The Calculus Bottleneck

What if someone told you that learners in high-school don’t actually need calculus as a compulsory subject for a career in STEM? Surely I would disagree. After all, without calculus how will they understand many of the topics in the STEM. For example basic Newtonian mechanics? Another line of thought that might be put forth is that calculus allows learners to develop an interest in mathematics and pursue it as a career. But swell, nothing could be farther from truth. From what I have experienced there are two major categories of students who take calculus in high school. The first category would be students who are just out of wits about calculus, its purpose and meaning. They just see it as another infliction upon them without any significance. They struggle with remembering the formulae and will just barely pass the course (and many times don’t). These students hate mathematics, calculus makes it worse. Integration is opposite of differentiation: but why teach it to us?

The other major category of students is the one who take on calculus but with a caveat. They are the ones who will score in the 80s and 90s in the examination, but they have cracked the exam system per se. And might not have any foundational knowledge of calculus. But someone might ask how can one score 95/100 and still not have foundational knowledge of the subject matter? This is the way to beat the system. These learners are usually drilled in solving problems of a particular type. It is no different than chug and slug. They see a particular problem – they apply a rote learned method to solve it and bingo there is a solution. I have seen students labour “problem sets” — typically hundreds of problems of a given type — to score in the 90s in the papers. This just gives them the ability to solve typical problems which are usually asked in the examinations. Since the examination does not ask for questions based on conceptual knowledge – it never gets tested. Perhaps even their teachers if asked conceptual questions will not be able to handle them — it will be treated like a radioactive waste and thrown out — since it will be out of syllabus.

There is a third minority (a real minority, and may not be real!, this might just be wishful thinking) who will actually understand the meaning and significance of the conceptual knowledge, and they might not score in the 90s. They might take a fancy for the subject due to calculus but the way syllabus is structured it is astonishing that any students have any fascination left for mathematics. Like someone had said: the fascination for mathematics cannot be taught it must be caught. And this is exactly what MAA and NCTM have said in their statement about dropping calculus from high-school.

What the members of the mathematical community—especially those in the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM)—have known for a long time is that the pump that is pushing more students into more advanced mathematics ever earlier is not just ineffective: It is counter-productive. Too many students are moving too fast through preliminary courses so that they can get calculus onto their high school transcripts. The result is that even if they are able to pass high school calculus, they have established an inadequate foundation on which to build the mathematical knowledge required for a STEM career. (emphasis added)

The problem stems from the fact that the foundational topics which are prerequisites for calculus are on shaky grounds. No wonder anything build on top of them is not solid. I remember having very rudimentary calculus in college chemistry, when it was not needed and high-flying into physical meaning of derivatives in physics which was not covered enough earlier. There is a certain mismatch between the expectations from the students and their actual knowledge of the discipline as they come to college from high-school.

Too many students are being accelerated, short-changing their preparation in and knowledge of algebra, geometry, trigonometry, and other precalculus topics. Too many students experience a secondary school calculus course that drills on the techniques and procedures that will enable them to successfully answer standard problems, but are never challenged to encounter and understand the conceptual foundations of calculus. Too many students arrive at college Calculus I and see a course that looks like a review of what they learned the year before. By the time they realize that the expectations of this course are very different from what they had previously experienced, it is often too late to get up to speed.

Though they conclude that with enough solid conceptual background in these prerequisites it might be beneficial for the students to have a calculus course in the highschool.

The logician, the mathematician, the physicist, and the engineer

The logician, the mathematician, the physicist, and the engineer. “Look at this mathematician,” said the logician. “He observes that the first ninety-nine numbers are less than hundred and infers hence, by what he calls induction, that all numbers are less than a hundred.”

“A physicist believes,” said the mathematician, “that 60 is divisible by all numbers. He observes that 60 is divisible by 1, 2, 3, 4, 5, and 6. He examines a few more cases, as 10, 20, and 30, taken at random as he says. Since 60 is divisible also by these, he considers the experimental evidence sufficient.”

“Yes, but look at the engineers,” said the physicist. “An engineer suspected that all odd numbers are prime numbers. At any rate, 1 can be considered as a prime number, he argued. Then there come 3, 5, and 7, all indubitably primes. Then there comes 9; an awkward case, it does not seem to be a prime number. Yet 11 and 13 are certainly primes. ‘Coming back to 9’ he said, ‘I conclude that 9 must be an experimental error.'”

George Polya (Induction and Analogy – Mathematics of Plausible Reasoning – Vol. 1, 1954)

On mathematics

Mathematics is regarded as a demonstrative science. Yet this is only one of its aspects. Finished mathematics presented in a finished form appears as purely demonstrative, consisting of proofs only. Yet mathematics in the making resembles any other human knowledge in the making. You have to guess a mathematical theorem before you prove it; you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies; you have to try and try again. The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.

George Polya (Induction and Analogy – Mathematics of Plausible Reasoning – Vol. 1, 1954)

Unreal and Useless Problems

We had previously talked about problem with contexts given in mathematics problems. This is not new, Thorndike in 1926 made similar observations.

Unreal and Useless Problems

In a previous chapter it was shown that about half of the verbal problems given in standard courses were not genuine, since in real life the answer would not be needed. Obviously we should not, except for reasons of weight, thus connect algebraic work with futility. Similarly we should not teach the pupil to solve by algebra problems which in reality are better solved otherwise, for example, by actual counting or measuring. Similarly we should not set him to solve problems which are silly or trivial, connecting algebra in his mind with pettiness and folly, unless there is some clear, counterbalancing gain.
This may seem beside the point to some teachers, ”A problem is just a problem to the children,” they will say,

“The children don’t know or care whether it is about men or fairies, ball games or consecutive numbers.” This may be largely true in some classes, but it strengthens our criticism. For, if pupils^do not know what the problem is about, they are forming the extremely bad habit of solving problems by considering only the numbers, conjunctions, etc., regardless of the situation described. If they do not care what it is about, it is probably because the problems encountered have not on the average been worth caring about save as corpora vilia for practice in thinking.

