. So it is dangerous for you to be pragmatical alone without hearing me as an idealist because it can make you blind. And also it will be dangerous also for me without learning your notions because it can make me empty.

Education is for ALL. The teacher should take responsibility both the winner and the looser. It is very-very bad and inappropriate behavior for the teacher to urge the looser to give applause to the winner while the prize is only for the winner. The teacher should also consider the psychological conditions of his/her looser students, because their fate as the looser is also because of the teacher's act/behavior/schema. So again, in education, I prefer to promote COLLABORATION rather than COMPETITION.

All that I strive to prove that some of your notions are not fit with the nature of education, and so that I disagree with you.

Behnaz Herbst, MSc.OCTWe need to "teach" in the ways that the brain "learns"! In many cases, the teaching happens, the learning may not! I wish for our school system to become more brain - friendly. Also, our students need to be taught how cognition takes place, how memory works, how they can focus, and retain their attention for longer periods of time. They need to be convinced that their brain can change and adapt and that their perceived inability is really a myth. If we could achieve this, they might be upset from home, but when in class, they will learn! There is no way they won't!

William GalinaitisPeople learn new things all the time when the need is there and they are mentally capable. To be really clinical about it, I can sent up an experiment which shows the innate curiosity of an average person (motivation) and their ability to learn something new, when the stimulus is correct.

Marsigit Dr MA@Behnaz Herbst, MSc.OCT: I am worry that your much pay attention on manipulating students' brain is also really a myth?

@William Galinaitis: In my perception, stimulus-response psychology is out of date. I prefer to use various approach.

Patricia FreyBecause when basic math was taught, it was mostly taught by generalists who did not understand even the basic math! Consequently, they taught a bunch of rules and procedures to follow without thinking! How easy is it to remember a bunch of rules and the order in which they are to be performed, if you have no understanding of the basis of those rules?
Dr Patricia Frey

Marsigit Dr MA@Patricia Frey: I prefer to use the sentence "...when basic math was learned by the student...". Regrettably, what you meant by specialist may still perceive to teach bunch of rules and procedure although with thinking. In my perception, it is very difficult to connect younger (primary school) with a bunch of rules and procedure; however, it can be a certain struggling. I prefer to introduce basic math using concrete object surrounding them.

Marsigit Dr MA@Elias Gourtsoyannis: I am more concerned about LEARNING MATH rather than TEACHING MATH. So I am more concerned about how the student learn math rather than WHO TEACH MATH. Hence, theories of learning math should come first; it should come before theories of teaching.

Elias Gourtsoyannis@Behnaz. This is the view of the 'embedded mathematics' program advocated by George Lakoff and Raphael Nunez in their their book. I will provide a full reference in a moment. They do seem to overstate their case. They claim that 'brain research' has confirmed their view. This can alienate some practitioners, however. Not enough is known as to what the 'brain' actually does. Aristotle, for example, thought it was some kind or refrigeration system for the blood. Perhaps our present state of knowledge will prove just as outdated, some day!

@Bill. I liked your joke on 'sending up' an experiment. It is always healthy to question 'objective' procedures!

@Pat. I agree with you. Mathematics should always be taught by Maths
graduates. However, the reality is that, until this happens, most primary
mathematics teachers would be trained practitioners. Increasingly, however,
in some countries, teacher training includes basic mathematical skills.
With the right trainer, student teachers can, and do, experience something
of the flavor of true mathematical thinking and processes. And, later, they
can pass it on to their pupils!

@Marsigit. Sorry. I just run out of time in editing my previous comment. I had to resubmit it. But, I noted your comment. And, I do not disagree!

Elias Gourtsoyannis@All. The reference is: 'Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into being'. By George Lakoff, and Rafael E. Núñez 2000,ISBN 0-465-03770-4. There is a Wikipedia article on it. I am not sure as to its accuracy. But, I have read the book itself.

Marsigit Dr MAThank Elias Gourtsoyannis for the information

Elias GourtsoyannisΔεαρ Marsigit. You can call me Elias!

Behnaz Herbst, MSc.OCTElias, thanks for your comment, but I was not referring to enactivism or the theory of embodied mind, brought forward by Humberto Maturana. I am simply stating that there are better ways to teach math. We don't know a lot of things about how the brain works, but we know some stuff and we better start using these facts. For instance, we should teach in 15 minute sessions with a couple of minutes of rest in between and repeat every hour of teaching after an hour, the next day (a 5 minute review), in one week, and then in one month if we want the information to be submitted to the long - term memory. We need to teach our students that when it comes to learning, brain cannot multitask, so they should not be watching TV and texting their friends while doing their homework! In a recent study, children who jogged for 30 minutes 3 times a week showed significant academic improvement compared to those who did not exercise. Physical activity is a must when it comes to cognitive performance. It would be nice if the neuroscientists and educators would collaborate and do real - life research together. Don't you think?

Elias Gourtsoyannis@Μπενάζ. 'Νους υγιής, εν σώματι υγιεί' - a healthy mind resides in a healthy body. Alan Turing and a friend developed a version of chess. Where you jog around the house in between moves. And, by the time you return, your opponent has to complete her next move!

Bradford Hansen-SmithArt, you state “Education has its problems today. Competition is not one of them.”
Teaching to the test is certainly competitive when the results are used to determine who gets the prize, from individual students, to teachers, to school board, to country ratings.

