Teaching of proof in the UK and US Table of Contents 1. Introduction ............................................................................................................................ 3
2. Literature Review................................................................................................................... 4
2.1 The concept of proof ........................................................................................................ 4
2.2 The validity of different proof schemes ........................................................................... 5
2.2.1 Externally based proof schemes .................................................................................... 4
2.2.2 Empirical proof schemes ............................................................................................... 6
2.2.3 Analytic proof schemes ................................................................................................. 5
2.3 Distinguishing between Empirical Arguments and Proof ................................................ 6
2.4 Children’s ability and development in constructing proofs ............................................. 8
2.5 The teacher’s role with proof ......................................................................................... 12
2.6 Teacher’s conception of proof........................................................................................ 13
2.7 A summary of what constitutes a proof from the reading of the literature .................... 14
2.8 An analytic framework of reasoning and proving .......................................................... 15
3 The United States’ and United Kingdom’s National Documents On Primary and Secondary
Mathematics ............................................................................................................................. 18
3.1 United States’ National Documents on Primary and Secondary Mathematics .............. 18
3.2 The United Kingdom’s National Documentation for Primary and Secondary
Mathematics ......................................................................................................................... 19
3.3 Comparing the two curricula using the framework ........................................................ 20
3.3.1 Identifying a pattern .................................................................................................... 20
3.3.2 Making a conjecture .................................................................................................... 23
3.3.3 Proof ............................................................................................................................ 24
4 Conclusion ............................................................................................................................ 27
References ................................................................................................................................ 30 Page 1 of 33 Page 2 of 33 1. Introduction According to several authors (Chazan, 1993; Healy & Hoyles, 2000; Sowder & Harel, 1998;
Stylianides, 2008b) there are many students of all levels of education that have difficulty with
the concept of proof. I therefore chose to focus on this aspect of the school mathematics
curriculum in the comparative analysis.
In this essay I will be comparing and contrasting the mathematics curricula of the United
Kingdom and the United States in the context of the teaching of proof. The rationale for
comparing both the USA and England is that they both have similar TIMSS scores in terms
of their mathematics achievement for the 4th Grade which were 541 and 542 respectively, and
for the 8th Grade mathematical achievement for the USA and UK were 509 and 507
respectively. Both countries scored above the TIMSS centre point of 500. In terms of their
TIMSS scores for reasoning, in the 4th Grade, USA scored 525 and England scored 531, and
in the 8th Grade the TIMSS scores for reasoning for USA and UK were 503 and 510
respectively (TIMSS) (Mullis, Martin, Foy and Arora, 2012).
I will be examining the structure of the national curricula regarding proof, what the national
documents say about proof, and the amount of content and priority given to it. The literature
review consists of the following:
•
the concept of proof as defined by various researchers;
•
discusses the validity of different proof schemes;
•
distinguishes between an empirical argument and a proof;
•
discusses children’s ability and development in constructing proofs;
•
the teacher’s role with proof;
Page 3 of 33 •
the teacher’s conception of proof;
•
a summary of what constitutes proof from the literature;
•
and finally discusses a framework for the comparison between the two countries
based on constructs in the previous discussion.
This essay attempts to answer the following questions in terms of analysing the teaching of
proof in the United Kingdom and the United States:
1) Does the approach to teaching in upper primary education as reflected in the national
documents support the expectations for the development in the learning of proof in lower
secondary school according to the framework of Stylianides (2008a)?
2) Is there a difference in the mathematical ideas children are expected to prove formally as
indicated by the national documents of the UK and the US?
2. Literature Review
2.1 The concept of proof Hersh (1993) asserts that a proof is a cogent argument which is judged by qualified
arbitrators. Hersh explains that there are three different meanings of proof. Firstly,
according to the root meaning in English the word ‘proof’ means to demonstrate the truth or
existence of something by evidence or argument, and proof originates from the Latin word
‘probare’ which means to test, approve, or demonstrate. Secondly, in mathematics the word
‘proof’ has a meaning in common practice: ‘an argument that convinces qualified judges’
(p.391).
According to Stylianides (2007b) in the context of students’ learning a ‘proof’ is a
mathematical argument that is composed of three components: the first component consists of
Page 4 of 33 ‘a set of accepted statements’, which are true and do not need any further justification. The
second component is known as ‘modes of argumentation’ which uses forms of reasoning
which are valid, which are known to, or within the conceptual ability of the individuals in the
classroom. The third component is known as ‘modes of argument representation’ which
consists of forms of expressions which are relevant and known to, or within the conceptual
level of the classroom individuals.
2.2 The validity of different proof schemes Harel and Sowder (1998) define a ‘proof scheme’ as
what convinces a person, and to what the person offers to convince others. ‘Proof
scheme’ as used here refers to justifications in general, so it should not be interpreted
narrowly in terms of mathematical proof in its conventional sense. One’s proof
scheme is idiosyncratic and may vary from field to field, and even within mathematics
itself (p.275).
