Bruner (1960) introduced the theory of ‘scaffolding’; in that children build upon information they have already mastered. In 1966 he stated there were three phases of learning: enactive, using concrete equipment to aid learning, iconic – using pictoral representations and symbolic using abstract representations and language. He suggested that the three phases were integrated not discrete stages. These phases are extremely apparent in the progression of children through the mathematics curriculum.
From this, his spiral curriculum theory in which he stated; “We begin with the hypothesis that any subject can be taught in some intellectually honest form to any child at any stage of development. ” This has been one of the most important theories I have immersed myself in during my research into formative assessment. I choose to mention it here as Bruner suggested that even the most complex of material could be given to young children as long as it is presented at a stage appropriate for that age. Revisiting and building upon prior knowledge can increase the complexity of a subject, e. . fractions in mathematics.
I recently planned and delivered the unit of fractions to year 3 and found myself using this model, building upon children’s prior knowledge from Y1/2 and increasing the complexity to thirds, fifths, fractions of equivalency, comparing, ordering addition and subtraction of fractions from the Year 3 objectives. The National Curriculum shows Bruner’s spiral curriculum by introducing complex topics such as fractions early on in Year 1 at an age and stage appropriate level, which is revisited throughout the Primary school years with increasing complexity.
Jean Vygotsky, did not refer to stages, he believed there was “a fundamental role of social interaction in the development of cognition” Vygotsky, (1978) and that development varies between countries and cultures. Vygotsky developed the Zone of proximal development (ZPD), in which the transition from what is known, to what is not known is guided by a knowledgeable person. The ‘what is not known’ is too difficult for the child to master on their own but with guidance and encouragement the skill can be mastered and the knowledge learnt.
Jean Piaget (1936), began his work when he became interested in why children gave wrong answers to questions that required logical thinking. Piaget wanted to know how children grasped concepts such as number, time, quantity etc. He believed that stages of development are universal. Bruner and Vygotvsky theories were not set into stages, they saw cognitive development as a continuous process, building upon previous knowledge and experience. Vygotsky concerned himself with the social setting around children and how this contributes to the learning of children, Piaget failed to mention this in his theory.
Piaget’s data are renound for their unreliability. Firstly he collected the data alone, without any help and the number of children he used in his studies was small. Vygotsky proposed that language and thought develop together, Piaget’s thought were the opposite of this in that he theorized that thought proceeds language. Benjamin Bloom (1956) developed the hierarchy of thinking; a taxonomy of six levels within the cognitive domain that is still widely used today when considering questioning.
Questions are constructed to reflect the level of thinking within each level. Bloom stated that levels of thinking are hierarchical and in order to reach the higher levels of thinking, children must acquire a mastery of the lower levels. The hierarchy is shown in fig 1. Sanders (1966) further sub categorized the comprehension level of the taxonomy into two categories, Translation and Interpretation. This created a seven level taxonomy, which gives more definition to the thinking levels, particularly when we talk about questioning in mathematics.
Bloom’s questioning could be assumed to be ‘the sensitive guidance’ needed in Vygotsky’s Zone of Proximal Development. A teacher using questioning based on Bloom’s taxonomy and hierarchical thinking is leading the child through the ZPD to master the skills and knowledge being learnt. In regards to assessment, Bloom states that where assessment of higher order cognition is present there is no need for assessment of lower orders of thinking. Blooms’ colleague Lorin Anderson, along with David Krathwohl revisited the taxonomy in the 1990’s.
Anderson et al (2001) changed the hierarchical terms from nouns to verbs, as this better reflects the actions of the process of thinking and asking questions. This is shown in figure 1 below: Figure 1: Comparison of Bloom’s hierarchy (1956) with Anderson and Krathwohls revised version (2001), note the change from noun to verb form (Wilson, L. O. 2001) Combining the theories of both Bruner and Bloom; using a spiral curriculum and a hierarchy of questioning we can teach deep thinking and how to learn to use questions from the very beginning of their schooling.
An example during the topic of fractions was a KWL prior knowledge assessment. This asks the children what they already know, Would like to know, and at the end of the topic we completed the What I have Learnt. Using questioning to get the children thinking at the beginning of a topic makes them consider; what can I remember from earlier in my schooling? I then then asked the children what would you like to know, encouraging the children to direct and take responsibility for their own learning.
At the end of the unit/ lesson I asked the children ‘what have you learnt’, assessing what new knowledge had been acquired from my teaching and reflecting on whether formative assessment by marking, questioning, self-assessment and mini quizzes had been effective. Anderson et al made the hierarchy less rigid, enabling teachers to build up the hierarchy of skills by revisiting and combining categories in their questioning. In schools today the revised version of Bloom’s Taxonomy (Anderson et al, 1990) is used and is categorized into the following: remember, understand, apply, analyze, evaluate, and create.
When the revised version of Bloom’s taxonomy is used, the teacher can assess using questioning in the classroom the children’s learning and progress against the objectives taken from the national curriculum. This I find has been the more useful version of Bloom’s taxonomy in planning and delivering my questions in mathematics as it allows me to combine categories within my questions.
For example, within mathematics planning I have used questions that ascertain the prior knowledge such as Can you see a pattern? Interpretation) leading on to why do you think that? A table of questions I used in my fractions topic is given in appendix 6. Richard Skemp (1976) documented his theory of instrumental and relational learning. He explained that instrumental learning is the ability to perform a procedure, whereas relational the ability to explain a procedure. He argued that the two modes of learning were different.
However if we compare his theory to Bruner and Bloom, they disagree. Bruner’s scaffolding and spiral curriculum theories are clear and give unambiguous theory and rogression of learning. It is evident in practice that this works. Building in elements of Bloom’s revised taxonomy of questioning, with questions that are more fluid and the hierarchy a little more flexible is reflected in my pedagogy. Skemp’s theory in that performing and explaining procedures are two different abilities and modes of learning does compute. In order to explain what you are doing, you first need to be able to have the knowledge and understanding so the model of a hierarchical taxonomy is more appropriate.
