Some children were completely silent at the start of the intervention and many others fell silent when asked even the simplest of mathematical questions.
‘He just answered the questions with a single word; he couldn’t put any mathematical language into a sentence.’
Numbers Count teacher
Almost every child soon began to talk about the mathematics. Teachers attributed this, in part, to the child’s enjoyment of it, partly to the child’s need to engage in a duologue, and partly to the teachers’ own specific modelling of mathematical language to give children the tools with which to express themselves.
By the end of the intervention, children were more willing and able to explain their mathematical strategies, not just to their Numbers Count teachers but also to their teachers and classmates in class mathematics lessons and to their parents at home.
‘We were doing ‘time’ in the classroom, which isn’t one of the things Omar has worked on in Numbers Count, but he was so confident in himself that he started actually teaching the other children in his group. He believed that he could do it.’
Class teacher
This not only indicated greater self-confidence but also a far greater desire, than at the start of the intervention, to engage in mathematical thinking and dialogue – in short, a more positive attitude towards learning mathematics.
Discussion
This has been the first large-scale study of a mathematics intervention in England. While previous studies showed impressive outcomes with small numbers of children, there is always the danger that the “small-study effect” (Bell 2011) will be diluted when an intervention is rolled out nationally to thousands of children in a wide range of contexts across a large number of schools, implemented by many different teachers who have learned about the intervention from a variety of trainers, none of whom were its authors. Numbers Count avoided such a dilution: the 7956 children’s number age gain of 14.0 months after 3 months, with an effect size of .85, was at least as strong as the outcomes from previous small-scale studies of similar interventions.
The large numbers of children involved also enabled a more thorough investigation than previously possible of the impact of a mathematics intervention on children with different background characteristics. This study has
confirmed Dowker’s (2008) and Torgerson et al’s (2011) initial findings of no significant differences. All groups, irrespective of gender, free school meals’ status, special educational needs’ status, ethnic background, home language, and season of birth, achieved number age gains in Numbers Count of at least 13.0 months, with effect sizes of at least .8. The greatest gains were made by children who spoke English as an additional language;
feedback from participants suggested that this may be because one-to-one support enabled them to communicate more effectively with their teachers than in the classroom. While children with special educational needs made the least progress, indicating that their needs may have been more complex, their number ages still improved by an average of 13.0 months.
Children’s attitudes towards mathematics also improved significantly, with an effect size of .72. Their parents and teachers reported a marked increase in enjoyment of mathematics, confidence, and positive learning behaviors that boded well for children’s continued success in learning mathematics after leaving the Numbers Count
intervention. Program data indicates that they continued to make good progress in class mathematics lessons over the next 6 months (Every Child a Chance Trust, 2010).
A qualitative analysis revealed six distinct but overlapping manifestations of children’s increased confidence.
Three of these (enjoyment, “having a go”, and talking about mathematics) had been addressed in the teachers’
training as part of the development of a “numerate child” outlined in the overview of Numbers Count at the start of this chapter. The other three (concentration, resilience, and independence), however, had not been explicitly targeted. Their emergence can be interpreted as a reminder that it is important for an intervention to help children to cope with the inevitable difficulties that are inherent in learning mathematics, and as an indication that Numbers Count succeeded in this.
Attributing causes to the success of Numbers Count must be relatively speculative, but three factors may be suggested. The first is that its design incorporates the findings of previous research into the essential features of successful interventions. Dowker’s (2004) recommendations that interventions should be implemented individually and at an early age, should begin with a detailed diagnostic assessment, and should address distinct components of arithmetic are all implemented in Numbers Count. Torgerson et al. (2011) reported that observations of Numbers Count lessons found all the elements of teaching and learning that they expected to find in a successful
intervention. Numbers Count is perhaps most different from other interventions in the specific emphasis that teachers place on fostering children’s enjoyment of mathematics and effective and learning behaviors, reflected in the strong attitude gains that they made during the intervention.
Secondly, Numbers Count teachers engaged in a thorough professional development program to inform and support their teaching. Torgerson et al (2011, para. 7.3.6) reported that “many head teachers took an interest in the training and reported that it was the best they had ever seen.” The key contents of the program were a combination of practical training in the delivery of Numbers Count, study of early mathematics curriculum and pedagogy, and reflection on teachers’ developing experience and understanding. Trainers aimed to develop a collaborative learning culture which would underpin teachers’ continuous reflection and learning as they investigated the needs of each child and considered how best to meet them.
Finally, Numbers Count was underpinned by a rigorous quality assurance system. Each school’s receipt of funding for its teacher was dependent on adherence to strict national standards for the selection of children, the
deployment of the teacher, quality of teaching, and support by school management. Adherence was monitored by the training leaders through frequent school visits and through analysis of data about each child’s progress. The
training leaders gave individual support to any teachers and schools that fell short of expected standards and outcomes. At the same time, feedback from teachers and schools was used to refine and improve Numbers Count.
