Texbooks as the Source o[ Leaming Mathematics
Textbooks as the Source of Learning Mathematics
Fou-Lai Lin
Department of Mathematics College of Sciences
INTRODUCnON
This paper de祉s with 也em位n factors that contribute to the va1idity and reliability of 也e school mathematics textbooks, namely, readabili旬， fonn of writing, sequence of topics, motivation, application, and exercise.
In sections 2 and 3, we will explain 血e problems which arise from these factors, and propose some suggestions for improving textbooks in tenns of these factors.
As a practica1 example
,
we will differentiate the writings of the topic“In甘oductionto Probability" in three different textbooks in section 4.
In the lastparagtaph, section 5, we discuss the authorship of textbooks briefly.
At the beginning section, we wi11 describe the role of the textbook in mathematics lessons and the important goa1s of mathematics textbooks.
Bulletín 01 National 自iwanNormal University No. 28
CONTENTS
Introduction . . . . . . .553
1.. The ro1e and the goa1s of mathema位cstextbooks . . . .555
1. The ro1e of the textbook in mathematics 1essons. . . . .555
2. Goa1s ofmathematics textbooks. . . .. . .556
3. Types ofwriting textbooks. . . ., • • • .558
II. Readab血 tyof textbooks. . . .559
1. Readability... .559
2. Factors affecting the readabi1ity . . . .559
3. Removing reading difficulties . . . .563
111. Principles of improving the textbook . . . 566
1. Form ofwritingjlogic of argument . . . 566
2. Sequence of topics. . . 567
3. Motivation... 568
4. Application... ". . . . 569
5. Exercise... 569
IV. Ana1ysis of some textbooks. . . . 570
1. Factor analysis . . . " . . . 570
2. Summary... 576
V. Epilogue (authorship) . . . .. ~ . . . ~ . . . 578
References. . . . 579
Texbooks ω the Source 01 Leaming Mathematics SECTION 1. THE ROLE AND THE GOALS OF MATHEMATICS TEXTBOOKS
1. The role of the textbook in mathematics lessons
In this paragraph, we will describe the role of the textbook in mathematics lessons in terrns of teaching methods, teacher, pupil, curriculum and c1assroom phenomena.
(i) Teaching methods in use at present, e.g. pupils working individually or in a small group, often require that the pupil gets ma也ematica1 meaning from the written word. In the past, conventiona1 c1ass-teaching might use a textbook only as a source of exercise. Now, due to the growth of mixed ab逝ty grouping and individual learning systems, textua1 materia1 has become the source of explanation and instruction as well as of exercise.
(ii) Every year, many student teachers begin teachíng mathematics in schools. Textbooks which are well written and suitable for 血e pupils will be the most appreciated gift to those inexperienced teachers.
Also
,
due to some existing reasons,
there is a certain percentage of unqua1ified mathematics teachers in every country. For example ，扭曲e U. K., according to the heads of mathematics departments in secondary schools, there are 12.7% of ma血ematics teachers who are not only unqua1ified but unsuitable to teach mathematics (Hall and Thom品， 1977). The report in Aspects of Secondary Education in Eng1and: Supplementary inforrnation on mathematics (1980) said that out of a1l mathematics teachers, 30% of them can only teach very low level mathematics, 30% of them can teach up to O-level and the other 40% can teach over O-level. According to the Cockcroft Report, Mathematics Counts (1982, p.l92), there are about 5000 secondary school mathematics teachers who have“
ni1" qua1ification. In addition, there are another 4000 mathematics teachers with a poor qualification. In an unqua1ified, non-effective mathematics teacher's class, the only chance that pupils can stiU learn mathematics well is for them to have a good textbook.(iii) Even in a qualified mathematics teacher's c1ass or a conventiona1 c1 ass-teaching c1ass, pupi1s still have to learn mathematics by reading a textbook in many circumstances; e.g. some pupils may not like their mathematics teache郎， so they don't want to pay attention in ma也ematicslessons; pupils may be absent in some mathematics lessons due to illness or being involved in some other activities, etc. To catch up with 也e 自st of the c1ass, they might haye to read the textbook by themselves.
Bulletin 01 National Taiwan Normal University No. 28
The textbook may a1so play the role of authority to pupils in a mathematics lesson. In many mathematics courses, pupils may have only one textbook, and, in addition, their teacher's teaching relies heavily on that textbook, then to them the textbook is equa1 to the course in this case. Another situation occurs when a textbook provides answers to exercises.
(iv) Actually, many secondary school teachers rea11y depend upon their textbooks, as Forbes (1970) said. If teachers just follow the textbooks to teach,
then 也e text has not only determined the content of the curriculum but also
也eprocess of the cur吋culum.
'"The textbook determines the curriculum" is a statement which can be made about mathematica1 education in many countries. In some countties, e.g. the Republic of China in Taiwan and Mexico, the textbook is an integral part of c.. nationàl curriculum, they cannot be considered apart. In others, e.g. Federal Germany, the U. K. and the U. S. A., where there are many textbooks available, the textbook chosen would appear to exert a great influence of the curriculum in - the individua1 c1assroom, as Howson (1981) said. So we may view the textbook as the
“
hidden" curriculum.The textbook is a1so the introducer of the new content in cUÍTÎculum. Whenever a new content is inc。中 orated 扭 the curriculum, textbooks have to be produced to introduce this content. Most teachers leam the new material through the textbook and then design their teaching.
。) The c1assroom, as Bishop (1980) described, is an arena for human interaction and negotiation as the sociologist sees it. The teacher and his pupils' can find many reasonable questions from a well designed textbook to negotiate, to work together-After they have worked out some problems, they share the feeling of achievement. This kind of feeling a1ways improves the relationship between teacherand pupi1 as well as between pupi1 and pupi1. Whenever pupi1s and their teacher have a c10se relationship, then lessons are sure to go better and better.
From different points of views
,
we have seen the importance of the textbook in mathematics lessons. There has been a wish to produce textbooks which pupils can read easily by themselves.2. Goals of mathematics textbooks
In Chi1dren Reading Maths (1980
,
p.26),
the authors st~te 也at acquisition。 f concepts, principles, skills, and problem-solving strategies are important goa1s which school mathematics textbooks aim for. Besides these four items
,
we thinkSS6-Texbooks as the Source o[ Leaming Mathematics culture and attitude are important too.
