Has strategy become the new algorithm?
Dianne Scouller M A (Hons); M Ed (Hons).
The Numeracy Development Projects are now well established in our schools. Published
reports record measurable successes, but is all really well in mathematics teaching and
learning in New Zealand schools? Are the successes at the level predicted or of the quality desired? This paper addresses some areas of concern about the balance between the knowledge and strategy strands of NDP, along with other related issues.
There is no doubt that the introduction of the Numeracy Development Projects has had a significant impact on teaching and learning mathematics in New Zealand schools. In the early years of the pilot programme there were promising improvements in achievement and understanding levels for both children and teachers, largely through the involvement of designated facilitators. Has the successful growth of the NDP been its own downfall in a sense?
For many decades mathematics educators have struggled to find ways to help children develop the ability to think mathematically, to make sense of numbers, relationships and symbols, in fact to develop mathematical and statistical literacy. New Zealand curriculum documents have recognised this need, for example Mathematics in the New Zealand curriculum (1992, p.11)was very explicit in this regard stating the goals as being to provide the opportunity for learners to “… develop the ability to think mathematically”. George Booker’s (1999) article Thinking mathematically: problem solving, sense-making, and communicating addressed this question in a way which recognised the complexities involved for both teacher and learner. More recently Neill (2008) addressed the same issues and in doing so raised some other fairly common concerns. He argued for a recognition of the need for the learning, indeed memorisation of basic facts accompanied by plenty practice, rather than a simplistic reliance on strategies. There is no denying the value of strategies, since the memorising of decontextualised facts is barren, to say the least, while at the same time strategies are useless in isolation from an acquired bank of knowledge to which they apply.
Here, in theory at least, the Number Framework has it right, with both knowledge and strategy presented as parallel paths. In the first Numeracy Professional Development Projects booklet it is stated: “It is important that students make progress in both sections of the Framework” (Ministry of Education, 2008, p.1). This paper asks the question as to how effectively that partnership of strategy and knowledge is outworked in the reality of our classrooms, while simultaneously raising some other associated issues around mathematics in our primary schools.
At this point a disclaimer is offered. Much of the concern expressed in this paper has arisen from observations of student teachers and their associate teachers during mathematics lessons, along with conversations with several young teachers managing their own classrooms. Should an in-depth, longitudinal research project be undertaken to investigate the issues raised, the results may be different.
One feature of mathematics teaching and learning in the past was the reliance on an algorithmic approach to arithmetic. The four operations were neatly boxed into procedures guaranteed to lead the student to the right answer if every step was followed accurately. Furthermore, examples to be worked through were often at a computational level far beyond that which was appropriate. Anyone born since the introduction of decimal currency in 1967 and general metric measurements from 1975 can count themselves blessed not to have been confronted with working out problems such as these:
£3-14s-7½ or 51lb-11oz ÷16 or + 5
Nonetheless, for many years the formality of this algorithmic approach predominated, leaving behind those who could not understand what was being done or how and why it ‘worked’. Those who succeeded were largely learners with good memories or those who were able to work out for themselves why they worked.
With the establishment of NDP in New Zealand any attempt to revert to such methods is scorned and loudly pilloried. A very disturbing instance arose early in 2009 where a young teacher was severely disciplined by his mentors for daring to write a problem on the board in the form
The intensity and dismissive nature of the rebuke was alarming.
It can not be denied that exclusive use of these traditional algorithms excluded large numbers of children from success and enjoyment of mathematics, so is not a desirable approach on its own. But was that due exclusively to the reliance on algorithms, or can responsibility also be laid at the feet of traditional methods of pedagogy and the prevailing attitudes to mathematics itself? There is no doubt that some mathematicians firmly reject any idea of an algorithmic approach. Holton (2002, p.23) declared categorically that the philosophical underpinnings of the New Zealand mathematics curriculum are in clear opposition to such an approach:
Right at the start we need to say that we do not see mathematics as a set of algorithms that
have to be learned by heart nor teaching as a process of transmitting knowledge.
His understanding is clearly that the formality of an algorithmic approach prevents children from enjoying mathematics or learning genuine life skills, so the emphasis must be on processes and problem solving.
Let’s look for a moment at this word “algorithm”. The Collins Concise Dictionary defines the word thus: a logical arithmetical or computational procedure which if correctly applied ensures the solution of the problem. If a learner is taught to resolve the problem by saying, e.g. 42 +35 40 + 30 =70 2 + 5 = 7 42 + 35 = 77
what is that but an algorithm? Certainly different from the first illustration, and probably considerably more helpful to the learner, but still an algorithm. Or if a problem is resolved by using doubling and halving, is this not an algorithm too? The answer must be ‘yes’ if the above definition is accurate. It is evident that in some of our classrooms the NDP strategies are being taught in the same rigid way as the rejected methods of the past, rather than being presented as a range of possible procedures designed to enrich understanding of knowledge and help build connections.
