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Ch.2: What Are the Levels of the Structures of Knowledge and Process for Mathematics?

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Ch.2: What Are the Levels of the Structures of Knowledge and Process for Mathematics?

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1. The Levels of the Structure of Knowledge

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1. The Levels of the Structure of Knowledge

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The levels of the Structure of Knowledge refer to how mathematical knowledge is organized into different levels.

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2.  The Levels of the Structure of Process

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2.  The Levels of the Structure of Process

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The levels of the structure of the Process refer to the sequential stages of mathematical thinking.

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The lowest level in the structure of knowledge and it includes rote memorization and does not guarantee conceptual depth of understanding.

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Topics and concepts refer to the specific areas or subjects and their underlying ideas, principles, and relationships within mathematics.

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c. The Generalizations and Principles Levels

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c. The Generalizations and Principles Levels

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Generalizations and principles represent a deep conceptual understanding that enables connections and broad applications.

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a. Mathematical Processes, Algorithms, Strategies, and Skills. 

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a. Mathematical Processes, Algorithms, Strategies, and Skills. 

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Mathematical Processes: Mental Actions used in problem-solving.

Algorithms: Step-by-Step procedures for solving problems.

Strategies: Deliberate approaches to solving problems.

Skills: Proficiency in executing mathematical procedures.

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b. The Iterative Stages When Learning  Mathematics

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b. The Iterative Stages When Learning  Mathematics

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Using math processes, strategies, and skills helps students better understand math concepts. As they learn mote concepts, they also learn new strategies and skills.

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i. Facts are Not Transferable and Includes Definitions

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i. Facts are Not Transferable and Includes Definitions

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Knowing a definition or a formula does not imply understanding.

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The Pythagorean Theorem is a^2 + b^2 = c^2

This is a memorized fact and it does not reflect conceptual understanding.

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A student who understands the principle of symmetry can apply it to mathematical concepts like similar right triangles. They share common ratios of corresponding sides.

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Generalization = Central Ideas or Lines of Inquiry

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i. 6 Mathematical Process Broken Down Into Skills and Strategies. 

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i. 6 Mathematical Process Broken Down Into Skills and Strategies. 

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1. Problem-Solving

2. Reasoning and Proof

3. Communication

4. Making Connections

5. Creating Representations

6. Investigating

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Solving 4 x 9

Mathematical Process: Reasoning

Algorithm: Standard Multiplication Steps

Strategy: Breaking down the problem (4 x 10 = 40 then subtract 4)

Skill: Recalling Multiplication Facts

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These tools help students learn and explore math concepts. 

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Symbiotic Relationship between The Structure of Knowledge and The Structure of Process

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Symbiotic Relationship between The Structure of Knowledge and The Structure of Process

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Knowledge and Process work together:

1. Knowledge guides problem-solving.

2. Problem-solving deepens knowledge.

3. They reinforce each other.

This connection helps students:

1. Understand math concepts better.

2. Improve problem-solving skills.

3. Become more confident in math.

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Concept Based Education: The Structure of Knowledge. 

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3. Macro, Meso, and Micro Concepts in Mathematics

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3. Macro, Meso, and Micro Concepts in Mathematics

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Macro, Meso, and Micro concepts in mathematics form a hierarchical structure, ranging from board principles to connected topics to specific details.

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Macro: Overall mathematical principles and concepts.

Meso: Connections and relationships between math topics.

Micro: Specific math facts, formulas, and procedures.

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Macro: Calculus

Meso: Differentiation and Integration

Micro: Slope, Rate of Change, Velocity, Limits, Derivative, and Summation

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The chapter of this book explains the Structure of Process.

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Another Example that is non-math related

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Content is the curriculum. Therefore, curriculum planning needs to focus on content.

Rationale

Curriculum planning needs to start with the big picture and work its way in. At a school level, this means considering what the purpose of your curriculum is - what some people call a 'rationale'. What is it that your curriculum should be focusing on teaching your students, and why?

Because we are talking about the big picture, a school's curriculum rationale will be broad in scope. For example, it might state:

We want our students to:

Learn the knowledge they need to understand the world around them.

Develop the skills they need to continue to learn and contribute to society.

Develop attributes that reflect our school values.

Achieve a portfolio of qualifications that open the doors they need them to open and that reflect their very best

Love learning for the sake of learning, viewing it as interesting, exciting and empowering.

A school's curriculum rationale sets the direction for the curriculum in its entirety: subjects, assemblies, clubs, committees, sports, shows, trips, volunteering and more besides. However, it is in subjects that students spend most of their time and in which most learning happens. Therefore, it is subjects that we are going to focus on here.

Levels of organisation

Thought about in terms of subjects, the curriculum exists at different levels of organisation: macro, meso and micro. Let's explore each of these.

Macro-curriculum

The macro-curriculum relates to the specific subjects taught, the time allocated to each, and their purpose.

The subjects taught and the time allocated to each are the timetable. Most school leaders have a good handle on this. However, too few have as good a handle on the purpose of subjects. They haven't really considered why particular subjects are included in their school's curriculum. What this means is that they don't have a good enough handle on the real substance of the curriculum: content.

Why are we including this?

Why is it that art, history and science are included in your curriculum? The answers 'because these seem like important subjects for students to study', 'because other schools teach them', or 'because we've always taught these' aren't really good reasons. They reflect a very superficial way to think about the curriculum. School leaders need to go much deeper than this. They need to be clear about what is being taught and why.

Clarity in what and why

For example, if a specified purpose of art in the curriculum includes to "develop students' cultural literacy", school leaders should be clear that there needs to be a focus on the history of art, including timelines and the specific work of particular artists. If this purpose isn't clear, then it isn't clear that this content needs to be taught.