Success, however, came at a price. Numbers Count was an expensive intervention, both because it was delivered individually to children by a qualified teacher and because it was underpinned by extensive professional
development and quality assurance. In a newly-prevailing climate of austerity, a national policy decision was taken to gradually withdraw state funding of the Every Child Counts initiative, which included the Numbers Count
intervention; if it was to continue, it should be as a marketed program which schools could choose to purchase if they believed that it provided value for money (Department for Education, 2011). Torgerson et al. (2011)
suggested that Numbers Count in its original form, for all its impact, was too expensive for most schools. Although analysts (Gross et al., 2009) had calculated that every £1 spent on an intervention such as Numbers Count would save the state between £12 and £18 in later spending, including special educational needs’ support, welfare, and the criminal justice system, less than £1 of this saving was realised by the primary schools that made the spending decisions. While a decision to invest in support for young children who struggle in mathematics is not a purely monetary one, cost is nevertheless an important consideration.
Numbers Count was, therefore, modified after the period of this study, to make it more flexible and more affordable for schools. Following the success of a small-scale trial of teaching Numbers Count to children in groups of two and three (Torgerson et al., 2011) and in line with Dowker’s (2004) finding that small groups can be effective, teachers were given the option of teaching up to three children together, once they had successfully learned to deliver Numbers Count to individual children; diagnostic assessment was still carried out individually, and teachers were trained and encouraged to make flexible decisions about group sizes to meet individual children’s needs. The initial training program for teachers was reduced from 10 to 7 days, while still retaining its essential features, and the number of school visits made by group leaders was also reduced. Additionally, the quality assurance system that tied school funding to adherence to national standards was replaced by a system of accreditation for schools and teachers who voluntarily adhered to standards that allow more flexibility while focusing on the key features that ensured children’s success in Numbers Count.
The early evidence is that the impact of Numbers Count actually increased during the first 2 years after these changes were introduced. Children’s average number age gains between entry and exit rose from 14.0 months in this study to almost 16.0 months after the changes. While most children were still taught individually, equivalent progress was achieved by the 25% whose teachers taught them in pairs.
This chapter has reported on children’s learning during the period of their Numbers Count interventions. The second aim of Numbers Count, that they would continue to make good progress subsequently, will be analyzed in separate studies. Initial results are encouraging, indicating that children who had completed Numbers Count in the first year of the intervention broadly kept pace with their peers over the next 4 years until they left primary school at age 11.
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Nick Dowrick
Nick Dowrick, Faculty of Education, Edge Hill University
Subject: Psychology, Cognitive Psychology, Educational Psychology, Developmental Psychology
Online Publication Date: Mar 2014
DOI: 10.1093/oxfordhb/9780199642342.013.014
Developing Conceptual and Procedural Knowledge of Mathematics
Bethany Rittle-Johnson and Michael Schneider
The Oxford Handbook of Numerical Cognition (Forthcoming)
Edited by Roi Cohen Kadosh and Ann Dowker
Oxford Handbooks Online
Abstract and Keywords
Mathematical competence rests on developing knowledge of concepts and of procedures (i.e. conceptual and procedural knowledge). Although there is some variability in how these constructs are defined and measured, there is general consensus that the relations between conceptual and procedural knowledge are often bi- directional and iterative. The chapter reviews recent studies on the relations between conceptual and procedural knowledge in mathematics and highlights examples of instructional methods for supporting both types of
knowledge. It concludes with important issues to address in future research, including gathering evidence for the validity of measures of conceptual and procedural knowledge and specifying more comprehensive models for how conceptual and procedural knowledge develop over time.
Keywords: conceptual knowledge, procedural knowledge, mathematics, iterative relations, instruction
Introduction
When children practise solving problems, does this also enhance their understanding of the underlying concepts?
Under what circumstances do abstract concepts help children invent or implement correct procedures? These questions tap a central research topic in the fields of cognitive development and educational psychology: the relations between conceptual and procedural knowledge. Delineating how these two types of knowledge interact is fundamental to understanding how knowledge development occurs. It is also central to improving instruction.
The goals of the current paper were: (1) discuss prominent definitions and measures of each type of knowledge, (2) review recent research on the developmental relations between conceptual and procedural knowledge for learning mathematics, (3) highlight promising research on potential methods for improving both types of knowledge, and (4) discuss problematic issues and future directions. We consider each in turn.
Defining Conceptual and Procedural Knowledge
Although conceptual and procedural knowledge cannot always be separated, it is useful to distinguish between the two types of knowledge to better understand knowledge development.
