Mathematical education in Israel is at a low ebb. In 1964 Israel took first place in the international tests in arithmetic for elementary school students. In 1999 it was in the 28-th place, among 38 nations, between Thailand and Tunisia. A recurring complaint from secondary school teachers is that students arrive from elementary school with very scarce knowledge of fractions.
It is hard not to relate the deterioration to the changeover which took place at the end of the 70-s. Almost overnight the textbooks were changed then in most Israeli schools, to books following the so called "structural approach". The developers of the method ascribe the failure to other factors, mainly the weakness of teachers. Clearly, had Israel gone from 28-th place to first they would have ascribed the success to the method.
The Israeli "structural approach" followed in spirit the "New Math" reform that occurred in the US in the 1960-s, and was already abandoned there by the time it reached the shores of Israel. The formal, abstract approach; the introduction of subtraction via addition and of division via multiplication; the use of the Cuisenaire rods and other modeling accessories; the teaching of each subject in its greatest generality - all these were taken from the "New Math". A typical example is the teaching of a non-decimal basis system before the decimal system itself: the Israeli books are probably among the last in the world containing this relic from the New Math.
As a possible remedy for the grim situation, the ministry of education decided to replace the elementary school math curriculum. A committee of 10 people was formed to write a new curriculum. As its head was appointed Prof. Pearla Nesher, who had been the leader of the "structural approach" reform.
The committee handed its recommendations for a new curriculum, called "Program 2000", towards the end of 2001. It still needs approval by the ministry of education.
The proposed curriculum follows very closely yet another reform that occurred in the United States - the 1989 "Standards". These are described in a later section. The main change in the proposed curriculum is a drastic cut in material. Basic topics are left out. Very little is left of the teaching of fractions; the standard algorithms for addition and multiplication are not included, as is also long division. A more comprehensive list is given below. Like the "Standards", the curriculum is constructivistic, namely it encourages self-discovery. Thus the assumption is that students will supply algorithms for addition and multiplication of their own, probably not the standard ones.
Where do 10 people find the courage to make such drastic decisions, which may affect the scientific and industrial future of the entire country? Probably from the thought that what is good for the US must also be good for us.
Unfortunately, the premise is wrong. The "Standards" reform has not done much good for American education. In fact, its results can be described as total failure. The American educational scene is still in turmoil, with the "math wars" that followed its adoption. California, which was first to adopt it, abandoned it in 1998. It was replaced there by a traditional, subject matter oriented curriculum, leaving didactics to the teachers. If Israel wishes to follow the steps of the US, it could take this curriculum as a model.
In a later section we shall describe the effects of the "Standards" in some detail.
So, if the curriculum is approved, we shall again be importing an educational reform after it has failed in its country of origin. And we shall be going, open eyed, into the strife that beset the countries that followed it.
The bulk of this article is a critical, detailed summary of the proposed curriculum. The aim is to make it more accessible to the public, and to highlight its underlying trends. It is written in English so that American experts can judge how close it is to the "Standards" reform, and to curricula constructed after its principles.
The main problems with the proposed curriculum are the irresponsible extensive cuts in material, and the import, en bloc, of a very dubious reform. But its line-by-line scrutiny below shows also that even when judged by its own goals and scope it is very poor. It is fragmented, has very little regard for coherence of the subject matter, and prefers external criteria to considerations pertinent to the inherent logic of the topics.
The bottom line of the article is that the curriculum should be halted. It is not that it needs modification, it needs complete thinking over. A panel of referees should be appointed, including mathematicians, computer scientists, scientists from other areas and leading figures from the Hi-Tech industry, who will decide whether the entire direction is feasible.
The summary refers to the version of the curriculum appearing on the internet, [here].
The summary is restricted to the curriculum itself. That is, it does not relate to the "philosophical", motivational part given by its authors. I also limited myself mainly to the arithmetic part, since this is the main body of knowledge the student should acquire in elementary school. Thus, the summary does not treat geometry.
Statistics, the newly introduced component of the curriculum, is discussed (quite briefly) in a separate section.
There are also separate sections on investigation, and on the use of calculators.
Mimicking the "Standards", the curriculum is written in the form of a matrix. The first column is that of the topics. The others are, in this order: mathematical insight, word problems, investigation and problem-solving, representations and accessories, processes, properties and links, concepts and terms, and mastery and performance ability.
The "topics" column is cited in this review in full. For the purpose of brevity, only a limited number of items were selected from the other columns of the matrix, but they should suffice to indicate the general trend. I have tried to be more detailed in the "insights" category, since it seems that it contains items which the authors deem to be important.
The standard ("long") multiplication algorithm (now replaced by self-invented algorithms based on the distributive law).
Long division (claimed not to be taught in practice also in the current curriculum).
Operations with large numbers (to be replaced by the use of calculators).
Fractions with denominator greater than 12.
Multiplication and division of fractions. These appear only once, under "investigation" in Grade 6, and even there not quite explicitly. In any case, they cannot be studied too thoroughly when denominators are bounded by 12.
Mixed numbers (like 5_2/3). The topic is not omitted by explicit declaration, but does not appear in the curriculum itself (it is mentioned in the comments attached to the curriculum.)
Measurements and weights.
Statistics, intended to constitute about 10% of the program.
Investigation. Of course, this is not subject matter as such, and hence not within the scope of an ordinarily defined "curriculum". But this curriculum does not restrict itself to subject matter. It tries to dictate a didactical approach.
Extensive use of the calculator and the computer (Excel).