Despite the ‘mechanical’ nature of algorithms, their very efficiency is an important aspect some mathematicians believe should not to be overlooked. Some mathematicians believe that when algorithms are coupled with understanding they acquire enormous power releasing learners to draw on cognitive resources necessary for deeper thinking (Akin, 2001; Raimi, 2002; Wu, 1999). There is no contradiction in the NDP Book 1 booklet for teachers, where it is clearly stated
Student should not be exposed to standard algorithms until they use part-whole mental
strategies. Premature exposure to working forms restricts students’ ability and desire to use
mental strategies… However in time, written methods must become part of a student’s
calculation repertoire… (emphasis added)
Basic fact knowledge is crucial. The Number Framework emphasises that the process of deriving
number facts using mental strategies is important in coming to know and apply these facts
(Ministry of Education, 2008, p.14)
There is a development in the learning of mathematics from the fundamental skills to more sophisticated ones which is only possible if those basics have been deeply imbedded in a student’s thinking. Many mathematics educators believe that such an approach is the only way to ensure deep and rich learning of knowledge, relationships and concepts.
Teach the grammar of math first: the facts gained inductively through observations of nature.
Then teach the logic of math: the ordered relationships of facts to each other, the abstract
principles and their applications. Students should be given the opportunity to discuss and
debate mathematical concepts… Finish with the rhetoric of math. Let the students
themselves relate mathematical principles to other areas and to the “real world”.
Nance (1996, p.71, punctuation original)
Yet, deep thinking and problem solving skills must not be neglected by aiming only at the refined development of basic skills (Kulm, 1991). The perceived dichotomy between skill acquisition and understanding of concepts is neatly summarised by Wu (1999, p.1)
This bogus dichotomy would seem to arise from a common misconception of mathematics…
that the demand for precision and fluency in the execution of basic skills in school mathematics
runs counter to the acquisition of conceptual understanding. The truth is that in mathematics,
skills and understanding are completely intertwined. In most cases, the precision and fluency
in the execution of the skills are the requisite vehicles to convey the conceptual understanding.
There is not “conceptual understanding” and “problem-solving skill” on one hand and “basic
skills” on the other. Nor can one acquire the former without the latter.
Let the main things be the main thing
So what are the goals of mathematics education in New Zealand? When the first full mathematics curriculum document Mathematics in the New Zealand curriculum (1992)was published, its main focus was on the problem solving approach. A clear rationale was presented, declaring the value of this approach to be the provision of a way for learners to learn to think mathematically, as mentioned above (MiNZC, 1992). By identifying one of the six strands of the curriculum as Mathematical Processes, with three substrands - problem solving, developing logic and reasoning, and communicating mathematical ideas - the Ministry of Education declared its belief in the need for students to develop the ability to think clearly rather than rely on memory recall alone. Similar goals are inherent in The New Zealand curriculum (2007, p.26).
These two disciplines [mathematics and statistics] are related but different ways of thinking
and solving problems. Both equip students with effective means for investigating, interpreting, explaining, and making sense of the world in which they live.
There is little doubt that the goals of the NDP in outworking these wider goals are valid and valuable in acknowledging that a rich understanding of number is paramount, forming a base on which wider mathematical understandings can be built.
One concern raised by Begg (2006) was that of potential overemphasis on number to the detriment of other branches of mathematical learning, despite a clear indication that “It is important that students can see and make sense of the many connections within and across these strands” (Ministry of Education, 2007, p.26). Has his counsel been heeded? Surely rich knowledge and a broad bank of strategies in number must flow over into other strands to enrich the learning in those areas and so to develop a conceptual understanding of the inherent relationships. Has the goal of a deep and rich mathematical literacy been taken captive by the emphasis on building the foundation?
As mentioned above, theoretically the NDP align both knowledge and strategy, but the challenge this paper poses is that in too many instances strategy overrides the development of appropriate knowledge. This should not be a surprise, as for many years now New Zealand has developed its curriculum on the Outcomes Based Education model which values process above knowledge. There is a certain irony in the declaration of New Zealand being a knowledge economy while our schools are consistently reducing content knowledge in favour of strategies and competencies. It should not be a surprise either that in many classrooms the learning of foundational mathematical knowledge has been taken captive by an undue emphasis on learning strategies.