First consider conceptual knowledge. A concept is ‘an abstract or generic idea generalized from particular instances’ (Merriam-Webster’s Collegiate Dictionary, 2012). Knowledge of concepts is often referred to as conceptual knowledge (e.g. Byrnes & Wasik, 1991; Canobi, 2009; Rittle-Johnson, Siegler, & Alibali, 2001). This knowledge is usually not tied to particular problem types. It can be implicit or explicit, and thus does not have to be verbalizable (e.g. Goldin Meadow, Alibali, & Church, 1993). The National Research Council adopted a similar definition in its review of the mathematics education research literature, defining it as ‘comprehension of
mathematical concepts, operations, and relations’ (Kilpatrick, Swafford, & Findell, 2001, p. 5). This type of knowledge is sometimes also called conceptual understanding or principled knowledge.
At times, mathematics education researchers have used a more constrained definition. Star (2005) noted that: ‘The term conceptual knowledge has come to encompass not only what is known (knowledge of concepts) but also one way that concepts can be known (e.g. deeply and with rich connections)’ (p. 408). This definition is based on Hiebert and LeFevre’s definition in the seminal book edited by Hiebert (1986):
‘Conceptual knowledge is characterized most clearly as knowledge that is rich in relationships. It can be thought of as a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete pieces of information. Relationships pervade the individual facts and propositions so that all pieces of information are linked to some network’ (pp. 3–4).
After interviewing a number of mathematics education researchers, Baroody and colleagues (Baroody, Feil, &
Johnson, 2007) suggested that conceptual knowledge should be defined as ‘knowledge about facts,
[generalizations], and principles’ (p. 107), without requiring that the knowledge be richly connected. Empirical support for this notion comes from research on conceptual change that shows that (1) novices’ conceptual knowledge is often fragmented and needs to be integrated over the course of learning and (2) experts’ conceptual knowledge continues to expand and become better organized (diSessa, Gillespie, & Esterly, 2004; Schneider &
Stern, 2009). Thus, there is general consensus that conceptual knowledge should be defined as knowledge of concepts. A more constrained definition requiring that the knowledge be richly connected has sometimes been used in the past, but more recent thinking views the richness of connections as a feature of conceptual knowledge that increases with expertise.
Next, consider procedural knowledge. A procedure is a series of steps, or actions, done to accomplish a goal.
Knowledge of procedures is often termed procedural knowledge (e.g. Canobi, 2009; Rittle-Johnson et al., 2001).
For example, ‘Procedural knowledge … is ‘knowing how’, or the knowledge of the steps required to attain various goals. Procedures have been characterized using such constructs as skills, strategies, productions, and
interiorized actions’ (Byrnes & Wasik, 1991, p. 777). The procedures can be (1) algorithms—a predetermined sequence of actions that will lead to the correct answer when executed correctly, or (2) possible actions that must be sequenced appropriately to solve a given problem (e.g. equation-solving steps). This knowledge develops through problem-solving practice, and thus is tied to particular problem types. Further, ‘It is the clearly sequential nature of procedures that probably sets them most apart from other forms of knowledge’ (Hiebert & LeFevre, 1986, p. 6).
As with conceptual knowledge, the definition of procedural knowledge has sometimes included additional constraints. Within mathematics education, Star (2005) noted that sometimes: ‘the term procedural knowledge indicates not only what is known (knowledge of procedures) but also one way that procedures (algorithms) can be known (e.g. superficially and without rich connections)’ (p. 408). Baroody and colleagues (Baroody et al., 2007) acknowledged that:
‘some mathematics educators, including the first author of this commentary, have indeed been guilty of oversimplifying their claims and loosely or inadvertently equating “knowledge memorized by rote … with computational skill or procedural knowledge” (Baroody, 2003, p. 4). Mathematics education researchers (MERs) usually define procedural knowledge, however, in terms of knowledge type—as sequential or
“step-by-step [prescriptions for] how to complete tasks”
(Hiebert & Lefevre, 1986, p. 6’ (pp. 116–117).
Thus, historically, procedural knowledge has sometimes been defined more narrowly within mathematics education, but there appears to be agreement that it should not be.
Within psychology, particularly in computational models, there has sometimes been the additional constraint that procedural knowledge is implicit knowledge that cannot be verbalized directly. For example, John Anderson (1993) claimed: ‘procedural knowledge is knowledge people can only manifest in their performance …. procedural knowledge is not reportable’ (pp. 18, 21). Although later accounts of explicit and implicit knowledge in ACT-R (Adaptive Control of Thought—Rational) (Lebiere, Wallach, & Taatgen, 1998; Taatgen, 1999) do not repeat this