(i) Concepts
Mathematics textbooks provide materia1 to expound concepts for the reader. One of the fundamenta1 numeric concepts in school mathematics is to know the concrete meaning of numbers, e.g. fraction, decima1, negative and imaginary numbers. Geometric materia1 iel1s us the concept of our living space.
A1gebraic material shows us many models which are the approximation of physica1 problems. Probability materia1 tel1s us how to make a reasonable prediction. Different means of a collection of data tel1 us some conc自te concepts about that data.
(ii) Principles
There are many fundamenta1 principles in school mathematics textbooks. “
The principle of deduction leads us to think axiomatically. The principle of mathematica1 induction convinces us to believe some facts are true for any natural number. To find the volume of a solid
, we apply Cavalieri's principle.
To enumerate the ways of arran斟ng subjects we apply the principles of addition and multiplication as wel1 as the.‘principle of pigeonholes.(üi) Ski11s
Skills mean the me曲。ds of solving problems. For instance
,
Euclidean a1gorithm, solving equations, method of approximation and method of translationa時 þowerfu1tools for later work and study.
(iv) Problem-solving strategies
Mathematics textbooks seIVe to develop 也e capacity of 血e human mind for the obselVation, selection, generalization, abstraction and construc位on of
models for use in solving problems in mathematics as wel1 as in other disciplines.
(v) Culture
Mathematics history is a crea世ve cu1ture. The prωess of construction numbers
,
the background stories of discovering theories as well as 血e stories of some greaf mathema位cians， such as Newton, Gauss, Euler, Cauchy, Napier, Descartes, Kepler, Fennat, Ga10is and Abel 缸。 good sources for the goal of cu1ture.557-Bul/etin of National Taiwan Normal University No. 28
(vi) Attitudes
During every mathematics lesson a child is not only learning, or fai1ing to learn, mathematics as a result óf the work he is doing but is also developing his attitude towards mathematics. Positíve attitudes assist the learning of mathematics; negative attitudes not only inhibit learning but very often persist into adult 1ife and affect choice of jo b ，的 MathematicsCounts says (1982, p.l 0 1). Puzzles and projects in textbooks always encourage us to do it.Motívations in textbooks arouse our curiosity of finding out something. If the textbooks can present mathematics in such a way as to continue to be interesting, enjoyable and inspiring then hopefully the readers wiIl develop positive attitudes towards mathematics.
3. Types of writing textbooks
For each of the goals listed above, written material features the Teaching
of those items, giving practice in the use of the items and providing materia1 to test the acquisition of those items. In addition to these, textbooks should also attempt to develop the pupils' vocabulary and their methods o[ writing
down mathematics.
The authors of textbooks use a wide variety of types of writing in an attemptto achieve those goa1s. Shuard (1979) classified those types of writing as: Exposition, Instruction, Examples and Exercises, Other types of Writing and Signals, and she anslysed many examples of textbooks by exhausting possíble representations in each type of writing.
In our present work, we wi1l analyse some examples of text in section IV in terms ofthe following factors:
(i) Readabi1ity
(ii) Form of writingjLogic of argument (iü) Sequence of topics
(iv) Motivation (v) Application (vi) Exercise
Texbooks as the Source of Leaming Mathematics SECTION 11. READABILITY OF TEXTBOOKS
1. Readability
Readability
,
according to Austin and Howson (1979),
means the matching of reader with materia1. Since written materia1 st血 remains major determinants of the curriculum to be followed, the method of teaching and a1so of the language used within the c1assroom. Not surprisingly many teachers wi1l often view the readabi1ity of texts as the major concern ofmathematica1 education.It is a general impression that ma也ematics is a very difficult subject for rnost chi1dren
,
as the results of the C.S.M.S. research irnples(Hart,
ed.,
1981,
p.209). Sometirnes students do poor1y in rnathernatics sirnply because they can't read well. In the Newrnan study (see Clernents, 1980), which was based on interviews of low achievers of sixth grade pupils,
there are 35% of errors occurred in the categories of reading and cornprehension. Furthermore, 47% of errors occurred before the pupi1s got to the point of using the process skills necessary to solve the problerns.One reason that pupi1s can 't read well is perhaps because the textbooks in specific content areas are beyond the pupi1's reading abi1i討的. In rnany rnathernatics lessons, pupi1s cornrnonly need to be able to read written instructions, carry out these instructions and learn frorn what they are doing. They need to be able to learn from the explanation of concepts, syrnbols and vocabulary which they find in the textbook. During 位lÌs learning process
,
very often pupi1s need their teacher's help to read. Unfortunately, a1so very often pupi1s are not satisfied. Usua11y textbooks have rnathernatica1 concepts ernbedded in prose that even trained teachers of rnathernatics find difficult to cornpletely process independent1y. For instance, an investigation by Elliott and Wi1es (1980) finds that rnore than a quarter of the rnathernatics teachers were thernselves unable to dea1 easi1y with a textbook which is rnarketed and widely used for the eighth grade level. We can irnagine 也ere are rnany pupi1s in those teachers' c1ass with difficulty in understanding their textbooks. Especially, pupi1s find reading rnathernatics textbooks to be different frorn reading other rnateria1s and often quite difficult, as Henney (1971) explains.2. Factors affecting the readability
In genera1, the factors found to affect the readabi1ity of rnathernatics writing are vocabulary
,
syrnbolic language,
syntax,
the flow of rneaning and the needBu/letin of National 血iwanNormal University No. 28
for specia1 reading techniques. Factors other than those a1so can affect readability. However
,
it is on those factors that most attention has been placed.(i) Vocabulary
There is no doubt that readability is affected by the choice of words. Understanding the meaning of words is essentia1 in leaming to read all types of materials. This is especially true in regard to ma血ematics textbooks
,
since in mathematics textbooks there exist three broad categories ofwords.(a) Words with a meaning only in mathematics
For instance, equation, hypotenuse, par叫lelogram， coefficient,
denominator, numerator, etc. These words cause difficu1ties because 也ey
rarely occur in pupils' experience.
(b) Words which have different meanings in mathematicallanguage and ordinary language
For instance
,
parallels in“
there are para11els between these situations" means similarities. In statistics, probable has a wider meaning, and we use it to describe all events which can happen, i.e. all events which in every-day speech are ca11ed probable or possible. Confusion is the difficulty with this kind of words.(c) Words which have the same meaning in mathematics as well as in ordinary
Janguage
For instance, similar, function, etc. Very often, “ordinary" words are mixed up with technica1 words and symbolism; the reader must a1ways be suspicious ofa word just in case its meaning is not ordinary.