Since the curriculum follows the "Standards" so closely, a few words of explanation should be given here about them to the uninitiated. They appeared in a book, published by the NCTM (the National Council of Teachers of Mathematics in the US) in 1989. They were intended to lead a revolution in mathematical education. No longer, goes the spirit of the book, is it the subject matter that should be at the focus of attention. The main objective is acquiring mathematical abilities like conjecture making and investigation, and "higher order reasoning".
Putting investigation before skills also implies the advocacy of the use of calculators. The student should be saved tiresome drilling. He should leap straight into the investigation of higher concepts, and the calculator provides the shortcut. The use of calculators in class is the focus of heated debates, in which mathematicians and computer scientists are mostly against it, and education researchers for it. In a later section we shall describe the details of this debate.
The proposed curriculum follows this trend, and advocates extensive use of calculators, and also of the computer - the use of Excel from early grades.
At the time of their publication, the Standards were heralded as the flagship of all educational reforms. California was of course first to join in. Within 5 years it dropped to almost the last place in the US in the achievements of its students in arithmetic [Loveless]. In the California state universities system the percentage of the students who had to take remedial math courses upon entrance rose from 23% in 1989 to 55% in 2000 (see James Milgram's testimony, [Milgram] and [here]). In Colorado three schools in which the parents insisted on the traditional, skills oriented teaching, were found to be much above average in achievements in their area. In a survey conducted in Los Angeles, 60% of the Grade 8 students did not master the multiplication table.
As Evidence for the failure of the reform kept pouring in, something unexpected happened: mathematicians and parents joined to fight the new approach. Internet sites were established for this purpose, the best known of which being Mathematically Correct and The Fordham Foundation. The new approach got the nicknames "whole math", and "fuzzy math". California returned in 1998 to a more traditional approach. The achievements of the students rose significantly since then (see [here]). The government retaliated by threats of budget cuts, which put the struggle in the focus of the public eye. Two hundred and twenty mathematicians and scientists, including seven Nobel prize winners, wrote an open letter to Richard Riley, then Secretary of Education, against the reform, and published it in the Washington Post (it can be found [here]).
These are the famous "math wars". They are not over yet. The education research people claim in response to the criticism that the new approach has not been given enough of a chance, that results will improve in time, when the teachers are better trained in the method. Judging by the experience of the "New Math", this is probably not going to be the case.
A prerequisite for valid judgement of a curriculum is defining the aims of elementary school mathematical education.
In what follows I restricted myself to what is to be learnt, rather than vaguely defined "abilities". One reason is that this constitutes a common solid base for discussion. The other is that about this part there is wide agreement. It is commonly accepted that at the end of elementary school the student should have a deep understanding of the four basic arithmetic operations.
This encompasses three different main bodies of knowledge. The first is understanding the meaning of the operations. What does it mean to add two numbers? To what real life situations does it correspond? Understanding the meaning of the operations includes also knowing the properties implied. For example - that adding 1 to the subtractor reduces the difference by 1. Or that multiplying and then dividing by the same number leaves you where you started. Thus, the principle of expansion of fractions is included in "meaning".
The second body of knowledge is how to perform the operations. This calls for deep understanding of the decimal system of representation of numbers, since it is in this system that the operations are performed.
The third body of knowledge necessary for mastering the four basic arithmetic operations is fractions. These are necessary in order to deal with division and ratios in a systematic way.
We are nearing the main objective of this article, namely the detailed analysis of the curriculum. Here is a preview of its conclusions.
The main question concerning this curriculum should be - are we allowed to cut material so extensively? Where, precisely, is the division of fractions to be studied? How are children who understand fractions only on an intuitive level (which is what the study of fractions of denominator up to 12 means) going to study algebra in high school? How are children whose understanding of the decimal system is hampered by not studying vertical multiplication and long division going to understand polynomials?
All this should have been thoroughly considered. I am not sure that it was.
But, as mentioned, the curriculum is defective also in other ways. An essential requirement from a curriculum is that it should coherently outline a direction. It should clearly enunciate the thought principles that are to be acquired through the six years of school, and indicate a logical string of ideas leading towards them. In this task the curriculum fails completely. The picture the teacher gets from it is that mathematics is a disorganized array of unrelated topics.
The main flaws of the curriculum can be summarized as follows:
Fragmentation. This is a necessary corollary of the choice of structure (borrowed from the "Standards"). Distributing the material and concepts among categories of educational aims results in non-linear development. There are no clearly defined sequences of evolving ideas.
Opting for breadth instead of depth results in low ambitions. In the 60-s, when Israel was first in the international tests, at the end of Grade 1 children knew all four operations up to 20. The proposed curriculum pushes this well into the second grade. The ambitions are lowered still more in higher grades.
Preference of external criteria to internal logic. This is most conspicuous in the breakup of the teaching of fractions in Grade 4, and then again in Grades 5, 6. In Grades 5 and 6 the order is done according to operations: a period devoted to addition and subtraction, first in ordinary fractions then in decimal fractions, and then a period of multiplication and division in the two. This, while ordinary fractions and decimal fractions belong to two quite distant conceptual realms. These breakups are possibly intended to follow the "spiraling" dictum, so loved by education researchers. But it completely misses the spirit of "spiraling".
Low priority given to the meaning of the arithmetic operations. The meaning of subtraction is not stressed. Fractions are not linked enough to division, to its meaning and to the laws resulting from this meaning. In this way it is indeed hard to develop the rules for multiplication and division of fractions. The preference of formality to meaning is a remnant from the "New Math".
Arbitrariness of the distribution of subjects among the different categories. It looks as if the committee heaped on its table a pile of subject matter pertinent to each topic, and then distributed it quite randomly among the columns. This brings us back to the first item in this list, but also shows how redundant, even harmful, is the "breadth" approach, the partition into different categories instead of weaving a consistent line of ideas.
