- Novalis (1772-1801)
But, first, I want to consider the possible objections to this proposal in order to put my proposed alternative into context.
Back to Basics
It seems peculiarly perverse in the current climate of mathematics education, particularly in the United States and Great Britain, to be proposing the abandonment of PPA. In the US the recently approved standards adopted by the California State Board of Education [California, 1998] which are a substantially revised version of standards issued by the California Academic Standards Commission [California, 1997] effectively mandate a PPA-only regimen in elementary school by banning calculators from all state tests in grades K-6. This back-to-basics approach1 is the antithesis of what this paper proposes.
In Britain a report [DFEE, 1998a] commissioned by the Secretary of State for Education, David Blunkett, of the new Labour government discourages ``as far as possible the use of calculators" in mathematics instruction for children through the age of 11. Although the final report [DFEE, 1998b], in response to widespread criticism of the preliminary report, does ameliorate this advice, it still espouses little or no use of calculators until ``the later years" of primary school (i.e. when children are 10 or 11). In fact, the current British government itself, even more than its predecessor, espouses a back-to-basics philosophy.
Although mathematics educators in the US have widely condemned the California standards and, similarly, mathematics educators in Britain are almost all (but not all - [Gardiner, 1998]) vehemently opposed to the government's position, many others in both countries are firm supporters of back to basics. Educators other than math educators, many parents and most politicians have always deplored putting calculators in the hands of young children 2 .
But perhaps those most strongly in favor of the new California standards are research mathematicians including many from prestigious institutions like Stanford and Berkeley. (In Britain the research mathematics community has been relatively quiet about [DFEE, 1998a and 1998b] but a significant cadre of that community are clearly opposed to having calculators establish any non-trivial beachhead in elementary school mathematics despite respected studies [Shuard, 1991] which show the efficacy of this approach.) The research mathematics community has good reason to be concerned about the state of pre-university American mathematics education, a topic I shall return to in the next section. Here I note only that the American research mathematical community has not understood very well the impact of technology on mathematics at the university level so it is hardly surprising that research mathematicians have generally, although there are some notable exceptions, utterly failed to understand its impact on the pre-university mathematics curriculum. Indeed, again with the caveat about exceptions, as a rule research mathematicians just don't know what they are talking about when they criticize what happens in elementary school [American, 1995-1997]. They have diagnosed the problem correctly - poor preparation for university mathematics - but they don't understand the causes of the problem or its cure3.
The Value of Learning PPA
Two factors have largely been the spur of the back-to-basics, anti-calculator movement:
1. The poor showing of American (and British) students on international comparisons [Schmidt et al, 1997]. Since such comparisons at the elementary school level focus on PPA prowess, if American and British students do badly, then the reason must be their failure to learn PPA properly4. And what is more likely to have caused this failure than the increasing use of calculators over the past quarter century? For secondary school comparisons, PPA does not play nearly so important a role (but, nevertheless, not a trivial one). However, it is often argued that failure to become proficient in PPA in elementary school inevitably hinders a student at all subsequent levels of mathematics.
2. The progressively poorer preparation of first year university students to study university mathematics. Given that this is the case - and a very wide spectrum of university mathematicians think it is - then might not the cause of this poor preparation be the use of trendy new techniques and materials - often calculators - in the teaching of school mathematics? Calculators are also often associated with a lack of rigor in teaching mathematics, a dumbing down of the curriculum.
My own belief is that, although the premises in both these arguments are correct, the conclusions are utterly wrongheaded and without basis in fact or plausibility. (I note, in particular, that, from teaching university mathematics for 30 years from 1965 to 1995, I fully subscribe to the premise in the second argument above. My students in general progressively knew less mathematics upon entering university and, as well and just as important, they knew progressively less about what mathematics is and what is important in it.)