(ii) Symbolic l~mguage
Mathematics textbook by its very nature has a strong symbo1ic orienta位on.
The existence of symbòlism is the most striking difference between mathematica1
text and ordinary prose.
A study by Ku1m (1973) showed that vocabulary is an important factor affecting readabili旬， but is far out-weighed by the difficu1ty of the symbolism of mathematics
,
especia11y in algebra.In a section on learning a1gebraic'symbols and syntax
,
Ausubel and Robinson (1969) stated 曲的 the same problems 晶 in leaming a second language are ìnvolved. The learner begins to 個nslate a1gebraic symbols into the na世ve lan那age of arithmetic. Sometimes,
even only a small change in familiar560-Texb()oks as the Source of Leaming Mathematics symbolism can cause confusion. Johnston (1981) provides an account of his experience, of the difficulties of understanding symbols. He says
he was familiar with expressions of the type y = a x
+
b and less familiar with y=
b + a x, . . . , the unfami1iar form y=
b + a x was sufficient to confuse him,"
The fol1owing characteristics of mathematica1 symbols a1ways cause trouble for pupi1s when reading textbooks.
There is litt1e redundancy in mathematica1 symbolism. 1n statements such as 4 、‘ .J 司、 ν 勻， h i ﹒‘、 =Fr 、 JEU -D AV 丸 OA =刊、=~= A)Ic i44+B AOUAU
There is little redundancy.
1n mathematics, the meaning of each individua1 symbol is vita1. The symbolic notation can carry not only exact, but also implicit meanings and these implicit meanings are likely to create many problems.
Furthermore, a great dea1 of mathematic<Jl symbolism is the interrelation-ship of symbols. Not only does each symbol have its own meaning, but this meaning is affected by its neighbouring symbols. Usua11y, we use a spatial structure in defining the meaning of symbols. For example, the meaning of symbo12 in
212，佑， J2，32, f(2), a2, R2, (2,3),1001 2
are not at' all consistent. If asked to find x y, when x
=
2 and y=
3, then 2 3certain1y is not the desired answer.
The non-linear way of reading symbols also disturbs the reader.
_ (3
+
X)2 = 49 ana3
+
XZ=
49are 80 different
,
but both say“
three plus X squared is 49".
The redundancy, implicit meaning, interdependencc, and non-1inearity 一 561 一
Bulletin 01 National 泊iwanNormal University No. 28
in reading of mathematica1 symbo1s all affect the readabi1ity of textbooks. (See Rothery et a1、， 1980, pp.48-49; Woodrow, 1976.)
(iii) Syntax
Sentence structure and sentence length - which reflect the reader's mem01y span - are important factors that affect the readability of textbooks
,
as Austïn and Howson (1979) view.Most of the mathematics textbooks are compact. As Watkin (1979) described, the princip1e characteristic of mathematica1 1anguage is terseness; i.e. the 1east possib1e number of words is used to deve10p each idea, so that to read to 1earn, one cannot miss a sing1e word. Because of the compactness, the text writers repeated1y use stylized syntactic forms such as
“
if and on1y if弋 “o叫y if',“
for almost every . . .". It takes time for the reader to become fami1iar with this style of writing.From Kane's (1968) point ofview
,
the grammar and syntax in mathematical Eng1ish are less flexib1e than in ordinary Eng1ish. The most obvious complication of syntax in mathematics texts is that sentences in mathematics are incorporated with the symbols. Ap. obse1Vation by Watkins (1979),
following the hypothesis that pupi1s will understand ma也ematics betier when concepts are rewritten with more common grammatica1 structures and without symbols, he confirmed that pupils learn better from treatments that use ordinary Eng1ish structure. Watkins conc1udes that symbols do not appear to help or to hinder students with limited background in mathematics and they may help more advanced students. Does mathematica1 Eng1ish give beginners the economy and the power that it gives mathematicians? It seems unlikely.(iv) The flow ofmeaning
The reader of mathematics texts needs to build a picture of the overa11 flow of meaning as the page
,
section or chapter develops.By re-arranging the flow diagram of the meaning units in a text, where the units are segmented according to the statements which are made in the text
,
the authors of“
Children reading Maths" c1aim that it is possible to re-wdhthe text in a more readable form.This means the now ofmeaning in the text does affect the readability.Texbooks as the Source o[ Learning Mathematics
(v) The need for special reading techniques
(a) Reading symbols
The reading of the symbols may cause problems. For example:
3+ 口= 8 and 3 + X
=
8present a different reading problem. The latter embodies an implicit question
“
what can 1 do to get the x?"I
~ ~
I
may 叫ike (~ ~)
and[~ ~
J
but the latter two, denoting a matrix, behavethe same way, and the former behaves quite different1y.
(b) Context Clues
In mathematics
,
more than in any other field,
leaming is built upon pr肘ious leamings. Lack of context c1ues a1so leads to further reading problems.(c) The active reading style ofmathematics
To read a mathematics text, we need a pen and some paper. It is not enough to sit back and be told things.We must ask the text questions, and try to answ.er them ourselves. We must fill in the gaps by ca1culating things, following the worked examples by trying thein ourselves and then peeping at the text bit by bit.
These factors are by no means exhausted. Some other factors, such 品
layout of the page, printing and format a1so can affect the readability.
3. Removing reading difficulties
(i) Vocabulary
The general them of most constructive advice on vocabu1ary can be summed up as follows:
Avoid difficult words if they are not rea11y essentia1 to the purpose of the text. If they are needed
,
then teach them thoroughly (Children Read Maths,
1980, p.97).τhe authors of SMP 11-16 course hope that the language of mathernatics
Bulletin o[ National Taiwan Normal University No. 28
should become part of the child's living vocabulary, related to their experience as a whole. If this be the case
,
the chiId must be secure at one level of work before going on further. The absorptíon of language and the ideas it expressed takes time. Thus, when introducing a new word use it several times and in different contexts,. if possible. In other words, do plan the introduction of new words.We can avoid using some words which are not real1y needed. For example
,
in a primary book one can often find“
table",“
chart",“
matrix",“
figure",
etc. a1l used to describe the same thing. Why not just use“
table" consistent1y.Brunner (1976) suggested that it may prove profitable for teache爪 and
thus textbooks a1so, to explicate specia1 uses of words in ma血emati品，
borrowed from English but used in their own way. Such as: if andonly if, some, any
,
exactly,
there exist and almost every,
etc.We may locate technica1 words and notations peculiar to mathematics from a glossary or dictionary and write a defmition and a sentence. For example:
Coefficients: The constants in a polynomia1 are ca11ed coefficients.
