Below are links to the other posts in this series. Scroll past them to read the article.
- Atomisation
- Overtisation
- Expansion and Context Shaping
- Cognitive load
- Review
- Lesson structure and schemes of work
- Speed principle
- Difficulty and Motivation
- Defining range/scope
- Categories for different types of knowledge
- Instruction for basic types of knowledge and fact systems
- Instruction for linked types of knowledge
- Instruction for routines
- Instruction for problem solving techniques
- "Real world" maths
- Prompts and scaffolding
- Correcting mistakes
- My take on the strengths and weaknesses of Direct Instruction
Categories for different types of knowledge
After atomisation, Direct Instruction then attempts to categorise each of these components according to common features. Each category then has a section detailing how best to convey this type of knowledge to students. In this post, I will describe and give examples for each category. In future posts, I will discuss the different teaching methods for each. Here is a list of the categories I will be detailing:
- Basic types
- Noun
- Non-comparative, single-dimension concept
- Comparative, single-dimension concept
- Linked types
- Correlated feature
- Single transformation
- Double transformation
- Complex forms
- Fact systems
- Cognitive routines
Basic Types
Noun
Nouns are labels for object classes (Cars or trees, for example). They are multi-dimensional concepts (for example, there are several features that an object must have in order to be considered a car: wheels, 2-7 seats, steering wheel, etc.). The boundary between positive and negative examples is not usually precisely defined (many people would disagree about at what point a car becomes a jeep, or when a car becomes a go kart, etc.).
I find that there are very few of these in maths and, where they do, the boundary of maths nouns is clearly defined. For example, a square is multi-dimensional (has 2 dimensions, all sides straight, four equal angles, 4 equal sides), but there is no disagreement in maths between positive and negative examples of a square.
For this reason, I do not think that the teaching method detailed in Direct Instruction applies much to the subject of maths and I will not be going through it in much detail.
Single-dimension non-comparative
Examples include: horizontal, improper and multiple. These are concepts that are defined by a single dimension (for example: a fraction is improper if the value of the numerator is larger than the value of the denominator. All other features of the fraction are unimportant for this definition.)
Single-dimension comparative
These concepts are also defined by a single dimension, but are about comparing two things. Examples include: getting steeper, increasing and getting heavier. Note that many non-comparative concepts can be transformed in to comparative ones (heavy → getting heavier, large → getting larger).
Linked Types
Correlated feature
Correlated features are two (or more) concepts where one is implied by the other (in the form if... then...). Examples include: if an equation has more than one variable then it cannot be solved on its own, if a shape is an enlargement then it is similar, if a fraction is improper then it is greater than 1).
Note here how teaching these as correlated features can make the links between topics explicit. This is an element I really like about DI.
Single Transformation
Instead of having a single-dimension which defines positive/negative examples, a transformation is a concept where you follow a single rule. An easy way to spot these is if the required student response is not the same every time (Is it a multiple? always has the answer yes or no. What is the value of the 7 in this number? can be a range of answers, depending on the number being displayed). Examples include: Add a numerator to make this fraction improper, 3+5 = ? and solving one step equations of the form ax=b.
I find that a lot of maths is made up of single transformations when you atomise the more complex concepts.
Double Transformation
Double transformations are a set of two single transformation concepts which share many similar features. Examples include: single-digit addition ↔ tens-digit addition, addition ↔ subtraction, 5 x 4 = ? ↔ 5 x ? = 25).
As I'm sure you can imagine, there is a large proportion of topics in maths which can be thought of as double transformations. There are two reasons to consider them as such:
- You can leverage the similarities to make the new concept easier to learn (its just like this, but...)
- You can spend time explicitly teaching the discrimination between the topics. In the past I did not spend nearly enough time considering the added difficulty of discriminating between similar-looking questions.
Complex forms
Fact System
A fact system is a series of facts which make up some larger 'whole'. Examples include: properties of 2D shapes and angle rules. When teaching a fact system, it is important to note that you should not be teaching the meaning of individual facts (these should be pre-taught as more basic types), but you are teaching their place as part of a whole system.
Cognitive routines
Most of the topics in maths that you will atomise in to several basic and linked types of knowledge are cognitive routines. They are built up from multiple different atoms (and smaller cognitive routines) and often require several steps and distinctions to successfully complete. Examples include: solving linear equations, adding fractions and finding the missing side using trigonometry.
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