I reason as follows:
1. There may seem to be no reason, in principle, why mixing PPA and calculators in an arithmetic curriculum should lead to poorer overall PPA performance and arithmetic understanding. It is, after all, agreed by almost everyone that the incessant drill-and-practice in arithmetic, which for many years was the hallmark of elementary school mathematics almost everywhere and still is in many places, does not instill much understanding of arithmetic, however much mechanical proficiency may be its result. Indeed, there is some evidence [Hiebert, 1986] that too much arithmetic drill-and-practice results in the message being lost to the medium. Which is to say that the message of why one does arithmetic is lost in the emphasis on computational accuracy. But it is the case that many teachers feel that the use of calculators in the classroom makes it more difficult for them to achieve PPA expertise. One might argue that, as this is the case, the reason is ineffective or prosaic use of calculators rather than imaginative use. Maybe. But my conclusion is that halfway houses are almost certain to be ineffective: Either elementary school arithmetic should consist of PPA without calculators or calculators without PPA.
Strangely, perhaps, the only people who are likely to agree with this proposition are the back-to-basics proponents although they, of course, will make a different choice than I would. Almost everyone who favors the use of calculators in elementary school classrooms appears to believe, perhaps out of a desire not to seem too radical in the face of the back-to-basics brigade, that this should be as a supplement to PPA or, in any case, that there should still be a major component of PPA in elementary school arithmetic. A typical opinion is: ``As one of the pro-calculator brigade, I, like many others, am also advocating that children must be fluent in ... pencil and paper methods" [Kitchen, 1998]. Support for both teaching PPA and using calculators will be found in studies that show, almost without exception, that the introduction of calculators into the classroom does not hinder the attainment of PPA skills and may even enhance them [Hembree, 1986]. But most (all?) of these studies were with teachers who were committed to trying calculators and felt comfortable using them. This is just not true for the majority of teachers who, more often than not, feel that calculators are a distraction even when they recognize that calculators are what almost everyone everywhere (in the developed world) uses to do arithmetic. Indeed, perhaps the crucial point is that children almost universally use calculators for arithmetic outside the classroom with only looming tests in PPA persuading them to practice PPA on homework or otherwise. In fact, it is the certainty that calculator use outside class does and will inevitably retard the acquisition of PPA skills which provides a telling argument against the back-to-basics regimen. Classical PPA instruction is doomed to relative failure in a world where arithmetic is almost universally done using calculators and where even the dimmest child will see that attaining skill in PPA has almost no value in non-academic pursuits.
So the issue is much less whether back to basics is a good idea as it is whether it can be successful no matter how much you might wish it to be. Even if it is true, as John Saxon once said [Saxon, 1990], that ``you must practice everything [in mathematics] to get it", can such practice be successful when children recognize that what they are practicing is not a useful life skill?
2. But if learning PPA is not a useful life skill, perhaps learning it is useful in later mathematics and other education and professional pursuits. First, it is clear that PPA is not useful in any professional pursuit. Although some professional mathematicians need to do considerable amounts of arithmetic, they almost always use calculators or computers when dealing with more than one or two digit numbers. And while scientists, engineers and other professionals may need to do quite a bit of arithmetic, almost without exception they do it with calculators and computers. So the argument in favor of learning PPA stands or falls insofar as this skill is necessary to learn subsequent mathematics.
Before going further, let me first demolish some straw people often used in arguments by the anti-calculator contingent. One is epitomized by the following quote from Jaime Escalante who has been justly praised for his achievements in secondary school mathematics teaching. Criticizing the NCTM Standards [NCTM, 1989], he said that ``Whoever wrote [the Standards] must be a physical education teacher... The use of the calculator in the classroom is negative for the learning process. All students will learn is how to push buttons." This is a common reaction to calculator usage in the classroom from people who know nothing about teaching mathematics in elementary school and even from those, like Escalante, who should know better. Not only is there no evidence whatever to support this position but, I submit, the curriculum I will outline later in this paper could not possibly be criticized on these grounds.