(ii) Symbolic language
Symbols should be introduced gradua11y and meaningfu11y.
Hearsee (1975) suggested that at an ear1y stage, the pupi1s' own notation may be the best vehic1e; standard symbols should. not be imposed rigidly from the start. The most natura1 usage is the best. For example
,
use a,
not f,
for acceleration;“
1-1 corrèspondence" not“
injection'\Kulm (1973) explained that in order for a symbolic expression to be understood, the student must not only recognize 血e symbol but he must a1so know the word or words that are symbolized.
Many students need extra practice in simple translation tasks such 品 writing or saying "x squared minus three x y plus y cubed" when give "x2
- 3 x y + y3 ". Textbooks should include more exercises like these.
Watkins (1979) obseIVes that symbols do not appear to help students with limited background in mathematics and they may help more advanced students. He finds that students learn better from treatments that use ordinary Eng1ish structure rather than mathematica1 English. The main difference of these two structures is the symbolism of ma值lematical Eng1ish. So he suggests that textbooks should be written' in a stvle c10ser to ordinary Eng1ish. And if the goa1' is for students to know ma血ematica1 1anguage， since it does help later on,
Texbooks as the Source o[ Leaming Mathematics then specia\ instruction may be necessary.
(iii) Syntax
To make the texts more readable, Austin and Howson (1979) a1so draw their attention to the need to use simple sentence constructions and avoid long sentences.
A1so 也ey suggest an a1terna世ve way of Simp1ifying textua1 material by providing children with specia1 reading instruction intended to help them approach and assimi1ate ma也ematica1 vocabulary and phraseology with greater confidence and faci1ity.
For exarr:ple, 'Call and Wig斟n (1966) eva1uated. the effectiveness of two teaching methods, teaching reading and teaching mathematics, on the teaching of second ye缸a1gebra. They assigned one c1ass to a norma1 re斟men(Math. teacher text and lectura1 without forma1 reading instruc世on) and another to an experimenta1 method (English teacher with limited training
,
text and lecture with specific instruction in translating words into mathematica1 symbols). Since the experimenta1 group performed better than the control group, the logica1 inference is that teaching reading, at least in this content area, has a significant influence on the student's acquisition of usable information.τnus， we shou1d write the text so that it can reflect the teaching procedure which is used in teaching reading, especia11y in a1ge bra.(iv) Diagrams
Draw the geometric pictures consistently
,
as Bishop (1977) suggests. For example,
in the following two figures:~
If we say b. ABC is a right 出ang1e and PQRS is a square
,
then we did draw 也e right angles in these two figures inconsistent1y.Each diagram can provide much information.τhe text should teach the reader to read and to obtain the useful infonnation. For instance
,
designing some questions in terms of the diagram for the reader to practice is one way bf teachingBulletin 01 National Taiwan Normal University No. 28
the reader to read the diagram.
Some other suggestions, such as setting up a system of references (each equation and diagram should be numbered), using bold-face or italics print for each new mathematica1 term or notation, can also make the text more readable.
SECTION llI. PR酬CIPLESOF IMPROVING THE TEXTBOOK
To make the text more readable, we discuss the readability in the previous section. In this section, we will draw our attention to a11 other factors which are listed in 3, Section 1.
(i) Form ofwriting/Logic ofargument
We distinguish two forms of writing: formalism and intuitionism
,
and prefer the intuitionism.In formalism
,
the way of writing usually follows the schema of: defmition•
propositions•
examples.This type of writing is based on the axiomatic deductive approach. One starts from some point and then builds the whole theory. It is well known that the process of discovering 也eory is not lik:e this. A def"mition makes sense only when it follows some concrete concepts which arise via analysis of problems. So the definition should not be the starting point. Actually ，也efoundations of
ma也ematics 訂'e still quite open. F. Klein (see K血泊， 1973, p.142) described that
“
In fact, mathematics has grown like a tree, which does not start at its tiniest root1ets and grow merely upward, but ra也er sends it roots deeper and deeper at the same tìme and rate that its branches and leaves are spreading upward . . We see, then, that as regards the fundamenta1 investigations in mathematics, there is no fma1 ending, and therefore on the other hand, no first beginning, which could offer an absolute basis for instruction."Readers feel that 血e .materia1 ofthis type of writing is too abstract. Certain!y mathematica1 thought operates by abstraction; màthematica1 ideas are in need of abstract progressive refmeme肘， axiomatiza組on， crysta11坦a位on. Yet, as Kline (1973
,
p.124) says,
the life blood of our science rises through its roots; these roots reach down in endless. ramification deep into what might be ca11edrea1i旬， whether this
“
rea1ity" is mechanics, physics, biologica1 form, economic behaviour, geodesy, or for what matter, . . .Rea1ity is the main characteristic of intuitionism. The fol1owing diagram,
Texbooks as the Source 01 Learning Mathematics
which is drawn by a mathematician and educator ca1led Allendoerfer (1962), describes intuitionism as a way of writing;
}ogical induction
propositions definitiori
organization
application
natural world natural wor1d
iιfor each topic
,
whenever it is possible,
start with some concrete physical examples. Analyse the problem, identify the concepts which arise from the problem, and so .motivate the leaming of some new processes or new material. Define new technical terms on1y áfter the terms are meaningful to the readers.An d ，的 Turnau (1980) suggested, introduce the symbolic formulae only after a .sequence of problems that would enable the student to grasp the rules implicit in those formulae.
(ii) Sequence of topics
Tumau (1980) recommended that the structure of the textbook should correspond to the spiral process of teaching. The spiral process of teaching means that the introduction of a topic is arranged with introductions of sub-topics according to the level of generality and formulization, with the use of means such as visualization and motivation to ensure that its functional and deep understanding is made possible. In the process a chi1d can acquire a limited amount of mathematies, iιeach sub-topic, which he feels at home with and cat) use rather than to have a large stock of mathematical notions and techniques
which he cannot bring to bear on any situation other than a standard examination question.