Another objection to our criticism may be that great mathematicians have been interested in problems which are admittedly silly or trivial. So Bhaskara addresses a young woman as follows: ”The square root of half the number of a swarm of bees is gone to a shrub of jasmine; and so are eight-ninths of the swarm: a female is buzzing to one remaining male that is humming within a lotus, in which he is confined, having been allured to it by its fragrance at night. Say, lovely woman, the number of bees.” Euclid is the reputed author of: ”A mule and a donkey were going to market laden with wheat. The mule said,’If you gave me one measure I should carry twice as much as you, but if I gave you one we should bear equal burdens.’ Tell me, learned geometrician, what were their burdens.” Diophantus is said to have included in his preparations for death the composition of this for his epitaph : ” Diophantus passed one-sixth of his life in childhood one-twelfth in youth, and one-seventh more as a bachelor. Five years after his marriage was born a son, who died four years before his father at half his father’s age.”

My answer to this is that pupils of great mathematical interest and ability to whom the mathematical aspects of these problems outweigh all else about them will also be interested in such problems, but the rank and file of pupils will react primarily to the silliness and triviality. If all they experience of algebra is that it solves such problems they will think it a folly; if all they know of Euclid or Diophantus is that he put such problems, they will think him a fool. Such enjoyment of these problems as they do have is indeed compounded in part of a feeling of superiority.

– From Thorndike et al. The Psychology of Algebra 1926

On not learning or con in the context

We will, we will, fail you by testing what you do not know…

We live in a rather strange world. Or is it that we assume the world
to be non-strange in a normative way, but the descriptive world has
always been strange? Anyways, why I say this is to start a rant to
about some obviously missed points in the area of my work. Namely,
educational research, particularly science and mathematics education
research.

In many cases the zeal to show that the students have
‘misunderstandings’ or are simply wrong, and then do a hair-splitting
(micro-genetic) exercise on the test the students were inflicted
with. Using terse jargon and unconsequential statistics, making the
study reports as impossible to read as possible, seem to be the norm.

But I have seen another pattern in many of the studies, particularly
in mathematics education. The so-called researchers spent countless
nights in order to dream up situations as abstract as possible (the
further far away from real-life scenarios the better), then devise
problems around them. Now, these problems are put in research studies,
which aim to reveal (almost in evangelical sense) the problems that
plague our education. Unsuspecting students are rounded, with
appropriate backgrounds. As a general rule, the weaker socio-economic
background your students come from, the more exotic is your study. So
choose wisely. Then these problems are inflicted upon these poor,
mathematically challenged students. The problems will be in situations
that the students were never in or never will be. The unreal nature of
these problems (for example, 6 packets of milk in a cup of coffee! I
mean who in real life does that? The milk will just spill over, the
problem isn’t there. This is just a pseudo-problem created for satisfying the research question of the researcher. There is no context, but only con.

Or finding out a real-life example for some weird fractions) puts many off. The fewer students perform correctly happier the researcher is. It just adds to the data statistic that so many % students cannot perform even this elementary task well. Elementary for
that age group, so to speak. The situation is hopeless. We need a
remedy, they say. And remedy they have. Using some revised strategy,
which they will now inflict on students. Then either they will observe
a few students as if they are some exotic specimens from an
uncontacted tribe as they go on explaining what they are doing or why
they are doing it. Or the researcher will inflict a test (or is it
taste) in wholesale on the lot. This gives another data
statistic. This is then analysed within a ‘framework’, (of course it
needs support) of theoretical constructs!

Then the researcher armed with this data will do a hair-splitting
analysis on why, why on Earth student did what they did (or didn’t
do). In this analysis, they will use the work of other researchers before
them who did almost the same thing. Unwieldy, exotic and esoteric
jargons will be used profusely, to persuade any untrained person to
giveup on reading it immediately. (The mundane, exoteric and
understandable and humane is out of the box if you write in that
style it is not considered ‘academic’.) Of course writing this way,
supported by the statistics that are there will get it published in
the leading journals in the field. Getting a statistically significant
result is like getting a license to assert truthfulness of the
result. What is not clear in these mostly concocted and highly
artificial studies is that what does one make of this significance
outside of the experimental setup? As anyone in education research
would agree two setups cannot be the same, then what is t

Testing students in this way is akin to learners who are learning a
new language being subjected to and exotic and terse vocabulary
test. Of course, we are going to perform badly on such a test. The
point of a test should be to know what students know, not what they
don’t know. And if at all, they don’t know something, it is treated as
if is the fault of the individual student. After all, there would be
/some/ students in each study (with a sufficiently large sample) that
would perform as expected. In case the student does not perform as
expected we can have many possible causes. It might be the case that
the student is not able to cognitively process and solve the problem,
that is inspite of having sufficient background knowledge to solve the
problem at hand the student is unable to perform as expected. It might
be the case that the student is capable, but was never told about the
ways in which to solve the given problem (ZPD anyone?). In this case, it might be that the curricular materials that the student has access
to are simply not dealing with concepts in an amenable way. Or it
might be that the test itself is missing out on some crucial aspects
and is flawed, as we have seen in the example above. The problem is
systemic, yet we tend to focus on the individual. This is perhaps
because we have a normative structure to follow an ideal student at
that age group. This normative, ideal student is given by the so-called /standards of learning/. These standards decide, that at xx age
a student should be able to do multiplication of three digit
numbers. The entire curricula are based on these standards. Who and
what decides this? Most of the times, the standards are wayyy above
the actual level of the students. This apparent chasm between the
descriptive and the normative could not be more. We set unreal
expectations from the students, in the most de-contextualised and
uninteresting manner, and when they do not fulfil we lament the lack
of educational practices, resources and infrastructure.

The Textbook League

I came across this site while reading an article, there are interesting reviews of textbooks used in schools. And some of these reviews are gory, splitting out the blood and guts of the textbooks and their inaneness. Hopefully, many people will find it useful, though the latest book that is reviewed is from about 2002. Perhaps one should do a similar thing for books in the Indian context, basically performing a post-mortem on the zombiesque textbooks that flood our schools.