Marsigit, the connections I see are when children are curious is that they will stay with what engages them for as long as they need to. Over time they develop self-discipline to stay with what is of most interest and not get diverted to that of less meaning and value. Learning to love what you do is a discipline of the self to that responsibility. Anytime one is deeply interested in what they do, originally growing out of curiosity, there is understanding that expands the conceptual context and system of logic that governs that particular activity.

Individual self-importance is not the same as giving value to yourself in the same way you value others. This is what makes collaboration possible.

Art DiVito@Bradford. I'm sorry, Bradford, but I regard "teaching to the test" as a construct brought by those who favor the collective and uniformity, ... not by those who value the individual and creativity. If you reject competition, then you reject assessment itself; education would reduce to absurdity. I just don't understand this desire to paint competition as some sort of negative. The White Sox just swept the Yankees. Doesn't that make almost all of us just a little happier this morning? : )

Bradford Hansen-SmithArt, I do not know about "almost all of us."
I am not a sports fan. Some team, political party, country, corporation, or individual in any field, winning over someone else dose not make my morning happier. The news is full of this sort of thing. I can certainly appreciate all that it takes for an individual or team to preform to excellence. I have done both in the competitive arena and find life to be much larger and more grand outside of the mind in competition.

Marsigit Dr MA@Art DiVito: Assessment is the most crucial problem in education. I totally agree with the assessment if it means to collect or record students' activities and achievements. However, it can be a big problem if it means to evaluate, because the next important question is who has the authority to evaluate? It will also no problem if the teacher himself evaluate his/her students, because the teacher is the people who knows the best about his students. The problems arise when evaluation is carried out externally or by external institution/board.
If it does so, philosophically there will be a huge reductions or simplification. Reduction or simplification is a kind of psychologically unhealthy partially dimension of life. If education from time to time always produce a simplified generation, then we will have a problematic generation. Look at directly to the phenomena in the society not only in the certain country but also in each country all over the world. So, according to my point of view, the best assessment should be supported by keeping-record (portfolio) both by the teacher and by the students themselves. The form of it can be authentic assessment or classroom-based assessment. Thank you

Marsigit Dr MA@Bradford: Because there important and strategic, so now I am spying your notions "the connections I see are when children are curious is that they will stay with what engages them for as long as they need to. Over time they develop self-discipline to stay with what is of most interest and not get diverted to that of less meaning and value. Learning to love what you do is a discipline of the self to that responsibility. Anytime one is deeply interested in what they do, originally growing out of curiosity, there is understanding that expands the conceptual context and system of logic that governs that particular activity. ". I may produce my comments later. Thank's

Marsigit Dr MA@Elias: The reference you indicated "Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into being'. By George Lakoff, and Rafael E." is very strategic and important. I have just read at a glance from the sample of excerpt. My first conclusion is that it is strategic and important evidences and then thus as theories of the origin of math concept. It is a very good illustration on the origin of math concept from the early stage up to the highest stage. I am still finding of what happened and comparing with other theories of how a certain student in a certain stage is to construct math knowledge, what kinds of math and what kinds of the limit or edge. As you know, according to realistic-ism, there are four stages to construct math: concrete math, concrete model, formal model, and formal math. In an easy way there are just two categories: horizontal math and vertical math.

Marsigit Dr MA@Art DiVito: I really wish to know about your perception of math? How you define math? What kinds of math? Do you have any particularity on how to develop Pure Math-Formal Math-Axiomatic Math? or Maybe School Math? That's all are really also my question to you. Is there any reference you may indicate that I can read? Thank's

Marsigit Dr MA@Bradford: After a moment I think them deeply and trying to reconstruct your ideas, I may produce the following comment. I agree with you about the connection between curiosity and self-discipline. The most important point is about SELF as the aspect of discipline. As you know, or as in the common-sense or at least it is my previous understanding , usually the term discipline is interpreted by something that coming from outside. I now understand, by relating with and imaging your activities with the students in searching the pattern of circle, that SELF-discipline ultimately come out from long engagement starting with curiosity. That the point that I really am enthusiastic also to expect about the emerging of SELF....discipline. Of course it needs the experienced adults to develop a scheme in such away that they are interested and not get diverted to that of less meaning and value.

The next most crucial problem is how the self-discipline leads to a certain responsibility. Responsible of what, how, when and where? To what extent that the degree of the stage of kids mental development come into the process of curiosity-selfdiscipline-responsibility? I think it will be very complicated psychological aspect of learning. Even it has not until the stage of understanding math concept and logical system.

