Good news. We’re giving ACARA, and our readers, the night off. No painful reading tonight; just painful reality.
Ofsted is the UK’s ACARA-ish organisation, although the “ish” hides the fact that Ofsted appears to be competent. Last week, Oftsed published a review into mathematics education. We’re sure we are missing something, because the review appears to be important, clear and correct.
Reportedly the work of Hannah Stoten,* the document lays out in a clear and methodical manner what a mathematics education entails, and thus the nature of a proper mathematics curriculum. Here is how the review gets going:
How the review classifies mathematics curriculum content
For this review, we have classified mathematical curriculum content into declarative, procedural and conditional knowledge.
Declarative knowledge is static in nature and consists of facts, formulae, concepts, principles and rules.
All content in this category can be prefaced with the sentence stem ‘I know that’.
Procedural knowledge is recalled as a sequence of steps. The category includes methods, algorithms and procedures: everything from long division, ways of setting out calculations in workbooks to the familiar step-by-step approaches to solving quadratic equations.
All content in this category can be prefaced by the sentence stem ‘I know how’.
Conditional knowledge gives pupils the ability to reason and solve problems. Useful combinations of declarative and procedural knowledge are transformed into strategies when pupils learn to match the problem types that they can be used for.
All content in this category can be prefaced by the sentence stem ‘I know when’.
When pupils learn and use declarative, procedural and conditional knowledge, their knowledge of relationships between concepts develops over time. This knowledge is classified within the ‘type 2’ sub-category of content (see table below). For example, recognition of the deep mathematical structures of problems and their connection to core strategies is the type 2 form of conditional knowledge.
Summary table of content categories considered in the review:
Declarative "I know that"
Facts and formulae
Relationship between facts (conceptual understanding)
Procedural "I know how"
Relationship between facts, procedures and missing facts (principles/mechanisms)
Conditional "I know when"
Relationship between information, strategies and missing information (reasoning)
Is this perfect? Of course not. It would be easy to nitpick over borderline calls. But as a basic guide to building and analysing a curriculum, it is beautifully simple and clear. As guides should be. And as ACARA’s Wheel of Death most definitely is not.
There’s plenty more we could quote. Like the whole damn thing. But we’ll restrain ourselves, and give just a few more. Here’s a note on “core knowledge”:
Foundational knowledge, particularly proficiency in number, gives pupils the ability to progress through the curriculum at increasing rates later on. The path of learning that begins with a diligent focus on core declarative and procedural knowledge is not a straight line, therefore, but a curve. This is a function of the curriculum’s intelligent design. For example, in countries where pupils do well, pupils are able to attempt more advanced aspects of multiplication and division in Year 4 if they have been given more time on basic arithmetic in Year 1. This may explain why successful curriculum approaches tend to emphasise core knowledge early on.
So, arithmetic skills are kind of important, especially early on. Who would have guessed?
Here’s an early comment on problem-solving:
Problem-solving requires pupils to hold a line of thought. It is not easy to learn, rehearse or experience if the facts and methods that form part of a strategy for solving a problem type are unfamiliar and take up too much working memory. For example, pupils are unlikely to be able to solve an area word problem that requires them to multiply 2 lengths with different units of measurement if they do not realise that the question asks them to use a strategy to find an area. They are also unlikely to be successful if they do not know many number bonds, unit measurement facts, conversion formula or an efficient method of multiplication to automaticity. Therefore, the initial focus of any sequence of learning should be that pupils are familiar with the facts and methods that will form the strategies taught and applied later in the topic sequence.
What’s this? Give the kids the knowledge and skills and techniques before having them embark on problem-solving? Are these people nuts?
One last one, on “positive attitude”:
Pupils are more likely to develop a positive attitude towards mathematics if they are successful in it, especially if they are aware of their success. However, teachers should be wary of the temptation to invert this causal pathway by, for example, substituting fun games into lessons as a way of fostering enjoyment and motivation. This is because using games as a learning activity can lead to less learning rather than more.
Some pupils become anxious about mathematics. It is not the nature of the subject but failure to acquire knowledge that is at the root of the anxiety pathway. The origins of this anxiety may have even been present at the start of a pupil’s academic journey. However, if teachers ensure that anxious pupils acquire core mathematical knowledge and start to experience success, those pupils will begin to associate the subject with enjoyment and motivation.
It’s hard to believe, but they seem to be suggesting that to get a kid to like mathematics you should get them good at it, rather than pretending they’re good at it. Crazy, crazy stuff.
Read the whole damn thing. We haven’t read it all yet, and yeah, we’ll probably find something in there that annoys us (because we’re that type). But we haven’t found it yet. It is a great, great document.
A couple of colleagues, Simon the Likeable and The Hot Dog Man, indicated that they were puzzled by the Review. They both wondered if perhaps what Hannah Stoten is saying isn’t really simple. Indeed, they are correct. In a nutshell, this is Stoten’s message:
The last 50+ years have been a complete screw up. Forget about them, and start again.
*) (Update 31/05/21) And Steve Wren. Efforts are already underway to kidnap them both. No one tell them.