The Web site of The Textbook League is a resource for middle-school and high-school educators. It provides commentaries on some 200 items, including textbooks, curriculum manuals, videos and reference books. Most of the commentaries appeared originally in the League’s bulletin, The Textbook Letter.

http://www.textbookleague.org/ttlindex.htm

What is a mathematical proof?

A dialogue in The Mathematical Experience by Davis and Hersh on what is mathematical proof and who decides what a proof is?

Let’s see how our ideal mathematician (IM) made out with a student who came to him with a strange question.

Student: Sir, what is a mathematical proof?

I.M.: You don’t know that? What year are you in?

Student: Third-year graduate.

I.M.: Incredible! A proof is what you’ve been watching me do at the board three times a week for three years! That’s what a proof is.

Student: Sorry, sir, I should have explained. I’m in philosophy, not math. I’ve never taken your course.

I.M.: Oh! Well, in that case – you have taken some math, haven’t you? You know the proof of the fundamental theorem of calculus – or the fundamental theorem of algebra?

Student: I’ve seen arguments in geometry and algebra and calculus that were called proofs. What I’m asking you for isn’t examples of proof, it’s a definition of proof. Otherwise, how can I tell what examples are correct?

I.M.: Well, this whole thing was cleared up by the logician Tarski, I guess, and some others, maybe Russell or Peano. Anyhow, what you do is, you write down the axioms of your theory in a formal language with a given list of symbols or alphabet. Then you write down the hypothesis of your theorem in the same symbolism. Then you show that you can transform the hypothesis step by step, using the rules of logic, till you get the conclusion. That’s a proof.

Student: Really? That’s amazing! I’ve taken elementary and advanced calculus, basic algebra, and topology, and I’ve never seen that done.

I.M.: Oh, of course, no one ever really does it. It would take forever! You just show that you could do
it, that’s sufficient.

Student: But even that doesn’t sound like what was done in my courses and textbooks. So mathematicians don’t really do proofs, after all.

I.M.: Of course we do! If a theorem isn’t proved, it’s nothing.

Student: Then what is a proof? If it’s this thing with a formal language and transforming formulas, nobody ever proves anything. Do you have to know all about formal languages and formal logic before you can do a mathematical proof?

I.M.: Of course not! The less you know, the better. That stuff is all abstract nonsense anyway.

Student: Then really what is a proof?

I.M.: Well, it’s an argument that convinces someone who knows the subject.

Student: Someone who knows the subject? Then the definition of proof is subjective; it depends on particular persons.Before I can decide if something is a proof, I have to decide who the experts are. What does that have to do with proving things?

I.M.: No, no. There’s nothing subjective about it! Everybody knows what a proof is. Just read some books, take courses from a competent mathematician, and you’ll catch on.

Student: Are you sure?

I.M.: Well – it is possible that you won’t, if you don’t have any aptitude for it. That can happen, too.

Student: Then you decide what a proof is, and if I don’t learn to decide in the same way, you decide I don’t have any aptitude.

I.M.: If not me, then who?

Mathematical Literacy Goals for Students

National Council of Teachers for Mathematics NCTM proposed these five goals to cover the idea of mathematical literacy for students:

  1. Learning to value mathematics: Understanding its evolution and its role in society and the sciences.
  2. Becoming confident of one’s own ability: Coming to trust one’s own mathematical thinking, and having the ability to make sense of situations and solve problems.
  3. Becoming a mathematical problem solver: Essential to becoming a productive citizen, which requires experience in a variety of extended and non-routine problems.
  4. Learning to communicate mathematically:  Learning the signs, symbols, and terms of mathematics.
  5. Learning to reason mathematically: Making conjectures, gathering evidence, and building mathematical arguments.
National Council of Teachers of Mathematics. Commission on Standards for School Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Natl Council of Teachers of.

Reflections on Liping Ma’s Work

Liping Ma’s book Knowing and teaching elementary mathematics has been very influential in Mathematics Education circles. This is a short summary of the book and my reflections on it.

Introduction

Liping Ma in her work  compares the teaching of mathematics in the American and the Chinese schools. Typically it is found that the American students are out performed by their Chinese counterparts in mathematical exams. This fact would lead us to believe that the Chinese teachers are better `educated’ than the U.S. teachers and the better performance is a straight result of this fact. But when we see at the actual schooling the teachers undergo in the two countries we find a large difference. Whereas the U.S. teachers are typically graduates with 16-18 years of formal schooling, the typical Chinese maths teacher has about only 11-12 years of schooling. So how can a lower `educated’ teacher produce better results than a more educated one? This is sort of the gist of Ma’s work which has been described in the book. The book after exposing the in-competencies of the U.S. teachers also gives the remedies that can lift their performance.

In the course of her work Ma identifies the deeper mathematical and procedural understanding present, called the profound understanding of fundamental mathematics [PUFM] in the Chinese teachers, which is mostly absent in the American teachers. Also the “pedagogical content knowledge” of the Chinese teachers is different and better than that of the U.S. teachers. A teacher with PUFM “is not only aware of the conceptual structure and the basic attitudes of mathematics inherent in elementary mathematics, but is able to teach them to students.” The situation of the two teacher is that the U.S. teachers have a shallow understanding of a large number of mathematical structures including the advanced ones, but the Chinese teachers have a deeper understanding of the elementary concepts involved in mathematics. The point where the PUFM is attained in the Chinese teachers is addressed. this Also the Chinese education system so structured that it allows cooperation and interaction among the junior and senior teachers.

Methodology

The study was conducted by using the interview questions in Teacher Education and Learning to Teach Study [TELT] developed by Deborah Ball. These questions were designed to probe teacher’s knowledge of mathematics in the context of common things that teachers do in course of teaching. The four common topics that were tested for by the TELT were: subtraction, multiplication, division by fractions and the relationship between area and perimeter. Due to these diverse topics in the questionnaire the teachers subject knowledge at both conceptual and procedural levels at the elementary level could be judged quite comprehensively. The teacher’s response to a particular question could be used to judge the level of understanding the teacher has on the given subject topic.