So I agree with you at the first stage, I am still thinking at the second stage; however, I still didn't agree with you at the last stage. I perceive that there is still a huge gap between self-discipline/responsibility and understanding of math concept and logical system. As you know that in mathematical understanding also consist of mathematical method and mathematical content as well as math attitude. I expect that self-discipline/responsibility may contribute to the math attitude; but they are still far away from math method, math concept and logical system. Thank

Elias Gourtsoyannis@Marsigit @ Bradford@Art

I believe Bradford you have a point. On the principle that Mathematics teaching cannot be that different than teaching drawing, painting, or sculpture. It is another one of the 'seven liberal arts' of the Hellenistic era. In the late Middle Ages, and the Renaissance, this was the standard format for 'university' education. Let us not forget that examinations and grades were a rather late invention. Until then, a student was judged by professors orally and sometimes in writing. 'Portfolio' work. Yet, there were dedicated scholars in many fields. Other than completely practical subjects. And, advanced scholarship went on. In the Islamic world, there was already a strong tradition of studies. Based partly on the Koran but also on Greek Classics in philosophy, mathematics and science. Some may even hold the view that the transmission of Greek learning to the west occurred via the medium of Arabic. I do not know if Arabs used grades. But, I am sure they, like the ancient and medieval scholars, held debates. If learning is based on a 'collegiate' approach, in the sense of 'community of scholars', lack of assessment does not necessarily imply lack of competition. I am sure it is just as easy to to arouse jealousy and ambition by 'winning' an important disputation. As an alternative to achieving high grades. Perhaps the most rewarding acclaim is the enthusiastic approval of one's peers!

Marsigit Dr MA@Art DiVito: I am interested with your claim " would reduce to absurdity".
I may interpret ABSURDITY as something not clear, not effective, not efficient, or even meaningless. In my opinion, this is very-very subjective claim. In the case of totalitarian government/institution/board, they perceive education as a tool or instrument to legitimate and achieve their interests; so they always expect that education should be very clear scheme, effective, efficient, and thus not reduce to absurdity. These also happened in the countries that implement the centralized curriculum. So for the people who are striving to promote decentralized-curriculum, your claim of absurdity can be very sensitive.

Marsigit Dr MAMarsigit Dr MA • @Elias: Your last description is inspiring me. However, I may smooth your notion by indicating that NOT TEACHING but LEARNING may not be differentiated by one's activities in drawing, painting, or sculpture.

Further, you seemed made the very technical reasons for the need of learning based on a 'collegiate' approach, or may I call it as 'small group discussion'. Really, it has a very deep of philosophical grounds.

As we know that there are various definition/assumptions about the nature of "KNOWLEDGE" i.e. the nature of "math". As I asked to Art DiVito, but it has not been responded yet, there are different ways to define math. Usually, Conservatives or Old Humanist, define MATH as a body of knowledge or as a body of structure. There are also the same definition made by Pure/Axiomatic/Formal Mathematicians. They usually do not much pay attention to what happen inside the learner.

However, if we read Paul Ernest on his Philosophy of Math Educ., Socio-Constructivist or even Liberalis define Math very differently. They define math as a CREATIVITY or PROCESS OF THINKING or even as SOCIAL ACTIVITIES. Accordingly, the nature of math can be perceived as social-activities. What then the implication to the practice of teaching? There absolutely need that in learning activities the teacher should give the chance the students to do/work math in a SMALL GROUP DISCUSSION.

I do agree with you that in the sense of 'community of scholars', lack of assessment does not necessarily imply lack of competition. Further, I may add that by COLLABORATIVE approach does not also necessarily imply lack of competition. However, jealousy, ambition, and rewarding acclaim are just the impacts of working in such a certain small group.


Elias Gourtsoyannis@Marsigit. Thank you for your kind comments.

A noted worker in the field of ancient mathematics is Serafina Cuomo. Her
main concern is to determine the self-image of ancient mathematicians from
the professional point of view. In other words, how did these ancient
mathematicians see themselves? As practical advisers, as
researchers-scholars, or what? Daily bills had still to be paid,
A well known book of hers can be found in

She has also written on Pappus, the Alexandrian mathematician.

Yogyakarta, 25 Agustus 2012


  1. Sumandri
    S2 Pendidikan Matematika D 2016

    Matematika ibarat sebuah bangunan bertingkat. Dia punya pondasi, rangka, lantai yang bertingkat, dan unsur-unsur lain yang melekat padanya. Secara struktural bangunan itu hanya dapat dipertanggungjawabkan keberadaannya sebagai bangunan apabila dibangun di atas pondasi atau landasan yang kuat. Dengan demikian, bangunan ini akan tetap kokoh dari masa ke masa meskipun diterpa badai dan taufan. Dalam matematika, landasan berfungsi untuk memperkokoh, menyokong atau menopang bangunan matematika. Selain itu landasan matematika juga sebagai sarana yang diperlukan untuk 'merenovasi' bangunan matematika itu sendiri.

  2. Saepul Watan
    S2 P.Mat Kelas C 2016

    Bismilahir rahmaanir rahiim..
    Assalamualaikum wr..wb...

    Artikel ini memaparkan keterkaitan antara ilmu filsafat dengan Matematika. Dalam proses pembelajaran di sekolah pembelajaran matematika hendaknya tidak didasarkan pada hafalan, melainkan proses pembelajaran yang menggunakan logika sehingga siswa mampu memahami bagamana proses dari setiap konsep yang diberikan. Guru pun hendaknya mampu menjelaskan kepada siswa bahwa matematika adalah milik mereka sehingga siswa memiliki kreativitas dalam pembelajaran. Selain itu, guru hendaknya mampu menerapkan berbagai pendekatan sehingga pembelajaran menjadi lebih menyenangkan.

