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Lowest level in the Structure of Knowledge

Rote memorization

Does not guarantee conceptual depth of understanding

Is defined by facts

Specific examples, locked in time, not transferable,

Includes definitions, symbolic formulae (ex: y = mx + b), and polygon names

Facts help students understand concepts, but are meant to be building blocks as knowing facts is more indicative of memorization and less of conceptual understanding

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Topics

Broader concepts that break down to specific micro concepts at the next level

Example: The topic "integration" is supported by the micro concepts of: limits, bounded area, summation

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Generalizations and principles are statements of conceptual understanding that let students make connections

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An equation that uses math symbols or variables to shown relationships

Is represented by facts in the Structure of Knowledge

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Statements that have been proven

They connect explanations to conceptual understanding

Represented by principles in the Structure of Knowledge

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Statements of conceptual understanding

Allow students to make connections between two or more topics

Principles are theorems (cornerstone truths)

Generalization example: Similar right-angled triangles share a common acute angle

Principle example: Theorems in calculus allow the evaluation of a definite integral without having to go back to the definition and take the limiting sum of the bounded area

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Skills make up algorithms

They are small operations or actions embedded in strategies that allow strategies to work

Skills underpin strategies

Examples: Creating a table of values, plotting a graph, using trial and error

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Specific to mathematics

A set of computation rules to produce a specified outcome

Example: PEMDAS

Consist of skills

Skills allow learners to follow a set of rules to come to a specified outcome

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Complex, sophisticated performances

Composed of strategies, algorithms, and skills

Broad techniques students draw upon when learning math

Support the understanding of concepts in a unit of work

Example: Problem solving

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Fundamental building block of math

"A series of skills and strategies that form mathematical techniques that a learner employs when faced with an unfamiliar situation"

Problem solving should be challenging, engaging, and pique learners interest enough to encourage pursuit of the presented situation

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Reasoning

The ability to make generalizations and provide explanations and justification for arguments

Leads to generalizations which provide interconnectedness of math knowledge

Gives learners opportunities to make mistakes

Leads to building mathematical memory to understand concepts

Proof

Driven by critical thinking skills and asking "why"

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Talking and writing about math

Presenting different methodologies provides learners with alternative perspectives and enables them to build a repertoire of strategies for problem solving

Involves: discussion, sharing ideas, collaboration ideally in the form of paired or small group work (not teacher-led whole class discussion)

Example: Number talks

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Refers to the learner's ability to:

See connections between facts and how they relate to one another

Make connections between symbols and procedures

Make connections between what they are learning and the real world

Connect new problems to old

See different concepts and how they relate to one another

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Representations include graphs, tables, geometric figures, etc.

Using a variety of representation methods allows students to learn in different ways and see connections

Purpose of representation:

To help learners organize, record, and communicate mathematical ideas

To select, apply, and translate different forms of representation to solve problems

To use representations to model and interpret physical, social, and mathematical phenomena

Pictorial representation

Includes drawings that contain irrelevant aspects of a problem

Schematic representation

Purposeful to solving the problem and are codes/ symbols to show relationships

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Being able to explore unfamiliar math situations

When investigating, students ask questions of themselves, furthering mathematical understanding

Includes the ability to be resourceful by extracting relevant ideas from research

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Give breadth of understanding

Posses the greatest transferability across most disciplines

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A further breakdown  of macro concepts

Example: Trigonometry

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Give us the supportive depth of understanding

Example: ratio, angles, similarity

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Examples of Macro, Meso, and Micro Math Concepts 

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Examples of Macro, Meso, and Micro Math Concepts 

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https://www.tes.com/magazine/archive/exploring-knowledge-structure-maths

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"Knowledge is produced within mathematics but policed within the mathematics curriculum," says Dr Cathy Smith.

"How do people decide what is or isn't maths? Largely through their school experiences."

She is explaining Basil Bernstein's work on how knowledge is formed, organised and reproduced; something that has particular resonance for current discussions in the edusphere about knowledge-rich curricula and teacher agency over curriculum decisions.

Is maths more like science or sociology? Smith suggests that, if we are convinced by Bernstein, it is in fact the latter.

Science is a hierarchical type of knowledge - when we learn new things, they tend to replace the old ones, overwriting existing ideas as new information comes to light.

Maths, says Bernstein, is a horizontal type of knowledge - new things add to what is already there without replacing them.

"Maths is made up of specialised languages like probability, algebra or graphs with which you can describe the world," says Smith. "But one language doesn't displace or disprove the other."
Quick read: Quick listen: Want to know more? Nudge theory: how it could help your class Podcast: the magic of unitising in maths Why maths needs its own Couch to 5K
So what is the knowledge structure of mathematics, and how does that relate to teaching it?