Sample

The sample for this study was composed of two set of teachers. One from the U.S., and another from China. There were 23 U.S. teachers, who were supposed to be above average. Out of these 23, 12 had an experience of 1 year of teaching, and the rest 11 had average teaching experience of 11 years. In China 72 teachers were selected, who came from diverse nature of schools.In these 72, 40 had experience of less than 5 years of teaching, 24 had more than 5 years of teaching experience, and the remaining 8 had taught for more than 18 years average. Each teacher was interviewed for the conceptual and procedural understanding for the four topics mentioned.

We now take a look at the various problems posed to the teachers and their typical responses.

Subtraction with Regrouping

The problem posed to the teachers in this topic was:

Lets spend some time thinking about one particular topic that you may work with when you teach, subtraction and regrouping. Look at these questions:
62
– 49
= 13

How would you approach these problems if you were teaching second grade? What would you say pupils would need to understand or be able to do before they could start learning subtraction with regrouping?

Response

Although this problem appears to be simple and very elementary not all teachers were aware of the conceptual scheme behind subtraction by regrouping. Seventy seven percent of the U.S. teachers and 14% of U.S. teacher had only the procedural knowledge of the topic. The understanding of these teachers was limited to just taking and changing steps. This limitation was evident in their capacity to promote conceptual learning in the class room. Also the various levels of conceptual understanding were also displayed. Whereas the U.S. teachers explained the procedure as regrouping the minuend and told that during the teaching they would point out the “exchanging” aspect underlying the “changing” step. On the other hand the Chinese teachers used subtraction in computations as decomposing a higher value unit, and many of them also used non-standard methods of regrouping and their relations with standard methods.

Also most of the Chinese teachers mentioned that after teaching this to students they would like to have a class discussion, so as to clarify the concepts.

Multidigit Multiplication

The problem posed to the teachers in this topic was:

Some sixth-grade teachers noticed that several of their students were making the same mistake in multiplying large numbers. In trying to calculate:
123
x 645
13

the students were forgetting to “move the numbers” (i.e. the partial products) over each line.}
They were doing this Instead of this
123 123
x 64 x 64
615 615
492 492
738 738
1845 79335

While these teachers agreed that this was a problem, they did not agree on what to do about it. What would you do if you were teaching the sixth grade and you noticed that several of your students were doing this?}

Response

Most of the teachers agreed that this was a genuine problem in students understanding than just careless shifting of digits, meant for addition. But different teachers had different views about the error made by the student. The problem in the students understanding as seen by the teachers were reflections of their own knowledge of the subject matter. For most of the U.S. teachers the knowledge was procedural, so they reflected on them on similar lines when they were asked to. On the other hand the Chinese teachers displayed a conceptual understanding of the multidigit multiplication. The explanation and the algorithm used by the Chinese teachers were thorough and many times novel.

Division by Fractions

The problem posed to the teachers in this topic was:

People seem to have different approaches to solving problems involving division with fractions. How do you solve a problem like this one?

1/(3/4) / 1/2 = ??

Imagine that you are teaching division with fractions. To make this meaningful for kids, sometimes many teachers try to do is relate mathematics to other things. Sometimes they try to come up with real-world situations or story-problems to show the application of some particular piece of content. What would you say would be good story or model for 1/(3/4) / 1/2 ?

Response

As in the previous two cases the U.S. teachers had a very weak knowledge of the subject matter. Only 43% of the U.S. teachers were able to calculate the fraction correctly and none of them showed the understanding of the rationale underlying their calculations. Only one teacher was successful in generating an illustration for the correct representation of the given problem. On the other hand all the Chinese teachers did the computational part correctly, and a few teachers were also able to explain the rationale behind the calculations. Also in addition to this most of the Chinese teachers were able to generate at least one correct representation of the problem. In addition to this the Chinese teachers were able to generate representational problems with a variety of subjects and ideas, which in turn were based on their through understanding of the subject matter.

Division by Fractions

The problem posed to the teachers in this topic was:

Imagine that one of your students comes to the class very excited. She tells you that she has figured out a theory that you never told to the class. She explains that she has discovered the perimeter of a closed figure increases, the area also increases. She shows you a picture to prove what she is doing:

Example of the student:

How would you respond to this student?

Response

In this problem task there were two aspects of the subject matter knowledge which contributed substantially to successful approach; knowledge of topics related to the idea and mathematical attitudes. The absence or presence of attitudes was a major factor in success

The problems given to the teachers are of the elementary, but to understand them and explain them [what Ma is asking] one needs a profound understanding of basic principles that underly these elementary mathematical operations. This very fact is reflected in the response of the Chinese and the U.S. teachers. The same pattern of Chinese teachers outperforming U.S. teachers is repeated in all four topics. The reason for the better performance of the Chinese teachers is their profound understanding of fundamental mathematics or PUFM. We now turn to the topic of PUFM and explore what is meant by it and when it is attained.

PUFM

According to Ma PUFM is “more than a sound conceptual understanding of elementary mathematics — it is the awareness of the conceptual structure and the basic attitudes of mathematics inherent in elementary mathematics and the ability to provide a foundation for that conceptual structure and instill those basic attitudes in students. A profound understanding of mathematics has breadth, depth, and thoroughness. Breadth of understanding is the capacity to connect topic with topics of similar or less conceptual power. Depth of the understanding is the capacity to connect a topic with those of greater conceptual power. Thoroughness is the capacity to connect all these topics.”

The teacher who possesses PUFM has connectedness, knows multiple ways of expressing same thing, revisits and reinforces same ideas and has a longitudinal coherence. We will elaborate on these key ideas of PUFM in brief.

Connectedness: By connectedness being present in a teacher it is meant that there is an intention in the teacher to connect mathematical procedures and concepts. When this is used in teaching it will enable students to learn a unified body of knowledge, instead of learning isolated topics.