  3. Wahyu Lestari
    PPs P.Matematika Kelas D

    Pencarian dasar matematika ini sejalan dengan mencari landasan filosofis pada umumnya. Aspek dasar matematika dapat ditelusuri melalui tapak sejarah filsafat dan sejarah matematika juga. Hersh, R. menjelaskan bahwa dasar matematika 1 memiliki akar kuno; para filsuf belakang Frege adalah Hilbert, Brouwer, Immanuel Kant. Filsuf balik Kant adalah Gottfried Leibniz; para filsuf belakang Leibniz adalah Baruch Spin¬oza dan Rene Descartes. Para filsuf balik semua dari mereka adalah Thomas Aquinas, Agustinus dari Hippo, Plato, dan kakek besar foundationism-Pythagoras. Istilah "dasar matematika" 2 kadang-kadang digunakan untuk bidang-bidang tertentu dari matematika itu sendiri, yaitu untuk logika matematika, teori himpunan aksiomatik, bukti teori

  4. Nurwanti Adi Rahayu
    S2 Pendidikan Matematika Kelas D 2016

    Ilmu yang ada dan mungkin ada di dunia ini memiliki dasar pemikiran yang menjadikan ilmu tersebut dapat digunaka
    Begitu halnya dengan disiplin ilmu matematika.
    Matemaika ibarat sebuah pondasi yang membutuhkan kerangka yang kuat.
    Pondasi, rangka, lantai yang bertingkat, dan unsur-unsur lain yang melekat pada matematika itu sendiri.
    Secara struktural bangunan itu hanya dapat dipertanggungjawabkan keberadaannya sebagai bangunan apabila dibangun di atas pondasi atau landasan yang kuat.
    Bangunan ini akan tetap kokoh dari masa ke masa.

  5. Yosepha Patricia Wua Laja
    S2 Pendidikan Matematika D 2016

    Berdasarkan link matematika adalah cabang dari filsafat yang mengkaji anggapan-anggapan filsafat, dasar-dasar, dan dampak-dampak matematika. Tujuan dari filsafat matematika adalah untuk memberikan rekaman sifat dan metodologi matematika dan untuk memahami kedudukan matematika di dalam kehidupan manusia. Sifat logis dan terstruktur dari matematika itu sendiri membuat pengkajian ini meluas dan unik di antara mitra-mitra bahasan filsafat lainnya.
    Tema-tema yang sering diperbincangkan di antaranya:
    • Apakah sumber pokok bahasan matematika?
    • Apakah status ontologis dari entitas-entitas matematika?
    • Apakah yang dimaksud dengan objek matematika?
    • Apakah sifat/karakter dari proposisi matematika?
    • Apakah kaitan antara logika dan matematika?
    • Apakah peran hermeneutika di dalam matematika?
    • Jenis penyelidikan apakah yang memainkan peran penting di dalam matematika?
    • Apakah tujuan dari penyelidikan matematika?
    • Apakah yang memberi pertautan antara matematika dan pengalaman?
    • Sifat manusia apakah yang berada di sebalik matematika?
    • Apakah yang dimaksud dengan keindahan matematika?
    • Apakah sumber dan sifat kebenaran matematika?
    • Apakah hubungan antara dunia matematika abstrak dan semesta materi?
    • Apakah matematika suatu bahasa yang mutlak dan universal?

  6. Dessy Rasihen
    S2 P.MAT D

    Dari percakapan mengenai foundation of mathematics diatas dijelaskan bahwa sifat matematika dapat dianggap sebagai kegiatan sosial dimana dasar matematika adalah dasar logis dan filosofis matematika atau penyelidikan matematika mendasari sifat matematika. Dasar matematika dapat dipahami sebagai studi tentang konsep-konsep matematika dasar dan bagaimana mereka membentuk struktur konsep yang lebih kompleks. Selanjutnya, juga dibahas mengenai peran seorang guru sebagai pendidik agar dapat mengkondisikan siswa ketika mulai mencoba mengkonstruk pengetahuan baru ke dalam pengetahuan yang telah dimilikinya.

  7. Nurwanti Adi Rahayu
    S2 Pendidikan Matematika Kelas D 2016

    Dalam matematika, landasan berfungsi untuk memperkokoh, menyokong atau menopang bangunan matematika.
    Selain itu landasan matematika juga sebagai sarana yang diperlukan untuk 'merenovasi' bangunan matematika itu sendiri.
    Dalam landasan matematika kita akan memfokuskan diri pada 2 unsur pokok, yaitu Logika Matematika (atau yang biasa disebut Symbolic Logic) dan Teori Himpunan (Set Theory).

  8. Supriadi / 16709251048
    Kelas C 2016 Pendidikan matematika – S2

    Dari artikel di atas salah satu inti yang saya dapat pahami dari percakapan di atas adalah "An educator is important in the moment when a learner is trying place new knowledge in to the context of their own understanding of the world. Sometimes I can "read" the student (ask them questions, have them try to explain a concept to others...) and provide the appropriate stepping stones for them to integrate the new material into their framework or modify it". dari pernnyataan tersebut peran seorang guru sebagai pendidik sangat dibutuhkan bagi siswa ketika para siswa mulai mencoba memahami sebuah pengetahuan baru ke dalam pengetahuan atau konteks yang telah dipahaminya. seorang guru sebagai pendidik dan fasilitator tentunya sangat dibutuhkan siswa untuk mengasimilasi dan mengakomodasi pengetahuan baru tersebut agar sesuai dengan konteks yang telah dimilikinya atau mengubah konteksnya sesuai dengan pengetahuan barunya.