Clarity in what students should be doing with knowledge

Clarity in the purpose of subjects also helps to bring about clarity in what it is that we want students to do with the knowledge they are taught. A school's curriculum should be knowledge-based, skills-orientated. Being able 'to do' things with knowledge is the whole point of learning it.

For example, the purpose of a science curriculum might include that students:

Describe and explain the big ideas and key concepts of science.

Discuss contemporary scientific issues.

Plan and carry out experiments safely.

Present information in reports and graphical formats.

Make predictions and generalisations.

Draw valid conclusions supported by evidence.

At a macro level, we need to be clear about what students are expected to do with knowledge so we can be clear about the sorts of learning activities we should be planning and what we should be assessing.

Meso-curriculum

The meso-curriculum relates to the topics and sub-topics taught withina subject. For example:

In science, a topic might be 'chemical changes'. Sub-topics might include 'rates of reaction' and 'acids and bases'.

In history, a topic might be 'democracy'. Sub-topics might include 'Ancient Greek democracy' and 'democracy in the UK'.

In effect, the meso-curriculum is a thematic content organiser.

In setting this out, it can be useful to provide a broad overview of what students will learn in topics or sub-topics. For example, the overview of an 'introduction to business' topic might read:

This topic focuses on helping students develop knowledge about the basics of business. In it, they will learn how businesses satisfy needs and create consumer wants. They will explore the contribution of local entrepreneurs to the community.

Again, this is about starting with a bigger picture and working your way in.

Micro-curriculum

The micro-curriculum relates to the specific content within topics or sub-topics, set out as specific knowledge, skills and experiences.

Knowledge and skills

For example, in a science topic on chemical reactions, specific knowledge might include:

Key vocabulary: 'chemical reaction', 'reaction rate', 'concentration', 'particle size', 'temperature'

Everyday examples of chemical reactions, including: iron rusting, burning, milk turning sour

Factors affecting reaction rate, including: concentration, particle size and temperature

In a history topic on Ancient Roman Civilisation, this might be set out as follows:
Julius Caesar

Defeats Pompey in civil war, becomes dictator, 'Veni, vidi, vici' ('I came, I saw, I conquered')

Cleopatra of Egypt

Caesar assassinated in the Senate, Brutus Augustus Caesar

Life in the Roman Empire

The Forum: temples, marketplaces

The Colosseum: circuses, gladiator combat, chariot races, roads, bridges, and aquaducts

Eruption of Mt Vesuvius, destruction of Pompeii

Persecution of Christians

Setting a curriculum out in this way should be infinitely more useful to teachers, students, parents and school leaders than setting a curriculum out like this:

I can use my knowledge of a historical period to interpret the evidence and present an informed view.

I can explain the similarities and differences between the lifestyles, values and attitudes of people in the past by comparing Scotland with a society in Europe or elsewhere.

By studying groups in past societies who experienced inequality, I can explain the reasons for the inequality and evaluate how groups or individuals addressed it.

Irreducible form

For it to be most useful in curriculum documentation, we should be trying to get the wording of content down to its most irreducible form.

For example, ' to describe factors affecting reaction rate, including: concentration, particle size and temperature' is reducible to: 'factors affecting reaction rate, including: concentration, particle size and temperature'. At the micro level, it is the content that matters most, not what students should be able to do with this. There might be all kinds of things we want students to be able to do with specified content: describe, explain, compare, contrast, predict... Where we don't need to, we shouldn't put limits on this in curriculum planning.

That said, there will be some areas of the curriculum where the relationship between knowledge and what students should be able to do with this - skills - are inextricably linked. This will tend to be when content relates to procedural knowledge rather than declarative.

For example:

in maths, 'Multiply decimal numbers by 10, 100 and 1000', for example, 0.1, 0.01 and 0.001'.

in music, 'Compose chord sequences on the keyboard or guitar in C major or A minor using mainly primary chords'.

In such cases, this is the irreducible form. You can't take the 'doing' away from the learning as you can with some other content. However, specificity remains important. Contrast the maths example with, 'Carry out calculations involving decimals.' A lack of specificity means that wouldn't be a particularly useful content statement.

Experiences

Some learning can only come about as a result of experiences. For example:

in science, observing the 'screaming jelly baby' experiment.

in music, listening to Mahler.

in English literature, watching a production of An Inspector Calls.

in history, going on a trip to a museum.

The learning that comes from such experiences can be difficult to articulate, but it is learning all the same. Experiences of this kind are subtly different from what we typically mean by 'pedagogy'. By pedagogy, we typically mean things like 'direct-interactive instruction', 'formative assessment', 'researching' and 'investigating'. There are good reasons why we shouldn't be specifying pedagogy in curriculum plans (which are discussed in The Teaching Delusion 2). However, by 'experiences', we mean something different.

The experiences we are talking about are those that have value in their own right. They are the types of experiences that:

1. enrich learning.

2. are enriched by learning - knowledge that students have already learned adds to the experience.

3. bring about learning in a way that no other approach can.

For example, students could learn about the screaming jelly baby experiment if a teacher describes this to them or they read about it themselves. However, neither approach can bring about the learning that comes from seeing it in action. If students have been taught relevant knowledge before watching the experiment, they will get more from the experience than if they haven't.

At their best, experiences of this kind complement and consolidate learning that has come in other ways. They also help create excitement and enjoyment, the importance of which we should never underestimate.

Taken from The Teaching Delusion 2: Teaching Strikes Back by Bruce Robertson, published by John Catt Educational.

Available at:

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A 5-Minute Guide to: Curriculum Planning - Macro, Meso & Micro

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Content is the curriculum. Therefore, curriculum planning needs to focus on content. Rationale Curriculum planning needs to start with the big picture and work its way in. At a school level, this means considering what the purpose of your curriculum is - what some people call a 'rationale'.