Sowder and Harel (1998) suggest that proofs can be categorised into three categories:
‘externally based proof schemes’, ‘empirical proof schemes’, and ‘analytic proof schemes’.
Figure 1: Summary of different types of proof schemes. (Sowder & Harel, 1998, p.671,
Figure1)
2.2.1 Externally based proof schemes Externally based proof schemes are outside sources which convinces the students and what
the students would use to persuade others. Such schemes would include authoritarian proof
Page 5 of 33 schemes, such as a textbook, a teacher statement, or the reliance on a classmate to justify a
particular result. A second type of externally based proof scheme is a ‘ritual proof scheme’.
This involves a student who judges the correctness of an argument entirely based on the form
of the argument as opposed to the accuracy of reasoning involved (Martin & Harel, 1989;
Sowder & Harel, 1998). The third type of an ‘externally based proof scheme’ is called a
‘symbolic proof scheme’ where symbols or symbol manipulations have no meaningful basis
in the context (Harel & Sowder, 1998). Sowder and Harel (1989) state that one side of
symbolic proof schemes is bad, while the other side is good. The bad side involves students
treating the symbols as though they have a meaning independent to the relationship to the
quantities in the scenario in which they were used. The good side of the ‘symbolic proof
scheme’ is that students can see the power of symbols, particularly in algebra. Sowder and
Harel state that a knowledgeable person can ‘unpack’ correct symbolic reasoning.
2.2.2 Empirical proof schemes The second category of proof scheme, ‘empirical proof schemes’ are justifications that are
made solely on the basis of particular examples. Sowder and Harel state that there are two
types of empirical proof schemes: the ‘perceptual proof scheme’ and the ‘inductive proof
scheme’. The ‘perceptual proof scheme’ involves students arriving at a conclusion based on
their perceptions of a single, and sometimes many drawings. The ‘inductive proof scheme’
(Harel & Sowder, 1998) also called ‘examples-based proof scheme’(Sowder & Harel, 1998)
is when students convince themselves and others by evaluating a conjecture through one or
more examples .
2.2.3 Analytic proof schemes Finally, the ‘analytic proof schemes’ involve mathematical proof, but the emphasis is on
students' thinking rather than formal written work produced by students. There are two types
Page 6 of 33 of ‘analytic proof schemes’ the ‘transformational proof schemes’ and the ‘axiomatic proof
schemes’. The ‘transformational proof schemes’ “involves the creation and transformations
of general mental images for a context, with the transformation directed towards explanation,
always with an element of deduction” (Harel & Sowder, 1998,p.276). Sowder and Harel
(1998) give an example of predicting the number of edges in a n-gonal prism. They suggest
that one may investigate triangular prisms, and quadrilateral prisms, etc, and arrive at a
conjecture that the number of edges is 3n edges since each of the two bases of the prisms
have n edges plus the n lateral edges. Sowder and Harel suggest that finding a pattern is
good, but seeing why a generalisation holds is even better. They use the example of the ngonal prism as a ‘transformational proof scheme’ so long as the student can perceive the
underlying structure behind the pattern. They further emphasise that “the argument is general
and involves reasoning rather than counting and pattern finding of an examples-based proof
scheme” (p.673).
Harel and Sowder (1998) suggest that an individual posseses an ‘axiomatic proof scheme’
when an individual understands that mathematical justification have originated from
undefined terms and axioms which are facts, or accepted statements without proof. Sowder
and Harel (1998) give an example from Euclidean geometry “with ‘point’ and a ‘line’ being
undefined and statements like ‘two points determine a unique line’ being assumed. A student
who is comfortable working with such a system is showing an axiomatic proof scheme”
(p.674).
2.3 Distinguishing between Empirical Arguments and Proof According to Stylianides and Stylianides (2009) an empirical argument is disparate from a
proof and the two modes of argumentation cannot be considered to be the same. Stylianides
and Stylianides argue that empirical arguments provide inconclusive evidence by confirming
Page 7 of 33 the verity only for a subset of all possible cases covered by the generalisation, in contrast
proofs provide conclusive evidence for its truth by considering all cases covered by the
generalisation, and proofs offer secure methods for validity of mathematical generalisations.
Several researchers (Balacheff, 1988; Chazan, 1993; Fischbein, 1982; Harel & Sowder, 1998;
Healy & Hoyles, 2000; Martin & Harel, 1989; Porteous, 1994) have found that many students
have difficulties in constructing, following, and writing a formal proof; these students also
lack understanding of how formal proof differs from empirical evidence, and in
implementing proofs to subsequently derive further results. Other researchers (Coe &
Ruthven, 1994; Healy & Hoyles, 2000; Sowder & Harel, 2003; Stylianides & Stylianides,
2009) have found that students both in secondary school and university are more inclined to
use or accept empirical arguments as proofs of mathematical generalisations.