  9. Primaningtyas Nur Arifah
    Pend. Matematika S2 kelas C 2016
    Assalamu’alaikum. Diskusi yang menarik di forum para profesional di Linkedln. Saya setuju dengan pendapat Prof. Marsigit ‘Penilaian adalah masalah yang paling krusial dalam pendidikan. Saya sangat setuju dengan penilaian jika itu berarti mengumpulkan atau mencatat aktivitas dan prestasi siswa. Namun, itu bisa menjadi masalah besar jika itu berarti untuk mengevaluasi, karena pertanyaan penting berikutnya adalah siapa yang memiliki wewenang untuk mengevaluasi? Juga tidak masalah jika guru mengevaluasi sendiri muridnya, karena guru adalah orang yang paling tahu tentang muridnya. Masalah timbul saat evaluasi dilakukan secara eksternal atau oleh lembaga / dewan eksternal.’

  10. Lana Sugiarti
    PPs Pendidikan Matematika D 2016

    Saya juga setuju seperti yang disampaikan oleh William bahwa seorang pendidik penting saat pembelajar mencoba menempatkan pengetahuan baru ke dalam konteks pemahaman mereka sendiri tentang dunia, sehingga guru seharusnya mampu merancang pembelajaran yang kontekstual dan sesuai dengan karakteristik siswa agar siswa mampu membangun pengetahuannya sendiri dan menemukan konsep dari yang dipelajarinya.

  11. Ahmad Wafa Nizami
    S2 Pendidikan Matematika D

    Foundation of mathematics adalah studi tentang dasar filosofis dan logisdan / atau algoritmik matematika, atau, dalam arti yang lebih luas, penyelidikan matematis tentang apa yang mendasari teori filosofis mengenai sifat matematika.( Leon Horsten (2007, rev. 2012), "Philosophy of Mathematics" SEP) Dalam pengertian yang lain, perbedaan antara dasar matematika dan filsafat matematika ternyata sangat kabur. Dasar-dasar matematika dapat dipahami sebagai studi tentang konsep matematika dasar (bilangan, gambar geometris, himpunan, fungsi, dan lain-lain) dan bagaimana mereka membentuk hierarki struktur dan konsep yang lebih kompleks, terutama struktur fundamental yang penting yang membentuk bahasa matematika. (Rumus, teori dan model mereka yang memberi makna pada formula, definisi, bukti, algoritma, dll.) Juga disebut konsep metamatematis, dengan memperhatikan aspek filosofis dan kesatuan matematika. Pencarian dasar matematika adalah pertanyaan utama filsafat matematika; Sifat abstrak dari objek matematika menyajikan tantangan filosofis khusus.

  12. Cendekia Ad Dien
    PPs Pendidikan Matematika Kelas C 2016

    Pondasi matematika merupakan sarana yang diperlukan untuk 'merenovasi' sistem dari matematika itu sendiri. Landasan matematika memfokuskan diri pada 2 unsur pokok, yaitu Logika Matematika dan Teori Himpunan. Walaupun, beberapa pemikir pada filsafat modern dari matematika menolak bagi keberadaan pondasi di dalam matematika, namun beberapa filsuf masih tetap menaruh perhatian kepada kegiatan kognisi manusia sebagai dasar bagi diletakkannya pondamen matematika. Mereka mencoba meletakkan dasar matematika pada kegiatan kognisi manusia, seperti yang dilakukan Immanuel Kant, bukan pada obyek di luar matematika.

  13. Resvita Febrima
    P-Mat D 2016
    Bidang pengetahuan yang disebut filsafat matematika merupakan hasil Pemikiran filsafati yang sasarannya ialah matematika itu sendiri. Filsafat sebagai rangkaian aktivitas dari budi manusiapada dasarnya adalah pemikiran reflektif (reflective thinking). Pemikiran relatif atau untuk singkatnya refleksi (reflection) dapat dicirikan sabagai jenis pemikiran yang rediri atas mempertimbangkan secara cermat suatu pokok soal dalam pikiran dan memberikannya perhatian yang sungguh-sungguh dan terus-menerus (the kind of thinking that consits in turning a subject over in the mind ang giving it serious and consecutive consideration). Suatu pendapat lain yang mirip merumuskannya sebagai pertimbangan cermat secara penuh perhatian beberapa kali terhadap hal yang sama (thinking attentively several times over of the same thing). Dalam sebuah kamus psikologi refective thinking dianggap sepadan denag logikal thinking (pemikiran logis), yakni aktivitas budi manusia yang diarahkan sesuai dengan kaida-kaida logika.

