Like Teaching - Assume m is positive

Quick wins from Direct Instruction: Dimensions of Difficulty

2020-07-14T11:32:00.000+01:00

This post was inspired by an episode of the Craig Barton podcast with Kris Boulton. Kris was acting as a salesman for Engelmann's Direct Instruction, but he didn't seem to have much to offer when Craig asked him for Direct Instruction 'quick wins' that teachers could implement and improve their teaching right away (besides atomisation and minimal variation, which followers of Craig Barton will already be very familiar with).

One of the things that has really helped me after reading Direct Instruction is the way that it breaks down, in great detail, the different elements that cause students to find a topic difficult. As an experienced teacher, I had already figured out some of these, but Direct Instruction really clarified my thinking and it has made me much more confident in predicting how successful students will be on a topic. This is vitally important to the Direct Instruction method, as designing every task so that over 90% of the students will be instantly successful is a core element of the method. I believe that these ideas would be even more useful for newer teachers. First of all, here is the list of difficulty dimensions:

Similarity to other topics - e.g. the letter 'd' is harder to learn than the letter 'o'. Because of its superficial similarity to 'b', an additional amount of problem differentiation is needed by the student (note that this only applies to whichever letter out of b and d that they learn last).
Method is shared with another topic - conversely, if the method is identical to that of another topic, the difficulty of the new topic is reduced. This is because you can teach the new topic as a "double transformation", where students convert the new topic into the topic they already know. e.g. 35% of 250 can be transformed into 35/100 of 250 if they already know fractions of amounts.
Prior knowledge required - each additional piece of prior knowledge required will increase the load on long-term memory.
The length of time elapsed since students last successfully practiced on that topic (or any prior knowledge topics).
Number of times students have successfully practiced the topic.
The number of steps in the method - Increases the load on working memory. This should include any decisions that are required, not just the overt written steps. e.g. for a basic trigonometry question you have the following steps: (1) decide whether it is trig or pythag, (2) decide which side is h, o and a, (3) work out which sides are used in the question, (4) decide whether to use sin, cos or tan, (5) remember the formula for the trig function, (6) substitute the correct numbers in to the formula, (7) rearrange the formula, (8) type the formula in to the calculator, (9) round the answer to the correct accuracy.
The context of the problem on the page (this includes how the question is worded as well as any real-life situation the problem is put in).
Blocked/Interleaved questions - Are the questions in a block of the same question type? Are they interleaved with other topics? How many other topics? Have they been primed to think about this topic earlier in the lesson (via instruction or earlier questions)?
Difficulty of response - How are students expected to respond to a question? These are some options going from easiest to most difficult:

Yes/No binary option
multiple choice
scaffolded response
single line answer (without scaffold)
multi-step process (with scaffold then without)
Reasoned explanation

As mentioned at the start, teaching so that each new topic has over 90% of students 'getting it' on their first attempt is central to DI. This is crucial for motivation, by setting up an expectation of success in the subject, and for the high pace required in the DI method. Knowing and considering these dimensions of difficulty is essential for pitching the work correctly so that this happens.

Though all of this is discussed in DI, I find the way that they consider each of these dimensions separately to be complicated and unwieldy. Which brings me to the 'win' that may take a bit more explanation...

My interpretation/application of the dimensions of difficulty concept

For each topic I teach I assign a sort of 'difficulty score'. Then, through knowledge of the students and testing of the class, I decide what is the maximum difficulty score I can give to that class, where I will still get >90% success (this varies wildly from class to class). If the 'difficulty score' of the new topic is too high, I find ways to reduce the score to an acceptable level. Though I've tried to codify it here, it is important to note that in my lesson planning I rarely work out the exact score, but consider each dimension to get a rough feel for the difficulty before deciding how many difficulty reduction steps I should take. Here is how I assign my score and how I reduce it:

Degree of difficulty	Points	How to reduce the difficulty
Similarity to other topics	Add 1 for every topic you must differentiate from this topic	Start with questions only on this topic (takes off all points added here) Avoid introducing them in lessons that are close together in time. Teach how to differentiate between this topic and one other topic at a time, then interleave questions with these two topics. (takes off all points but 1). (Obviously you would want to practice full interleaving eventually)
Method is shared with another topic	Take 1 away
Prior knowledge required.	Add 1 for each topic required (that is not completely secure: I would not include addition, for example, with some classes but would with some very low sets).	The initial problem set should use these skills as little as possible. Reteach the prior knowledge topic first until it is secure.
The length of time elapsed since any topics last practiced	Add 1 for every prior knowledge topic the class has not covered this term.	Revisit the topics through starters, during the mixed reviews or during other short tasks
Number of times any topics practiced	Add 3 if the topic is completely new, add 2 if they've seen it once before and add 1 if they've seen it twice before	Take 1 away each time students have successfully practiced the topic (recently)
The number of steps in the method	Add 1 for every required step in the method	Split the teaching up so that students first learn each step in isolation, then split the full method in to smaller, overlapping segments of steps, before teaching the full method.
The context of the problem on the page	Add 1 for every new way question is written	Restrict initial questions to have the exact same setup. Teach other ways of writing the same question in separate lessons. Teach a variety of contexts (separately) until students can correctly identify the whole range of contexts where this skill could apply.
Blocked/Interleaved questions	Add 1 for each question you are asking on other topics in between questions on this topic	Start with blocked questions, then interleave where roughly every other question is the new topic, then gradually add more questions in between the new topic. Prime students to think about this topic by having blocked questions before starting your interleaved review.
Difficulty of response	Add 1 each time you go down the list above	Start with the easiest type of question and build up as necessary.

I know that this sounds like quite an intense thing to do whilst planning for every lesson, but I find doing a rough version of this in my head takes less than a minute and it helps my planning immensely.

Principles of Engelmann's Direct Instruction: Categories for different types of knowledge

2019-08-05T08:50:00.000+01:00

This post is part of a series where I go in to detail on some of the main aspects of Direct Instruction as laid out by Siegfried Engelmann in this book.

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.

Principles of Engelmann's Direct Instruction: Speed Principle

2019-07-28T17:57:00.000+01:00

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
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

Speed Principle

Maintaining a high pace is central to the DI method. This is for several reasons:

There are multiple activities and repeated review built in to the schemes of work. Activities need to be short in order to cover the large amount of GCSE content in this way.
In order to get sufficient practice of a skill, within these short activities, the time a student has per question must necessarily be short.
If a component cannot be explained and grasped in <1 minute, it it likely too complex and will result in the failure of the initial instruction to several students. This is a sure sign that the component should be further atomised.
Time between each example and each student question needs to be very short in order to ensure that students are seeing the questions as a whole topic and making the correct conclusions about the range/scope of the topic.
If students get bogged down with the calculations, they will be unlikely to make inferences from one question to the next. In this case, the component should either be further atomised or taught as a routine.

Principles of Engelmann's Direct Instruction: Difficulty and Motivation

2019-07-26T12:45:00.000+01:00

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
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

Difficulty and Success

Ensuring success, in the acquisition of both individual components of knowledge and whole schemes of work, is at the heart of Direct Instruction. Every part of it was built and tested with this single criteria. Each activity in a DI program is tested and improved until it achieves over 90% success rate of all questions along all students. Even when doing remedial work to correct mistakes and learned misconceptions, DI takes great pains to ensure that students will successfully answer at least 2/3 of questions directed at them (more on this in the correcting mistakes section).