A related straw person is that anyone who advocates calculator use in elementary school must be in favor of dumbing down the curriculum by making it easier and less work. Such a claim is just plain false. I challenge anyone who reads this paper to its conclusion to claim that I am proposing an easier or less challenging, less demanding mathematics curriculum. Another straw person is that the use of calculators in elementary school must be associated with lack of attention to other traditional portions of primary school arithmetic such as place value, fractions and proportions. There have been some trenchant criticisms [Wu, 1998; Klein, 1998] along these lines of the standards promulgated by the California Academic Standards Commission. But all these traditional - and important - topics can and should be in a calculator-based elementary school curriculum. If it is true that often when calculators are widely used, other traditional topics are short-changed, then that is the fault of those who design and teach calculator-based curricula; it is not a necessary concomitant of calculator use. Again I challenge any reader of this paper to aver that I advocate any weakening of the non-PPA elementary school curriculum.
So is a facility with PPA necessary - or even desirable - for later study of mathematics? If a student arrives in secondary school, say in a first algebra course, unable to do PPA, is that student ipso facto disadvantaged compared to students with PPA skills? I take it t basis, it will be possible - even easy - to teach the division of polynomials.
Finally, a word about another delusion regarding mental arithmetic, namely that if you wish to stress mental
arithmetic in elementary school, it follows that you must discourage the use of calculators. In Numeracy
Matters [DFEE, 1998a] there is a praiseworthy emphasis on mental calculation (as there is also in
[DFEE, 1998b]). But this leads to the non sequitur that ``the importance of mental calculation
inevitably has implications for a judicious use of calculators" with ``judicious" meaning ``discouraging as
far as possible the use of calculators" with children up to age 11. Why isn't it as clear to everyone as it is
to me that mental arithmetic is more important when calculators are used in the curriculum than when
they are not? This is not just - or even mainly - for the efficiency reason given at the beginning of this
section. It is also because since, as we all know, the likelihood of egregious mistakes when using a
calculator (because of a wrongly selected button) is greater than with PPA, calculator users must be able
to estimate mentally the results of their calculations. Recently a teacher remarked on an email
group that ``the older generation possesses an instinctive feel for number which soon sniffs out a crazy
answer; this is a facility which the younger generation lacks". As this is true, it must be because this
``younger generation" has not learned to be good mental calculators and to use this ability to ``sniff out
Elementary School Mathematics without PPA
Doing away with PPA in elementary school may sound radical to some but, for those of you who cannot bring yourselves to embrace this idea, how would you feel about postponing any instruction in PPA until the sixth grade? That sounds pretty radical, too, doesn't it? But, in fact, it is an old idea tried successfully in the late 1920s by the school superintendent of Manchester, New Hampshire and recorded in a paper which should be read by all mathematics educators [Benezet, 1935-36; see also Gleason]. Benezet's idea seems to have died a quick death but perhaps you will agree that a PPA-less curriculum is not so far removed from this 70-year-old idea.
In any case, what I propose for the first, say, eight years of school (K-7; an implicit assumption is that
algebra should be an eighth grade subject) is the following:
1. A stress on mental arithmetic from the time when any arithmetic idea beyond counting (which, of course, is itself a mental activity) is first introduced. This means, in particular, that as each arithmetic operation is introduced, children will be expected to do mental calculations using that operation. Of course, as part of this, children will be expected to learn the addition and multiplication tables at appropriate times. In line with the paragraph above, I make no assumptions about precisely when any arithmetic operation should be introduced. It may well be that, as Benezet suggested and Gleason supported, there is no good reason to introduce arithmetic as early as is now customary. Clearly children should have many, varied, and substantial experiences with numbers from kindergarten on. This is much more important than their precise form.
2. Calculators should not only be allowed from kindergarten on but their use should be encouraged. Indeed, there is every reason why the use of calculators and the teaching of mental arithmetic should be interleaved because instruction in one will be reinforced by the use of the other and vice versa. Of course, it is obvious that the mental arithmetic portion of the curriculum must be assessed in a non-calculator environment as must also the application of mental arithmetic in problem solving. Otherwise, calculators should be generally allowed in all testing situations. The aim of calculator usage should not be the negative value of avoiding PPA but the positive one of providing exercises and problems which develop number sense and understanding of arithmetic. When decimals and fractions are introduced, calculators should be used to illustrate the relation between the two, what infinite decimals are and why they occur etc.