  14. Anwar Rifa’i
    PMAT C 2016 PPS

    Fondasi matematika adalah sebuah studi tentang dasar-dasar logika dan filsafat dari matematika atau dalam arti yang lebih luas, investigasi matematis mengenai konsekuensi-konsekuensi dari teori-teori filsafat dasar tentang hakikat dari matematika itu sendiri. Ada dua pandangan bagaimana memperoleh kebanaran matematika, pertama kebenaran matematika diperoleh murni menggunakan akal pikiran, kedua kebenaran matematika diperoleh berdasarkan pengalaman. Di dalam filsafat matematika, adanya pertentangan antara kaum rasionalis dan kaum empiris menimbulkan pengakuan mendalam akan sintesis Immanuel Kant bahwa matematika adalah ilmu yang bersifat sintetik a priori

  15. Ardeniyansah
    S2 Pend. Matematika Kelas C_2016

    Assalamualaikum wr. . wb.
    Matematika ibarat sebuah bangunan bertingkat dia punya pondasi, rangka, lantai yang bertingkat, dan unsur-unsur lain yang melekat padanya. Secara struktural bangunan itu hanya dapat dipertanggungjawabkan keberadaannya sebagai bangunan apabila dibangun di atas pondasi atau landasan yang kuat. Dengan demikian, bangunan ini akan tetap kokoh dari masa ke masa meskipun diterpa badai dan taufan. Dalam matematika, landasan berfungsi untuk memperkokoh, menyokong atau menopang bangunan matematika. Selain itu landasan matematika juga sebagai sarana yang diperlukan untuk 'merenovasi' bangunan matematika itu sendiri. Pengetahuan matematika di satu sisi bersifat subserve yakni hasil dari sintesis pengalaman inderawi, disisi lain bersifat superserve yakni mengetahui apriori sebagai hasil konsep matematika yang bersifat imannen. Dikarenakan dalam pikiran kita sudah terdapat kategori-kategori yang memungkinkan untuk memahami matematika tersebut.

  16. Annisa Hasanah
    PPs Pendidikan Matematika C 2016

    Pada artikel diatas bapak marsigit menyebutkan bahwa kedisiplinan berhubungan erat dengan rasa ingin tahu. rasa ingin tahu itu sendiri dapat dipicu dari kegiatan berkelompok siswa yang dapat meningkatkan motivasi siswa. tentunya hal ini juga harus didampingi guru agar kegiatan lebih bermakna.

  17. Syaifulloh Bakhri
    S2 Pendidikan Matematika C 2016

    Assalamu’alaikum wr.wb.
    Pondamen matematika adalah ide-ide dasar yang disampaikan oleh para filsuf. Maka landasan awal dari matematika adalah sebagai berikut:
    Pondasi matematika yang dibangun dengan LOGISISME Frege yang bermula dari Leibniz, dengan menempatkan logika berada di depan dalam mempelajari matematika. Selanjutnya dengan pondasi dari Kant, maka kebenaran matematika adalah "a priori tapi sintetik" (SINTETIK A PRIORI), yang diperoleh melalui sintesa dari berbagai pengetahuan a priori/abstrak, dengan memperhatikan KONSISTENSI-nya Newton dan sifat NATURAL dari Hume maka pondamen berikutnya ditambahkan oleh Rene Descartes dengan RASIONALITASNYA, yang semuanya bermuara kepada IDE-nya Plato sebagai BUAH DARI PIKIRAN.
    Dengan demikian pondasi matematika dibangun atas:
    1. Logisisme
    2. Sintetik Apriori
    3. Konsistensi
    4. Natural
    5. Rasional
    dengan memperhatikan benang merah yang selalu tersambung diantara Bilangannya Phytagoras dengan Teori ketidaklengkapannya Godel serta Relatifitas yang disampaikan oleh Hilbert.

  18. Sehar Trihatun
    S2 Pend. Mat Kelas C – 2016

    Diskusi yang dibahas dalam artikel ini mengenai kenapa dasa bermatematika seseorang itu lemah. Ada beberapa pendapat yang telah dikemukakan, salah satu diantaranya yang telah dikemukakan oleh Bapak Prof Marsigit yang memandang bahwa yang terpenting dalam pembelajaran matematika itu adalah prosesnya, bagaimana proses dan pengalaman belajar siswa dalam memahami matematika, dan bukan hanya menekankan pada penilaian-penilaian dari jawaban akhir yang diberikan siswa. Karena tidak setiap jawaban yang diberikan oleh siswa mencerminkan secara keseluruhan dari apa yang benar-benar dimiliki oleh siswa. Penilaian-penilaian yang objektif atau hanya melihat dan menilai dari jawaban akhhir siswa tanpa mempertimbangkan proses atau kegiatan bermatematika yang dialami siswa, bukanlah suatu hal yang bijak yang dapat dilakukan oleh guru. Penilaian seperti itu seolah-olah memberikan batasan dan mencap siswa ke dalam kategori-kategori yang sempit, padahal keadaan siswa atau kondisi siswa dalam mempelajari dan mengenal matematika itu sangatlah kompleks.