When I am planning a lesson now, I use the following rubric:

For an initial instruction activity, can I explain this in <1 minute of teacher direction so that >90% of success will be able to perform the task first time, without additional support?

If not, I need to atomise this further in to components that fit this criteria.

For review activities, am I confidant that students will remember this topic?

If not, I need a prompting activity before the review (to be faded in a subsequent lesson).

Motivation

Direct Instruction does not have much to say about student motivation.This, in my view, is one of its main shortfalls, as we all know how much of a factor student motivation is for successful learning. However, it is highly focused on achieving student success. This successful progress is a big factor in most of the popular current theories on motivation. I will demonstrate this below by looking at three theories in particular: the Expectancy-Value Theory, Maslow's Hierarchy of Needs, and Dweck's Growth-Mindset Theory.

Expectancy-Value Theory

According to this theory, a student's motivation is determined by:

How much they value the goal(s) that are set
To what extent they expect to succeed at these goals

These factors are said to multiply, so that if either value is zero (or very low), the overall motivation is zero:

= X

Expectancy is obviously heavily tied to your previous experiences of success and failure in that subject. Therefore if, as with the DI model, you are regularly and reliably making successful progress from lesson to lesson, your expectancy of success in future lessons will increase.

DI does not say anything about Value, and so other techniques should be added to a DI program in order to increase this. At some point I plan on making a blog post specifically about this.

Maslow's Hierarchy of Needs

The crux of this theory is that individuals’ most basic needs must be met before they become motivated to achieve higher level needs. I feel like this plays right to the biggest strength of DI. This is because, whilst enquiry-based learning, project-based learning, collaborative problem solving, open-ended, rich tasks, etc. all attempt to service a student's need for self-actualisation in one way or another, DI is an attempt to service the deeper, psychological need to build self-esteem. In my experience, it does this very successfully with a wide range of students.

Dweck's Growth-Mindset Model

This model states that some people believe their success is based on innate ability (with a fixed mindset) while others believe their success is based on hard work, learning, training and doggedness (with a growth mindset). This theory has been met with serious criticism and I only include it here because of its popularity within the profession.

A student's beliefs about what causes learning is formed by their experiences of their own and other's successful learning. In DI, a student gets to experience rapid and reliably successful learning. The learning even feels easy for them (so long as they follow along with each section). This means that students get to regularly experience the positive effect of practice on their abilities. This experience can be signposted to make sure that they appreciate it using several techniques:

Teacher praise
Pre and post-learning check-in tests*
Assessment score tracking for students
Dan Meyer's headache-aspirin set-up to learning*

*these are not included directly in DI, but I have found adding them useful

Addressing concerns

How does this fit with the theory of desirable difficulty?

This theory states that the best learning happens when students work on problems that are at the right difficulty - challenging but doable. On the face of it, this runs completely counter to the DI model, which aims to make each individual task very easy for students to replicate. Indeed there are several elements of the pure DI model that contradict desirable difficulty:

Individual tasks are designed to be (or seem to be) easy in all stages
Feedback is designed to always be instant (or as close as possible to this) whereas some studies have shown that delayed feedback is preferable in certain situations.
Differentiation is only present in DI in the section under 'correcting mistakes'

However, there are also many ways in which DI fits nicely with the theory of desirable difficulty:

Spaced retrieval practice/Interleaving: This is at the heart of every DI session (85% of each lesson spend on review)
The scheme of work remains unchanged; students still learn topics to the same level of difficulty as before - DI just gives a different approach on how to lead up to that level of difficulty.

To account for these discrepancies I have adjusted somewhat:

I usually do not add any differentiation to initial instruction (as I do want all students to be successful at understanding the central concept). However I do differentiate the expansion and context-shaping activities, as some learners will immediately see wider generalisations and range of applications than others. During these activities I will plan a range of difficulties that students can choose or be directed to.
For homework (not mentioned in DI) - it is important that it is accessible for all without additional support (it otherwise risks widening the achievement gap). The only way to do this and achieve desirable difficulty is with differentiation. This can be achieved by choice of task or the amount of scaffolding/prompting on the sheet.
I also try to give feedback instantly during initial instruction (as stated in DI, this is a particularly important time to make sure that students are not learning mis-rules) but will often use delayed feedback for other activities. I still find it difficult to know when to use different types of feedback, but here is a decision tree by improvingteaching.co.uk that aims to help:

Principles of Engelmann's Direct Instruction: Lesson structure and Schemes of Work

2019-07-22T08:51:00.000+01:00

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
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

Lesson Structure

Due to atomisation and the speed principle, the lesson can be split in to 3-5 minute activities (in practice I find this hard to achieve with expansion and cumulative review activities).

These activities should be mix of 15% initial instruction on new material and 85% review (including a combination of the different types of review listed here.

Here is an example lesson plan:

Starter: review of two recent topics, prompted by pre-written worked examples with similar features (this task needs to be doable without prompting from me as I need to be settling the class and taking register)
Prompted review of an older topic (I will pick either one that students have previously struggled with, or one that will be required for the topics to be learnt in coming lessons).
Initial teaching of a new topic, blocked practice followed by an expansion activity.
Re-teaching of last lessons new topic, alternating question practice followed by expansion.
Context-shaping or question discrimination activity on the new topics.
Cumulative review including the topics used in activities 2, 3, and 4.

Scheme of Work

A track is collection of skills grouped not by topic, but by similarity of the required student response. For example, ratio sharing, fraction of an amount, percentages of an amount and pie charts may all be taught in a single track. Each of these skills should be taught using highly similar language and highly similar overt responses. The aim is to reduce teaching time by making the connections in maths (that teachers often take for granted) as clear to students as possible.

Three or four of these tracks should be taught in parallel. That way, the interleaving of topics is woven right in to initial instruction. This is very unlike most teaching models, where interleaving, if used at all, is employed only during topic review/revision.

Each track chosen should be highly different (both visually and in terms of required student response) to avoid blending/confusion.

Principles of Engelmann's Direct Instruction: Defining range/scope

2019-07-19T13:28:00.000+01:00

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
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

Deciding on the range

Once you have atomised your curriculum in to components, you need to thing about the range of all possible examples of each of them. When doing this, you should think about how students will use this skill in the long term, not just about the current topic.

For example, if you were teaching "Solving one-step equations using opposite operations", you would need to include equations like:

and

as well as x, ÷, + and -.

It is important to note that this does not mean that these all must be taught straight away. Instead, you can show these as examples of the topic that "we haven't learned how to do yet". Then you can teach these sub-components at a separate date before blending them in to the other example types.

There are several advantages to this approach:

It effectively pre-teaches elements of several topics that will later build upon the component (in this example, trigonometry, formulae, solving quadratic equations, etc.).
It highlights the links between these topics by teaching the components of the topics that require the same response as a single component.
It allows students to see the full scope of the new skill and its potential applications. This also demonstrates the value of learning the skill.
By showing the full range and limitations early, you are avoiding stipulation where students under-generalise or over-generalise the skill or pattern.

Demonstrating range efficiently

This process often involves vastly broadening what may previously have been a simple component. In order to maintain a high pace on your activities, it is important to use an efficient method to demonstrate the full range of a component and its limitations. In DI, this is achieved by juxtaposing examples using the sameness principle and the difference principle.

Difference principle

In order to demonstrate the full range of applications of a component, you should juxtapose (put next to) examples and questions that are as different as possible (in a single dimension). This allows students to interpolate all the questions in between that would also work.