Multiple Perspectives: In order to have a flexible understanding of the concepts involved, one must be able to analyze and solve problems in multiple ways, and to provide explanations of various approaches to a problem. A teacher with PUFM will provide multiple ways to solve and understand a given problem, so that the understanding in the students is deeper.

Basic Ideas: The teachers having PUFM display mathematical attitudes and are particularly aware of the powerful and simple concepts of mathematics. By revisiting these ideas again and again they are reinforced. But focusing on this students are not merely encouraged to approach the problems, but are guided to conduct real mathematical activity.

Longitudinal Coherence: By longitudinal coherence in the teachers having PUFM it is meant that the teacher has a complete markup of the syllabus and the content for the various grades of the elementary mathematics. If one does have an idea of what the students have already learnt in the earlier grades, then that knowledge of the students can be used effectively. On the other hand if it is known what the students will be learning in the higher grades, the treatment in the lower grades can be such that it is suitable and effective later.

PUFM – Attainment

Since the presence of PUFM in the Chinese teachers makes them different from their U.S. counterparts, it is essential to have a knowledge of how the PUFM is developed and attained in the Chinese teachers. For this Ma did survey of two additional groups. One was ninth grade students, and the other was that of pre-service teachers. Both groups has conceptual understanding of the four problems. The preservice teachers also showed a concern for teaching and learning, but both groups did not show PUFM. Ma also interviewed the Chinese teachers who had PUFM, and explored their acquisition of mathematical knowledge. The teachers with PUFM mentioned several factors for their acquisition of mathematical knowledge. These factors include:

  • Learning from colleagues
  • Learning mathematics from students.
  • Learning mathematics by doing problems.
  • Teaching
  • Teaching round by round.
  • Studying teaching materials extensively.

The Chinese teachers during the summers and at the beginning of the school terms , studied the Teaching and Learning Framework document thoroughly. The text book to be followed is the most studied by the teachers. The text book is also studied and discussed during the school year. Comparatively little time is devoted to studying teacher’s manuals. So the conclusion of the study is that the Chinese teachers have a base for PUFM from their school education itself. But the PUFM matures and develops during their actual teaching driven by a concern of what to teach and how to teach it. This development of PUFM is well supported by their colleagues and the study materials that they have. Thus the cultural difference in the Chinese and U.S. educational systems also plays a part in this.

Conclusions

One of the most obvious outcomes of this study is the fact that the Chinese elementary teachers are much better equipped conceptually than their U.S. counterparts to teach mathematics at that level. The Chinese teachers show a deeper understanding of the subject matter and have a flexible understanding of the subject. But Ma has attempted to give the plausible explanations for this difference in terms of the PUFM, which is developed and matured in the Chinese teachers, but almost absent in the U.S. teachers. This difference in the respective teachers of the two countries is reflected in the performance of students at any given level. So that if one really wants to improve the mathematics learning for the students, the teachers also need to be well equipped with the knowledge of fundamental and elementary mathematics. The problems of teacher’s knowledge development and that of student learning are thus related.

In China when the perspective teachers are still students, they achieve the mathematical competence. When they attain the teacher learning programs, this mathematical competence is connected to primary concern about teaching and learning school mathematics. The final phase in this is when the teachers actually teach, it is here where they develop teacher’s subject knowledge.  Thus we see that good elementary education of the perspective teachers themselves heralds their growth as teachers with PUFM. Thus in China good teachers at the elementary level, make good students, who in turn can become good teachers themselves, and a cycle is formed. In case of U.S. it seems the opposite is true, poor elementary mathematics education, provided by low-quality teachers hinders likely development of mathematical competence in students at the elementary level. Also most of the teacher education programs in the U.S. focus on How to teach mathematics? rather than on the mathematics itself. After the training the teachers are expected to know how to teach and what to teach, they are also not expected to study anymore. All this leads to formation of a teacher who is bound in the given framework, not being able to develop PUFM as required.

Also the fact that is commonly believed that elementary mathematics is basic, superficial and commonly understood is denied by this study. The study definitively shows that elementary mathematics is not superficial at all, and anyone who teaches it has to study it in a comprehensive way. So for the attainment of PUFM in the U.S. teachers and to improve the mathematics education their Ma has given some suggestions which need to be implemented.

Ma suggests that the two problems of improving the teacher knowledge and student learning are interdependent, so that they both should be addressed simultaneously. This is a way to enter the cyclic process of development of mathematical competencies in the teachers. In the U.S. there is a lack of interaction between study of mathematics taught and study of how to teach it. The text books should be also read, studied and discussed by the teachers themselves as they will be using it in teaching in the class room. This will enable the U.S. teachers to have clear idea of what to teach and how to teach it thoughtfully. The perspective teachers can develop PUFM at the college level, and this can be used as the entry point in the cycle of developing the mathematical competency in them. Teachers should use text books and teachers manuals in an effective way. For this the teacher should recognize its significance and have time and energy for the careful study of manuals. The class room practice of the Chinese teachers is text book based, but not confined to text books. Again here the emphasis is laid on the teacher’s understanding of the subject matter. A teacher with PUFM will be able to choose materials from a text book and present them in intelligible ways in the class room. To put the conclusions in a compact form we can say that the content knowledge of the teachers makes the difference.

Reflections

The study done by Ma and its results have created a huge following in the U.S. Mathematics Education circles and has been termed as `enlightening’. The study diagnoses the problems in the U.S. treatment of elementary mathematics vis-a-vis Chinese one. In the work Ma glorifies the Chinese teachers and educational system as against `low quality’ American teachers and educational system. As said in the foreword of the book by Shulman the work is cited by the people on both sides of the math wars. This book has done the same thing to the U.S. Mathematics Education circles what the Sputnik in the late 1950’s to the U.S. policies on science education. During that time the Russians who were supposed to be technically inferior to the U.S. suddenly launched the Sputnik, there by creating a wave of disgust in the U.S. This was peaked in the Kennedy’s announcement of sending an American on moon before the 1970’s. The aftermath of this was to create `Scientific Americans’, with efforts directed at creating a scientific base in the U.S. right from the school. Similarly the case of Ma’s study is another expos\’e, this time in terms of elementary mathematics. It might not have mattered so much if the study was performed entirely with U.S. teachers [Have not studies of this kind ever done before?]. But the very fact that the Americans are apparently behind the Chinese is a matter of worry. This is a situation that needs to be rectified. This fame of this book is more about politics and funding about education than about math. So no wonder that all the people involved in Mathematics Education in the U.S. [and others elsewhere following them] are citing Ma’s work for changing the situation. Citing work of which shows the Americans on lower grounds may also be able to get you you funds which otherwise probably you would not have got. Now the guess is that the aim is to create `Mathematical Americans’ this time so as to overcome the Chinese challenge.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