(Citations from the curriculum are italicized. All non-italicized parts, with the exception of headings, are my own comments, reflecting my own opinion.)
Natural numbers up to 100: counting and counting objects, reading and writing of numbers, parity, estimates, whole tens. The operations of addition and subtraction: addition and subtraction up to 20 (according to the student's abilities), 0 and 1 in addition and subtraction, the link between addition and subtraction, word problems, adding and subtracting whole tens.
Under the heading insight there appear, among others, counting, counting by twos, threes etc. completing sequences. "Greater than" and "smaller than", "in between", parity.
Under representations and accessories there appear for example the Cuisenaire rods, and the calculator (!). Under the same category there appears the collection of tens, which is a notion rather than accessory, and should definitely not be here. I presume that the authors are referring here to a paricular device they have in mind for the concretisation of collection of tens.
The mastery column contains knowledge of the facts of addition and subtraction up to 10, writing addition and subtraction exercises up to 20 and their solution using accessories.
The notions of greater than and smaller than, while not appearing in the topics themselves, appear under three different columns: "insight", "concepts" and "mastery".
The basic flaw of the curriculum, the neglect of the requirement of outlining a logical direction of ideas (as opposed to didactic approach), is already apparent here. The curriculum fails to indicate a thread of ideas, and to distinguish the primary concepts from those of secondary importance.
What should be attained by the end of the first year? The acquisition of rudimentary concepts about numbers: counting; left, right; larger and smaller. Then, a first acquaintance with the arithmetic operations. And finally, a first systematic encounter with the decimal system. All these are "topics", not skills or "mastery". Each should have been mentioned as a heading, under which subsidiary conceptual structures should have been arranged in some logical order. Instead, the curriculum gives only fragments of each. And those are distributed at random among the various categories. Basic structures like order ("greater than" and "smaller than") are piled up in the same column together with "completing sequences", or "parity". What message does this deliver to the teacher who is trying to construct a program for the entire year? That mathematics is a pile of disconnected ideas, which can be taught in any random order.
A good example is the notion of "parity". It can and should be used as an introduction to the operations of multiplication and division. This context should at least be mentioned. In the curriculum it appears as an isolated topic, and for some unclear reason under "insights".
As to arithmetic operations, one problem of the curriculum for Grade 1 is that it does not stress their "meaning" aspect. Addition and subtraction are first mentioned as "addition and subtraction up to 20", implying that it is only the calculation that matters. Worse than that: while there is some indirect referral to the meaning of addition (in the "word problems" category), the meaning of subtraction is not mentioned at all. There are three mentions of the connection between addition and subtraction: both the "topics" and the "investigation" columns include the link between addition and subtraction, and the "processes" column includes rudimentary understanding of the invertibility (my guess is that the authors meant "being inverse") of addition and subtraction. One can assume that it is intended that subtraction is to receive its meaning from addition - an approach inherited from the "New Math".
The weakest part of the curriculum for Grade 1 is the treatment of the decimal system. The obvious is implicitly admitted, that the foundations of the decimal system should be laid already in Grade 1. But the reference to it is fragmented: only adding and subtracting whole tens is mentioned under "topics". This is indeed important, but it does not lay foundations. What are the first steps towards the understanding of the decimal system? And where are they leading? The curriculum fails to lead the way here.
Finally, two points pertinent to the entire curriculum. The first is that taking a chain of ideas and distributing its links among various categories necessarily results in fragmentation and incoherence.
The second point is that disorderly heaping of material dictates minimalist aims. Linear and systematic development of ideas can lead further. Grade 1 students can comprehend the meaning of multiplication, and even division. They can perform these operations up to 20. They can also be exposed to explicit formulation of the distinct values represented by the two digits in a two digit number. All this was achieved in first grades in the 60-s, when Israel took first place in the international tests in arithmetic.
Natural numbers: two and three digit numbers, reading and writing, order among numbers. Addition and subtraction: continuation, up to 100. Beginnings of multiplication and division: the meaning of multiplication and division. Multiplication up to 100.
Under insight there are mentioned: counting with jumps, finding the consecutive number and predecessor of a given number, counting by tens in large numbers, the digits as representing sets of tens and hundreds, adding tens to a two digit number, the number line.
With respect to multiplication and division, the "insight" column contains: multiplication table (by heart), multiplication up to 50, by heart, division into equal sets.
Under investigation there are: investigating the properties of large numbers (?), research into the 100 board, constructing boards and models. With regard to multiplication and division: algorithms invented beyond those learnt in class, remainders upon division by 10, multiplication of a two digit number by a 1 digit number by non-standard algorithms. Example from everyday life of partition into equal sets.
Under representations there are: additional means of representation to the Cuisenaire rods, means of modeling ones, tens and hundreds, both discrete and continuous, the 100 board, the number line.
Under processes: the link between the organization of tens and hundreds and their writing as two and three digit numbers, the link between odd and even numbers, the role of 0 in two and three digit numbers.
The curriculum for Grade 2 is slightly more coherent than that of Grade 1. Two main lines of ideas emerge, and can be figured out from it: one is the principle of the decimal system and rudiments of calculating in it, and the other is the meaning of multiplication and division. Still, these are not explicitly stated, and the stages leading to them are not clearly presented. Instead, they are dispersed between various categories of educational aims. The link between organization of tens and hundreds and their writing as two and three digit numbers is not a process or property (under which title it appears), but a main idea of the decimal system. It should be included in the "topics" part. These are not "side effects" of the topics, but the main issue.