For example:

From example 1-2 and 2-3, only one dimension has changed (the number on the right-hand side), but it has changed in drastic ways:

Between 1-2, the answers turn from natural numbers to negative decimals.
Between 2-3, students must now divide a decimal by 5.

Between 3-4, a new dimension changes (the co-efficient of x becomes a power).

Through the use of four examples, we have showed how the same technique works in several different situations, and allowed students to infer how to apply the method to many questions in between.

Sameness principle

In order to demonstrate the limitations of a component, you should juxtapose pairs of positive/negative examples that are as similar to each other as possible. This allows students to see the exact boundary of where the component stops working and extrapolate to other questions that this skill will not work for. Also, because "any sameness shared by both positive and negative examples rules out a possible interpretation", every dimension that has not changed between the examples cannot be the cause of whether a technique works or does not.

For example:

Each question now looks highly similar, with many features remaining the same, but 2 can not be solved as a one-step equation. From this students will quickly infer that only equations with single variables can be solved using this technique.

By applying the sameness and difference principles, you can effectively and efficiently show the full range and limitations of a skill.

Principles of Engelmann's Direct Instruction: Review

2019-07-19T12:47:00.000+01:00

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
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

Review

According to Direct Instruction, only 15% of each session should be on new content, with the rest dedicated to review. This seems miniscule to me and I have not managed <40% let alone 15% whilst trying it last term. However, I have massively upped the amount of review that I now plan for in my lessons. This has given me much more scope for the reminding and interleaving of topics and it seems to have made a positive impact on student confidence and grades.

Types of review

Reteach: In the follow-up lesson from initial instruction, reteach the component with a set of examples. This is made feasible as instructions now take <1 minute (see atomisation).
Blocked practice: Over the next few lessons, practice the skill in a set of questions on this topic only. This may start prompted, with worked examples or some scaffolding, but these prompts should fade until students can perform on the questions without prompts.
Question discrimination: Where there is another skill whose set-up looks highly similar to the skill just learnt (say worded LCM questions and worded HCF questions), teach how to differentiate between the two and then present a random set of questions on the two topics. Note that to save time, you may get students to only perform the discrimination and perhaps the first step or two of the working.
Cumulative review/Interleaving: A review of up to 6 topics. These topics will change between each review session, allowing students to refresh their memory on a wide range of skills. The topics should include:

The two most recently taught topics
One topic that is highly similar in set-up to the most recent topic
One topic that students have struggled with in the past (and has been re-taught)
Two other topics not reviewed recently

In order to keep track of which topics have been reviewed recently, I use an excel spreadsheet (I - initial instruction, R - review, S - reviewed but struggled with):

Building to cumulative review

Also mentioned in DI is a step-by-step approach for building up to the full cumulative review. I have found this particularly helpful with lower-ability sets. Before I have avoided doing unprompted cumulative reviews with these classes as I knew that students would not be able to remember the topics (indeed they often forget everything from the lesson on the day before!).

Steps to build to cumulative review:

Prompted, blocked questions
Unprompted, blocked questions
Questions alternated between the new topic and a mix of highly different, secure topics
Continue number 3, but add more and more questions in between each question on the new topic
Full cumulative review (after teaching discrimination with highly similar topics)

The idea is to do two or three of these in a single activity, slowly building up the complexity as students become more comfortable. At first you may do 1-2, then 1-2-3, then 2-3, then 3-4, then 4-5, then only 5 (or you may build it up much faster or slower, depending on the performance of the class).

DI also recommends avoiding moving on to new topics whilst students are still struggling with the cumulative review.

Example of question discrimination

I find SSDD problems to be a great place to start thinking about which topics are similar to the one taught. For some classes, I know that I can give them some SSDD questions immediately after initial instruction and they will be fine with just this. However, for many students, particularly with English as a foreign language, I find it useful to first teach the discrimination as a correlated feature before they work up to the SSDD problems themselves. So with pythagoras for example:

First define the correlation: "I know that I can use pythagoras because I can make a right-angled triangle where I know two sides"
Now show various example questions, some on pythagoras and some on similar-looking topics (like trig). The aim is not to answer the questions, but to decide when pythagoras is applicable. Demonstrate the first few before students have to answer.
The answer is either:

"Yes" Why? "Because I can make a right-angled triangle where I know two sides"
"No" Why? "Because I cannot make a right-angled triangle"
"No" Why? "Because I do not know two sides"

Practice this until they are firm.

Notice here that I defined it as "When can I use pythagoras?" rather than "When should I use pythagoras?". Otherwise the correlation would become "I know that I should use pythagoras because I can make a right angled triangle where I know two sides and I want to work out the third side". This is a further example of atomisation - a way of splitting up the "When should I..." into more easily digestible components - to facilitate high success rate of initial instruction.

Therefore a final instruction plan would be to teach "When can I..." then "When should I..." before finally actually answering SSDD type problems.

Practicalities and Addressing Concerns

This may seem like it is going to slow down the teaching of new topics drastically. This will be addressed in the next section on "lesson structure". I have found some slow-down, but it is not drastic. This effect may also reduce over time as students are (hopefully) more likely to remember the topics from last year when learning new topics which build on them in the next.

DI recommends choral response for most of their activities. I think this only really works with classes of up to 10 students. Any more than this and it is hard to hear what everyone says and to make sure everyone is taking part. I like to do these reviews on mini-whiteboards, as this allows me to quickly assess everyone and to give instant feedback on misconceptions.

It is important to keep a high pace during these activities, so with longer, more complex procedures, I may ask students: "What's the first step?" or "What's the next step in the working shown?"

Principles of Engelmann's Direct Instruction: Cognitive Load

2019-07-17T15:09:00.000+01:00

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
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

Cognitive Load Theory

The basic idea of Cognitive Load Theory is that everyone has an inherent limit to how much information can be stored in your working memory at one time, therefore students will struggle to grasp concepts when their working memories are overloaded with data.
Though Direct Instruction does not reference Cognitive Load Theory directly, it still has a lot to say about ways to reduce the cognitive load for students, particularly during initial instruction.

Reduce clutter

Remove all elements of a problem that are non-essential to the skill, allowing students to infer what is essential/non-essential to the concept. This means changing this question:

Abby, Charles and David share out 32 sweets in the ratio 2:5:9. How many more sweets does David get than Charles?

in to this question:

Share 32 in the ratio 2:5:9

This also means removing non-essential elements from diagrams and avoiding the use of 'real-life' pictures or situations (more on this in the "real-world" maths section)

The wording/setup principle

Every example and practice problem that students do during the initial instruction should have the exact same setup, right down to the wording and the font of each question. In practice, along with the "single dimension transformations" discussed below, this looks a lot like Craig Barton's Variation Theory problems:

Single dimension transformations

As seen in the problem set above, only a single aspect of the question changes from problem to problem. This reduces cognitive load as there is a single aspect to concentrate on between questions. Craig Barton points to other benefits:

By holding constant as much as possible and varying one element, we can direct students’ attention to that element that has varied. Any change (or lack of) in the answer may then be attributed to the change in the element. Moreover, because each question or example is related to the one that preceded it, students are able to form expectations as to what the answer will be. I call this process reflect, expect, check. This can lead to significant moments of revelation and discussion when these expectations are not realised, compelling students to think more deeply about the processes involved, instead of just cruising through an exercise on autopilot.