2.4 Children’s ability and development in constructing proofs According to Piaget children of the ages eleven to thirteen are able to cope with some formal
operations, but they are not able to write exhaustive proofs (Inhelder & Piaget, 1958; Piaget,
1928). Children aged eleven to twelve and beyond, are capable of what Piaget describes as
‘hypothetico-deductive reasoning’ which means that a child is capable of deductive reasoning
based on any assumption (Stylianides & Stylianides, 2008). However, Piaget’s claim that
learning proofs is contingent on the child’s age has been questioned as many authors have
found evidence to disagree with this claim. Hazlitt (1930) reported that there was no
relationship between age and logical reasoning due to a child’s lack of experience of
advanced knowledge. Others researchers (Ball & Bass, 2003; Maher & Martino, 1996; Reid,
2002; Stylianides, 2007b; Stylianides & Stylianides, 2008; Zack, 1997) have argued that
there is evidence in mathematics education research that suggests that even primary school
children who are placed in supportive classroom environments are able to use deductive
Page 8 of 33 reasoning to construct arguments and proofs. Stylianides (2007a) suggests that research into
the learning of proof might provide a conceptualisation of proof in primary school which is
appropriate for young pupils and which can be connected to how proof is understood in the
secondary school. This will be a good foundation on which teachers of mathematics can
develop a consistent and cohesive program for the teaching of proof across all grades.
According to Ball, Hoyles, Jahnke, and Movshovitz-Hadar (2002) most mathematics
teaching in primary school consists of too much emphasis on arithmetic concepts,
calculations, and algorithms. According to Ball et al (2002) when primary school pupils
enter secondary school they are suddenly expected to understand and write proofs
predominantly in geometry.
For example below (Figure 2) is an example from a UK Year 7 textbook where student have
perform a geometric proof:
Figure 2: Geometric proof (New Maths Frameworking Year 7 Pupil Book 3, p.35)
These researchers found substantial empirical evidence that this curricular pattern is prevalent
in many countries (Ball et al., 2002). Ball et al. (2002) examined how mathematics reasoning
develops in a class of third graders (equivalent to year four in the UK) by scrutinising the
pupils’ notebooks, tests, and interviews with the pupils. They evaluated pupils’ growth in the
skills and dispositions towards mathematical reasoning. An interesting case was when these
third grade pupils had to derive a proof for the conjecture that an odd number plus an odd
Page 9 of 33 number equals an even number. Some of the pupils had long lists of examples such as 3 + 5
= 9, 9+7 = 16, etc., and these examples were sufficient for certain pupils that they had found
a proof. However, a day later the class derived a proof; the pupils used tally sticks to
represent an odd number as a number than can be grouped into two pairs with one left over.
The pupils showed that if you add two odd numbers the two remaining tally sticks would
form a new pair of tally sticks (group of two tally sticks) resulting in an even sum.
Figure 3: odd number + odd number = even number
(Ball et al., 2002, p.910, Figure 1)
Ball et al. say that the pupils were able to share definitions of odd and even numbers enabling
them to establish a logical argument based on number structures. Furthermore, the pupils
were convinced that the proof would work for an infinite set of cases. Ball et al. highlight
that in this particular instance the pupils had developed the ability to construct, examine, and
consider arguments using their prior mathematical concepts. According to Bass (2009) a
‘generic proof’ represents an argument which is formed to support a particular claim and uses
one’s intuition while following the reasoning. The argument seems to be generic rather than
individualistic of a particular example. Bass further elaborates by stating that generic proofs
are prevalent in elementary classrooms where the jargon and notation which are required to
facilitate the precise general description of the ideas to be utilised are still not fully
developed. Bass (2009) refers to ‘Bernadette’s generic proof’ which was similar to the proof
Page 10 of 33 that the pupils devised discussed in Ball et al. (2002). Bernadette adds seven tally sticks with
another seven tally sticks to end up with fourteen tally sticks. However, Bass emphasises that
Bernadette turned to the class and did not look or point at the board. Bernadette’s exposition
was completely generic and she did not refer to the numbers seven and fourteen. In her
explanation she stated that if you add two odd numbers together you can add the ones
remaining on their own and it would always result in an even number. Bass argues that
Bernadette is articulating generically about any pair of odd numbers and not specifically
about the seven tally sticks on the board. Bass (2009) advocates that despite Bernadette not
using any algebraic notation, or formal definitions of odd and even numbers. She was able to
devise generic formulation of her mathematical thinking and was able to construct what one
can see as a generic proof of the conjecture that an odd number plus an odd numbers is an
even number.