3. Nothing above should be meant to imply that mental arithmetic and calculators should be the only tools with which to teach arithmetic. Manipulatives and other arithmetic models (e.g. the area model for multiplication) should continue to play an important role. As well, as already noted, teachers could certainly use the algorithms of PPA to illustrate arithmetic operations as seems useful and convenient. And, in any case, pencil-and-paper as a recording and experimental medium should continue to play an important role in elementary (and secondary) school mathematics. For example, there should be no objection, in principle, to doing 2-digit by 2-digit multiplication by writing down two 1-digit by 2-digit products done mentally and then adding these mentally.
4. To reiterate a point made earlier, children should be expected to work hard and to think hard in all their mathematics instruction. Nevertheless, I believe that a mental arithmetic-calculator regimen can accomplish all the goals of the best imaginable PPA curriculum with less total instructional time because, at least, although lots of drill-and-practice will be required to create good mental calculators, the effort involved should be less than is usual now in PPA drill-and-practice.
5. To my mind one of the major advantages - and attractions - of doing away with PPA and replacing it by a mental arithmetic-calculator curriculum is the additional time which would be made available to study other mathematics in elementary school. Of course, it is true that mathematics other than PPA is now taught in elementary school, some geometry, for example. But the dominance of PPA in the elementary school curriculum is such that not too much more than lip service is usually paid to other topics and children seldom take away from elementary school any significant mathematical skills or knowledge beyond PPA. But, of course, not just geometry but other mathematics - probability, an early introduction to aspects of algebra and others you could name - are well within the intellectual reach of elementary school students. Introduction of these topics into the elementary school mathematics curriculum would prepare students for secondary school mathematics far better than they usually are now and might, in time, mute the criticisms of university mathematicians about the mathematical knowledge of entering university students.
6. A word about abstraction. I mentioned earlier that one criticism of doing away with PPA is that
PPA provides students with an early introduction to abstraction. But so similarly does mental arithmetic
although I doubt very much that young children sense the idea of abstraction at all when they are first
introduced to numbers. But abstraction is one of the most important ideas in mathematics and not at all
beyond elementary school students, an assertion proved by the facility which many elementary school
students develop for computer programming where abstraction in the form of variables may not be overt
but is certainly inherent. I mentioned just above an early introduction to algebra in elementary school
mathematics. Certainly this is possible, perhaps in fifth grade, more likely in sixth grade and certainly in
seventh grade. What secondary school or university mathematics teacher would not agree that even if
all elementary school mathematics achieved was the most elementary notion of variable, this would
provide a wonderful entree to secondary school mathematics?
The above is not - and is not intended to be - a complete elementary school mathematics curriculum. It
is rather an attempt to convince you that a PPA-less elementary school mathematics curriculum could be
developed - actually, I believe, easily developed - which would prepare students for secondary school
mathematics far better than they are today not just in the US but anywhere (and far better than any
proposed curriculum in the US would achieve). Moreover, I believe it would - anyhow could - do this
without the boredom which so many students now feel in elementary school mathematics and which
must lose a shocking proportion of the mathematical talent inherent in American students when they first
Secondary School and Later Mathematics
What would be the impact of a PPA-less elementary school mathematics curriculum on secondary school and university mathematics? As the intent is to have students better prepared for secondary school mathematics than they are now, the impact would be entirely positive. In particular, better prepared students would allow the secondary school curriculum to introduce useful new subject matter - statistics, discrete mathematics etc. - as is now being tried in a number of places, without abandoning traditional subjects. But how would the mathematics style proposed above affect what is taught in secondary school mathematics and how it is taught?
University mathematicians who oppose the use of calculators in elementary school do so mainly because as quoted earlier, of the belief, which I share, that you ``cannot have understanding without technique". I have argued that, at the level of arithmetic, that technique and the understanding which follows from it can be achieved through mental arithmetic. What is the relevant analogy to secondary school (and later) mathematics?