Atomisation

Discussed in more detail here, atomisation, or the splitting of skills down to their component parts, also means that students are learning a single concept at a time. This means there is less information that needs to be stored in their working memory at any one moment.

Addressing concerns

Students won't be able to generalise to similar questions which are worded differently.

This is taught as a separate skill during expansion activities

Students won't be able to repeat the process when a questions is not preceded by several highly similar questions

This is also taught separately during review and context shaping activities.

So no context? No treasure hunts, code-breakers, funny pictures practical demos or anything?! Won't students find maths incredibly boring?

Most models for motivation put a high emphasis on 'perceived ability to succeed' as a main contributor to overall motivation. Atomisation is a fantastic tool for facilitating regular and obvious successful progress. I have found my students have enjoyed my lessons much more this year because they can 'feel' the progress they are making.
All of those activities are possible to fit in to a Direct Instruction program, and should only be strictly avoided during the initial instruction phase. Because each component is small, each initial instruction phase should only take a few minutes. This frees up time to do the code breaker/treasure hunt as an expansion or review activity in a future lesson.

In the Direct Instruction model, the aim of initial instruction is to teach the central concept only, whilst avoid the learning of misrules. Separating the concept, question identification and recall and teaching them as separate skills is a further tool for reducing cognitive load.

Principles of Engelmann's Direct Instruction: Expansion and Context-Shaping

2019-07-15T10:05:00.000+01:00

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
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

Expansion:

Expansion takes place after initial teaching, where things have been simplified to the only the essential features (see cognitive load). Expansion then introduces more variety, showing how questions can be worded differently and still mean the same operation

One type of expansion activity would be 'implied conclusion' tasks. These are tasks that you use the current skill (possibly alongside other skills which are already firm) to solve new problems. An example of this is ordering fractions after being taught how to compare fractions.

Matching ("circle all the fractions greater than two thirds") and "here is the answer, make up the question") are other types of expansion activities.

Context Shaping

Context shaping is the name given in Direct Instruction for demonstrating the range of contexts where a skill can be used.

This may be covered in a single activity or multiple activities, depending on the number of different possible contexts and the amount of difficulty added with each new context.

DI recommends juxtaposing highly different contexts one after the other (before interleaved practice). This maximises the chances that students will interpolate and infer any contexts in between those shown.

Both expansion and context shaping activities should happen as soon as possible after students are firm on the initial teaching. This is so that students do not stipulate by viewing the skill in too narrow a way.

Example:

These questions from https://www.map.mathshell.org show a variety of contexts for using direct proportional reasoning:

*Note, there is also some question discrimination here as students need to decide if the numbers are in direct proportion or not.

I would likely split this in to several activities in order to reduce the difficulty of individual sessions. In order to juxtapose highly different contexts I might show them in this order (though other orders would also work):

Speed/cycling and driving
Lines
Similar shapes/triangles
Maps
Money and Petrol

Reflections

Textbooks generally do this in their exercises. However, I find the structure that DI gives to these activities helpful for phasing in the wider activities. So for a high ability set, I may just give them the textbook and let them get on with it and, for a low ability set, I can expand their understanding in small stages over several days. This allows students of all abilities to feel successful and to deepen their understanding of a topic, something I definitely struggled to get right in the past!

Principles of Engelmann's Direct Instruction: Overtisation

2019-07-12T11:38:00.000+01:00

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
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

Overtisation:

Overtisation means giving a visible outcome to every cognitive decision students need to make. This goes further than the standard steps students are usually asked to write to "show their working". These visible outcomes could be oral, written or physical but they must be 'assessable' in that they show you whether a student is understanding that step. There are two reasons to do this:

Students get a visual or physical hook to remind themselves of what they need to do next. This allows students to get a 'feel' for when they are doing the right steps (much like you do when learning a physical skill like throwing).
The teacher gets clear indications of exactly when and where students are making mistakes.

Fading:

After the steps have been embedded in to a cognitive routine and the cognitive routine has been developed to a good level of understanding, you allow the students to 'covertise' these steps. This means they start to show less and less working to the point of minimal model answers. This is necessary to keep the pace up for later practice/assessment.

Example:

Before answering , students:

first read out the question (to assess understanding of the symbols for multiplied fractions)
write: to show what operations they will perform.

Once embedded, students would remove these steps and go straight to an answer.

It is worth noting that Engelmann often goes way further than this, having 10 whole steps to complete when 7 + 3 is the question!

Unanswered Questions:

Here are a few reflections I have had about covertisation that I still do not have the answer to:

How far do you go? 10 steps for single-digit addition seems like overkill, but maybe wouldn't be for some classes. My guess is that this comes down to teacher judgement.
When do you covertise? Engelmann goes in to some detail, but it is not clear to me how I judge this for a whole class. Again, at the moment I am using my personal judgement.
There is a trade-off between overtisation and speed of practice: the more steps you take, the longer you spend on each question. This surely needs to be considered when thinking about doing it.

By the way, you may be thinking "what about unassessable skills, like critical thinking and decision making? Does DI fall short of teaching these things?" I will be addressing this in the instruction for problem solving techniques section.

Principles of Engelmann's Direct Instruction: Atomisation

2019-07-10T13:06:00.000+01:00

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

Atomisation

Atomisation, to me, is the part of Direct Instruction that I found both the easiest to implement and the most impactful in terms of student outcomes. It is the splitting up of a skill or topic in to its indivisible component pieces of knowledge. It is important that these components each have a overt, assessable outcome (more on this in overtisation).

Each component skill can then be categorised and taught separately before being put together to learn the target skill or topic. There are several advantages to doing this:

It reduces cognitive load as students only have to concentrate on one aspect at a time.
It is easier to correct mistakes as there is only one component for students to misunderstand.
With a narrower focus to the topic, students can explore the full range of this component quickly. This means that they are more likely to get a deep understanding of the component.
It allows for a high percentage of student successes, building confidence in their abilities.
It gives you, as a teacher, a better understanding of how difficult a topic will be for students.
It helps to avoid stipulation, the learning of misrules about a topic.
It makes behaviour management easier.

Example:

Let's take a look at right-angled trigonometry:

In the past, I would probably still have done some splitting up of this topic in to smaller parts. Perhaps by working on missing angles before missing sides, or by looking at only sin before cos and tan. However, I don't think I really understood how complicated trigonometry can be until I atomised it. This is how I would split up the components of basic trig:

Deciding when trigonometry can/should be used
Deciding which equation to use

Labelling the sides of triangles as opp, adj, hyp
Remembering the three trigonometry triangles
Write down 9 equations from the three trigonometry triangles

Using compound measure triangles to create equations

Solving the equation

Substituting values in to a formula

Using algebra as a variable

Solving one-step equations involving inverse trig functions

Applying inverse functions to a fraction
Using algebra as an unknown

Using the trig and inverse trig buttons on a calculator

Combining the above steps in a clear routine

Now let's look at how this affects each of the advantages above.

