VIC MATHS NOTES

5 Ways to Use Worked Examples

19/5/2023

I recently wrote a blogpost for Think Forward Educators on worked examples. Enjoy the snippet below or read the full thing here.

If you put a problem in front of a student that involves new or novel information or skills that you are not expecting them to know, either they solve the problem (with time and lots of effort, by chance, or because they have been shown by someone else), or they can't solve the problem in which case we need to show them how.

We can streamline this process by showing students how to solve the problem from the get-go. This avoids, in extreme cases, the "guess what's in my head" game where students blindly guess because they have no clue, as well as possibly disenfranchising students who (rightly) don't know how to complete the problem when other students are able to (possibly from being shown beforehand). We can show students how to do something by using a worked example.

The worked example effect describes how we can enhance learning "by studying worked examples to problems rather than by trying to solve the original problems" (Ayres, 2012). We use worked examples to leverage the knowledge of experts who know how to complete a skill to show a novice how to complete that same skill. This involves breaking the skill into clear, concise, and manageable steps that the novice can then replicate.

Read the rest of this blogpost here.

Cognitive Load Theory and Engelmann: A Starting Point for a Common Vocabulary

20/11/2020

Engelmann and Carnine's Theory of Instruction has a reputation for being somewhat inaccessible to many who attempt to read it. It's big, it's long, quite a number of examples are maths related, and lastly (and probably most crucially) they use their own terminology when referring to specific ideas.

Over the past few years in particular, a number of ideas from Cognitive Load Theory have become more widely known and the terminology used has spread further. What I've found from reading Oliver Lovell's book Sweller's Cognitive Load Theory in Action is that a number of the ideas in Theory of Instruction relate back to the ideas in Cognitive Load Theory. That creates an opportunity, where the ideas in Theory of Instruction can be linked to ideas in Cognitive Load Theory to create a more widely shared vocabulary. Below I've used quotes from Ollie's book to compare Cognitive Load Theory with Engelmann and Carnine's Theory of Instruction.

MathsConf24 Presentation

4/10/2020

Yesterday I had the pleasure to present virtually at the LaSalle's MathsConf24 on Introducing Coordinate Members to a Set, an idea from Engelmann and Carnine's Theory of Instruction. If you're interested in seeing more about it, you can find the slides here.

How I Teach Sketching Transformed Graphs

26/7/2020

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Pdf copy of this post is available here for better formatting and math text.

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Teaching how to sketch a transformed graph of a function requires students to

know the shape of the untransformed graph
know key points of the untransformed graph to transform
know what transformations (dilations, reflections, translations) apply what operations (multiplication, multiplication by negative one, addition) to which variable (x, y)
know how to determine the transformations from the equation of the function y=f(x) → y=Af(b(x+c))+d

For most functions and most transformations this hasn't been a major hurdle to deal with. There is, however, one transformation, for one set of functions that has caused me and my students grief every year. And every year I've modified and overhauled how I've taught it to make easier for students. It is horizontal translations for circular functions (sine, cosine, and tangent).

Variation Theory and Engelmann: A Comparison

19/7/2020

In this post, I aim to look at the similarities and differences between the sequences and programs as described by Engelmann and Carnine in Theory of Instruction compared to the sequences and exercises described as variation theory. This post is in no means meant to claim that one is better than the other, only to show the similarities and differences between the two to prompt further thinking and discussion.

This post is broken down into parts:
Part 1: Examples and Non-Examples or Positive and Negative Examples
Part 2: Minimally Different Questions for Transformations and Correlated Features
Part 3: A Short aside about Essential and non-essential features

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Printable version here. It has much better formatting and maths text than Weebly will handle.

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Thoughts about Arbitrary and Necessary Knowledge

2/7/2020

It was during Stuart Welsh's session during MathsConf23 that I came across the notion of arbitrary and necessary knowledge. Intrigued, I decided to look up and read the article Arbitrary and Necessary Part 1: a Way of Viewing the Mathematics Curriculum by Dave Hewitt.

Dave Hewitt creates a distinction between two types of knowledge in the mathematics curriculum:

knowledge that you cannot arrive at by yourself, you need an external source of information (words, symbols, notation, and conventions). That is, arbitrary knowledge.
knowledge that, given the right prior knowledge, you can arrive at (or work out) by yourself because it is necessarily true (properties and relationships). That is, necessary knowledge.

This distinction is later used to describe how the two types of knowledge could be taught

Blog Separation

4/4/2018

Just thought I'd make a note to say that I split up the blog into different parts, one for each year level (and subject for VCE) so that I could make the update tickers at the top of each page. This main blog will mainly be used for major updates to the whole site or other blog posts that I decide to write in the future.

Methods 1&2 Update

19/2/2017

Methods units 1 and 2 are now fully updated with all the changes from units 3 and 4 and completion of all parts of the study design. Technology notes are coming at a later date.

Year 11 General
Euler's Formula for Planar Graphs updated
Purchase Options updated
Orders of Magnitude and the Log (Base 10) Scale split into Orders of Magnitude, and The Log (Base 10) Scale

Year 12 Further
Euler's Formula for Planar Graphs updated

Year 12 Methods
Dependent Events updated
Determining the Equation of Exponential Functions updated
Inverse Relations and Functions updated
Linear Function updated
Logarithmic Functions updated
Polynomials updated
Power Equations updated
Probability updated
Sketching Circular Functions updated
Sketching Linear Functions updated
Sketching Logarithmic Functions updated
Substiution and Expanding Updated
Transformation of Functions updated
Trigonometry updated

Methods U3/4 Completed Overhaul

29/1/2017

All the notes for year 12 methods have been updated and finished where they were not before. The missing notes for calculators are coming soon, as are the notes on how to use the simulations and the simulation for sample proportions.

I will update all the complete modules, for those that have been updated since, in the coming weeks now they won't be changing as much.

The year 12 notes will start to trickle down into the lower years from now so keep an eye out for those.

Year 12 Specialist
Anti-Differentiation and Integration updated
Kinematics - Straight Line Motion updated
Sample Distribution of Means updated

Year 12 Further
Linear Relations updated
Sketching Linear Graphs added

Year 11 General
Linear Relations updated
Sketching Linear Graphs added

Year 12 Specialist Overhaul

15/1/2017

The latest to receive a complete rework is units 3 and 4 of Specialist Mathematics. The notes for calculator usage for specialist are still in the works so keep an eye out for when they do get completed.

I will endeavour to have units 3 and 4 Mathematical Methods reworked before term 1 starts.

Year 12 Further
Log (Base 10) Scale updated
Relations of the Form y=kx^n updated
The Normal Distribution (Large) added
The Normal Distribution updated

Year 11 General
Directed Numbers added
Estimation added
Order of Operations added
Orders of Magnitude and the Log (Base 10) Scale updated
Percentage Change Applications updated
Percentages and Percentage Change added
Ratios and Rates added
Scientific Notation added
Exact and Approximate Answers, and Rounding added