So contrary to Piaget’s assertion that only pupils over twelve years of age are capable of
deductive reasoning Bass (2009) and Ball et al. (2002) have documented examples which
question Piaget’s claim. Moore (1994) suggests that in the U.S students only meet proof in a
one year course in geometry in secondary school. Moore further states that even in
undergraduate courses students primarily do calculus problems which rarely involves any
formal proofs and the majority of formal proof have been omitted from the lower level
university mathematics curriculum. Moore highlights that at several other universities
students are expected to write proofs in real analysis, abstract algebra, and other advanced
courses without being given any guidance or instruction in how to write formal proofs.
Moore has clearly shown that to be able to write formal mathematical proof is not age related
as students at university still have difficulty in constructing proofs since students are not
exposed to proof until when they are at university. This also contradicts Piaget’s claim that
to be able to write proof it is contingent on a student’s age.
Page 11 of 33 2.5 The teacher’s role with proof According to Hanna (1995) part of the problem in students’ learning of proof is that
constructivist teaching approaches has meant that pupils are left to their own devices in
understanding the logical nature of proof.
Many researchers (Blum & Kirsch, 1991; Hanna, 1995; Leron, 1983; Moore, 1994;
Movshovitz-Hadar, 1988) advocate that it is important to have a teacher that instructs pupils
to reason logically and to introduce the students to proof. These researchers posit that the
teacher’s role is paramount in facilitating students to identify the structure of proof, to present
arguments, and to know the difference between correct and incorrect arguments. Hanna
(1995) says that constructivism has a deleterious effect on the teaching of proof since it
disregards the importance of the teacher in the classroom.
Other researchers have emphasised the importance of teachers’ knowledge of proof (Martin
& Harel, 1989; Simon & Blume, 1996; Stylianides & Ball, 2008). Martin and Harel (1989)
state that the teacher’s perspective on proof is crucial. Since in primary education limited
emphasis is placed in the curriculum, the teacher will be the main facilitator of children’s
experience of proof and verification in the classroom. Knuth (2002b) argues that teachers’
knowledge, subject knowledge, and beliefs has significant impact on teachers’ classroom
teaching and how they implement the requirements of the NCTM Principles and Standards of
school mathematics.
Martin and Harel further state that teachers’ understanding of what comprises of
mathematical proof is essential even if they do not teach proof. Martin and Harel argue that
if teachers in elementary school mislead their students to believe that a few examples will be
sufficient for a proof. Consequently, when pupils are in the secondary school the pupils will
inevitably find proof difficult since they have not been taught correctly what a proof looks
Page 12 of 33 like. Since teachers have to follow the curriculum documents when planning and teaching
lessons their conception of proof and how it will be strongly influenced by the curricula of
their respective countries.
2.6 Teacher’s conception of proof There has been a significant demand on school mathematics teachers to enhance the role of
proof in the classroom and this requires a great deal of effort on their part (Chazen, 1990;
Jones, 1997; Knuth, 2002a). In the United States, there were substantial changes made in
school mathematics curricula and teachers’ teaching practices regarding proof. According to
the Principles and Standards for School Mathematics (NCTM, 2000) proof has a prominent
emphasis throughout the entire school mathematics curriculum. It is expected to be taught to
all students throughout the compulsory years of the pupils’ mathematics education (Knuth,
2002b). The situation for mathematics teachers has been exacerbated since mathematics
teacher education and professional development programs have not sufficiently prepared
teachers to fulfil the requirements of teaching proof successfully (Knuth, 2002b).
Knuth (2002b) interviewed some secondary school mathematics teachers about what
constitutes proof in school mathematics and their conceptions of proof. Knuth (2002b)
reported that several teachers in secondary school whom he had interviewed thought that
proof did not have an essential part in the school mathematics curriculum. Other teachers
believed that proof should only be taught to pupils in advanced mathematics classes or those
students who will pursue a mathematics-related undergraduate degree. Knuth states teaching
informal proofs to students is important to many. One teacher in particular stated teaching
lower ability classes that empirical argument would be suffice as a proof: “When they say I
noticed this pattern and I tested it out for quite a few cases; you tell them good job. For them,
that’s a proof. You don’t bother them with these general cases” (p.76).
Page 13 of 33 Most teachers asserted that a “proof is a logical or deductive argument which demonstrates
the truth of a premise” (p.71). Knuth says that most teachers could distinguish between
formal and informal proofs and Knuth further states that the teachers’ descriptions of proof
were based on conformity of following certain conventions in writing proofs. Many teachers
emphasised that proofs had to be written in the two-column format (i.e. when proof
statements are written in one column and the justifications are written in the second column)
and considered this to be the essential component of writing a proof. Many of these teachers
considered that explanations and empirical arguments as informal proofs, hence, verifying
that these did not constitute formal proofs. These teachers might be said to have an
‘algorithmic’ understanding of proof in the school context.