Here, too, the distribution of the principles to be learnt among the different categories is artificial, and worse than that - it serves to blur the direction. For example, the different digits as representing units, tens and hundreds is one of the two main ideas about the decimal system (the other being gathering tens). Putting it under "insight" (as if it is a by-product of understanding the rest) is misleading. The role of 0 in the decimal system is also a main idea, not a "process or link".
So, as in Grade 1, the curriculum does not give a sufficiently clear statement of what are the aims, and what are the means to achieve them. This is the result of the confusion stemming from putting general abilities before thought principles.
The columns of "representations" and "mastery" are here completely redundant, not adding anything new.
As to the topics themselves: again, the ambitions are minimalistic. It is possible already in Grade 2 to have preliminary acquaintance with fractions, with denominator up to 4.
A side remark, concerning meaning: it is important to stress the difference in meaning between 2 times 3 (namely 3+3) and 3 times 2 (namely 2+2+2). This becomes important later. A related omission is that of the terms "subtractor", "multiplier", "multiplicand", "divisor", "divisee". They do not appear anywhere in the entire curriculum!
The decimal structure of natural numbers, addition and subtraction up to 10,000, multiplication and division up to 100, fractions (rudiments).
Under insight there appear: place value, familiarity with the number structure, the quantitative value of a number, sequences with differences of 10 and 100, estimates, operations with numbers above 1000 without an algorithm, multiplication of two and three digit numbers by one digit numbers using the distributive law, various invented algorithms for the four operations in large numbers, and with regard to fractions, only: partition into equal parts, qualitative discussion of the relation between the part and the whole.
Under processes there appear: conversion and decomposition within a given number, a transparent (?) algorithm for vertical addition and multiplication.
Under mastery there appear, for example: the ability to give examples requiring multiplication, discerning between addition and multiplication, knowledge of the multiplication table up to 10 times 10.
As mentioned above, the Grade 2 curriculum is more coherent that that of Grade 1. With this exception, the further the curriculum advances the less structure it has. The reason is clear - the more complex the ideas are, the more they demand linear, step by step treatment, which is rendered impossible by the choice of the latitudinal presentation. From Grade 3 and on concepts, principles and skills float freely between the various categories, with little logic to justify the classification. No conceptual principle is developed in any coherent way.
Thus, for example, the principle that multiplying by 10 adds 0 at the end of the number appears under "investigation". This is important for vertical multiplication, and should be studied in connection with it. Systematic treatment of it is not opposed (as its placement in "investigation" may suggest) to active learning. It can be reached by interactive discussion in class. The difference is in the coherence, and the placing of the principle in context, a task that demands the intervention of a teacher.
The choice of material, apart from being limited as usual, is basically correct. Grade 3 is indeed the right year for an in depth study of the decimal system. But almost no principle related to calculations within the decimal system is studied coherently, or formulated properly.
The most obvious omissions are the standard algorithms for addition and subtraction, which are replaced by self-invented algorithms. This may serve as a good example for the difference between "investigation" and consistent teaching of principles. The vertical algorithm for addition is based on a principle that is not mentioned at all in the curriculum. Its first case is: the units digit of the result depends only on the units digits of the numbers added. The tens and hundreds digits will not affect it. This is why we start adding from the right, and not from the left - the rightmost digit of the result can be known right away, and will not have to be changed later on in the algorithm. This principle can also be studied in a fragmented way: letting the children find what is the last digit in examples like 23+34, or 25+28, and guess the rule. But studying vertical addition makes them realize how this principle goes over to the tens, and then the hundreds. In this way they learn the general, abstract principle, not just its first case.
But the curriculum's shortcomings appear even before that, in more basic principles, like conversion and decomposition. These are placed under the category of "insights". The fact is that they are main constituents of the operations with the decimal representation, and should appear under "topics". While having to be known before learning the algorithms, they must also be mentioned as part and parcel of the algorithms, not as separate "insights".
Again, the non-linear development of ideas results in reduced ambitions. Grade 3 is where long multiplication should be taught. This means the standard, vertical algorithm. As already mentioned, this algorithm is in fact not taught at all.
As to fractions, the curriculum says almost nothing. Besides the very few words cited above in "insights", it only mentions a few accessories. Thus there is very little to criticize. Apart from the obvious, that in Grade 3 the children are ripe to understand quite a bit about fractions, and the curriculum could be much more ambitious. And even if not, it could be more specific about the first steps towards understanding fractions.
The simple fraction: meaning - as part of a whole and as part of quantity. Different names for the same fraction (only fractions with denominator less than 12 (my guess is that it means including 12) are treated). Comparing fractions (according to certain points of reference, like "greater than 1/2", larger than 1/2" etc.) The natural numbers: acquaintance with "large" natural numbers (with no restriction), operations on natural numbers, order of operations, estimates, fluent use of the calculator.
The other categories are richly detailed in this grade. The partition, as usual, is arbitrary. Thus, for example, comparing the quantities corresponding to fractions as operators: what is larger, 1/2 of 10 or 1/2 of 40? appears under "insights", while the larger the denominator is the smaller the fraction under "processes and links", and classification of fractions as larger than 1, equal to 1 or smaller than 1 in "mastery". The impression is that the authors had a list of items, in which they could not find any logical order, and thus they threw them at random into the various boxes.
Under "insight" there appear also: discussion of the meanings of the simple fractions without formal calculations, understanding the difference between the fraction as a number and as an operator, finding the whole by unit fraction (in drawings), fractions larger than 1 on an intuitive level. With regard to natural numbers: feeling for "large" numbers, links between the four arithmetic operations, estimating the results of operations performed by a calculator or otherwise, using the calculator, including use of the memory, familiarity with the limitations of the calculator with respect to large numbers, rounding numbers.