The time from a lower form to a higher form of a topic is a critical factor
Bulletin of National Taiwan Nonnal University No. 28
of 甘le spiral process. Pupils forget the concepts andstatements which were taught too long ago、 So ， if they wait too long for the next sub-topic to show
坤， then it might be needed to re-teach the previous sub-topic again.
Turnau (1980) suggests two principles of writing text in terms of the spira1
organization of topics.
(1) Ideas and statements which were 恤 troduced in a lower form can be re討sed
and elaborated at a higher level of genera1ity and formuIization, a10ng with their application to a new range ofproblems.
(b) New ideas and statements should not be elaborated
“
up to the end", in their fina1 correct formulation, wi血a11 the most 加portant applica位onsshown or even trained. On the contrary
,
their ela:boration must be open to further formulizations and genera1izations as well as to new applications.(iii) Motìvation
Motivation is often decisive in respect of the effectiveness of the text.
If a subject has any va1ue
,
then the student must be able to appreciate its importance immediately; as Whitehead put in his article “Aims of Education": / whatever interest attaches to your subject matter must be evoked here and now, . . . (see Kline, 1973, p.151). And the need is to motivate 血 rough the interests aroused by reading as well as through the activity stimulated by the text.The choice of sources of motivation should be made veηr carefully and tested in practice. Here are some suggestions.
(a) Rea1 problems are good sources
Pupils live in the rea1 world. and, like a11 human beings, either have some curiosity about real phenomena or can be far more readily aroused to take an interest in them than in abstract mathematics. The genuine motivation wil1 be the one that arouses interest 加 the pupils. Rea1 problems,
as some limited experience has shown
,
will do thìs.Note 也at we don't mean that children are 加terested in practica1
problems only. Conversely, as Turnau (1980) 0 bseIVed，也e interest for purely abstract situations shou1d also be taken into account.
Motivation for the non-ma血ematician cannot be mathematica1. It
is pointless to motivate complex numbers for the general pupils by asking for solutions of x' + 1 : O. Since .pupils may not wish to solve the former equation? (See Kline
,
1973,
p.150.)-Texbooks as the S，οurce of Leamillg Mathematics (b) Application of non-mathematica1 situations is a1so a good source. Motivation
does not a1ways call for preceding the treatment of a mathematica1 topic by a rea1 problem. It is sometimes more convenient to introduce a mathematical topic first and then apply it to a non-mathematica1 situation. For example, the parabola as a curve may be taught as just.a locus problem or the graph of quadratic function. But then the uses of the parabola in focusing and directing light and radio waves should surely be presented. (c) Puzzles, games or other deVices serve at particular age levels and should
be applied as many pupi1s enjoy them.
(iv) Application
By internal and externa1 applications, we mean applications in mathematics and in non-mathematics respectively.
Due to their limited experience, writers, usually find it easy to present the interna1 applications in their text. But mathematics is not an isolated, self-sufficient body of knowledge. Knowledge is a whole and mathematics is part of that whole. Mathematics does not develop apart from other activities and interests. To separate learning into mathematics, science, economics and other subjects is artificia1. Each subject is an approach to knowledge, and any mixing or overlap where convenient and pedagogica11y useful is desirable and to be welcomed. Thus the text should present the relationship between mathematics and other subjects. Many of these relationships can serve as applications.
Professor Moore, (see K1ine, 1973, p.147) in his paper
“
On the Foundations of Mathematics", even recommended combining mathematics and science at the high scho011evel. In this way we might arouse in the 1earner“
a feeling that mathematics is indeed a fundamental rea1ity 'Üf the domain of thought, and not merely a matter of symb01s and arbitrary rules and conventions".Whether or not ma血ematics should be combined with science or presented as a part of man's efforts to understand and master his world would give students a historically and currently va1id reason for the great importance of mathematics, and this is application.
(v) Exercise
One muin reason why pupils do poorly in mathematics is they ure not doing their exercises. Learning by doing exercises is a basic method of the study of ma thema tics
-Bulletin of National Taiwan Normal University No. 28
result or a part of the resu1t."
These suggestions are excellent principles for use in dlsigning exercises for discovery training.
SECTION IV. ANALYSIS OF SOME TEXTBOOKS
In this section, as an example, we wi11 ana1yse briefly the topic of introduc-tion of probabi1ity in the series of textbooks SMP, SMG and MT (M and T indicate the authors ManSfield and Thompson respectively) in the U.K. in terms
of the six factors we discussed in the last two sections. The main references corresponding to each text are:
SMP, Book, E, Chapter 4 Experiments (pp.48-52) and Chapter 7 Probabi1ity (pp.79-92).
SMG, Book 2, Chapter 4 Introduction to Probabi1ity (pp.179-190).
MT, Book 2, Chapter 4 Pie Charts, Histograms, Probabi1ity, Pasca1's Triang1e, The Normal Distribution Curve (pp.68-77).
1. Factor analysis
(i) Readability
(a) Some hard terms appe訂 in these texts. For instance,
We a1so distinguish between
“
very probably" and“
'not very probable" events by a numerical method explained below. (MT,
p.68)The、 term
“
numerica1 me也od" is one of the most important methods in mathematics. But without any further explanation, pupils wi11 fmd it difficult to gain any understanding of this term. It is better to introduce a sequence of examples of probability frrst and then explain that the representation of probability is one example of numerica1 method.The terms "statistical experiment" and ''statistical evidence" (MT, p.69) are also too mathematica1 for pupils aged 12 years.
There is a word "random" in 2
,
Exercise 4,
SMG (p.184). Random is an imporlant and high level concept in probabi1ity theory. In this introductory topic,“
random" is avoidable.(b) The most important concept in the introduction of probability is
“
equally likely".τbree texts introduce this concept quite different1y.MT introduce this concept as a mathema位ca1 assumption 'and use
Texbooks as the Source 01 Learning Mathemiztics
3) Translate the mathematical language “Equa11y like1y" into oramary 1anguage
“
fairness".There are many examp1es, in pp.88-90, whiçh are designed to demonstrate the concept of fairness with visua1 materia1 - bar charts.
(c) SMG introduce the term
“
ordered pair" through a perfect situation.Two dice are thrown, . . . the possib1e outcomes can be shown as an array of ordered pairs, the first number being that uppermost on the first dice, the secOnd number being that on the second dice.