Harel and Sowder (1998) found that 80% of pre-service teachers considered ‘examples-based
proof schemes’ to be proofs. Harel and Sowder argue that at elementary and secondary level
mathematical instruction is predominantly examples-based. They argue “that instructional
activities that educate students to reason about situations in terms of transformational proof
schemes are crucial to students’ mathematical development and they must begin in an early
age” (p.280).
2.7 A summary of what constitutes a proof from the reading of the literature The literature suggests that a proof maybe constructed by students in a classroom and is a
mathematically based argument or a set of assertions which support or reject a mathematical
claim. The statements it uses are ones which are accepted by members of the classroom, or
any group of people (mathematicians) considering the proof. It also involves forms of reason
known to and accepted by members of the classroom, and expressed in a written format
familiar to the classroom members as defined by Stylianides (2007b). I suggest that it is not
Page 14 of 33 necessary for proof constructed by students to be written in an algebraic format and to follow
rigid mathematical conventions of notation. In the case of the third grade student who proved
that an odd number plus an odd number equals an even number, this proof is valid as a proof
as it is not a specific example (‘empirical argument’) as she demonstrated its generalisability
across all odd numbers. Such a proof is what would be defined as a ‘transformational proof
scheme’ (Sowder & Harel, 1998) rather than one based on one or several examples ‘empirical
argument’ (Stylianides, 2007b). The literature on proof has led me to the conclusion that
proof in school mathematics either falls under the category of ‘empirical arguments’ or
‘transformational proof schemes’. The literature also suggests that proof should be taught
developmentally i.e. we should not go straight to formal proof, but should start with informal
proof in primary school in beginning with inductive methods of calculations before learning
formal algorithms.
2.8 An analytic framework of reasoning and proving Stylianides (2008a) suggests that new knowledge develops and passes through different
stages and that the development of proof is usually the final encounter. Stylianides
emphasises a number of activities in the development towards proof:
Stylianides describes four aspects of the mathematical component of a proof:
• identifying a pattern (two types: ‘definite pattern’ and ‘plausible pattern’);
• making a conjecture;
• providing a non-proof argument (two types: ‘empirical arguments’ and ‘rationales’).
•
providing a proof (two types: ‘generic examples’ and ‘demonstrations’);
Page 15 of 33 Figure 4: The analytic framework of reasoning and proving (Stylianides, 2008a, p.10,
Figure1)
According to Stylianides (2008a) ‘identifying a pattern’ is a major challenge in mathematics
as students need to develop their abilities to make generalisations on the basis of
mathematical structures as opposed to the evidence offered by a few examples (empirical
generalisations). There are two categories of ‘identifying a pattern’ which are ‘definite
patterns’ and ‘plausible patterns’. A ‘definite pattern’ is a pattern in which the student can
provide conclusive evidence for the selection of the pattern this could be the nth term. In the
example of the hexagon train, the perimeter of train n would be 4n + 2 (see figure 5 below).
In this case we can say how each train pattern is determined for example: the first train has
one hexagon and each subsequent train will have one more hexagon added on. If there was
not structure, hence, if the student could not determine whether train 6 had 6 hexagons this
would be considered to be a ‘plausible pattern’.
Figure 5: The hexagon train problem (Stylianides, 2008a, p.10, Figure 2)
Page 16 of 33 A ‘conjecture’ is a reasoned hypothesis about a general mathematical relationship which is
based on incomplete evidence. Stylianides (2008a) defines ‘providing a proof’ as “a valid
argument based on accepted truths for or against a mathematical claim” (p.11). There are
two categories of ‘proof’ which are ‘generic examples’ and a ‘demonstration’. “A ‘generic
example’ is a proof that uses a particular case seen as representative of the general case”
(p.11) synonymous with Sowder and Harel (1998)’s ‘transformational proof scheme’ and “a
‘demonstration’ is a proof that does not rely on the ‘representativeness’ of a particular case”
(p.11-12) synonymous with Sowder and Harel (1998)’s ‘axiomatic proof scheme’.
‘Providing a non-proof argument’ involves “an argument for or against a mathematical claim
that does not qualify as a proof” (p.12). There are two types of ‘non-proof arguments’ which
are ‘empirical arguments’ and ‘rationales’. An ‘empirical argument’ is an argument which
uses incomplete evidence for a certain mathematical claim in the sense that the student will
check only a subset of all possible cases of the claim. ‘Rationales’ are neither proofs nor
empirical arguments. Stylianides (2008a) defines a ‘rationale’ as an argument that “does not
make explicit reference to some key accepted truths that it uses, or if it uses statements that
do not belong to a set of accepted truths of a particular community” (p.12).