"Maths has organising power," she says. "It has a strong grammar, not like nouns, verbs and adjectives, but in the sense that it organises the world in relatively precise ways."

One surprising consequence of this is that the way maths works, with its own evaluative rules contained inside it, could make it a particularly empowering subject.

"If students have access to the recognition rules (how do you know what is right and what is good?) and the realisation rules (how do you make maths for yourself?) of maths, then the teacher is not the only authority in the classroom - the students can also take some of that authority."

Smith is senior lecturer in the School of Mathematics and Statistics at the Open University, and leads the mathematics education team there.

She's an expert in maths teacher education - in particular, but not restricted to, post-16 teaching and learning.

"Post-16 mathematics is one of my passions," she says, smiling.

The bedrock of learning

In the podcast, we talk about the distinction between mathematical (subject) knowledge - what she calls the bedrock under the landscape - and knowledge of mathematical pedagogy, which is the trees, the grass, the more visible structures that are overlaid on the top.
Dr Cathy Smith mentions: Lucy Rycroft-Smith mentions:
"The maths part is knowledge of curriculum and how it fits together and the progressions; the pedagogy part is examples, useful language, misconceptions etc," she says. "And it is so important to consider which of them you are foregrounding and how to pivot between them when helping teachers examine their own practice."

We also talk about issues of control in the maths classroom; the arbitrary and the necessary in maths teaching; why it is important not to waffle; Stem (science, technology, engineering and maths) sentences; the reason behind Smith's Twitter handle (@Itchymaths); implicit and explicit classroom rules; and the secret club of subject association members.

Plus, the usual jokes, suggestions for further reading, and some particularly thorny questions from Twitter...

Lucy Rycroft-Smith produces the Mathmatips podcast for Tes

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Exploring the knowledge structure of maths

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"Knowledge is produced within mathematics but policed within the mathematics curriculum," says Dr Cathy Smith. "How do people decide what is or isn't maths? Largely through their school experiences." She is explaining Basil Bernstein's work on how knowledge is formed, organised and reproduced; something that has particular resonance for current discussions in the edusphere about knowledge-rich curricula and teacher agency over curriculum decisions.

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https://www.sagepub.com/sites/default/files/upm-binaries/59758_Erickson_Chapter_4.pdf

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https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1030&context=mathmidsummative

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https://www.splashlearn.com/math-vocabulary/algebra/formula

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What is Formula in Math?

The formula is a fact or a rule written with mathematical symbols. It usually connects two or more quantities with an equal sign. When you know the value of one quantity, you can find the value of the other using the formula. It helps to solve the questions quickly. In algebra, geometry and other topics, formulas are used to simplify the process of reaching the answer and saving time.

Simple Formulas in Math

Pythagorean Theorem is one of the examples of formula in math. Besides this, there are so many other formulas in math. Some of the mostly used formulas in math are listed below:

Basic Formulas in Geometry

Geometry is a branch of mathematics that is connected to the shapes, size, space occupied, and relative position of objects. In geometry, we use formulas to find the dimensions, perimeter, area, surface area, volume, etc., of the different shapes. The 2D shapes are flat and have only two dimensions, length, and width. Some basic 2D formulas for listed below:

The 3D objects are solid objects with three dimensions, length, width, and height or depth. For example, cube, cuboid, sphere, cylinder, and cone. Formulas for volumes of basic 3D Shapes are listed below:

Basic Algebraic Formula

Algebraic formulas build foundation for various topics of mathematics like equation, polynomials, trigonometry etc. Here are some most commonly used algebraic formulas.

Arithmetic Mean Formula

Arithmetic mean (average) = $\frac{Sum of values}{Number of values}$
Real-Life Applications of Math Formulas
It is important to understand that math formulas are part of every area of your life. The meaning of formula in math is to express information symbolically concisely, and they are derived after several decades of research. We use them widely in construction, architecture, engineering and more. Whether you realize it or not, we use algebraic formulas to plan our schedule and do our tasks simply. Geometry formulas like area, perimeter and the Pythagorean theorem is commonly used in the construction of different types of structures or buildings. We use algebraic formulas in the areas of financial planning and in fields of computer science. Formulas of algebra are used in medicine to measure drug dosage depending on the age and weight of an individual. In real life, we need formulas to solve most of our calculation-based problems.

Solved Examples
Example 1: Find the perimeter of a square with a side of 5 units.
Solution: Perimeter of a square = 4 × side = 4 × 5 = 20 units.