(1,1) (2,1) (3,1)... (Examp1e 2, SMG, p.184)
(ii) Fonn ofwriting
(吋 Basically ， the three textbooks are written in the way of intuitionism. For examp1e, the structure of defining theoretica1 probabi1ity in these texts is:
examp1e (p)
•
definition under this examp1e (q)•
genera1 definition (r)More precise1y, the flows of definition of probabi1ity in three texts are as follows:
MT: p
•
q•
r•
explain the difference between events with probability 5/8 and 3/8SMP: p
•
q•
five exercises as app1ication qf q•
rSMG: start with many exercises. of the following structure. Exposition/ Instruction of the sample space, but without the termino10gy.
•
Figure out a fraction according to an imp1icit ru1e which is induced by the definition of probabi1ityThenp
•
q•
r•
more examples.(b) Due to the difficulty of the mode1 of norma1 distribution, MT introduce this mode1 by on1y giving the formu1a and 1istipg the processes of using this formulato find the approximate probabi1ity.
This type of writing is often used whenever a high leve1 or complicated formula is going to be introduced. Let pupi1s accept a formula without rea1 understanding is sorhetimes necessary. For example, the Fundamenta1 Theorem of Algebra is usua11y introduced to high school pupils without proof. But only when it is necessary 也en we apply this type of writing. Introduce the ]Jlodel of normal distribution right next to the introduction
of probabi1ity is not suitable
,
hence it is not necessary to apply this type - 571 一Bulletin of National Taiwan Normal University No. 28 of writing here.
(iii) Sequence of topics
(a) SMP is a typical example of the spiral process of arrangement of topics. The experimental probability and theoretical probability are introduced in two separated chapters. The prerequisite, bar charts, was introduced in Chapter 9, Book B. Further levels of probability theory, inc1uding applica-tions, are introduced in the later books, in particular:
Problems and their .soIution sets (apply the example of probability toexplain the solution set), Chapter 11, Book F; and Probability (tree diagram and consecutive experiment), Chapter 13, Book H.
Before bar charts are applied in Chapter 7, Book E, to recall and develop pupil ideas more securely about this concept
,
SMP arrange an Interlude following Chapter 4, Book E,
to make sure that pupils are ready to apply it.(b) The flow öf topics which relate toprobability in MT series is:
Probability, Pascal 's TriangIe, The Normal Distribution Curve, Chapter 4, Book 2; and Probability (formal definition of Probability space, Markov Chain), Chapter 10, Book 4.
The mathematical level of the above two chapters are quite different,
so naturally they are separated.
The prerequisite and applications of introduction of probability are inc1uded in the same Chapter 4, Book 2. However, the mathematicallevel of the normal distribution curve is much higher than 由e introduction of probability. With these topics taken simultaneously it is debatable whether it is possible to teach in a meaningful way. With this example
,
we see 也atthe sequence of topics in MT is not arranged in spiral process.
(iv) Motivation
(a) The motivation for the introduction of probability in MT is quite artificial and mathematical. MT tell the reader what is to be done and how to do it. MT start with an artificial problem to distinguish the word
“
probable" and“
possible" .When we catch a bus we say it is probable that we shall arrive at our destination roughly on time, but it is possible 也at we will be three hours 1ate." (MT, p.68)
Texbooks as the Source of Learning Mathematics
it as ordinary words.
“Tf we make suitable assumptions about the event . . . we . . . obtain a theoretica1 probability . . . For instance, if we assume that the two coins considered above are equally likely to come down heads or tails. (MT, p.70)
SMG introduce this concept in two stages:
1) Using everyday life examples to lead 也e student to think of this concept. For example:
Why is tossing a coin considered to be a fair way of deciding which team should choose ends in a game of footba11 or hockey? (Exercise
1~3 ， SMG, p.l80)
Why is throwing a dice considered to be a fair way of making a score in a game ofchance? (Exercise 24
,
SMG, p.182)In the above two examples
,“
equa11y likely" is subsfìtuted by ordinary language“
fair way".2) Explain this concept with an example:
We say that the chance or probability of a head tuming up when a coin is tossed is 泊. In theory, when the coin is tossed, there are two possible outcomes: head or tail. These are equally likely to happen (SMG, p.182)
Following these two stages
,
the authors of SMG assume that pupils now understand this concept and wi11 be able to use it later on.SMP introduce this concept spira11y. Three stages are inc1uded:
1) State a reason to introduce 也is concept and then assume this concept as a condition in the definition ofprobability. The reason is as follows:
If the dice is a perfect cube (and is not loaded!) then we may reasonably suppose that we are equally likely to throw any one of the six numbers on it. 的MP，p.79)
Also, eva1uate the pupil 's understanding of this concept by an open question.
A football match can end in one of three ways a home win (助，
a draw (D), or an away win (A). Is it sensible to say that the probabi1ity of a home win is 1/3?
2) Emphasise this concept 詛 a whole section
,
iιsection 2. Equally likely outcomes (pp.81 ~87)， with many examples,
counter examples and exercises are also inc1uded.Bulletín of Natíonal Taíwan Normal University No. 28
And then tell the re早der how to distinguish the events which are
“
very probable" or “notveηr probable".Another artificia1 everyday 1ife example which is applied to i1lustrate the concept of probability
is:
The sta tement 曲的 if you died of lung cancer then there is a 9/10 probability 血的 you were a1so a smoker, is based only on statistical evidence, and not on detai1ed understanding about the medical con-nections between smoking and lung cancer. (MT, p.69)、
(的 Themotivation for the introduction ofprobability ín SMG is some exercises in which they ask the reader to compute some fracíions, such as total number of heads/total number of possible outcomes. SMG hope that 也e
reader wi1l1earn from this practice.
Some everyday life examples are given to lead pupils to think of the concept of equally líkely
,
as we quoted in (í) (b) of this sectioß.(c) SMP desígn four projects 扭曲e paragraph 2.2 (SMP
,
p.52) to motivate the experimental probability. These projects díscuss some problems which are fami1iar to the reade蹈， such as the fraction of people in a fixed grOUp with birthdays in a particular month,
the fraction of left-handed people in 也e c1ass, as well as an open圖ended problem which concerns a survey of research.Visual material - bar charts - are used to illustrate the concept of fairness. This is a concrete realization of the concept of equally likely.