I will be using this framework of comparing the mathematical components of reasoning and
proof when comparing the national documents of the UK and the US. I will go through the
curricula of the United Kingdom and the United States and classify the examples used in the
documents as one of the following: ‘identifying a pattern’ specifying whether they are
‘definite patterns’ or ‘plausible patterns’, ‘making a conjecture’. ‘providing a proof’
specifying whether they are ‘generic examples’ or ‘demonstrations’, or ‘providing a nonproof argument’ specifying whether they are ‘empirical arguments’ or ‘rationales’ when
comparing the two countries’ curricula.
Page 17 of 33 3 The United States’ and United Kingdom’s National Documents On Primary and Secondary Mathematics 3.1 United States’ National Documents on Primary and Secondary Mathematics The United States’ national document outlining the mathematics curriculum and
recommended teaching approaches is called the Principals and Standards for school
mathematics. The Principles and Standards is based on six core principles which are:
‘equity’, ‘curriculum’, ‘teaching’, ‘learning’, ‘assessment’, and ‘technology’. The NCTM
(2000) delineates these core principles as the following:
•
Equity. Excellence in mathematics education requires equity—high
expectations and strong support for all students.
•
Curriculum. A curriculum is more than a collection of activities: it must be
coherent, focused on important mathematics, and well articulated across the
grades.
•
Teaching. Effective mathematics teaching requires understanding what
students know and need to learn and then challenging and supporting them to
learn it well.
•
Learning. Students must learn mathematics with understanding, actively
building new knowledge from experience and prior knowledge.
•
Assessment. Assessment should support the learning of important
mathematics and furnish useful information to both teachers and students.
•
Technology. Technology is essential in teaching and learning mathematics; it
influences the mathematics that is taught and enhances students’ learning
(p.11).
The NCTM (2000) outlines standards for school mathematics for each grade from prekindergarten through to Grade 12 for the following areas of mathematics: Number &
Operations, Algebra, Geometry, Measurement, Data Analysis & Probability, Problem
Solving, Reasoning & Proof, Communication, Connections, and Representation.
Page 18 of 33 Having briefly described the United States’ national documentation I will now look at the
United Kingdom’s documentation.
3.2 The United Kingdom’s National Documentation for Primary and Secondary Mathematics In the United Kingdom there is a National Curriculum. There are four Key Stages, in
primary school (children aged 5 to 11) follow Key Stages 1 and 2. In secondary school
(children aged 11 to 16) follow Key Stages 3 and 4. The Qualifications and Curriculum
Authority (2007) state that Key Stage 1 and 2 is arranged into 3 attainment targets which are:
•
Ma2 Number and algebra
•
Ma3 Shape, space and measures
•
Ma4 Handling data
The Qualifications and Curriculum Authority (2007) state that at Key Stage 4 there are four
attainment targets namely:
•
Mathematical processes and applications
•
Number and algebra
•
Geometry and measures
•
Handling data
At Key Stage 2 students do not formally meet proof, but explore a variety conjectures and
problems where they have to justify their reasoning. However, at Key Stages 3 and 4
students meet a variety of formal proof.
Page 19 of 33 In addition the UK has guidance from the DfEE (1999) and DfEE (2001) to help teachers
decide the type of mathematical problems that students should encounter during each year in
their respective Key Stage. In contrast, the NCTM (2000) is just one document that is used
throughout the United States for the different areas of mathematics for each respective grade.
3.3 Comparing the two curricula using the framework 3.3.1 Identifying a pattern In relation to ‘identifying a pattern’, in primary education I will compare US standards for
grades 3-5 with the UK ‘learning outcome’ for year 4 (grade 3)
United States (Grades 3 to 5) Standard:
•
“Develop and evaluate mathematical arguments and proofs”
•
“Select and use various types of reasoning and methods of proof” (NCTM, 2000,
p.188)
Example
Figure 6: The doubling pattern (NCTM, 2000, p.189)
The NCTM (2000) suggests that even though the students’ explanations are constrained to
particular examples. Some students are actually developing arguments which could lead to
more generalised conclusions. The NCTM argues that: “although none of these third graders’
arguments is stated in a way that is complete or general, they are beginning to see what it
Page 20 of 33 means to develop and test conjectures about mathematical relationships.” (NCTM, 2000,
p.189).
United Kingdom (Year 4) Outcome:
“Recognise and extend number sequences formed by counting on and back in steps of any
size, extending beyond zero when counting back” (DfEE, 1999, p.16).
Example
Figure 7: Identifying the next terms of a sequence (DfEE, 1999, p.16)
The Qualifications and Curriculum Authority (2007) states that students are expected to
“search for a pattern in their results; develop logical thinking and explain their reasoning.”
(p.7)
In relation to ‘identifying a pattern’, in early secondary education I will compare the US
standards for grades 6-8 with the UK ‘learning outcome’ for Year 7 (grade 6).
