In our previous blog, Ruth and I discussed the use of the discovery method. I want to revisit that blog, and the knowledge object 1 + 1 = 2, and cast a material lens1 on the nature of curriculum knowledge. As Ruth points out, this approach is particularly relevant to the early years because of the material nature of the learning environment.
I contend that just as an architect or engineer designs the material world, so a teacher designs learning. For the purpose of this blog, I am going to argue that it is not the teacher, the learner or cognition that determines learning but the nature of the knowledge object2 and its relationship with the experienced world.
Thus, I design a learning experience to demonstrate the point (see Figure 1).
The nature of 1 + 1 = 2
Mathematics is an apriori symbolic representation3 of the experienced world. It is possible for students to learn its syntax without application; however, I assume that learning mathematics in the early years involves both 1. learning the symbols and their relations and 2. applying them to concrete objects.
1. Direct instruction: controlled abstract learning
The last blog recommended direct instruction to structure the session and teach the symbols and formulas of mathematics. The question is, what experiences or prior schema can teachers use to teach children the semantics of abstract symbols? The answer is likely to be few; the relational logic of mathematical symbols is not inherent to nature, other than the deliberate teaching of it by significant others. Thus, it makes sense to deliver initial explanations of the symbols 1 + 1 = 2 using direct instruction.
The disadvantage of abstract symbolic logic is that it places a heavy load on working memory. Teachers can manage load by employing constructivist techniques like scaffolding, CPA (concrete, pictorial, and abstract) and/or using delivery methods that leverage fast processing in cognition and are relatively effortless, like role-play.
2. Role-play: efficient experiential learning
When teaching abstract symbolic processing, teachers need to decide which concrete objects are to be leveraged to maximise prior learning; we selected apples. We also discussed a tea party and the exchange of goods. The more a young learner is likely to experience the concepts, the more likely those concepts will be remembered.
Learning appears uncontrolled using role-play, but it is not; teachers employ it to link to pre-existent schemas. It does not need the same instructional control as direct instruction, but it does require significant expertise to ensure that learning is taking place.
As Ruth points out EY practitioners recognise the role of the learning environment as “the third teacher”, which supports “effortless” experiential learning. The human brain is able to make sense of the social world without having to do too much work.4
Constructing cognitive bridges
The nature of the knowledge object, 1 + 1 = 2, necessitates differentiating between controlled learning and learning that is more complex and rich in social information. Ofsted’s recent Bold Beginnings report highlighted the difficulty of moving too quickly to abstract symbolism in mathematics.5 Fortunately, the two processes are already theorised in the field of social psychology (Spunt, 2015);6 however, there are no definitive answers offered by psychologists who increasingly refer to the four analytical tools: awareness, efficiency, intentionality and controllability.7
A learner receiving direct instruction is aware of the learning process, but learning is likely to be slower and less efficient. Similarly, a learner engaged in role-playing is barely aware of the learning process, and the teacher retains less control over it; however, it is more efficient in the sense that learning is fast and leverages pre-existent schemas. Researchers have found that children remember new information with associated meanings better than without, reasoning that retrieval is better when both episodic (hippocampus) and semantic (neocortical areas) memory systems are utilised
Identifying the nature of the learning helps to develop an appropriate delivery method (s). As Ruth notes, the skill required by teachers is to understand how to control learning using direct instruction, and to facilitate it when engaging in role-playing. The skill of knowing how to wait, watch and wonder before intervening (or not) to move the learning on and consolidate understanding, knowledge retention and future retrieval.
Iterative design: Securing learning and ensuring retrieval
The design of any educational learning experience requires an iterative process that involves a learner re-engaging with the knowledge object in conjunction with formative assessment. Iteration ensures that learning coheres to the knowledge object by a process of continuous interaction and practice.8
Iterative design inherits the constraints inherent to the knowledge object and, therefore, the design of such a process depends upon the nature of 1 + 1 = 2. It cannot be prescriptive because it is also dependent on how successfully learning has been delivered. A key factor will also be the extent to which learning is re-enforced by everyday experiences and the nature of cognition. Focus, in this case, has to be on abstracted symbolic processing and the retrieval of abstract knowledge.
Finally, once the learning experience process is complete, whether because learning is achieved or because of time constraints, then a process of summative assessment can determine future directions.
This blog designs a learning experience determined by the nature of a knowledge object. It offers a material approach to knowledge, but avoids biological reductionism;9 incorporates play but does not regard it solely in terms of human discourse and subjective experience.
It argues that the nature of a knowledge object determines the process required to deliver a learning experience as well as the cognitive processes that store and retrieve the knowledge object.
1. On the material, the radical empiricism of William James offered the view that “it begins in the midst, in a mess of relations” (Manning, 2015, p 54–55). Curriculum knowledge is at the centre of a “spatial and temporal web of …dependencies” (Manning, p 11), which does not mean that all relations have equal influence, but nor can you extricate it from the material world.
For Žižek, by contrast, ‘Materialism means that the reality I see is never “whole”—not because a large part of it eludes me, but because it contains a stain, a blind spot, which indicates my inclusion in it’. I also consider Heidegger’s thinking on Kant’s Copernican revolution, Harman’s speculative realism and Bhaskar and Bryant’s turning around of the transcendental question: How does reality have to be structured so that our cognition of reality is possible?
2. In this blog, I use knowledge objects and curriculum objects interchangeably. Graham Harman writes interestingly on objects defining them as:
(…) an object is anything that cannot be entirely reduced either to the components of which it is made or to the effects that it has on other things.
3. Kant, in a critique of pure reason, describes mathematics thus:
Before all, be it observed, that proper mathematical propositions are always judgements a priori, and not empirical, because they carry along with them the conception of necessity, which cannot be given by experience.
Bertrand Russell described mathematics as:
The fact that all mathematics is symbolic logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself
4. Suggested reading, Spunt 2015 (follow the link below):
The human brain is often able to make sense of the social world without having to do too much work because social stimuli are automatically initiated and run to completion without much if any, conscious intervention (Bargh & Chartrand, 1999; Gilbert, Pelham, & Krull, 1988).
5. Where mathematical teaching was most successful:
(…) leaders ensured that progression in mathematical concepts from the very beginning frequently used practical equipment to support children’s learning of new concepts. Only after much practice and rehearsal with concrete resources would teachers move children on to representing their understanding through visual images and models.
6. Dual processing theory (follow the link below):
The distinction between automatic processing and controlled processing is foundational to a family of theories known as dual-process theories (Chaiken & Trope, 1999; Sherman, Gawronski, & Trope, 2014).
7. New thinking in dual processing theory (follow the link below):
The social cognitive neurosciences provide little evidence for a definitive view of cognitive processing. Dual processing theory is useful to help us think about how different learning experience methods are processed cognitively; however, more recently researchers have considered that such processing is best-described using four characteristics: awareness, efficiency, intentionality and controllability.
8. I see the learning experience and Rosenshine’s instructional principles as broadly compatible, if not the same. The Iterative phase roughly equating to Rosenshine’s points 10:15
9. Ofsted defined learning, in their recently delivered curriculum workshops, as “an alteration in long-term memory”.