Reason and Faith – Misconceptions in Science Education

Reason does not work in matters of faith. But it may have a chance at clearing misconceptions.

via Tehelka

Truly so. In case of my field of study, namely science education research, it may be the other way round. The classic studies in science education aim at identifying the misconceptions that the learners have regarding a particular subject and then finding a mechanism by which they could be addressed.

This was a very simple but very basic presentation of  what most studies try to achieve, though the methodology may be different. There are some studies which present us with a conceptual framework so that all the responses and the problems with the learners can be seen in light of a theoretical construct. This they say will enable us to make sense of what we see in the classrooms, and what is present as representation in the learners mind. What I think they are trying to say is that we need to get to the conceptual structures that lead to formation of the misconceptions.

Now mind you that many of these misconceptions in science are very stubborn and people are very reluctant to give them up. The reason may be that many of these misconceptions come from direct factual experience in the real world. And from what I know about Philosophy of Science, we might want to make a case that all science is counter-intuitive to our everyday experience. This would explain why misconceptions in science arise. But would this case explain all the known misconceptions?

Let us do some analysis of how a particular misconception might arise.There can be two different reasons for a misconception to arise, if we adhere to deductive logic. That is to say we assume that we have a set of starting statements that are given, whose authenticity is not questioned. And from these set of statements we make certain deductions regarding the world out there. Now there can be two problems with this scenario, one is that the set of statements that we are taking for granted might be wrong, the other is that in the process of deduction that we have followed we made a mistake. The mistake is learnt only when the end result of our analysis is not consistent with the observations in the real world. Or it might be even the case that the so called misconception will lead to a correct answer, at least in some cases.  In these cases we have to resort to more detailed analysis of the thought structure which lead to the answers. Another identifying characteristic of the misconceptions is presence of the inconsistencies across different areas known to the learners. Whereas they might get a particular concept clearly and correctly, in applying same thing for another concept they just might revert to a completely opposite argument and in doing this they do not realise the inconsistency.

We will be clearer on this issue when we talk with a few examples. Suppose that we have a scenario in which we are trying to understand the phenomena of day and night, its causes and consequences. A typical argument in our class goes like this:

How many have seen the Sun set?

Almost all hands would go up, then comes the next question:

How many have seen the Sun rise?

Almost same number of hands go up, excepting a few, who are late risers like me. Some of the more intelligent and the more knowledgeable would say,

“Wait! Sun doesn’t rise and set, it is the Earth that is moving, so it causes the apparent motion of Sun across the sky, the start and end of which we call as day and night. So in conclusion the Sun doesn’t rise and set, it is an illusion created by motion of Earth.”

To this all of the class agrees. This is what they have learned in the text-book, and mind you the text-book represents truth and only truth, nothing else. It is there to dispel your doubts and misconceptions and is made by a committee of experts who are highly knowledgeable about these things. Now let us continue this line of reasoning and ask them the next question in this series.

Does the Moon rise? If so, does it rise everyday?

The responses to this question are mixed. Most of them would say that it does not rise, it is always there, up in the sky. Some would gather courage and say that it does rise.

Does the Moon set?

Again to this the response is mixed, and mostly negative. Most of them are adamant about the ever presence of the moon in the sky. The next question really upsets them

Do the stars rise and set?

Now this question definitely gets a negative response from almost all of them. Even the more knowledgeable ones fall. They have read different parts of the story, but have not connected them. They tell you the following: “No the stars do not move, they are there all the time.” They also tell you that there is something called as the fixed stars and this is in the text-book, which cannot be wrong. And when asked:

Why are we not able to see the stars during the day time?

They tell you “Of course you cannot see the stars during the day time. This is because our Sun, which is also a star, is too bright and the other stars too far away and hence are dim. So our Sun’s brightness, overwhelms the other stars, and hence they are not visible during the day time, but they are there nonetheless. In the night time, since the Sun is no longer visible, the stars become visible. Have you never noticed that during the evening twilight the stars become visible one by one, the brighter ones first. Whereas in the morning the brightest are the last ones to disappear.”

Of course, the things said above and the reasoning given sounds good. So much so that the respondents are convinced that they understand how things work, and have an elaborate reasoning mechanism to explain the observed things, in this case the formation of day and night and appearance / disappearance of stars during night and day respectively.

You ask them:

Don’t you think there is a problem with what you have just said?

“Where is the problem?”, they tell you. “We just explained scientifically how things are in heaven.”

Then you open the Pandora’s box,

“Well you have just said that the Sun doesn’t move really, it is the Earth that moves, and hence we see the apparent Sun rise and Sun set.”

Then they say, “Yes, that is the case. The Sun doesn’t move, but the Earth does.”

You ask, “How do you know this? Do you see that the Earth is moving?”

They say, “The textbook tells us so ” Some of the more knowledgeable ones say that “Galileo proved that the Earth moves and not the Sun. Since we are on Earth, we see only apparent motion of the Sun.”

You say: “But wait, just now you said that the Moon does not move, it is always in the sky. Also you said that the stars do not move, they are there all the time. Now if the Earth moves, then all these bodies should also move, if only, apparently.Then the stars must also move, just like the Sun does, do not forget that Sun is a star too! So other stars should also just set and rise like the Sun, and so should also the Moon!”

Or you can argue just the opposite: “I claim that it is the Sun that moves, Earth does not move. Isn’t it a lot more easier to explain this way, why we do see the Sun moving, because it moves. And we anyway do not see Earth moving! How will disprove me?”