"Investigation" includes multiplication of two digit numbers in various ways, based on the distributive law.
Under "representations and accessories" there appear (with regard to fractions) representation of fractions in objects, in drawings and in number tables, and with regard to natural numbers: the number line, concretization means for the decimal system, including the abacus, the "disks" -- and then: the calculator, the computer.
Under "mastery" there appear: identification and labeling of part of a whole in models encountered in class, finding part of a quantity, finding a fraction larger than a given fraction (larger than a half, smaller than a half), comparison of two fractions in simple cases (having equal numerators or equal denominators). With regard to natural numbers there appear: addition, subtraction and multiplication of numbers up to 1000 in ways familiar to the student, not necessarily the traditional algorithms, (operations with large numbers are to be performed using the calculator).
Let me start with the unit on natural numbers, since logically it should come first. This unit is unclear with respect to both timing and purpose.
As to timing: it is not clear why it is separated from the bulk of the study of the decimal system (in Grade 3) by the unit on fractions.
As to purpose: One declared intention is that of dealing with large numbers. This is of course important. It is known (see e.g. ) that performing operations with large numbers enhances abstract understanding - intuition ceases to suffice there. For example, vertical multiplication of three digit numbers requires better understanding and higher abstraction than that of two digit numbers, since it requires generalization of the rules.
The trouble is that this goal is missed by the curriculum. It is more of the same as in Grade 3. First, the operations are not performed on really large numbers: multiplication is extended only to that of two digit numbers by two digit numbers. Second, as was already mentioned, the standard multiplication algorithm is not taught at all. Multiplication algorithms are relegated to "investigation".
It is interesting to speculate what will that mean. Guidance from the teacher will still be necessary, of precisely the same kind as children receive on their way to the standard multiplication algorithm. So, the difference will not lie there. It will lie in reduced structure - the use of the less structured algorithm of "opening up brackets", writing 23*45 = 20*40 + 20*5 + 3*40 + 3*5 (this is actually indicated under "insights", see above, and under "investigation" in Grade 4). While learning this algorithm does require the understanding of important principles, it renounces the structure offered by the standard algorithm, meaning cumbersomeness, and hence probably less drilling.
There are many other stated goals of this "natural numbers" unit, all basically unrelated to each other: order of operations; estimates; a return to the meaning of the operations (under both "insights" and "processes" there appears links among the four operations), sequences, criteria for divisibility. Of these, "estimates" is indeed important. But it should be part of the performance of operations, and should be stated as such. The teachers should be encouraged to ask for estimates before the performance of each exercise. "Links among the four operations" is too vague. "Criteria for divisibility" is a nice topic, but learning it without explanation of the reasons is not at all useful. Today the criteria for divisibility by 3 and 9 are taught without explanation (in fact, hardly any teacher knows the reason behind them). Putting this topic under "investigation" may mean even less explanation (say, not even explaining the criteria for divisibility by 2, 4 and 5).
It is necessary to clearly state the main goal of the unit (which should be the performance of operations with large numbers), and state explicitly that the less central status of the other topics. The picture as is now is a disorganized array of subjects.
We are now reaching fractions, the other unit of Grade 4 (appearing first). It is not surprising that the treatment of fractions is the weakest point of this curriculum: it is an abstract and difficult subject, demanding coherent treatment. The teachers need guidance in structuring it. Such guidance is not provided by the syllabus, and assuming that the books are written in the order dictated by the syllabus, it will not be found also there.
What is the starting point of the teaching of fractions? Probably most natural is the choice of a unit. One takes a certain body, or set, and declares it to be the "unit", the whole. A third (say) of it is then one part of the whole, when it is divided into three equal parts. Three out of the four or five meanings of fractions come then relatively easily: (1) The number of parts in a division (2/3 is two parts in the division into three equal parts) (2) The fraction being an operator (2/3 of a certain body or set is obtained by taking two thirds of it when this body is the "unit") (3) The fraction as a number (the number 2/3 is two thirds when the unit is taken to be the number 1). A fourth meaning - 2/3 is the result of dividing 2 by 3, is obtained with slightly greater effort. A fifth meaning, that of being a ratio, can be postponed.
So, if the syllabus wishes to be a guide, it should start from "choosing the `whole' unit", and develop the ideas from there. What is actually done is putting discussion of the various meanings of the fraction in "insights", and then going into various unrelated topics: the possibility of representing the same fraction in different ways (appearing under 3 different categories), tackling fractions in real life contexts, and mainly - comparison of fractions. The latter appears in 8 out of the 30 items of this unit(!), probably reflecting a preference of the authors to the representation of fractions as points on the real line (in my view, one of the less insightful representations of fractions, since it does not easily reflect on the multiplication and division of fractions).
The picture that the teacher, and consequently also the student, gets is that of a disorderly array of facts about fractions, not a line of thought leading somewhere. This "somewhere" is the systematic treatment of division. To the student the aim can be presented as the ability to divide 1 by (say) 3, and later also any number by 3 (this meaning of the fraction is introduced in the syllabus only in Grade 5).
The syllabus for this grade contains four units, which are in order of appearance: fractions, the decimal fraction, natural numbers and ratios. Later, in Grade 6, there will be a similar sequence of units: fractions, decimal fractions, and ratios. Here the fragmentation reaches its peak. And it is here that its origin is most transparent - the lack of understanding of the underlying logic, and the preference of external criteria to it. The order chosen by the syllabus is: fractions - addition and subtraction; decimal fractions - addition and subtraction; fractions - multiplication (the very little of it that the syllabus contains); decimal fractions - multiplication and division. The linking element of the units is thus the nature of the operations, not the type of numbers treated.