(v) Applica的ons
(a) MT introduce two sub-topics, Pascal's Triangle and the Normal Distribution, right next to the iníroductíon of probability 品 its app1ications. Both are internal applícations.
Introducing Pascal's Triangle as an application ofprobability is interest-ing. Yet, there is an easier way to introduceit, i.e. by listing the coefficients in the expansion of(x + l)n, n
=
0,1,2,3,.The normal distribution is one of the most important models in probabi1ity theory. But it is very hard for pupils of age 12 to apply theír just learned and 1imited knowledge of probabi1ity to see the importance of this mode1.
(b) In SMG, the paragraphs 4 Calculating expected frequencies and 5 Probab也，ty
of certain success and certain fai1ure, are intema1 app1ications of the
Texbooks as the SOU1臼 o[Leaming Mathematics introduction of probabi1ity. The mathematioa1 1evels of these applications are c10se to the introduction ofprobabi1ity.
To find the probabi1ity of the sum of the two numbers when two dice are thrown (Example 2, SMG, p.184) is an intema1 application too. Authors of SMG also take this chance to introduce the concept - the ordered pair.
(c) The four projects in paragraph 2.2, p.52, SMP, are very interesting applications. They are talking about game, people's birthdays in a particu1ar month
,
proportion of left-handed people and a survey of research respectively. A11 those are very practica1 and meaningful to pupi1s. The last one, a survey of research, is quite open. Pupi1s do have a chance to discover some facts by themselves.Appling the concept of equa11y likely to the ordinary language -fairness and testing it by visua1 materia1 of bar charts is an excellent arrange-ment.
(vi) Exercise
(a) In SMP, the four projects in paragraph 2.2 (p.52) and the five problems in Exercise C (pp.90-92), are of above average difficu1ty. To do these problems, pupi1s need to apply some processes which are not described in that chapter, or are a result of the logica1 understanding of 也e concepts in that chapter. For example, the method of working out 也e proportion of left-handed people is not described in 血e content; with the bar chart to judge the fairness, pupi1s have to understand the concept of equally likely. In project 4 (p.52), pupi1s were asked to do a survey of research, this is an open回endedquestion and is good for discovery training.
The remaining 31 problems in these two chapters are either easy or average. The proportions of the two levels ïs about even.
(b) In SMG
,
there are 33 problems in Chapter 4. Exercises5 and 6 (pp. 188-190) are applications of the probabi1ity 由 eory in 也is chapter. 1/3 of them are easy, the other 2/3 are about ave:rage. In Exercise 4, there are 20 problems. Out of these 20 problems the proportion of easy,
average and hard levels is about 2,4, 1.Exercise 1. Tossing a coin
,
Exercise 2. Throwing a dice and Exercise 3. Two more experiments are 扭也e main prose. Each of 也em inc1udes severa1 problems,
and some problems require the teacher and pupi1s to workBulletin of National Taiwan Normal University No. 28
together. These exercises seIVe as exposition. As Shuard (1979) says
,
exposition is quite often placed within an exercise, making it essentia1 for a reader to work through the exposition with paper and pencil. Such exposition is not distinguished from other exercises, whose purpose is practice. Exercise 1, 2 and 3 in SMG are in this category.(c) In MT, we consider Exercises 4b (p.71) and 4c (p.76). Because of the high mathematical level of the normal distribution CUIVe, the problems which concern this concept are very hard. For example:
1) Use the normal distribution CUIVe to obtain an estimate of the theoretica1 probability that the following events happen:.
(i) when 9 coins are tossed, 6 heads and 3 tails appe缸;(ii) . . . .
2) In each case
,
say how many times the event can be expected to occur when the coins are tossed 100 times.3) In each case, perform an experiment to see how close the theoretica1 probability is to the experimental probatility after 100 throws. (p.76) This problem requires a very advanced concept of approximation. The other six problems in ExercÏse 4b and 4c are either easy or average. The proportìon of easy and average is roughly even.
2. Summary
(i) MT inc1ude some hard words which the readers may just skip when they read them.
As to the concept of equally likely, MT let it happen naturally. They did not plan to emphasize 泣，as we had examined in IV (i) (b).
The normal distrib~tion CUIVe and the examples for mbtivation favour on1y the readers who ha1e a good background of understanding of the 血eory
of probability.
They did not inc1ude enough problems in exercises.
This bòok, still a persona1 opinion, is suitable for more able pupils or for teachers as a reference book.
(ii) The word
“
random" appears many times in the problems of Exercise in SMG. This word is avoidable in this introductory level.SMG apply some everyday life examples which are familiar to the readers to motivate the concept of equa11y likely.
Exposition is designed as an exercise. Pupils can practice and learn the materia1. The problems in exercise are quite enough.
Texbooks as the Source 01 Leaming Mathematics
We distinguish three levels of problems in 血e exercise
,
easy average and hard,
according to the requirement of the level of understanding of doing them. A problem which is designed for pupi1s to practice and to be familiar with the formula may only require the level of instrumental understanding. For example, what is the area of a field 20 cms by 15 cms? This is an easy one.If we replace 15 cms by 15 yards in the above example, and ask pupi1s to answer. Skemp (1976) quotes that the reply was
“
300 square cms". The teacher asked“
why not 300 square yards?" Answer“
Because area is a1ways in square cen time tres'\Skemp (1976) explains 也前， to prevent errors like the above,pupi1s need another rule to help them understand that both dimensions must be in the same unit. The problems which require the level of relational understanding to do it are the average one.
To answer
“
what is the area of the shaded part in the following square ABCD?"A D
2
B C
pupi1s have to develop their logica1 understanding. Such a problem is a hard one.
In each exercise
,
how many problems of different levels should be offered depends on the range of ability of the readers. In principle, there should be as many as possible of all those three levels of questions so 由at pupi1s can practice whenever theÿ wish.Including some problems in exercise for discovery training is necessary.
Anthony (1973) confmned that in comparing the effectiveness of discovery training and didactic training, if the criterion is the ability to make further discoveries
,
then the discovery training should be more effec世ve than the didactic training. For example, some basic rules or facts of geometric figures can be presented in an exercise for discovery training.Polya (1965
,
pp. 104-105) suggests that“. . .
let the students discover by themselves as much as is feasible under the given circumstances. . . . let the students actively contribute to the formulation of the problem 也at 血ey have to solve afterwards. . . . before 也e students do a problem, let them guess theBulletin of National Taiwan Normal University No. 28
Most of 也e average ability level of pupils wi11 appreciate their leaming by practicing.