It will hardly surprise anyone who has read this far that, just as I favor full use of technology in elementary school, I similarly favor its full use in secondary school and universities - graphing calculators, symbolic calculator-computers and computers. But how then will students achieve technique - algebraic technique is crucial here - if they are allowed full use of calculators? As Edward Effros has said : ``I can state categorically that if a student cannot factor $x^2 - 9$ instantly it is extremely unlikely that [the student] will pass calculus". I agree. The path to algebraic technique in a world of calculators is essentially the same as the path to arithmetic technique, namely mental algebra. Surely it is not unreasonable to expect students of algebra to do a fair amount of polynomial algebra mentally. Moreover, even as normally taught now, tasks like completing the square are essentially mental tasks which use pencil and paper as the recording medium. Going beyond Effros I would expect students to be able to factor a variety of three term quadratics mentally. And, of course, as with mental arithmetic, there should be calculator-free assessment of mental algebra.
More generally, just as attaining number sense is the essential goal of teaching arithmetic, attaining symbol sense should be the essential goal of algebra. Symbol sense is harder to define than number sense but it includes such things as being able to predict the form of the result of a symbolic calculation (what is the degree of the product of two polynomials?), the ability to select the most appropriate of several equivalent forms (e.g. polynomial, completed square or factored form of a quadratic), having judgment about the reasonableness of a result (if there are six times as many students as professors, do we express this as 6P=S or 6S=P?), developing competency in engineering symbolic expressions (given some points in the plane, find an algebraic or trigonometric function which passes through or close to all of them) etc. (See [Arcavi, 1994] for a good discussion of symbol sense). Students who have this symbol sense, no matter how much they use graphing and symbolic calculators, will be well-prepared to study calculus.
Although university mathematicians have been slower to use technology in their teaching than scientists
and engineers, they have started to come around in recent years. Still too many feel about symbolic
mathematical systems (aka computer algebra systems), such as Mathematica and Maple, the way they
feel about calculators for elementary school children. One result is that despite so-called calculus
reform, the aim of most college calculus courses still seems to be to create a student-machine in which
functions are fed to its maw and derivatives and integrals emerge at the other end. And this in spite of
the demonstrable failure of calculus courses to produce students with more than a mechanical grasp of
the subject. Whether or not you agree with my long held belief that discrete mathematics should play a
role co-equal to calculus in the university curriculum [Ralston, 1981], there can be no excuse any more
for calculus courses which do not make full use of technology and which teach students to do mentally
much of what they now do mechanically.
Could My Proposal Work?
It could - almost - be implemented tomorrow. A detailed curriculum could be developed, textbooks could be written, lessons planned etc. Let's even be optimistic that politicians, parents, mathematicians - all those antediluvian groups - could be convinced of the rightness of abolishing PPA. But are elementary school teachers themselves ready for such a curriculum. I fear - but would happily be proven wrong - that even most excellent elementary school teachers would have serious difficulties teaching such a curriculum. This is mainly because their preparation to teach elementary school mathematics has been woefully lacking and has prepared them to teach little aside from classical PPA. But also it is because many (most?) teachers attracted to teaching in elementary school are less interested in mathematics than in any other subject; indeed, too often elementary school teachers suffer from mathophobia.
How many elementary school teachers, for example, are prepared to teach two-digit mental arithmetic? How many of them have the knowledge to teach the other aspects of mathematics which would enter the curriculum if PPA were replaced by a more demanding but less time-consuming mental arithmetic-calculator curriculum? Some of these problems could be alleviated with the current cadre of elementary school teachers with appropriate in-service training. For future teachers much better - and intensive - pre-service mathematics education would be necessary.
But so long as mathematics is low on the interest scale of current and prospective elementary school teachers, it is unlikely that much progress will be made this way. People with a talent for mathematics and an interest in it are just unlikely to be attracted to elementary school teaching since most of the elementary school curriculum involves, as it should, language-based subjects. This is true even of elementary school science which is almost entirely descriptive.