Imagine the cognitive load of weaker students trying to process all that at the same time!
Students may have a misconception about any one of these components. This misconception may only occur for certain types of questions and it may not be visibly shown in their working. This makes it very difficult to diagnose mistakes. However, if you are only teaching how to "Write down 9 equations from the three trigonometry triangles", you minimise the number of different misconceptions students can have (either they get the order wrong or they mess up the sin(angle) part).
In the past, I may have scaffolded trig for my students by looking at an easy subset of the problems (e.g. only sin, only finding hypotenuse). This can easily lead to students learning cognitive shortcuts that do not work for all questions. Another common difficulty with trig is the relative nature of opposite and adjacent sides (the fact that these switch when seen from the perspective of different angles). However, if you teach "Labelling the sides of triangles as opp, adj, hyp" as a separate skill, you can explicitly show all variations of this and students can practice and be assessed on it in depth in a matter of minutes.
Before, I have despaired at the thought of teaching trig to a lower ability class! In the past I might have tried whilst only expecting 50-60% of students to succeed (cementing students' self-images of low intelligence) or skipped the topic altogether. However, atomisation gives me a route map to build up their knowledge whilst ensuring the success of >95% of the class on each component.
I now have a much more detailed idea of how difficult the topic will be for students and the potential pitfalls they may have. Combined with my knowledge of the class, I can decide for each component whether they need to be assessed, recapped with review activities/homework or completely retaught/taught from scratch.
Stipulation, or the learning of misrules, can be very difficult to remedy once embedded in to their practice. As mentioned in point 2, atomisation helps to spot and correct these misunderstandings during initial instruction.
Teaching trig is complex. To explain it takes at least 5 minutes, even without students' questions and other interruptions. Getting 5 minutes of undivided attention from some classes is a big ask. However, all of the components listed can be explained in under 1 minute. I have found that students are much more willing to give you their attention when you ask for it in short, efficient bursts like this.

Unanswered questions:

Here are a few reflections I have had about atomisation that I still do not have an answer to:

Is it ok to combine components sometimes in order to speed up the teaching of a topic?
When is it ok to combine components?
With which types of components is it ok to combine them?
How different would this look with different classes?

Other practicalities:

It is important to show the full range of possible questions that can be asked of any particular atom to avoid students learning cognitive shortcuts. These shortcuts can cause serious problems if embedded in their learning as they can be very difficult to unlearn.

If you cannot teach the full range of possible questions in a single session, you should still show the full range, with students marking off the questions that "we don't know how to do yet".

Exploring the difference between averages

2017-05-19T09:59:00.000+01:00

Two tasks designed to get pupils thinking about the different averages and how they are affected by new numbers added to a set. It preempts this lesson on which average you should use and why. Its also quite a nice bit of cross-curricular thinking.

Task 1 - Averages and probability (editable files here)

Task 2 - Match the averages to the description

Folding Fractions

2017-05-15T21:48:00.000+01:00

Skills Practiced:

Spatial reasoning
Fractions of shapes
Multiplication of Fractions
(Potentially) Pythagoras

Material:

Paper cut in to squares
Display Board

Instructions

Each student gets a square piece of paper.
Ask if anyone can fold the square into a shape that is 1/4 of the original size. Ask them to explain how they know that it is exactly 1/4.
Project the fractions on the picture below on to the board.

Students aim to make those fractions from the original square.
Every time they get an answer, they must convince their partner that it is correct before they put their name on it and blue-tack it to the board. Multiple different answers are allowed for each fraction
Below this (or on another board(, students put up any other fractions they can find.

Example work

Extensions and Variations:

Find the same fractions as above, but now your final shape must be a square.
https://www.youcubed.org/task/paper-folding-fun/

Area and Arc Length of Sectors Puzzle

2017-05-09T17:50:00.000+01:00

Probing questions about finding the mean from grouped data

2017-04-02T12:57:00.001+01:00

Estimating the mean from grouped data is one of those topics that all my students can do in lesson, but tend to forget in tests. Because it is quite an abstract process, students often learn the steps of the method without really understanding what is happening.

To improve this, I have developed two resources:

The first can be found here and it enables students to learn how to find the mean from a table without you saying a word! It is very differentiated and I have used it successfully with students at KS3 and KS4.
The second is the slide below. The idea is to get students to think about how the steps they are calculating relate to their final answer.

The False-Positive Paradox as a Class Activity/Discussion Point

2014-06-06T09:27:00.000+01:00

There is a test that screens for some rare disease (only 1 in every million people is infected). It is 99% accurate (it gives the correct answer of positive or negative 99% of the time). You go for the test and to your dismay it returns a positive result!
What is the chance that you actually have the disease?

You may think that its pretty likely you have the disease. However, from the information above, the actual probability that you have the disease is still very unlikely (1 in 10'102). This is the basic premise of the false-positive paradox.

I don't really like that it is named as a paradox, because it is not actually in any way paradoxical, just counter-intuitive. It works because there are so many more non-infected people (999'999) that each have a 1% chance of getting a false positive than the number of infected people (1) who has a 99% chance of getting a true positive result.

This counter-intuition can cause exactly the sort of cognitive-disonance that can be so useful in getting people to realise that their current understanding needs improving, but in its current form its pretty wordy, inaccessible and abstract. I wanted to make it in to a class activity. Instructions below.

How to run the Class Activity/Discussion Point:

As students enter the room, give them each a slip of paper with a random word on it. Tell them not to show anyone else.

Once they're settled, tell them that there is a (make-believe) disease going round school that only affects about one student per class. In this class the one student, the one with the word banana on their slip of paper, is infected (Don't tell anyone if its you or not!).

Tell them you have a way of checking if anyone has the disease, but its not always accurate. To perform the test, you will ask the student if they are infected. Before answering, the student will roll a dice and will tell the truth if they roll a 1, 2, 3, 4 or 5, but they will lie if they roll a 6.

Perform this test on some students until someone claims to be infected.

Ask the class how likely is it that this student is actually infected. Ask for justifications. Most will say very likely, some may evan say 5/6.

Perform the test on a few more students until you get a few more claiming to be infected.

But they can't all be 'very likely'! What is going on here?

By the way, make sure the students know the disease is not real!

Would this work in your classroom? Could it be improved? At the moment its just an idea and I'd like to know what you think.

Half-Full

2014-03-31T14:00:00.003+01:00

Lesson Sketch: Probability - a matter of life and death

2014-03-18T13:32:00.000+00:00

"In 1999 a woman in the UK, Sally Clark, was convicted of murdering her two children. They had both seemingly died from SIDS (more commonly known as cot death), an unexpected but natural cause of death.

One of the main pieces of evidence against her was the testimony of a Professor and expert on child-abuse. He used the probability of one child (in a family with no known factors that might increase the chance of SIDS) dying of SIDS (1 in 8543). He then squared this fraction to get the probability of BOTH children dying from SIDS being about 1 in 73 million.

The jury saw this number as the probability of an innocent explanation and, since it was so low, found her guilty of the murders. Unfortunately the professor who testified was not an expert on probability and miscalculated. The jury also did not understand the meaning of the statistic being calculated and drew the wrong conclusions from it.

In this series of lessons we will look at why this probability of 1 in 73 million is both wrong and irrelevant to the case. We will also look at how probability could have been properly used in this case."

This is the introduction to a series of lessons on probability.This case will frame the students' work on probability throughout the unit and not one student will ask "when am I ever going to use this?"

There are many options on how open or closed a project this is. Because I was short on time (being interviewed) I first asked them if they could spot any problems in the probability (useful for showing progression) before working in quite a closed manner, as a class, through section 1**.

Section 1 - Improbable events never happen

Introduce expected frequency of events. After students understand the general concept and are able to calculate the expected frequency of different events. Get them to work on the questions below:

The probability that both babies in a family of two will die of cot-death is 1/73'000'000.
There are about 340 million families with two children in the world.
1) How many families would you expect both babies to die of cot-death?