This goes against the grain of the subject, and the internal logic. While fractions constitute a tight body of ideas, and their addition and subtraction are closely related to their multiplication and division, there is no kinship in ideas at all between addition of fractions and addition of decimal fractions. Similarly, there is no affinity between multiplication and division of fractions and their counterparts in decimal fractions. The set of ideas needed for the understanding of decimal fractions belongs to the realm of the decimal system. The understanding of fractions demands deep understanding of the meaning of division - a very distant set of ideas. There is no inherent justification for the order chosen by the syllabus. It only adds to the fragmented impression, and severs the logical string.
We shall list the items of the syllabus separately for each unit, and comment on each separately.
Simple fractions (with denominator up to 12): meaning - the fraction as the quotient of division, the fraction as a point on the real line, comparison of fractions, expansion and contraction, common denominator, addition and subtraction.
Under insights one can find: discussion of the meaning of the simple fraction (without calculations), the fraction as part of a whole, the fraction as a quotient, order of magnitude of fractions, estimates of sums and differences, addition and subtraction of two "easy fractions", finding a fraction between two given fractions.
Processes contains: many names to a fraction, links between the numerator and denominator of a fraction, use of the associative and commutative laws for quick addition and subtraction of fractions, multiplication of a fraction by an integer as repeated addition or as finding a part of the integer.
A logical development of the subject could lead to understanding fractions with arbitrary denominators, not only smaller than or equal to 12. It could later be used to study multiplication and division of fractions, which will be so important in algebra in high school, and which the syllabus renounces so easily. Instead, the syllabus chooses to disperse ideas here and there.
This time let me give just one example. In processes we find multiplication of a fraction by an integer as repeated addition or as finding a part of the integer. This means that 6 * 1/3 can be viewed as
(1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3) = (1/3 + 1/3 + 1/3) + (1/3 + 1/3 + 1/3) = 2
and it could also be viewed as a third of 6, namely 6 divided by 3, which is again 2. In the syllabus it appears as an isolated idea. In fact, this is one of the stages leading towards the multiplication of fractions. Had the syllabus made an effort to place it within a logical string of ideas, it could lead to the study of multiplication. As it is now, the students will learn an isolated fact, and will not understand the motivation for it. They miss the opportunity to place it in a more general framework.
This is just one of dozens of examples in the syllabus, where throwing an idea at random into a certain category results in missing the opportunity to link it to others.
Meaning, comparison, addition and subtraction
In insight one finds: conversion between simple fractions and decimal fractions in simple cases, discussion of the meaning of the decimal fraction without formal calculations, identification and labeling of a part of a whole (fraction) as a decimal fraction, comparing decimal fractions.
The Processes column contains among others: extending the place value system from natural numbers to decimal fractions, the density of the decimal fractions, the effect of adding 0 in different places of the decimal fraction.
It suffices to note that this table repeats to a large extent, column by column, the parallel table on fractions, from the first unit. The conceptual difficulties in decimal fractions are totally different from those of the ordinary fraction (decimal fractions are much easier than fractions, for those who understood the decimal system). The two belong to two quite different realms of ideas. This shows again the proclivity of the authors to external criteria, rather than the internal structure of the subject matter.
Another indication of the non-systematic development of the subject is that the first step in the development, extending the place value system from natural numbers to decimal fractions, appears as one of the items in "processes". How is a teacher supposed to know where to start?
Operations, expansion and going into them in depth.
Estimates of results of operations, estimates of quantities, rounding of large numbers, feeling for large numbers, the effect of changing one of the factors on the outcome of an exercise.
Multi-stage problems - continuation, writing mathematical expressions for more complex problems, problems involving large numbers.
Investigation: generating and investigating sequences and tables of numbers, the investigation of a structure or regularity in a process?
Representations: calculator, computer.
This is a strange unit, a hybrid between the statistical part of the syllabus, the search after "number sense" and the meaning of the operations. It is true that a feeling for large numbers is necessary, but it should appear in some context, not by itself. I would put it in a unit on powers - something that is definitely within reach of Grades 5 and 6 students. Children do understand powers of 2 and of 10. Powers of 10 are an opportunity to speak about large numbers and the ways of writing them.
As to complex word problems: word problems should appear throughout the six years. It is possible to plan them so that they are always related to the subject currently studied.
Ratio, finding ratios, comparing ratios, finding a missing datum in the equality between ratios (the so called "rule of three").
Informal experience with situations of direct ratios, identification of situations with direct ratios, the constant ratio between the edges of similar shapes, finding simple ratios, comparing simple ratios.
Drawings for the representation of quantities, cubes disks and other objects, electronic sheets.
The multiplicative or additive relation between pairs of numbers having the same ratios [??]
This is an important subject, and the syllabus does get a few of its constituents. Again, what is missing is a more systematic treatment. The unit should come under the "fractions" umbrella: the ratio is one of the meanings of the fraction. Questions about finding a part of a whole are really questions about fractions.
Another remark, on a possible systematic approach: it is advisable to formulate a concept on reduction to a unit: if a car travels 80 kms in 2 hours, how many does it travel in one hour? This is the key to many problems about ratios, particularly those falling under "the rule of three", and should be explicitly stated.
Here, again, there are four units. They are: simple fractions, decimal fractions, ratios and percents. The flawed logic of halving each of the first three units and distributing them between Grades 5 and 6 has already been discussed.
Simple fractions (fractions with denominator not exceeding 12): multiplication and division, addition and subtraction (review).
Insight: performance by heart of very simple multiplication and division excercises, discussion of the growing or diminishing of a number as a result of its being multiplied by a fraction, finding part of a whole and the whole by its part in very simple cases, estimates, calculating by heart the product of an integer by a simple mixed number.
