# LinkedIn Groups

**Group:**Math, Math Education, Math Culture

**Discussion:**Why do you think people's basic math skills are weak?

**Marsigit Dr MA**• Outside perception is sometime not good to the students, because they tend to judge rather than to facilitate.

**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 Winters**:

Is 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 Nistoran**:

I'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 MA**:

The 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.

Thanks

**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 Galinaitis**:

Agree 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 Northridge**:

I 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 Patterson**:

I 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, MBA**:

My 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 Mills**:

Yes 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.Thank

**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 Mills :**

With 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-Smith :**

The 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 DiVito**:

Argh. 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 thetime, I did not even know trig identities. I revised until late night fromthe textbook. Next day, I took part. Several schools. It was a big multiple-choice test. Together with some additional questions. The topprize 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 competitionwas 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, 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 MA**:

Art 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.OCT**:

We 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 Galinaitis:**

People 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 Frey :**

Because 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?

**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 MA :**

Thank Elias Gourtsoyannis for the information

**Behnaz Herbst, MSc.OCT:**

Elias, 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-Smith :**

Art, 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-Smith**:

Art, 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 "....education 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 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. Thank's

**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, presumably! She has also written on Pappus, the Alexandrian mathematician.

**Marsigit Dr MA**:

@Everybody: Having much discussion with many peoples in this forum, I then conclude that the PROBLEMS of math education/math teaching are mostly COMING from THE ADULTS peoples, teachers, policy makers, educationist, university professor/lectures, ..etc and ARE NOT COMING FROM the YOUNGER, or students.

Bradford Hansen-Smith :

Bradford Hansen-Smith :

Marsigit, yes it is necessary to have adult support, adults do not develop discipline for students, self-discipline is not imposed or taught. It is the students responsibility to themselves to take what is of interest to explore and develop. Adults must be able to demonstrate the value of disciplining oneself, that the student may understand it is critical to the success of their own work in any discipline and fulfillment in all areas of life.

The human animal is lazy and tends to go along with the herd, to react more than to in-act. Self-discipline is to action from within to control our animal nature and to find greater value in our human nature. We have responsibility to our individual human potential along with a duty to be of service to humanity in discovering and developing our individual and unique gifts. This does not happen without self-control. Potty training young children is certainly learning self-discipline. That is a conscious mind function and is a natural place to begin to discipline the mind for a lifetime of discernment about ones own life. This leads to understanding what is appropriate to a given situation; we might say a precondition towards wisdom, which we all have to limited capacity.

I do not know much about the psychology of learning and at what stages math and “logical” systems “should” be introduced, but it does not seem complicated if we are clear on the purpose of self-discipline as necessary to progress individually and collectively. When a child first starts to draw, these are images, symbols that represent systems of connections in the child’s mind. A rudimentary understanding of patterned connections is being developed. Because we do not recognize children have experience in pattern recognition, they don’t have the language to express it, we think they don’t know anything. Pattern recognition is built in to animal perception as the means to survive our environment and with the human mind to be able to discover the grandeur and the unseen forces of this and other universes. My sense is that a math attitude comes from the unforgiving position of mathematics itself. We are not born with a disposition to not like math. You have stated this clearly in your last post. The problem with math must lie with mathematicians and the people that teach it.

**Marsigit Dr MA :**

@Bradford: I do agree with your all elaborations in the first, second and third paragraph. The more I discuss with you, the more some useful aspects of students' learn mathematics I found.

I also do agree with you that we are clear on the purpose of self-discipline as necessary to progress individually and collectively. You made very good illustration by saying that "Because we do not recognize children have experience in pattern recognition, they don’t have the language to express it, we think they don’t know anything". It is coming from the last phrase that all problems of math educ are arising. Even anywhere all over the world, most teachers starting their teaching by assuming that "their students don't know anything".

I also agree with you on student's (or generally human) Pattern Recognition, because

since long time ago even Immanuel Kant has elaborated that the pattern of recognition is something embedded and being self developed by and inside the individual human being as of the initial form of every knowledge. Inside the pattern of course there are categories of knowledge. I even wish to say that due to the importance of categories inside individual human, I may claim that all kinds of knowledge (math) is really a kind of category. In learning art or constructing math shapes, the adults (teacher) just only be able to peep from outside, and only get a small portion of the aspect of student's categories.