United States (Grades 6 to 8) Standard:
“Students should have frequent and diverse experiences with mathematical reasoning as
they:
•
Examine patterns and structures to detect regularities;
•
Construct and evaluate mathematical arguments.” (NCTM, 2000, p.262)
Page 21 of 33 Example
Figure 8: Line Segments (NCTM, 2000, p.266-267, Figures 6.34 and 6.35)
The standards for Grades 6 – 8 have evidence of ‘identifying a pattern’ (in the context of
making mathematical generalisations) in terms of the line segment example (see Figure 8
above) since pupils are providing conclusive evidence of how to select a pattern i.e. providing
a formula for the nth term for the line segments. Since the nth term of the line segment does
not use any explicit reference to some key established truths within mathematics this would
be considered to be a ‘rationale’ (in the context of proof) since the students have not
developed a proof which is generalisable.
United Kingdom (Year 7) Outcome:
“Suggest extensions to problems, conjecture and generalise; identify exceptional cases or
counter-examples.” (DfEE, 2001, p.32)
Example
Figure 9: Matchstick shapes (DfEE, 2001, p.32)
Page 22 of 33 The matchsticks shapes problem is an example of ‘identifying a pattern’ which is a ‘definite
pattern’ since the students can provide conclusive evidence for the selection of any pattern
i.e. the nth term of the sequence. The example also illustrates what the Qualifications and
Curriculum Authority (2007) suggest that students should be able to “form convincing
arguments based on findings and make general statements.” (p.144)
3.3.2 Making a conjecture In relation to ‘making a conjecture’, in primary education I will compare the US standards for
grades 3-5 with the UK ‘learning outcome’ for year 6 (grade 7).
United States (Grades 3 to 5) Standard:
•
Make and investigate mathematical conjectures
•
Develop and evaluate mathematical arguments and proof (NCTM, 2000, p.188)
The NCTM (2000) suggests that the students in grades 3 to 5 should be able to:
reason about the relationships that apply to the numbers, shapes, or operations they
are studying. They need to define the relationships, analyze why it is true, and
determine to what group of mathematical objects (numbers, shapes, and operations) it
can be applied. (p.188 - 189)
Example
Figure 10: Fractions Conjecture (NCTM, 2000, p.191-192)
Page 23 of 33 In the case of ordering fractions, which is aimed at Grades 3 to 5, (see Figure 10 above) there
is some evidence of ‘providing a conjecture’ when Patrize raises the point that: “if you make
the top number, the numerator, higher than a half of the denominator, but you don’t make it
the same as the denominator like 5/5 ‘cause then it will be a whole” (NCTM, 2000, p.191).
United Kingdom (Year 6) Outcome:
“Recognise odd and even numbers and make general statements about them” (DfEE, 1999,
p.18)
Example
Figure 11: Conjectures about odd and even numbers (DfEE, 1999, p.19)
This would come under what Stylianides (2008a) describes as ‘making a conjecture’ since
students are using incomplete evidence about a general mathematical relation.
In relation to ‘making a conjecture’, in early secondary education I will compare the US
standards for grades 6-8 with the UK ‘learning outcome’ for year 7 (grade 6).
United States (Grades 6 to 8) Standard:
•
“Formulate generalisations and conjectures about observed regularities”
•
“evaluate conjectures” (NCTM, 2000, p.262).
The NCTM (2000) emphasises that “students should discuss their reasoning on a regular
basis with the teacher and with one another, explaining the basis for their conjectures and the
rationale for their mathematical assertions” (p.262).
Page 24 of 33 Example
Figure 12: Gauss’s method (NCTM, 2000, p.264, Figure 6.33)
According to the NCTM (2000) this example provides “formulating a plausible conjecture,
testing the conjecture, and displaying the associated reasoning for evaluation by others”
(p.264).
United Kingdom (Year 7) Objective:
“Present and interpret solutions, explaining and justifying methods, inferences and reasoning”
(DfEE, 2001, p.30)
Example
Figure 13: Square totals (DfEE, 2001, p.30)
This is an example of ‘making a conjecture’ since students are using incomplete evidence to
investigate a general mathematical relation.
3.3.3 Proof In relation to ‘proof’, in secondary education I will compare the US standards for grades 9-12
with the UK ‘learning outcome’ for year 9 (grade 8).
United States (Grades 9 to 12) Standard:
•
“Develop and evaluate mathematical arguments and proofs”
Page 25 of 33 •
“Select and use various types of reasoning and methods of proof” (NCTM, 2000,
p.342)
Example
Figure 14: Pythagoras’ Theorem Proof explained (NCTM, 2000, p.301, Figure 7.7)
This was the only ‘proof’ i.e. a ‘generic proof’ that I could find in the NCTM (2000). This
proof is an algebraic explanation to facilitate students with the visualisation of the proof. The
problem is not directly asking the students to proof Pythagoras’ Theorem by themselves.