(iii) SMP use many paragraphs to emphasize the most 加portant cοncept of equa11y likely in the introduction of probabi1ity.
The sequence of topics in SMP is arranged spira11y. The examples for motivation are familiar to pupi1s too.
SMP cover .manyproblems of easy, average and hard levels. Different ability levels ofpupils can find different levels ofproblems to do in this book.
SMP is good for a very wide ability range of pupi1s to use. For instance, it wil1 be suitable for a mixed ability c1ass.
V. EPILOGUE
Who writes the textbook? This is another factor of the textbook.
School teacher, university mathematician or educator, who is the most suitable person to write a textbook?
In his artic1e
“
Textbooks and Teacher Education", Howson (1980) suggests that the textbook should be written by a team of school teachers and mathematica1 didacticians. He explains his view as follows:“
In some countries, textbooks are usually written by teachers, either individually or in groups: although 世s often guarantees a high degree of‘
teachability', it may mean that mathematica1 errors are present - for which the teacher must be on his guard - or that there might be insufficient mathematica1/
pedagogical structure built into the books. At the other extreme, a book written by a university mathematician may be accurate and possess an inherent and consistent mathematica1 structure, but may present the teacher with difficult didactical problems."Kline (1973, Chapter 10) has the 'same point of view as Howson except that Kline emphasizes that the university mathematicians should act as consultants. They can contribute their views on the syllabus of school mathematics. But they shöuld not dominate the curriculum of school ma thema tics.
MT are a production of some mathematicians. In the chapter we have examined, we found it very mathematica1. The important concepts ot
ma血ematics in the topic are mentioned, but aæ not iritroduced in an easier and more acceptable way.
The principa1 authors of SMP are school teachers. They accept the advice
Texbooks as the Source of Leaming Mathematics
on points of fundamenta1 mathematics given by some mathematicians (s閉， Acknowledgmen尬， Book A, p.ix, SMP). It is well known that $MP series of textbooks is well accepted by many schools in the U.K.
There is one fma1 thing to say. We believe 血at the teacher is a1ways the most important participant in a ma血ematics lesson. A poor teacher with a good textbook may teach poor1y
,
whereas a good teacher will overcome the deficiencies of any textbook and teach successfully.REFERENCES
1. Aiken, L. R. (1972). Language Factors in Learning Mathematics. Math. Ed. Reports, ER1C, Columbus, Ohio, October 1972.
2. A11endoerfer, C. B. (1962). ‘'The Narrow Mathematician." Am. Math.
Monthly. 69, June-July 1962, pp.461469.
3. Anthony, W. S. (1973).
“
Learning to Discover Ru1es by Discovery."Journal of Educational Psychology. 64,3,325-328.
4. Austin, J. L. and A. G. Howson, (1979).
“
Language and Mathematica1 Education." Educational Studies in Mathematics. 10 ， 2 ，正61-193.5. Ausubel, D. P. and F. G. Robinson (1969). School Learning: An
Introduction to Educational Psychology. New York: Holt
,
Rinehart and Winston. 6. Bishop, A. J. (1980). Classroom Dynamics. Paper prepa自d for 1ntemationa1 Project on Basic Components in the Education of Mathematics teachers. Rieste, F. R. G. Apri1.7. Bishop
,
A. J. (1977).“
1s a picture worth a 也ousand words?"Mathematics Teaching. 81, Dec. 32-35.
8. Brunner, R. B. (1976) Reading Ma也ematica1 Exposition. Educational
Research. 18,3, June, 208-213.
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“
Reading and Mathematics."Methematics Teacher. 59, 149-157.
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“
Reading and Content in Technica1-Vocational Education." Journal of Readtng. 20,
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14-17.11. Clements, M. A. (1980). “Ana1yzing Children's Errors on Written Mathematica1 Tasks." Educational Studies in Mathematics. 11, 1-21.
12. Cockcroft, W. H. (Chairman) (1982). Mathematics Counts. London: H.M. S.O.
13. D. E. S. (1980). Aspects of Secondary Education in England:
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mentary Information on Mathematics. LonJon: H. M. S. O.
14. Ellio仗， P. G. and C. A. Wiles (1980). ''The Print Is Part of the Prob1em." School Science and Mathematics. 80
,
3742.15. Forbes, J. E. (1970). Tex tbooks and Supplementary Materials. The Teaching of Secondary Schoo1 Mathematics
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Thirty-Third Yearbook of the N. C. T. M.,
Washington D. C. 89-109.16. Hall J. -C: and J. B. Thomas (1977).
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Mathematics Department Headship in Secondary Schoo1," JournaZ.of British Education Administration Society. 5,2.17. Hart, K. M. (ed.) (1981). Children 's Understanding of Mathematics:
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Ab血ty through Reading Instruction." Arithmetic Teacher.‘ 18,223-229.
19. He閃閃， J. (1975).
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Notation and Language in Schoo1 Mathematics."The Mathematics Gazette. 59
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A.-G. (1980). "Textbooks and Teacher Education" (draft). 21. Johnston,
P. (1981).“Making it Algebraic." Mathematics Teaching.97
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The Readabì1ity of Mathematica1 English."Joumal of Research in Science Teaching. 5,296-298.
23. Kline
,
M. (1973). Why Johnny Can 't Add - 1現e Failure of the New Math. New York: St. Martin's Press.24. Kulm, G. (1973).
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Sources of Reading Difficulty in Elementary Algebraic Textbooks~" Mathematics Teacher. 66, 7, Nov., 649-652.25. Mansfield, D. E. and D. Thompson (1963). Mathematics: A New Approach. Books 2 and 4. London: Chatto and Windus.
26. Polya, G. (1965). Mathematical Discovery. John Wiley and Sons,
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,
H. (1979). Language and Reading in Mathematics. Paper for Keele mee曲唱，Jan.29. Skemp
,
R. R. (.1976). "Relational Understanding and Instrumental Understanding." Ma them" ttcs Teachtng. 77,
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31. School Mathematics Project (1970). Books A, E and H. Cambridge: Cambridge University Press.
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,
A. E. (1979). ‘The Symbols and Grammatica1 Structures of Mathematical English and the Reading Comprehension of College Students."Journal for Research in Mathematics Education. 10,3,216-218.
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