The probability of winning the lottery is 1/14'000'000.
About 7.5 million people play the lottery each week
2) How many winners would you expect to get:
i) Each week
ii) Each month
iii) Each year
iv) In 10 years

The probability of being born with 11 fingers or toes is 1/500.
3) What other information do you need to estimate the number of people in Bristol who were born with 11 fingers or toes?

4) Look at your answer to question 1. How is it related to the court case mentioned earlier? Does the answer to question 3 help you decide whether the woman was innocent or guilty?

5) Since the probability that both babies in a family of two will die naturally is 1/73'000'000, does that mean that the probability that both were murdered is 72'999'999/73'000'000? Explain why/why not?

I then got students to discuss question 4 in pairs before sharing with the group.
Concepts to get across:

Probability is an estimate of frequency and whether something will happen or not depends a lot on the number of trials.
Probability can not say for certain whether she is guilty or not as unlikely events do happen (though we can't be sure that's what happened here).
The 73 million outcomes will include mostly families where neither child has died or a single child has died.

You could also go in to relative frequency here. This may help students understand where the 1 in 8543 statistic might have come from.

Section 2 - Dependent and Independent Events

Here you need to go in depth about the difference between dependent and independent events. Pupils need to be able to distinguish between the two, have some understanding about how the outcome of one event can change the probability of other events (e.g. if it rains today I am less likely to hang my washing out) and how to draw a probability tree from conditional probability problems.

What does this have to do with the case? Well, the probability of 1 in 8543 can only be squared if the death of the two children are independent events (otherwise the probability of the second death would be a different fraction). The causes of SIDS is not entirely known, but it is highly plausible that there could be genetic or environmental factors that would be common to the two children. This would mean that if there were a SIDS death in a family, then further SIDS deaths would be more likely than otherwise.

Students can work on these questions to see this in action*:

1a. The probability that a family's first baby will die of cot-death is 1/8543.
If the first baby dies of cot-death, the probability that the second will also die of cot-death is 1/442.
If the first baby DOES NOT die of cot-death, the probability that the second will die of cot-death is 1/11002.
Draw a probability tree to show this information.

b. What is the probability that:
i. Neither child dies of cot-death?
ii. One of the children die of cot-death?
iii. Both children die of cot-death?

c. Using the number of families in the world with two children from the previous lesson (340 million), calculate how many families would you expect both babies to die of cot-death?

d. Why might the probability that the second child will die of cot-death be affected by whether the first child has also died of cot-death?

Section 3 - Further Conditional Probability

Now we get in to Bayes' theorem and the real reason why the 1 in 37 million statistic is simply irrelevant to the case:

As seen in question 5 from section one, the probability of 1 in 73'000'000, even if it were correct, is not the probability of innocence. It would instead be the probability that if you picked any random family of two children, both children would have died AND the cause of death was cot-death. This is not appropriate to use in this case because we already know that both children died.

Again, pupils could first work on standard conditional probability questions until they are comfortable with the function and usage of the above formula. Then pupils are going to calculate the probability that both children have died of cot-death GIVEN that both children have died. They can also compare this to the probability that both children were murdered GIVEN that both children have died:

The probability that both children in a family of two will be murdered is 1 in 10 million.
1. What is the probability that both children will die of cot-death OR murder (use the probability calculated in the previous session).

2. What is the probability that both children have died of cot-death given that both children have died of EITHER cot-death OR murder?

2. What is the probability that both children have been murdered given that both children have died of EITHER cot-death OR murder?

3. How do these two probabilities compare?

4. How does this change your view of the case?

5. Does this help you decide whether the woman was innocent or guilty? How?

Wrapping it up

A great way to recap all of this would be to get the students to prepare a letter to send to the judge of the case explaining the ways in which probability has been misused in this case and in what ways the calculations could be improved upon.
What is particularly great about this is that when they are done you can compare their letters to the letter written by the The Royal Statistical Society voicing their concerns (here) or the more specific letter written by Professor A.P. Dawid (of the R.S.S.) for her (successful) appeal (here)

Comments and Thoughts

*Although many of the figures used above are directly from the court case, and some are estimates based on other data I was able to find online, some of the data is currently unavailable and therefore is made up and should be used purely for illustration purposes.

**How I would prefer to run it:

Start each class recapping previous work and ideas on the case
Introduce the topic of the lesson and work on standard questions of that topic
Ask students: "how this topic might be related to the case? What data would we need to apply today's skills to this case?"
Give them the data they ask for and let them do the calculations.
Ask students: "How does this new information change your view of the case? Does this help you decide whether the woman is innocent or guilty? Why?"

There are many other problems with the probability used in the case including (but definitely not limited to):

Some environmental factors were included in the calculation, which make the probability of cot-death less likely, but many other factors in this individual case (which could increase the risk of cot-death) were not included.
Cot-death is not the only alternative cause of death to the theorised double-murder, but the probability of these other causes of death were not included.
The study where the 1 in 8543 figure came from was not intended for use in a criminal trial and instead is looking at possible causes of cot-death. It does not, therefore, directly apply to this case.

I thought for some time about the ethics of using a real, living person's tragedy as the basis for a Maths lesson. However, since this particular miscarriage of justice is due solely to the misunderstanding of basic probability by many of the involved parties, its use to encourage a better understanding of this topic seems appropriate. I wouldn't want to involve this in a quick and easy pseudo-context question. Instead it should be handled with the importance and gravitas it deserves.

Here are some other links about this case:

Wikipedia
UnderstandingUncertainty.org
Berkeley University - This one is my favourite. Though it is more complex than some of the others, it has a lot more information about the lack of data in some areas and the assumptions that need to be made in order to make the calculations work. Interestingly, this paper argues that the probability of double-murder is much higher (over 99%) than stated in any of the letters endorsed by the R.S.S. (the point still stands, however, that statistics, if used in court at all, should be verified and reviewed by professionals).
Bad Science

The True Cost of Gaming: Adjusting for Inflation

2013-12-19T08:29:00.000+00:00

I was inspired by this image:

by Auir2blaze

If I was American, I would just blank out some of the numbers and call it an activity. Being from the UK, I had to do a bit of leg-work:

There are plenty of intriguing questions here, such as:

Which Xbox was cheapest?
Which is the most expensive console ever?
Are consoles cheaper to buy in the US or the UK? (using the original image as a reference and a conversion rate taken from the internet)
How can we have percentages that are more than 100%?
What did SEGA do wrong? How did it lose the console war? (Speculation)

Answers:

Link to table on a google doc

Improving Functional Skills in Mathematics

2013-12-12T18:22:00.000+00:00

As mentioned in a previous post, I have been thinking lately about how to attack Functional Skills Mathematics to improve results. I have decided on a 4-pronged approach.

Prong 1 - Improving literacy in maths

I have already made a post with ideas and resources for this, which you can find by clicking here.

Prong 2 - Improving problem-solving skills

As functional skills is so focussed on long, involved questions. Students need to feel confortable deciding what steps they need to take and set out their work so that they can follow it and check for mistakes.