Investigation: discovering the algorithms of multiplication and division, sequences of numbers obtained by applying regularity (?), chains or repeated operations, links among operations.
Representations and accessories: objects, drawings, fraction rulers, calculator, computer, the number line.
Processes: between every two fractions there is another fraction, multiplying by a number smaller/larger than 1 reduces/enlarges the multiplied number, division by a number smaller/larger than 1 enlarges/reduces the number.
Could you imagine a syllabus of Linear Algebra, or Calculus, which is not developed linearly but distributes its different ingredients among various categories of intellectual activities, and quite arbitrarily at that? Fractions is a deep subject, in principle not very different from Linear Algebra. It is not surprising that it cannot be developed very far in this method.
The origin of fractions is in the division of natural numbers. This link should be stressed and used to deduce the properties of fractions. This is hardly done in the syllabus. Dwelling on this link and the development of a few basic rules concerning division (e.g. what happens to a quotient of two natural numbers when the divisor is multiplied by 3?) could make it possible to go much further.
A side remark: the curriculum advocates the use of the calculator for the study of simple fractions. I am curious to know what do the authors have in mind! How can the calculator be used for dealing with simple fractions? Especially those whose denominator does not exceed 12?
Multiplication and division, addition and subtraction (review), connections between decimal fractions and simple fractions.
The rest of the columns basically repeat their "fractions" counterpart, as cited above. The senselessness of this stratagem was already commented upon.
Here the syllabus contains an error - this is a repetition, with very slight changes, of a unit from Grade 5.
Basic concepts, simple calculations, links between different representations of numbers.
Insights: the meaning of the use of percents in everyday life, passing from representations of 1/2, 1/4, 1/10, 3/4 as simple fractions to decimal fractions, percents and vice versa, estimating the percentage, by-heart calculations of simple percentages.
Investigation contains, among other things: finding the whole quanitity by a given percentage of it (not necessarily by computation).
Representations: fraction and decimal fraction rulers, gridded squares, calculator, computer.
Among Processes one can find: the percent as a special ratio of part of 100, connection between ratio and percent, representing percents as decimal fractions and ordinary fractions, the equivalence between a% of b and b% of a.
For a student who understood the notions of the decimal fraction and the ratio, percentages are a simple subject, with no new principles at all. After all, the percent is one special fraction. Indeed, the unit adds nothing to its predecessors, a fact well exemplified by the inclusion of the curiosity that a% of b and b% of a are the same (but why is this a "process", "property" or "link"?) Actually percentages could be used to introduce new principles, for example - after 20% tax the price was so and so, what was the original price? As it is, this unit is unnecessarily expanded.
The statistics part of the curriculum is given jointly for Grades 1 and 2, then for Grades 3,4 and for Grades 5,6.
The topics are divided into two: processes and analysis tools.
The processes are identical for all three pairs of grades: collection, organization and presentation of the data, and the drawing of conclusions.
The analysis tools are:
Grades 1,2: classification of data by characteristics and different criteria.
Grades 3,4: same, plus representation of data by criteria (range, mode (most frequent value), median, and criteria found by the students), testing for frequencies and trends.
Grades 5,6: average, median, mode, relative frequency, range.
The concepts part for Grades 5,6, which can also serve as an indication as to how far the curriculum goes, contains: diagram, table, axes, categories, average, abnormal data, frequency, median, relative frequency (expressed as a fraction and as a percentage), sample, population.
The rationale behind the teaching of statistics in elementary school is that statistical data are encountered in real life. This is true, and there is no doubt that knowing statistical concepts is helpful. The question is whether the gain is worth 10% of the studies time, as is the declared intention of the syllabus.
There is very little depth in the concepts studied, which means that a lot of activity will take place in class, but the insights gained from it will be shallow. In particular, when the fact is taken into account that the intention of the syllabus is that most calculations will be performed with the aid of the computer.
The question is also what part of the statistical concepts is really meaningful to a non-professional statistician. Take, for example, the notion of the "median". For the statistician the importance of the median is that it is less liable to influence by a single datum than the average. The child cannot appreciate this point. It is clearly not going to be reached in class (and is definitely not mentioned in the curriculum). So, for the child the calculation of the median remains a meaningless, not very difficult exercise.
All this is general criticism, and does not relate to this particular curriculum. As to the curriculum itself - in this case the structure suits the subject. The study of statistics is supposed to be done in an "investigative" fashion, and there is no linear development of ideas. Hence, in this case the structure is not detrimental to the purpose.
Following the "Standards", stress is put in this curriculum on investigation. Anybody who ever tried to teach in elementary school knows that lecturing there is not really feasible. The picture of the traditional teacher as "pouring information into the child" is just not the general case. Any reasonable teaching is interactive.
What is, then, the innovation in "investigation"? In what way does it differ from interactive discussion in class? The difference does not lie in the amount of active learning. It lies in the depth of the discussion, and in its structure.
Most items in the "investigation" column are nothing but material which is taught in class also without the title "investigation". Sometimes they are even main concepts. The most obvious example is the discovery of algorithms for addition, subtraction and multiplication of numbers beyond 10. Here are some other examples. In Grade 5 one of the items is finding as many ways as possible for comparing fractions, for finding a common denominator, for adding and subtracting two fractions. In Grade 1 one finds: the connection between addition and subtraction, addition and subtraction with an unknown, sequences.
Sometimes the topics are less central, like finding the even and the odd numbers in the 100 board (from Grade 1 again). Only a small part of them are of a third type, namely isolated activities, like linking the different parts of a Tangram to their part of the total area.