There is really only one solution. This is to have much, if not all elementary school mathematics taught by specialist teachers much as subjects like music and art are. Some schools have headed tentatively in this direction by appointing a mathematics specialist whose task is to give advice and help to all the teachers. Another partial solution is team teaching in which the mathematics teaching is done by those teachers who are most comfortable with mathematics. Both these ideas are all to the good but I believe that, inevitably, they cannot be fully effective in implementing the curriculum described in this paper. Even elementary school mathematics specialists seldom have the training and breadth of mathematical knowledge and perspective to teach or advise on a mental arithmetic-calculator curriculum.
I conclude that specialist mathematics teachers who do all the mathematics teaching in elementary
school, from at least the third grade on if not earlier, are the only solution if the curriculum is to be
radically changed. This is clearly a long term solution. But I predict that no great progress will be
made in the mathematical education of American elementary (and, therefore, also secondary) school
students until this solution is implemented . Until it is, the hand-wringing at the poor performance of
American students in international comparisons and the disappointment of university teachers with the
mathematical preparation of their students will be a continuing phenomenon of the American educational
Do We Need Research?
Of course, we do. We always need research. But the immediate question is: Does any proposal in this paper require that research be done before that proposal should be tried out? Or, put another way, a guiding principle of educational innovation should be, as in medicine: Above all, do no harm. So is it possible that students ``deprived" of the opportunity to learn PPA could be harmed thereby?
I submit that the answer is, no. Since no one argues any longer that knowledge of PPA is a useful skill in life (or, for that matter, in mathematics), the question is only whether such ``deprivation" could leave students without the understanding or technique necessary to study further mathematics. We may not know how much mental arithmetic the average child may reasonably be expected to learn - and we should surely be doing research to find out - but I do not see how you could argue that an elementary school mathematics curriculum built around mental arithmetic and calculators, which was demanding for students and which introduced them to considerably more of mathematics outside arithmetic than is usual today, could possibly do them any harm . Indeed, a curriculum emphasizing number sense in this context could hardly help but prepare students for secondary school mathematics at least as well as they are now.
Yes, much of the argument in this paper has been ex cathedra but I make no apology for that. My arguments are, in fact, far better based on experience and research than the arguments of those who wish to continue to subject children to a PPA regimen. So, yes, more research by all means but there is no reason whatsoever why the ideas in this paper should not be tried before further research is done.
It is also worth noting that nothing in this paper is at variance with
any modern theoretical, experimental
or practical research on the teaching or learning of mathematics (see,
for example, Kamii ). Indeed, I would expect the many
proponents of a constructivist approach to mathematics to applaud a
curriculum which emphasizes
having children construct their own understandings of arithmetic through
The reform of mathematics education is a much debated topic currently, particularly in the United States. What has been called the ``math wars" [Becker and Jacob, 1998; Jackson, 1997; Ross, 1998] pits mathematicians, mathematics educators, other educators, parents and politicians against each other in a bewildering array of contexts in which curriculum, pedagogy, teacher training, textbooks and technology are all subjects of controversy. One irony of all this, in my view, is that, while the crucial decisions that need to be made concern elementary school mathematics, the chief mathematical protagonists are professional mathematicians, who rarely understand the issues with respect to elementary school mathematics, and the National Council of Teachers of Mathematics which, although it is a group whose membership and interests cover both elementary and secondary school mathematics, is dominated by university mathematics educators and secondary school teachers. The result is that elementary school mathematics too often gets short shrift or misunderstanding or worse in the debates about mathematics education.
Some of the debate about mathematics education concerns such matters as how much focus there should be on problem solving in the school mathematics curriculum, whether mathematical pedagogy can and should be influenced by recent work on learning theory and on a constructivist approach to mathematics, and on whether new curriculum proposals give sufficient weight to traditional topics like algebra and trigonometry. I have not discussed these matters here because I claim that the proposals made in this paper are equally apt whichever side you take on any of these questions.
My belief is that a mental arithmetic-calculator curriculum taught by specialist mathematics in
elementary school would not just be a better curriculum than is now taught (virtually) everywhere but
that, more strongly, unless this approach to elementary school mathematics is adopted, major
improvement in the mathematical performance of American students at the elementary school, secondary
school or university levels will remain a chimera.
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