Then the grumbles start. They have never thought about this. They knew the facts, but never connected them. This lead to the misconceptions regarding these things. They were right in parts, but never got a chance to connect the dots, metaphorically speaking.The reason for these misconceptions is the faith in the text-books, but if the text-books fail to perform the job of asking them the right question, where the reasoning alone can get rid of many of the misconceptions.

If we choose the alternative question, of challenging them to disprove that the Earth is stationary, almost most of them are unable to answer the question of disproving that the idea that the Sun moves and not Earth. They would suggest that we can see this from the satellite in the sky (Can we really?).

Most of us take the things for granted and never question many (or as in most cases, any) of them. And many times the facts are something we do not question. We say that “It is a fact.” This statement basically posits that the information which we think is out there can be unquestionable. But there are many flavours of the post-modern philosophy which challenge this position. They think that the facts themselves are relative, that is to say that one culture has different science than another one.  But let us leave this, and come back to our problem of the stars and the Sun and Moon.

Lets put out the postulates for the above arguments and try to deduce deductively the results that were obtained.

Claim 1: Sun doesn’t move.

Claim 2: Earth moves.

Observation 1: We see the Sun moving across the sky daily, it rises and it sets.

Explanation 1:  Since the Earth moves, and the Sun is stationary, we see that Sun moves apparently. This apparent motion of the Sun is seen as the Sunrise and the Sunset by us. This is what causes the day and night.

But we can have Observation 1 explained by another set of claims, which is exactly opposite, namely, that the Earth doesn’t move but the Sun moves.

Claim 3: The Sun moves.

Claim 4: The Earth does not move.

Explanation 2: Since the Earth does not move, and the Sun does, we just see the Sun passing by in the sky, around the Earth. This causes day and night.

We see that Explanations 1 and 2 are both valid for Observation 1, if the claims 1 and 2, 3 and 4 are true then the respective deductions from them, in this case the Explanations 1 and 2 respectively are also true.So in this case the logical deduction is correct, provided that the Claims or assumptions are correct. But this process does not tell you whether the claims themselves are true or not. But both set of assumptions, cannot be true at the same time. Either the Earth moves or it does not, it cannot be in a state of both. If at all we had an explanation which came from these assumptions which did not correspond with the observations, but was logically deducible, then we can question the assumptions or premises as philosophers call them.

Of course, the things said above and the reasoning given sounds good. So much so that the respondents are convinced that they
understand how things work, and have an elaborate reasoning mechanism

We can have one example of this type.

Assumption 5: Stars do not move, there are so called “fixed stars”.

Assumption 5: During the day time the Sun is too bright, as compared to the other stars.

Now in this case combining Assumption 5 (A5) with Observation 1 (Ob1) we would get the following:

Explanation 3: The stars are too dim as compared to Sun, hence we cannot see them during the day time, but they are present. Hence they do not move.

In Explanation 3 (E3) above the deduction has a problem. The deduction does not follow from the assumption. This is the other problem in which we talked about above.

Most of the people who would suggest these responses have mostly no background in astronomy. Even then the basic facts that Earth goes round the Sun and not the other way round are forced upon them, without any critical emphasis on why it is so. Neither are they presented at point with the cognitive struggle of another view point, namely the geo-centric view. So presenting the learners with opportunities that will make them observe things and make sense of the explanations in light of the assumptions that were made, will enhance the reasoning and help them to overcome some of their misconceptions.

But there is another observation which can be made of the skies. And it can be either done in the classroom with the aid of Free Softwares like Stellarium. After the round of above questions, we usually show the class the rising of the stars from the east. In a darkened room with a projector the effect is quite dramatic for those who have not witnessed such a thing before. So you can show the class, just as the Sun rises, all other celestial bodies like the Moon and the stars also must rise and this is an observed fact.

Observation 2: The stars and planets and the Moon also rise and set everyday.

So how do we make sense of this observation, Ob2 in the light of the assumptions that we have.

Assumption 6: Sun is a star.

Explanation 4: We observe that Sun moves during the day, from East to West. Sun is a star, hence all other stars should also move.

Now why this should be the case will be different for the geo-centric and the helio-centric theories. In case of H-C theory the explantion is simple. The Earth moves hence the stars appear to move in the opposite direction. And this applies to all the objects in the sky.

Since the Earth moves all other celestial objects will appear to move. In case of G-C theory we have to make an assumption that the
stars are “fixed” on some imaginary sphere, and the sphere as a whole rotates.

But coming back to the misconceptions, it is just the ad-hoc belief that the stars do not move (“fixed stars”) in conjunctions with another observation that in presence of too bright objects dim objects cannot be seen leads to belief that the stars are immobile and do not rise and set as the Sun does. There is another disconnection from another fact that they know, or are told in the textbooks, that  the apparent movement of the Sun is caused by the actual movement of  the Earth. There is no connection between these two facts which is  made explicit.

We think that providing opportunities for direct observation aided by software, Stellarium in this case, which help in visualizing the movements of celestial bodies will help in developing the skill of reasoning and explaining an observed phenomena.

Examinations: Students, Teachers and the System

We think of exams as simple troublesome exchanges with students:

Glance at some of the uses of examinations:

  • Measure students' knowledge of facts, principles, definitions, 
    experimental methods, etc
  • Measure students' understanding of the field studied
  • Show students what they have learnt
  • Show teacher what students have learnt
  • Provide students with landmarks in their studies
  • Provide students with landmarks in their studies and check 
    their progress
  • Make comparisons among students, or among teachers, 
    or among schools
  • Act as prognostic test to direct students to careers
  • Act as diagnostic test for placing students in fast 
    or slow programs
  • Act as an incentive to encourage study
  • Encourage study by promoting competition among students
  • Certify necessary level for later jobs
  • Certify a general educational background for later jobs
  • Act as test of general intelligence for jobs
  • Award's, scholarships, prizes etc.