  19. Sehar Trihatun
    S2 Pend. Mat Kelas C – 2016

    Dari diskusi ini juga, ada beberapa pandangan yang mengungkapkan bahwa mereka tidak terlalu mempermasalahkan dan merasa sah-sah saja apabila kita hanya melakukan penilaian dari hasil jawaban siswa. Karena mungkin saja jawaban siswa itu juga sejatinya mencerminkan apa yang dipahami siswa dalam matematika. Bagi mereka yang lebih setuju untuk melakukan tes-tes kemampuan siswa yang hanya melihat dari respon siswa, akan lebih mudah untuk mengajarkan matematika kepada siswa melalui hafalan, karena dengan menghafal dan mengingat matematika, siswa akan dapat mempelajari matematika yang lebih tinggi. Dengan diajarkan untuk menghafal matematika, siswa akan dapat menjawab dengan tepat dari pertanyaan-pertanyaan matematika yang diajukan, misalnya saja ketika siswa ditanya berapa 6 dikali 7, maka jawaban siswa haruslah 42. Mereka menganggap bahwa jawaban 42 yang dilontarkan oleh siswa tersebut dapat menjadi indikator bahwa siswa sudah dapat menguasai materi perkalian karena jawaban 42 itu adalah benar. Sebaliknya, jika siswa tidak menjawab 42 maka siswa tidak paham mengenai konsep perkalian. Padahal, mengetahui kemamouan matematika siswa tidak hanya dapat dilihat dengan satu atau beberapa respon yang dikemukakan oleh siswa, dan belajar matematika bukanlah hanya dengan menghafal matematika saja, melainkan bagaimana siswa mengkonstruksi atau membangun pemahaman matematika mereka. Karena dengan membangun pengetahuan matematika, sejatinya siswa diarahkan untuk dapat mengetahui dan memahami konsep yang ada di dalam materi matematika itu sendiri.

    S1 Pendidikan Matematika I 2014

    Fondasi matematika merupakan studi tentang filsafat matematika namun secara lebih luas. yang menarik bagi saya adalah pendapat penting dasa bermatematika, proses adalah seuatu hal yang dituju, bukan hasil akhirnya, saya sependapat dengan Pak Marsigit dimana pentignya perhatian terhadap proses pembelejaran, siswa jika memaknai, memahamai, dan mengikuti setiap proses pembelajran dengan baik maka akan mendapatkan pemahaman yang sebenarnya, namun jika hanya memandang hasil akhir maka siswa terkadang akan melakukan segala cara untuk mendapatkan hasil akhir yang bagus, penyimpangan pun sering terjadi seperti mencontek dll.

  21. Desy Dwi Frimadani
    PPs Pendidikan Matematika Kelas C 2016

    Foundation atau pondasi sebagai pengokoh dasar dari matematika. Matematika memiliki dua unsur dasar atau dua pondasi yaitu Logika Matematika (atau yang biasa disebut Symbolic Logic) dan Teori Himpunan (Set Theory).

  22. Heni Lilia Dewi
    PPs Pendidikan Matematika Kelas C 2016

    Pondasi dalam hal ini berkaitan dengan bagaimana caranya membangun matematika dalam pengetahuan siswa. Saya sependapat dengan Prof. Marsigit bahwa fokus terhadap bagaimana belajar matematika harus lebih diperhatikan daripada mengajar matematika. Hal ini berarti bahwa kita perlu memikirkan cara bagaimana agar siswa dapat belajar matematika dengan baik, yaitu siswa yang berproses, daripada memikirkan bagaimana cara atau metode mengajar matematika. Jadi, fokus pada bagaimana belajar matematika harus didahulukan sebelum penagajaran matematika.

  23. Windi Agustiar Basuki
    S2 Pend. Mat Kelas C – 2016

    Landasan matematika erat kaitannya dengan epistemology matematika yaitu bagaimana membelajarkan matematika kepada peserta didik. Dalam pembelajaran matematika sesorang mengontruksi matematika melalui proses adaptasi dan organisasi. Perkembangan struktur mental seseorang bergantung pada pengetahuan yang diperoleh siswa melalui proses asimilasi dan akomodasi. Penalaran matematika adalah penalaran induktif dan deduktif

  24. Luki Slamet Purwoko
    S1 Pendidikan Matematika I 2014

    Bicara tentang matematika di sikolah merupakan hal yang sangat menarik untuk dikaji. Sepertihalnya dalam perbincangan yang didokumentasikan oleh Pak Prof. Matematika di sekolah bukanlah untuk dihafal Namun diresapi hingga pori-porinya. Matematika sekolah hendaknya berupa aktivitas matematika khususnya untuk matematika sekolah untuk SD dan SMP. Matematika bukanlah hafalan atauhal-hal yang berbau normal. Saya tekankan bahwa matematika sekolah itu matematika aktivitas. Yaitu dengan melkukan aktivitas sehingga siswa langsung merasakan objek dari matematika secara langsung bukan secara toeritis. Karena matematika teoritis atau matematika sebagai ilmu itu untuk orang dewasa.

  25. Muh Ferry Irwansyah
    Pendidikan Matematika PPS UNY
    Kelas D
    Dalam artikel ini terdapat masalah bagi guru yaitu bagaimana mengungkap kedalaman matematika pada siswa. Sampai saat ini masih terdapat kriteria keberhasilan mengajar matematika, setelah mendengar klaim siswa bahwa matematika benar-benar milik mereka. Secara alami tidak mungkin untuk mengukur kemajuan matematika siswa dengan menggunakan pendekatan atau kriteria tertentu. Dalam mempromosikan paradigma baru dari belajar yaitu membangun aktivitas dimana saja dan kapan saja, tidak banyak bergantung pada guru. Akibatnya, mengukur siswa kompeten matematika juga dalam cara dimana saja dan kapan saja, yaitu terus menerus dan menggunakan berbagai pendekatan misalnya portofolio. Uji kriteria benar-benar berbahaya bagi siswa karena itu adalah tindakan pengurangan atau menyederhanakan karakteristik siswa. Jadi tidak ada pilihan bagi para guru untuk mengakui, kepercayaan dan memberdayakan siswa dalam hal memfasilitasi kebutuhan mereka dalam belajar matematika sebagai upaya mereka untuk membangun kehidupan mereka sendiri yaitu matematika.