Neill (2008) does not deny the importance of strategy, nor does this writer, but misunderstanding of its role in the learning of mathematics has given rise to the concerns being expressed. When a teacher lacks the very fluency of understanding he/she wants to impart to the class then it is very easy to use the support material as text books rather than seeing them as the source of ideas and strategies from which a framework for lessons can be formed. This easily leads to a reliance on strategy as the goal of the learning, rather than as a tool with which to develop the understanding of the interrelatedness of mathematical facts and concepts.
It has long been recognised that individual teachers have enormous impact on the quality of learning in their pupils. Those with poor mathematical knowledge are more likely to use a rules-based approach whereas the teacher who has a degree of mathematical fluency is more likely to teach conceptually (Shulman, 1986, cited in Brown & Baird, 1993). More recently Hattie (2005) showed from a meta-analysis of a large number of studies that the teacher’s influence on learning is second only to what the learner brings to the situation himself. This is supported by Ward & Thomas (2008) in their contribution to the 2007 review of NDP, where they recommend targeted professional development for teachers. They raise an interesting issue of teachers’ content knowledge not necessarily correlating with pedagogical content knowledge, suggesting even that a mathematically knowledgeable teacher may still struggle with the balance of knowledge and strategy.
Knowledge versus process (strategy)
Mention has already been made of a perception that the knowledge of basic skills and conceptual understanding are separate from one another. A learner who demonstrates a deep and rich conceptual understanding could be said to have developed a level of mathematical literacy, the goal of the NDP. This leads back to the issue of balancing knowledge and strategy in our programmes to give our students the best possible chance to reach that goal.
In exploring the role of memorisation of basic facts Neill (2008) suggests the need for a learner to have instant recall of these facts in order to work on the problem in hand without the need to redirect attention to access the needed facts. The same can be said of strategies. Each learner needs to have a bank, a repertoire of strategies on which to draw in order to maintain focus on the problem needing solution. So it is clearly not one or the other, but a matter of both truly working together, the knowledge being the material on which the strategy operates.
There is nothing new in this assertion, for many decades ago Skemp outlined advantages and disadvantages of an ‘instrumental’ approach focussing on learning facts and rules, and a ‘relational’ approach focussing on strategies. He believed strongly that depth of conceptual understanding comes only with the recognition of the interdependence of the two approaches. “Knowing how [rules] are inter-related enables one to remember them as part of a connected whole, which is easier” (Skemp, 1976, p.23).
There are some who even believe there is still a place for an element of drill and practice to consolidate the knowledge and skills base in order for learners to develop an automatic response. Akin (2001) contends that any hope of developing real depth of facility or even joy in the art of a subject is vitally dependent on the establishment of a deeply embedded foundation. This is seen to be possible only as a result of constant practice and memorisation, despite a widespread reluctance among parents and educators to include expectations of memorisation from children. Although he acknowledges the challenges associated with deep thinking, he also supports the attitude that algorithms and routines are vital resources which children must be able to access almost automatically.
Neill’s (2008, p.19) concern that “Today there has been a tendency to emphasise strategies” needs something added to it, namely, “at the expense of knowledge and to the point of losing sight of the goal.” No doubt many will challenge such an assertion, but again personal observations of many classrooms in a wide range of schools has provided enough ‘evidence’ for this concern to be voiced. Many a classroom of Year Five and Six classes has been visited where children are only just beginning to learn multiplication tables. Would anyone seriously consider leaving the learning of the alphabet this late?
Raising questions about appropriate pedagogy is another discussion which is not new, but one which has been dominated in recent years by reforms which have rejected the traditional knowledge transmission approach in its totality. Certainly teacher-centred classrooms can be lifeless and dull but they don’t need to be, as Hattie’s (2005) findings clearly show. Current approaches to teaching in general, and mathematics specifically have focussed on child-centred, constructivist pedagogy which has stripped teachers of the opportunity to actually teach, making them merely facilitators. This movement is based on the dubious claim that since knowledge is changing so rapidly we need not actually learn anything, but only know how to access it. Consequently our curricula have developed a major focus on thinking skills to the detriment, or even exclusion of memorising and retaining valuable content knowledge. As Quirk (1998) pointed out, such an approach ignores the fact that remembered content is a vital foundation for all understanding and thinking. This belief in the structure and growth of deep understanding was the basis on which Ausubel developed his subsumption theory of developing schemata (Kearsley, 2004).
Rowe (2007) expressed deep concern at professional ignorance of the impact of effective instruction in creating general education effectiveness, claiming that pre-service trainees are rarely taught how to actually teach, but are instead immersed in constructivist, enquiry-based approaches. His plea is for a balanced approach which recognises the strengths and weaknesses of both approaches. He sees the situation where teachers are not effectively using direct instructional methods for the learning of foundational skills and knowledge as an abdication of professional responsibility. Both Rowe (2007) and Quirk (1998) recognised the implications of this lack for marginalised and underachieving children.