There is no need to read all that list; I post it only as a warning against trying to do too many different things at once. These many uses are the variables in examining business, and unless we separate the variables, or at least think about separating them, our business will continue to suffer from confusion and damage.

There are two more aspects of great importance well known but seldom mentioned. First the effect of examination on teachers and their teaching –

coercive if imposed from the outside; guiding if adopted sensibly. That is how to change a whole teaching program to new aims and methods – institute new examinations. It can affect a teacher strongly.

It can also be the way to wreck a new program – keep the old exams, or try to correlate students’ progress with success in old exams.

Second: tremendous effect on students.

Examinations tell them our real aims, at least so they believe. If we stress clear understanding and aim at growing knowledge of physics, we may completely sabotage our teaching by a final examination that asks for numbers to be put in memorized formulas. However loud our sermons, however intriguing the experiments, students will be judged by that exam – and so will next years students who hear about it.

From:

Examinations: Powerful Agents for Good or Ill in Teaching | Eric M. Rogers | Am. J. Phys. 37, 954 (1969)

Though here the real power players the bureaucrats and (highly) qualified PhDs in education or otherwise who decide what is to be taught and how it is evaluated in the classroom. They are “coercive” as Rogers points out and teachers, the meek dictators (after Krishna Kumar), are the point of contact with the students and have to face the heat from all the sides. They are more like foot soldiers most of whom have no idea of what they are doing, why they are doing; while generals in their cozy rooms, are planning how to strike the enemy (is the enemy the students or their lack of (interest in ) education, I still wonder).  In other words most of them don’t have an birds-eye-view of system that they are a focal part of.

Or as Morris Kline puts it:

A couple of years of desperate but fruitless efforts caused Peter to sit back and think. He had projected himself and his own values and he had failed. He was not reaching his students. The liberal arts students saw no value in mathematics. The mathematics majors pursued mathematics because, like Peter, they were pleased to get correct answers to problems. But there was no genuine interest in the subject. Those students who would use mathematics in some profession or career insisted on being shown immediately how the material could be useful to them. A mere assurance that they would need it did not suffice. And so Peter began to wonder whether the subject matter prescribed in the syllabi was really suitable. Perhaps, unintentionally, he was wasting his students’ time.

Peter decided to investigate the value of the material he had been asked to teach. His first recourse was to check with his colleagues, who had taught from five to twenty-five or more years. But they knew no more than Peter about what physical scientists, social scientists, engineers, and high school and elementary school teachers really ought to learn. Like himself, they merely followed syllabi – and no one knew who had written the syllabi.

Peter’s next recourse was to examine the textbooks in the field. Surely professors in other institutions had overcome the problems he faced. His first glance through publishers’ catalogues cheered him. He saw titles such as Mathematics for Liberal Arts, Mathematics for Biologists, Calculus for Social Scientists, and Applied Mathematics for Engineers. He eagerly secured copies. But the texts proved to be a crushing disappointment. Only the authors’ and publishers names seemed to differentiate them. The contents were about the same, whether the authors in their prefaces or the publishers in their advertising literature professed to address liberal arts students, prospective engineers, students of business, or prospective teachers. Motivation and use of the mathematics were entirely ignored. It was evident that these authors had no idea of what anyone did with mathematics.

From: A Critique Of Undergrduate Education. (Commonly Known As: Why The Professor Can’t Teach?) | Morris Kline

Both of the works are about 50 years old, but they still reflect the educational system as of now.

A parable on…

A Parable

Once upon a time, in a far away country, there was a community that had a wonderful machine. The machine had been built by most inventive of their people … generation after generation of men and women toiling to construct its parts… experimenting with individual components until each was perfected… fitting them together until the whole mechanism ran smoothly. They had built its outer casing of burnished metal and on one side, they had attached a complex control panel. The name of the machine, KNOWLEDGE, was engraved on a plaque  set in the centre of the control panel.

The community used the machine in their efforts to understand the world and to solve all kinds of problems. But the leaders of the community were not satisfied. It was a competitive world… they wanted more problems solved and they wanted them solved faster.

The main limitation for the use of machine was the rate at which data could be prepared for input. Specialist machine operators called ‘predictors’, carried out this exacting and time consuming task… naturally the number of problems solved each year depended directly on the number and skill of the predictors.

The community leaders focussed on the problem of training predictors. The traditional method, whereby promising girls and boys were taken into long-term apprenticeship, was deemed too slow and too expensive. Surely, they reasoned, we can find more efficient approach. So saying,  they called the elders together and asked them to think about the matter.

After a few months, the elders reported that they had devised an approach that showed promise. In summary, they suggested that the machine be disassembled. Then each component could be studied and understood with ease… the operation of machine would become an open book to all who cared to look.

Their plan was greeted with enthusiasm. So, the burnished covers were pulled off, and the major mechanisms of the machine fell out… they had plaques with labels like HISTORY and GEOGRAPHY and PHYSICS and MATHEMATICS. These mechanisms were pulled apart in their turn… of course, care was taken to keep all the pieces in separate piles. Eventually, the technicians had reduced the machine to little heaps of metal plates and rods and nuts and bolts and springs and gear wheels. Each heap was put in a box, carefully labelled with the name of the mechanism whose part it contained, and the boxes were lined up for the community to inspect.

The members of the community were delighted. Their leaders were ecstatic. They ‘oohed’ and ‘aahed’ over the quality of components, the obvious skill that had gone in their construction, the beauty of designs. Here, displayed for all, were the inner workings of KNOWLEDGE.

In his exuberance, one man plunged his hand into a box and scooped up a handful of tiny, jewel-like  gear wheels and springs. He held them out to his daughter and glancing, at the label on the box, said:

“Look, my child! Look! Mathematics! ”

From: Turtle Speaks Mathematics by Barry Newell

You can get the book (and another nice little book Turtle Confusion) here.

 

Why children hate maths…

          Today, because it is the 15th Monday of your 5th grade year,
          you have to do this sum irrespective of who you are or what
          you really want to do; do what you are told and do it the
          way you are told to do it.

 From: The Children’s Machine by Seymour Papert