United Kingdom (Year 9) Objective:
•
“Combining understanding, experiences, imagination and reasoning to construct
knowledge”
•
“Posing questions and developing convincing arguments” (Qualifications and
Curriculum Authority, 2007, p.140)
Examples
At Key Stage Three (lower secondary school Years 7 to 9) there are many examples where it
is specifically written that students have to ‘prove’ particular mathematical generalisations
shown below in Figures 15 to 20 all which are ‘generic proofs’:
Page 26 of 33 Figure 15: Diagonals of a polygon (DfEE, 2001, p.35)
Figure 16: The Garden Path Proof (DfEE, 2001, p.35)
Figure 17: Some algebraic proofs (DfEE, 2001, p.35)
Figure 18: Proof of expanding brackets (DfEE, 2001, p.119)
Figure 19: Proof of the product of two odd numbers (DfEE, 2001, p.121)
Page 27 of 33 Figure 20: Proof of Pythagoras’ Theorem (DfEE, 2001, p.121)
There is clear evidence from the NCTM (2000) that the upper elementary (Grades 3 to 5)
builds the foundations of mathematical generalisations namely identifying patterns, making
conjectures, and providing non-proof arguments (rationales) as highlighted above. There
does not seem to be evidence of what type of proofs that students should be competent to do
or examine during Grades 3 to 5, and Grades 6 to 8. The NCTM (2000) does not specifically
ask students to perform any formal proofs.
Even though the examples of reasoning and proof at upper primary in the UK does not have
any proof in it. However, it provides the basic skills of making generalisations, finding nth
terms and evaluating conjectures which provide the essential skills at key stage three for
proof.
4 Conclusion Documents from both the United Kingdom and the United States suggest that pupils will be
provided with the skills of making mathematical generalisations and making conjectures at
the upper primary level where, for example, students can find the next pattern in a sequence.
The upper primary national documentation from both countries did not have evidence that
pupils are expected to provide formal proofs. However, the United Kingdom’s national
Page 28 of 33 documentation did provide more examples of mathematical generalisations such as working
out the nth term of arithmetic sequences, and writing algebraic formulae and equations of
various scenarios in comparison to just two examples mention in the NCTM (2000). Both the
examples in the national documentation of the two countries came under what Stylianides
(2008a) describes as ‘rationales’ or ‘non-proof arguments’. The United Kingdom’s national
documentation had more vivid examples of how to provide the rudimentary skills for
encountering proof in lower secondary school in comparison to the United States’ national
documentation.
The United Kingdom’s national documentation for lower secondary school had clear
examples of formal proof that students are expected to be able to perform for example: the
proof of Pythagoras’ Theorem, proof of divisibility, the proof of expanding two brackets via
geometric representation, as well as the proof of the product of two odd numbers in
comparison to the United States’ national documentation which had no evidence of asking
students to perform formal proof at lower secondary level, but only contains examples of
‘non-proof arguments’, ‘rationales’, and an example of a proof of Pythagoras’ Theorem (see
Figure 14).
The comparisons I have made above are based on national documents which may or may not
be being presently used by teachers when preparing their work in classrooms. In the UK, the
national curriculum is in a state of flux and the document referred to is no longer statutory.
Therefore, below I have included examples of proof from current mathematics textbooks
from the United Kingdom and the United States. The UK textbook clearly expects students
to perform a proof where-as the US textbook only asks students for a pattern for the diagonals
and not a proof.
Page 29 of 33 Figure 21: Proof from UK GCSE Linear Higher Student Book (Pledger, K, 2010, p.143)
Figure 22: Example of reasoning from US Secondary Textbook (Davis, 2009, p.234)
These examples appear to support the conclusion made in reference to the national
documentation. Since the UK’s national documentation has vivid examples of ‘proof’ for
different problems in contrast to the United States’ NCTM (2000) examples merely being
‘identifying patterns’ or ‘non-proof arguments’ could explain why the UK TIMSS score for
reasoning was higher for both fourth grade and eighth grade in comparison to the United
States TIMSS reasoning scores.
One way to expand this study would be to compare UK teaching of reasoning and proof, as
suggested by national documentation, with countries which gained high TIMSS scores for
reasoning. For example we might look at Hong Kong (TIMSS Reasoning score for 4th and
8th grade are: 589 and 580) or Singapore (TIMSS Reasoning score for 4th and 8th grade are:
Page 30 of 33 588 and 604). Hong Kong and Singapore have consistently been ranked at the top of
international secondary mathematics comparison articles. Comparing Singapore’s national
documentation would clearly provide some insight to how we could improve the content and
the teaching and learning of reasoning and proof in the United Kingdom.
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