In promoting the wise use of instructional methods Rowe (2007) cites a large body of evidence to support his claim that many cases of underachievement are not the results of cognitive or behavioural challenges, but rather a clear lack of actual teaching and academic challenge. His claim is supported by Quirk (1998) who also believes that children from socio-economically deprived areas have difficulties in school mainly because of the shallowness of content which offers no challenge or inspiration. Recent history has provided several examples to support these claims.
In the 1970s and 1980s Marva Collins worked in the ghettoes of Chicago with African American children, who at five and six years of age had been expelled from school as emotionally and intellectually incapable of learning. Her rigorous academic curriculum led to all her pupils successfully graduating either to high school or into the work force (Collins & Tamarkin, 1982). One major longitudinal study undertaken in the USA is worth discussing, because the results indicate clear success from the application of a curriculum model with a pedagogical focus on direct instructional methods. Project Follow Through was introduced in 1967 with the aim of finding effective ways to counteract the negative effects of poverty on educational outcomes. The project ran until 1995, involving 75,000 children in 180 sites, and 22 sponsors who were given the opportunity to prove that their model of pedagogy and curriculum would in fact successfully raise the learning standards of the children in the study.
The curriculum model mentioned above was that which the University of Oregon sponsored - the Direct Instruction model (DI). Their interpretation of the findings of this study have led them to claim that the data unambiguously proves that their model alone consistently showed significant gains across a wide and variable range of sites. This behaviourally oriented model is a highly controlled and structured program emphasising children’s learning behaviour and mastery of learning (Grossen, 1995/1996; Lindsay, 2004). The advocates of DI also claim that children gain significant cognitive and affective benefits along with the development of academic skills and knowledge. Both Grossen, (1995/1996) and Lindsay, (2004) claim that political embarrassment at the success of this model has lead to it being ignored by educational authorities. Although there are aspects of this particular model which could justifiably be challenged, it can not be denied that the focus on teacher lead instruction led to successful learning for the children.
There are several other groups in the USA currently choosing to reintroduce this instructional approach, including one group of charter schools who have introduced a program called the Knowledge Is Power Program (KIPP). This was set up for much the same reason that Marva Collins began her school, namely deep concern at the poor academic achievement levels of disadvantaged children, again mostly urban African-American and Latino children. Beginning at grade five, these college-preparatory schools set a rigorous program of both extended class time and compulsory homework. Rigorous academic work is balanced with extracurricular programs of music, sport and outdoor field trips. In the 2003/2004 school year large numbers of children from disadvantaged homes, who attended KIPP schools, showed extraordinary progress in reading, language and mathematics (Mathews, 2005). There is a growing body of evidence that the methods used by KIPP schools and others using rigorous, knowledge-rich curriculum models which emphasise teacher directed instructional methods significantly raise the academic achievement levels of minority children, dramatically reducing the proficiency gap between them and those in other state schools (Kersten, 2007).
A caution may be appropriate here, as no programme is ever totally successful, and transferring from one context to another raises its own challenges. However these examples, along with the research of the scholars cited above, offer ample evidence to suggest a very strong case for refocusing our curriculum and reconsidering our approaches to pedagogy, especially in our mathematics classes.
This paper is primarily addressing approaches to the strategy strand of the NDP, but that can not be seen in isolation. Numeracy is only one part of mathematics, and mathematics is only one subject of the entire school curriculum, but the issues addressed with regard to mathematics also apply to all other areas of the curriculum. An option has been presented to consider reinvestigating the value of some aspects of a knowledge-based, teacher-lead curriculum model. In particular this paper poses a challenge to those who teach mathematics at any level to reassess the balance of knowledge and strategy/process. Recognition must be given to the importance of evidence provided by decades of empirical research in support of the need for a rich knowledge base on which learners can build a repertoire of strategies in developing the mathematical and statistical literacy NZC claims as its goal. Along with this is the need to be honest about the extent to which current approaches are failing many of our children. There certainly are successes to be applauded, but far too many children are still failing to acquire the academic and life skills necessary for them to move in to fulfilling career paths.
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Ken Rowe’s article is an extremely well balanced approach to issues raised in this paper, as he willingly acknowledges the strengths and weaknesses of both a constructivist approach to teaching maths as well as advocating for a reassessment of the inherent value of teacher directed instruction.
Dianne Scouller is currently the Head of the School of Education at Laidlaw College in Henderson, Auckland, and is currently completing a PhD on a knowledge-based approach to curriculum development. She has been a teacher of mathematics in New Zealand schools since 1968.
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