3-Act Maths - I find this way of working to be an ideal way to introduce problem-solving to students in a non-threatening way. Starting with just the actual question and no constraints, variables or other information means that problems don't even look mathematical at first and students feel free to take guesses, make assumptions and bring in outside knowledge. Getting students to work out 'the information they would like in order to work out the answer' means that, when you bring those variables in, they are no longer confusing and unclear, but are now the very tools the students were asking for. I've used 3Acts from Dan Meyer's list, problems over at Finding Ways to Nguyen, and picture and video prompts from 101qs. I also sometimes just take a book question, skip all the set-up to the question and just ask the actual question (e.g. How long would it take me to cycle to Edinburgh? Can we get enough rain-water from the college roof to have toilets that run on the recycled water? etc.).
Structuring of workings out - e.g. using the problem scaffolder from the previous post on literacy. The idea of this is to help students to think clearly about the steps they are making and what they are trying to calculcate from each step.
Discussion of problem-solving strategies - I made some posters last year which I have up on the wall. Before tackling a problem I ask "which strategy could we use here? Why will that help?", when students are stuck I ask "which strategies have you tried? which might you try next?", when they finish I ask "which strategies did you use? did anybody use a different strategy?" Getting students to a point where they're comfortable with using these strategies and are starting to get a feel for when each one is applicable can only help their confidence when faced with a difficult problem.
Problem Posing - I originally got this idea from Matt Ives (blog post about it here). Basically the idea is to move focus away from getting the answer and move it on to thinking about the steps you would take and what information you would need. Matt Ives' way of achieving this: don't have any numbers, so pupils can't work out the answer, instead rewarding students who work out the steps they would hypothetically take to find the solution. He has a whole raft of them here on scribd and they've worked very well in the past, once students get away from trying to find an answer.

Prong 3 - Improving mathematical skills

Regular lessons - In amongst the literacy and problem-focused lessons, I will also include regular skills lessons. These will come before more complicated problems that involve these skills (along with others) so that students get to see them in context.
Starters and Homework - I will also include general basic skill revision in lesson starters and homeworks.
Fill in the blanks on calculation tables are a great way to make repetitive practice of a skill more interesting. I wrote a post about it here.

Prong 4 - Putting it together (and improving clear working)

Exam questions - The previous prongs will then be put in to practice when we do past-paper questions in class and for homework. I will link the problems to the work we have done on literacy and problem solving and discuss the skills that will be needed to solve the question.
3-Act questions rewritten as past paper questions and vice-versa - After solving a 3-Act problem, I will show a version of how I think it would look like in an exam paper. Hopefully this should make students feel empowered that they've just solved such a difficult-looking question and will help them see the link between the problem-based lessons and the actual exams. After practicing an exam problem I will get students to try to imagine how it would look as a 3-act problem. Not only does this reinforce the link between the two, but help students decide which part is the question, the constraints, the variables, etc.
Peer marking - Will help students to see the importance of clear and methodical working as well as to think about different problem-solving strategies that others have used in their working.

Secret prong 5 - Improving Engagement

Overarching all of this is what I feel that I actually do best as a teacher, which is to show students that maths can be enjoyable and engaging. I achieve this in the following ways:

Improving accessibility - 3-Act maths are great for this, as are other rich tasks, and the fill-in-the-blanks calculation tables. This is because of the multiple entry points to the questions, and the lack of immediate reading and comprehension required. If students' literacy can be improved, then the more wordy questions will become less and less daunting.
Improving confidence - As students get more and more comfortable with multi-step problems they will see that there are things that they can achieve in maths. This never becomes obvious to students who only practice the basic, underlying skills, no matter how good they get at them.
Improving relevancy - By this I don't necessarily mean 'real' maths, but I mean maths that students can imagine themselves using in their lives. This is where are a focus on problem-solving strategies can be helpful, because even though the problem itself might not relate strongly to them, being able to use the strategies is applicable by anyone.
The strength of my sparkling personality? ...

Inspiring Mathematical Applications: Randall Munroe answers all

2013-12-06T12:21:00.000+00:00

As a Maths teacher, I'm guessing you'll already be familiar with Randall's awesomely nerdy web-comic, xkcd:

Maybe you're also aware of his "What If?" site. If you're not, I'd like to introduce you to it. Each week Randall Munroe, a physicist and former NASA employee, answers question that people post to him online. Though the questions are usually strange ("How fast can you hit a speed bump while driving and live?") and sometimes ridiculous ("If you suddenly began rising steadily at one foot per second, how exactly would you die? Would you freeze or suffocate first? Or something else?"), Randall tries to answer them as accurately as he can, backing up his statements with citations and calculations.

Though, the maths sometimes gets quite complex, the funny topics and humourous pictures combine to make it much more approachable than it otherwise would be.

I love to read them just for myself, but I've also started bringing them in to some of my classes. Last week I found lots of ideas and resources for improving literacy in maths. This week, one of the ideas I tried was about scaffolding the problem, specifically splitting a question in to its important parts (The actual question, the information given, the constraints, what calculations are needed, etc.). As a plenary I asked them to guess the answer to, "Is it possible to build a jetpack using downward firing machine guns?", then to read through the What If? article about it, whilst highlighting the different components of the text (The question, the information gathered from research, the calculations, the constraints, etc.).

It worked very well, and the students really saw the link between the work they'd just done and the work done in the article. I'm hoping that this link will help them to feel empowered (because the techniques they are using work on much more complicated questions) rather than disenchanted by the complexity of the maths.

Improving Literacy in Maths

2013-12-03T13:15:00.000+00:00

I've been thinking, lately, about how to improve literacy in Maths. If you would like to read the strategies I have so far come up with, skip to the first heading.

For the first time this year I am teaching Functional Skills Mathematics (Levels 1 and 2). The scheme of work I have for these classes is pretty bare-bones and is very skills-based. I am finding it hard to stick straight to the scheme of work, as most of these skills are actually more basic than what my students have already seen when attempting their GCSE and are made even simpler as there is no non-calculator test.

Historically, many students fail the level 2 course and when I asked my colleagues why, they all said that it was down to poor literacy skills. This is understandable when the questions all look like this:

Obviously literacy isn't the only difficulty with this question, so I have decided to teach Functional Skills with a four-pronged approach. I will go in to detail about the other prongs of my plan in a separate post... Prongs.

Prong one: Ways to Improve Literacy

1. Scaffolding questions and correct working

This can be done in different ways.

So far I have been getting them to write titles for each seperate part of their working before starting any calculations, so that they can then split the work in to more managable chunks.
In future I will try using a problem scafforlding sheet like this:

Problems with clue cards would also help to split up a question and promote discussion and team-work. It works like this: Each group gets a simple question. Each individual in the group also gets a different card with a single piece of information that is needed to answer the main question. Members of the group have to decide how to use and combine the information sheets they are given. Here is an example (For lots more, click here):

Question sequencing: All the correct working for a set of problems is written down on separate tabs of paper, but their order has been mixed up. Students must put the working in to a correct order.