So, what is the difference going to be? In the mode of learning, which is going to be less verbal, and more of the form of individual activities. "Hands on instead of minds on", as an American opponent put it.
What are the likely results? There is no need to speculate. It is enough to watch the consequences in the US.
This is another focus of controversy. In fact, closely related to the investigation - skills debate. The proponents of the use of the calculator wish to save the student the tedious, long calculations, and let him leap straight into research. The opponents believe that calculations have their value. They contain non-trivial mathematical principles concerning the decimal representation of numbers. A fact which is perhaps better understood by mathematicians than by education researchers is that behind the decimal system there lies a very deep idea. This is witnessed by the fact that it was not discovered by the Greeks, although they founded so much of modern mathematics. In fact, it is only about 1000 years old. Assimilating its principles is one of the main goals of mathematical education in elementary school, and it can be done only via the practice of its (non-trivial!) algorithms.
In a poll conducted by by the mathematician Stephen Wilson among 80 mathematicians, not even one advocated the use of the calculator.
According to surveys of the TIMMS (the body running the international mathematical tests), elementary school students from the countries occupying the first five places in the tests hardly ever use the calculator. In 10 among the last 11 countries they use it regularly [Kline].
The calculator is advocated in the curriculum as means for achieving estimates for results, prior to the actual performance of the calculations. This reflects basic misunderstanding. Estimates are indeed highly important. But it is getting the estimate that matters, not knowing it.
From my own personal experience and that of my friends, the effect of the use of the calculator in elementary schools is highly detrimental to arithmetic skills. Seventh grade students will take out a calculator to compute 13 times 4. In an exam at the Technion a student from a prestigious faculty asked for a calculator to calculate 7 times 8.
Let me summarize the issues under debate:
Investigation versus skills.
The use of the calculator.
Linear development of the topics.
As to the first issue, it seems that some education researchers have romantic views of what it means to do mathematics. They think of it in terms of "creativity", or "deep ideas". Mathematicians know that they have to acquire technical skills and meddle with technical details - original ideas come later, by themselves. The aim education people have in mind is that of active learning, which is indeed a valid educational value. But "investigation" is not more active than verbal, interactive discussion. It is just less structured.
As to the use of the calculator, it appears that mathematical educators are missing two points. For one, they do not appreciate enough the depth of the decimal system and its algorithms. They do not understand that some of the deepest principles that should be acquired in school lie there, not in "higher order thinking". Second, they do not understand that getting an answer, or an estimate, without actually performing the calculation, is not worth much. It is the climbing to the top of the mountain that is important, not being there.
The third point is a matter of order of preferences. Of course, mathematical education researchers are also for consistent, step-by-step development of subject matter. But they put other values before it. The curriculum studied here exemplifies this point again and again.
The key term in all three issues is respect. Mathematicians have more respect to the internal logic of the subject matter.
To the best of my knowledge, hardly any. A planned review day was cancelled. The subject committee was the only body that was supposed to examine it. A subcommittee was formed to do the job. As far as I know, they made no essential criticisms.
The link with the standards, and the very problematic consequences of the adoption of the standards in the US, were not discussed, and were probably unknown to most members of the subject committee. The subject committee includes two research mathematicians, but as far as I understand they did not participate in the subcommittee. The curriculum has revolutionary implications: the renouncing of very basic bodies of knowledge, assumed until now to be necessary preparation for high school. And yet, it was not given to any mathematician, scientist or leading figure in the industry for thorough refereeing. Explanations were not given as to how education is to continue from that point, without the missing fundamental components.
Worse than that. The entire educational system is behaving as if the curriculum has already been approved. A very costly program of professional development of teachers is undertaken these days (wearing the mantle of "professionalization"), a large part of which is devoted to training in the new curriculum.
There is another important point: the committee's duty to provide full information. The authors claim that they used as their guides the 1989 Standards, Standards 2000 (a follow-up to the standards, written in a reparatory attempt), and curricula from Holland, England, New Zealand, Japan and Singapore. The truth is that the curriculum is a mirror image of the Standards and of syllabi written in its spirit. I am sure that it has non-empty intersection with the curricula of Japan and Singapore (countries that won second and first places, respectively, in the international tests), as it does with any other curriculum. But it does not share their spirit. The stress on investigation, the massive cuts in material, the non-linear development of the concepts, statistics, the use of the calculator and computer - all these are taken straight from the Standards reform. If the curriculum takes new elements also from other countries, it is because they also imported the reform.
But if this is so, it was essential that the committee should have informed the public about it. Some members of the committee are well aware of the developments in the US ensuing the adoption of the Standards. They should have included a report about them. The curriculum has a long preface (also borrowed from the "Standards"), full of nicely formulated principles, regarding its philosophy. This would be just the right place to mention the precise amount of indebtedness of the curriculum to the "Standards", their history, and the "math wars". It must be clear that if the Standards are imported, the "math wars" are sure to follow, and that the results may be similar to those of the import of "whole language".
The preface would also be the right place to discuss the debate about the calculator, which is introduced without the slightest hint as to the controversy around it.
[Kline] David Kline, American Federation of Teachers: Should We Curb Use of Calculators for Younger Students? American Teacher, March 2001.
[Milgram] James R. Milgram: Written testimony before House committee on education and the workforce. Feb 2, 2000, http://www.house.gov/ed_workforce/hearings/106th/ecyf/fuzzymath2200/milgram.htm
[Loveless] Tom Loveless, A Tale of Two Math Reforms, The Politics of the New Math and the NCTM Standards. Presented in Curriculum Wars: Alternative Approaches to Reading and Mathematics, conference held in Harvard, October 1999.