  26. Ratih Eka Safitri
    PPs Pendidikan Matematika C 2016

    Dari Percakapan diatas saya setuju dengan pernyataan oleh Old Humanis, mendefinisikan matematika sebagai tubuh pengetahuan atau sebagai badan struktur. Ada juga definisi yang sama yang dibuat oleh Pure / aksiomatik / matematika formal. Mereka biasanya tidak banyak memperhatikan apa yang terjadi di dalam pelajar berbeda dengan pendapat Paul Ernest yg mendefinisikan matematika sebagai kreativitas atau proses berpikir atau bahkan sebagai kegiatan sosial. Dengan demikian, sifat matematika dapat dianggap sebagai sosial-kegiatan. sebenarnya kedua pendapat tersebut saling mendukung tergantung dari bagian mana pandangan matematika secara menyeluruh.

  27. Lihar Raudina Izzati
    P. Mat C 2016 PPs UNY

    Salah satu unsur esensial matematika adalah bagian yang melandasi bangunan ilmu Matematika itu, yang dewasa ini disebut landasan matematika. Landasan matematika tidak hanya berfungsi sebagai penyokong atau penopang bangunan matematika, tetapi juga sebagai sarana yang diperlukan untuk membangun dan mengembangkan matematika itu sendiri. Misalnya bisakah teorema-teorema atau rumus dipercaya kebenarannya tanpa pembuktian formal atau penalaran?

  28. Ahmad Bahauddin
    PPs P.Mat C 2016

    Assalamualaikum warohmatullahi wabarokatuh.
    Selama 2.000 tahun fondasi matematika tampak sangat solid. Elemen Euclid (sekitar 300 bce), yang menghadirkan satu set argumen logika formal berdasarkan beberapa istilah dasar dan aksioma, menyediakan metode eksplorasi rasional yang sistematis yang mengarahkan matematikawan, filsuf, dan ilmuwan ke abad ke-19. Bahkan keberatan serius terhadap kurangnya ketegasan dalam pengertian Sir Isaac Newton tentang fluks (turunan) dalam kalkulus, yang diajukan oleh empiris Inggris Anglo-Irlandia, George Berkeley (antara lain), tidak mempertanyakan dasar-dasar dasar matematika. Penemuan pada abad ke-19 dari geometri alternatif yang konsisten, bagaimanapun, memicu sebuah krisis, karena ini menunjukkan bahwa geometri Euclidean, yang didasarkan pada asumsi aksiomatis yang paling intuitif, hampir tidak sesuai dengan kenyataan sebagaimana yang dipercaya para ahli matematika. Ini, bersama dengan penemuan berani ahli matematika Jerman Georg Cantor dalam teori yang ditetapkan, menjelaskan bahwa, untuk menghindari kebingungan lebih lanjut dan dengan memuaskan menjawab hasil paradoks, diperlukan landasan baru dan lebih ketat untuk matematika.

  29. Kunny Kunhertanti
    PPs Pendidikan Matematika kelas C 2016

    Dasar dari matematika adalah studi tentang dasar logis dan filosofis matematika. Penyelidikan matematika adalah yang mendasari teori filosofis tentang sifat matematika. Dasar matematika dapat dipahami sebagai studi tentang konsep-konsep matematika dasar dan bagaimana mereka membentuk hiraki yang lebih komplek.

  30. Ujang Herlan Permana
    S1 Pendidikan Matematika I 2014

    Pembelajaran di dalam kelas yaitu berusaha memahami konsep suatu materi, namun banyak ditemukan guru yang mengajar di dalam kelas lebih kepada hafalan dan latihan soal karena pembelajaran saat ini mengacu kepada nilai atau bisa dibilang kepada UN, tanpa memperhatikan bagaimana siswa mengkonstruk penegtahuannya untuk memhami konsep, maka dari itu konsep dasarnya pun kebanyakan siswa tidak paham yang berakibat lemahnya kemampuan dasar matematika siswa. Maka dari itu pentingnya pemeblajaran yang realistik yang mempu memberikan pembelajaran yang bermakna bagi siswa sehingga konsep dasarnya tertanam pada long term memory siswa.

  31. Wahyu Berti Rahmantiwi
    PPs Pendidikan Matematika Kelas C 2016

    Aktivitas otak yang berkaitan dengan perilaku dan pembelajaran yang berasal dari faktor genetik dan lingkungan. Beberapa guru mengajarkan matematika dengan hafalan dan pengulangan mekanis, dan bukan mengembangkan kemampuan pemahaman dan logika siswa. Sehingga yang tetjadi siswa hanya mampu memecahkan masalah yang sangat mirip atau hampir mirip dengan soal yang pernah diberikan oleh gurunya. Jika soal yang diberikan diubah tataletaknya saja maka siswa akan kesusahan dalam menyelesaikannya. Hal yang demikian harus kita hindari dengan tidak memberikan konsep di awal pembelajaran dan selalu memberi contoh dan bukan contoh pada materi. Model pembelajaran dibuat semenyenangkan mungkin agar siswa mampu mengembangkan kemampuan pemecahan masalah, kemampuan logika dan kemampuan komunikasi di dalam kelas.