Solution: Using the laws of exponents, we get

Since the bases are the same and equating the powers, we get -7 + n = 2

Therefore, n must be 2 + 7 or 9.
Example 3: Find the area of a square with a perimeter of 28 cm.
Solution: Perimeter of the square = 28 cmSo, the length of each side must be 28 ÷ 4 or 7 cm. Therefore, Area of square = 7 ² = 49 cm ².

Practice Problems

Frequently Asked Questions

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What is Formula in Maths? Meaning, Definition, Examples, Facts

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The formula is a fact or a rule written with mathematical symbols. It usually connects two or more quantities with an equal sign. When you know the value of one quantity, you can find the value of the other using the formula. It helps to solve the questions quickly.

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https://divisbyzero.com/2008/09/22/what-is-the-difference-between-a-theorem-a-lemma-and-a-corollary/

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What is the difference between a theorem, a lemma, and a corollary?

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I prepared the following handout for my Discrete Mathematics class (here’s a pdf version). Definition — a precise and unambiguous description of the meaning of a mathematical term. It charac…

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https://link.springer.com/chapter/10.1007/978-1-4020-8872-8_10

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Abstract

In chemistry education, micro-macro thinking using structure-property relations is considered as a key conceptual area for students. However, it is difficult but challenging for students and teachers. In this chapter, we have redefined this domain in terms of a coherent set of philosophical, substantive and pedagogical substructures. Starting from the philosophy that chemistry should be considered as a human activity, scientific and technological developments are interrelated with issues in society and part of our cultures. In many communities of practice in society, knowledge is regarded as a tool necessary for performing the activities of those practices. Learning chemistry can be seen as participation in relevant social practices. Within this vision, we have selected tasks belonging to authentic chemical practices in which structure-property relations were explored in different sub-domains (biochemistry, inorganic material science and organic polymeric material science). Within the substantive substructure, meso-structures are essential to iterate between the macro- and the sub-microscopic level. Interrelating structure-property relations connect student learning of these chemical concepts to the contexts of their everyday lives and to contemporary science and technological issues. Using this way of macro-micro thinking, two units for teaching structure-property relations were designed. These units focus on macro-micro thinking with steps in between: what we have termed 'meso-levels'. The results of the conceptual analysis of structure~- property relations and how these relations are used in macro-micro thinking are discussed. We also present a first exploration of students' learning of authentic tasks, focusing on their conceptual development.

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References

Meijer, M. (2007). Ontwerp en Evaluatie, 1e ronde chemie van een klevend network [Design and Evaluation of 1 st unit 'Chemistry of sticky network']. Utrecht: Utrecht University, Freudenthal Institute for Science and Mathematical Education. Document Doc07-01. Available online: http://www.cdbeta.uu.nl/onderzoek/publicaties.php [cited 03-10-2007].

Pavlin, J. (2007). Using students' intuitive notions about 'structure', 'property' and 'structure-property relations' for the learning of macro-micro thinking. Ljubljana: Univerza v Ljubljani, Pedagoška fakulteta, Fakulteta za kemijo in kemijsko Tehnologijo & Utrecht: Freudenthal Institute for Science and Mathematics Education (Unpublished Master thesis). Document Doc07-02. Available online: http://www.cdbeta.uu.nl/onderzoek/publicaties.php[cited 03-10-2007].

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Author information

Authors and Affiliations

Universiteit Utrecht, Utrecht, The Netherlands

Marijn R. Meijer

Corresponding author

Correspondence to Marijn R. Meijer .

Editor information

Editors and Affiliations

1. 1 Croswell Cottages, Goose Lane, Mayford Woking, Surrey

Prof. John K. Gilbert

2. 1 Croswell Cottages, Mayford, UK, GU22 0NW

Prof. John K. Gilbert

3. Curtin University of Technology Science & Mathematics Education Centre, Perth, WA, 6001, Australia

Prof. David Treagust

Copyright information

© 2009 Springer Science+Business Media B.V.

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Cite this chapter

Meijer, M.R., Bulte, A.M.W., Pilot, A. (2009). Structure-Property Relations Between Macro and Micro Representations: Relevant Meso-levels in Authentic Tasks. In: Gilbert, J.K., Treagust, D. (eds) Multiple Representations in Chemical Education. Models and Modeling in Science Education, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8872-8_10

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# data[].attributes.attachment_link.title
Structure-Property Relations Between Macro and Micro Representations: Relevant Meso-levels in Authentic Tasks

# data[].attributes.attachment_link.description
In chemistry education, micro-macro thinking using structure-property relations is considered as a key conceptual area for students. However, it is difficult but challenging for students and teachers. In this chapter, we have redefined this domain in terms of a coherent set of philosophical, substantive and pedagogical substructures.