You are also in the same position with me that the problems with math must lie with mathematicians and the people that teach it. As I said that that PROBLEMS of math education/math teaching are mostly COMING from THE ADULTS peoples, teachers, policy makers, educationist, university professor/lectures, ..etc and ARE NOT COMING FROM the YOUNGER, or students. So It needs for the adults to reflect or even more i.e. to overhaul/reconstruct their past theories of teaching/education. At any chance I and also you and other people who care about it, need to introduce and promote any aspect of progressive, innovative theories of teaching/educ, to liberate our younger generation in order to build/develop/construct their own life. We as an adults have a function to facilitate their needs to learn.

What kinds of overhauling/reconstruction are needed? Ebbutt and Straker (1995) differentiated between University Math and School Math. They define the NATURE OF SCHOOL MATH as: 1) a search of pattern and relationship, 2) problem solving activities, 3) investigation activities, 4) that math is a means of communication. Thank's

**Bradford Hansen-Smith**:

@ Marsigit, Not knowing the work of Ebbutt and Straker, my observation about the points you mentioned1-4 are as follows. This is not about math per say, it is about learning and education. I see no NATURE OF SCHOOL MATH. What math is we make to our liking as we teach it. The diversity of ideas and viewpoints in these discussions give some indication of this.

Your four points: 1. Pattern recognition is a skill for greater understanding about the universe. We seem to teach patterns as an end as they relate to math. Relationships can only be understood to have value in larger context than themselves. 2, To identify problems means we have some idea of what is comprehensibly appropriate, to know when something is out of alignment, to know the adjustments and changes necessary. Prior to solving a problem we must observe where we are and our place towards purpose that presents a problem. Math problems rarely relate to social, economic, philosophical, ethical, and moral issues that urgently need our attention. This is problem solving 101 and math with a sense of responsibility could serve towards a greater purpose if desired. 3. Even infants are investigating where they are, which is pretty much arrested by formal education because there are no formulas for investigation, curiosity is always personal. 4. If math is a means of communication then let’s approach it as such and stop teaching as it it were separate as the only elevated capacity to human understanding.

What kind of reconstruction is needed is beyond my knowing. I am communicating my observations with hope that in our view of math being about relationships between unit parts, we will come to realize what is missing is an understanding that parts are first in relationship to unity of the whole, which determines the relationships between those parts.

**Marsigit Dr MA :**

@Bradford: That's it. You have elaborated the four points of the nature of school math very succinctly. As to compare directly with what the origin, herewith I expose what Ebbutt and Straker (1995) said :

As a search for pattern and relationship, mathematics can be perceived as a network of interrelated ideas. Mathematics activities help the students to form the connections in this network. It implies that the teacher can help students learn mathematics by giving them opportunities to discover and investigate patterns, and to describe and record the relationships they find; encouraging exploration and experiment by trying things out in as many different ways as possible; urging the students to look for consistencies or inconsistencies, similarities or differences, for ways of ordering or arranging, for ways of combining or separating; helping the students to generalize from their discoveries; and helping them to understand and see connections between mathematics ideas.

Creativity in mathematics lies in producing a geometric design, in making up computer programs, in pursuing investigations, in considering infinity, and in many other activities. The variety and individuality of children mathematical activity needs to be catered for in the classroom. The teacher may help the students by fostering initiative, originality and divergent thinking; stimulating curiosity, encouraging questions, conjecture and predictions; valuing and allowing time for trial-and-adjustment approaches; viewing unexpected results as a source for further inquiry; rather than as mistakes; encouraging the students to create mathematical structure and designs; and helping children to examine others’ results

Mathematics can provide an important set of tools for problems- in the main, on paper and in real situations. Students of all ages can develop the skills and processes of problem solving and can initiate their own mathematical problems. Hence, the teacher may help the students learn mathematics by: providing an interesting and stimulating environment in which mathematical problems are likely to occur; suggesting problems themselves and helping students discover and invent their own; helping students to identify what information they need to solve a problem and how to obtain it; encouraging the students to reason logically, to be consistent, to works systematically and to develop recording system; making sure that the students develop and can use mathematical skills and knowledge necessary for solving problems; helping them to know how and when to use different mathematical tools.

Language and graphical communication are important aspects of mathematics learning. By talking, recording, and drawing graphs and diagrams, children can come to see that mathematics can be used to communicate ideas and information and can gain confidence in using it in this way. Hence, the teacher may help the students learn mathematics by: creating opportunities for describing properties; making time for both informal conversation and more formal discussion about mathematical ideas; encouraging students to read and write about mathematics; and valuing and supporting the diverse cultural and linguistic backgrounds of all students.

You may look at my work containing their work at the following:

http://staff.uny.ac.id/system/files/pengabdian/marsigit-dr-ma/marsigitmateri-workshop-qitepphilosophymatheduclesson-study-teamfinal.pdf

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