2. Building vocabulary and making connections

This could include:

Word-walls: Where mathematical words can be written with their definitions and/or diagrams and grouped according to connections with other words.
Connect-two: You have a list of words at the top (eg. Percentage, fraction, TAX, Chocolate). You pick different pairs of them and explain in what way they are connected and give examples.
Word-splash: Similar to connect-two, but the words are on the board and you pick people to make a connection.
Anticipation guide: A table where the first column consists of statments and/or questions (eg. A decimal number is a number less than 1). The second and third columns give a space where students decide whether they think the statement is true or false and why. The second column gets their opinions before working on the subject and the third column gets it after working on the subject (good evidence of progression here too!).
Tarsias: These are jigsaws where students match a question to an answer (or a word to its definition). Mr Barton has loads available here. You could also print off a blank tarsia and get students to create one.
3 facts and a fib: Students write down three facts and a lie about a maths object, word or topic. They try to make the lie believable enough that when they pass it to their neighbour, the neighbour can not work out which statement is the lie.
Definition map: to the example below I would add a space for illustrations of the word or concept:
Taboo: The classic game game be adapted easily to a nice plenary activity - give them a maths password (eg. Mutually-exclusive) and a list of taboo words that they are not allowed to use (eg. Probability, events, same time, both). The chosen student then has to describe the password, without using any taboo words, to the class until someone can guess it.
Pictionary: Similar to taboo except that you make drawings to represent the word.
The Mathematics Assessment Project has many matching activities designed to promote discussions and expose misconceptions, as well as Professional Development modules to help you use them effectively. Some of them are excellent.
Think, pair, share: Students work on their own on a question. In this time they may do some calculations but should focus more on organising the information and making notes. Next they get in to pairs. First they should take it in turns explaining their thoughts (one talker, one listener) so far, before working together to answer the question. Finally, they join another group and explain how they got to their answer (again taking it in turns so that both groups get to explain their work).
Odd one out: Take three different mathematical objects (eg. triangles) and get students to decide which is the odd one out, then discuss their decision with their partner. This works best if each object could be the odd one out for a different reason (isoceles, right-angled, different area, different perimeter, etc.). That way different students get a different answer and their discussions lead others to see the objects in another way.
Carroll diagrams: Two-way tables where students have to sort objects in to the correct section that matches the description of that word. Here is a good example.

3. Peer marking and constructive discussion

Marking each others' work will help them understand the improtance of clear working, with titles, explanations,etc. Discussing how a correct answer could be improved to make it more clear would also help with this.
Always, sometimes, never true: Students read a statement and decide whether it is always true, never true, or sometimes true. They then justify their answer and compare results with their peers (examples can be found here).
Rally coaching - Students work in pairs. They take it in turns with one person answering a single question, whilst the other coaches. Coaching could simply be helping to solve the questions or could be more complex, with rules like, "avoid telling them what to do. Try to ask a question that will help them to figure out the next step."
Many of the activities from section 2 would also apply here.
Bowland Maths have a Professional Devolopment activity (module 3) described as "Fostering and Managing Collaborative Work: How can I get them to stop talking and start discussing?" I have not been able to look at this yet as there is no sound on my computer at school, but I will update the page when I do. I have already enjoyed and would recommend some of the other PD modules on this site though.

4. Filtering information from text

Students highlight the different parts of a question in different colours (question, given figures, constraints, etc.)
Students write the gist of a block of text using only 20 words. Students could create an individual version, then a version in pairs, before writing a final one in their books.
There is a video series (click here) with worksheets (click here for the worksheets), where you are asked to fill in information as you hear it on the videos.

Credits

Much of these ideas (and plenty more ideas!) have come from:

"Growing the Connection Between Mathematics and Best-Practice Reading and Writing Strategies" by Jennifer Kosiak, Sue Schumann, Ann Harry and Bonnie Jancik (available here)
"WHAT’S LITERACY GOT TO DO WITH IT? Literacy in the Math and Science Classroom" by Blair Covino and Barb Mazzolini (available here)
"Literacy in Maths" by The Highland Curriculum for Excellence (available here)
Other sources that have been linked above: Cumbria Grid for Learning, Mr Barton, Mathematics Assessment Project, Nrich, Bowland Maths and Channel 4 Learning.

Random glitch on an online weather map creates a vector field:

2013-11-30T13:24:00.000+00:00

beautiful, no?

Building understanding of the equals sign

2013-11-21T19:36:00.001+00:00

I recently read a paper called the Importance of Equal Sign Understanding in Middle Grades, where it is argued that most middle-schoolers view the equals sign as a kind of signifier for "That's where the answer goes" (3+4=?), rather than understanding the deeper, relational meaning (3+?=3+29+10 or y=2x+1), and that this significantly hinders their understanding of simple algebra problems.

I still see this misunderstanding whilst currently teaching college-age students. This is worrying.

Ways that understanding can be taught:

Dave Orritt has created a lesson with resources looking at questions of the form 3 + __ = 2 + 9, etc. of varying difficulty. I would have liked a little progression such as: 5 + 21 = 13 + __ 5 + 21 + 154 = 13 + __ + 154
Nrich has a nice problem that involves changing equivalences, so that even before you get on to generalising, you can look at how the equals sign still applies as you move around the grid.
Here are two (first and second) similar web-apps dealing with balancing equations. They are made to be used to help learn solving equations, but you could also just let students try things out to see what happens to the equation.

Explore linear graphs starting with finding values of x and y that make the equation x + y = 5 (or any other) true, then plotting them all as points on a graph at the front of the class. (What happens to x as y increases? etc)
Building up equations. Can't remember where I found this, but its the best version of its like that I've seen:

Which could then be followed up with students building their own similar webs.
Practicing simple, worded and otherwise, formulae; including substitution of different variables and rearranging. You could also ask if different forms (e.g. F=ma, F/a=m, Fm=a) are always, sometimes or never the same and to justify their answers.

Equation chains - e.g. 3 + 2 = 9/? + 2 = (2+?)/3 + 2 = 25/?

Hands-on equations or something similar

How would you do it?

These are some initial suggestions but I'm wondering if anyone has any other ideas.

How to create enagaging, functional maths problems in 2 minutes *UPDATED

2013-11-11T10:25:00.000+00:00

Step 1: Take a table of calculations that have already been done:

e.g. Bills, receipts, timetables, answer keys to textbook functional maths tasks.

Step 2: Progressively blank out more and more information:

Here the aim is to create questions that get harder as you work down the page. Often that means either blanking out more of the boxes or blanking out the initial boxes from which the calculations are made (creating a reverse, "here's the answer, what's the question?", problem).

Step 3: Profit?

For some reason, my pupils really seem to enjoy these. One possible reason for this is that it makes any random table in to a kind of logic puzzle. Mainly though, I think my functional maths groups get a bit sick of your standard functional maths problem, which looks like this:

Lots of good, and relevant, Maths here, but its terrifyingly wordy, there are so many constraints thrown at you at once, and you have to keep flipping between the question, your working and a data-sheet.

Blanking out a table can't replace these types of questions, as this is exactly the type of question that they will be tested on in their functional maths exams, but it is a good way to introduce a complicated set of calculations in a non-threatening way. The further constraints and wordy questions can be introduced later.

Optional extras:

Before you show the table, get pupils to think about what its going to contain/what calculations will be done (e.g. "If you were trying to work out how much you'll be getting paid at the end of the month, what pieces of information are important? If you knew these things, what calculations would you do?").
Use this as an intro, to get pupils accustomed to the table and calculations involved, then bring in the wordy questions as a follow-up.
Get the pupils to come up with the constraints for the more complex problems (e.g. "Write a list of steps to take for getting up and taking the bus to school. How much does each one take (roughly)? When do you want to get in to school? Which bus should you take?")

(instead of this question)
Involve pupils in the process - I haven't tried this one yet, but it could be done when they have completed a more textbooky problem. Get them to tipex over some of the numbers in their calculations in order to create a problem set (as in steps 1 and 2 above), then swap books with their neighbour and try to fill in the tipexed blanks.