Tuesday 28 September 2021

A new type of numeracy approach (that is actually old)

 A single sentence by Lakartos gets me thinking about what we are actually trying to achieve when we attempt to teach adults and children mathematics.

Speaking of the argument he is about to make in his book, Lakatos states: 

[The book's] ... modest aim is to elaborate the point that informal, quasi-empirical, mathematics does not grow through a monotonous increase of the number indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations. [Emphasis mine]

In other words, mathematical prowess does not primarily improve by adding the knowledge of algorithms and formula to our strategic repertoires.  Thus, if you decided as an adult to learn mathematics (and I hope you do!) you would not really benefit from attempting to memorize all the methods taught in your maths books.  Ah, okay, what should you do then?

You would improve your use of mathematics substantially by developing your ability to make guesses (speculating) and then criticizing the guess.  That means improving the quality of HOW you make guesses, and improving the quality of your criticisms.  This requires argument, proof and refutation - the building blocks of mathematics.

This is very different from how things are reflected in the NZQA numeracy standards and how content is taught in the adult sector.  We could argue about the differences between numeracy and mathematics, in particular that numeracy ought to be concerned only with giving people the skills to complete a job.  However, this argument generally hits the rocks when we realise that 'mathematical understanding' and 'agency' are still pivotal to improving numeracy.  

Anyway, if you want to improve your own, your learners', or your children's math skills then think about how you might improve their ability to guess and then to criticize their guess.  Because it is these two skills that help the learner to 'tack' in a zig zag way toward correct answers.

Friday 5 February 2021

Error Analysis continued: Multiplicative thinking and financial literacy.

Error Analysis continued: 

Multiplicative thinking and financial literacy.

Thanks for your comments regarding Jenny’s response to the investment question. Most of your comments were regarding the language used. This is a little bit of a red herring. Like many of you noted, in the adult numeracy domain, language is always intertwined with mathematical thinking. Real world situations do not look like the problems presented in school-based problems. In fact, those school problems may have messed you up! 

See here for more info if you are interested: https://www.youtube.com/watch?v=oqZ1Xg5tUiA&t=4s

Language comprehension is of course a factor in this error, but there is another reason that most ‘non-maths educators’ (if I may be so bold) are unaware of. I’ll explain below if you are interested. There are two parts below, explaining the misunderstanding, and what to do about it. Feel free to just read the part that you are interested in.

 The misunderstanding that led to the error

If you have attended my numeracy toolbox series, you will be familiar with some of the conceptual ‘Big ideas’ that we cover. These are big conceptual ideas that learners develop over time. They include additive thinking (counting, adding and subtracting), multiplicative thinking (multiple groups of units, multiplication and division), and proportional reasoning (fixed relationships between quantities, ratios, rates etc.).

Each of these ‘big ideas’ is considered a new conceptual step that is different in type from the previous. While they are deeply connected, children tend to develop these understandings somewhat hierarchically, that is, they first make sense of additive thinking and then begin to develop an understanding of multiplicative thinking, then develop an understanding of proportional reasoning.

The research argument is that learners need a good understanding of previous concepts to be able make the leap to the next. This is because the concepts become progressively more complex and must be built on previous understandings. Many adults have disengaged from numeracy before consolidating their understanding across these domains. This means many of the adults in our sector are still developing their understanding of multiplicative or proportional reasoning (and additive for that matter).

A second key point is that real world numeracy situations often require different types of understanding. For example, counting stock or cash handling might require additive skills, a painter calculating the area of a room to purchase the correct amount of paint might require multiplicative skills, and mixing 10 litres of a 25:1 petrol and oil fuel mix for a chainsaw might require proportional reasoning.

Adults yet to develop their conceptual understanding of these domains will struggle with tasks that require it (unless tools or instructions remove the need for mathematical thinking). That is what is happening in the error shown in the picture. Jenny is applying her additive reasoning skills to the task, but the task is multiplicative in nature.

It is worth pointing out here, that if you are analysing the demands of a numeracy workplace task, or a financial programme you will have to identify additive, multiplicative and proportional reasoning aspects and build in supports (or learning plans) if learners demonstrate a need. For example, if you are running a financial literacy programme and are discussing compounding interest, PE rates, return on investment or the compounding cost of service fees on Kiwisaver accounts, and many of your audience are still developing their understanding of multiplicative thinking, they are unlikely to fully understand the concepts. Hence why embedding multiplicative thinking into your programme can increase learning outcomes and understanding of all content.

In short, many adults will struggle with multiplicative concepts and will need direct support. A lack of support might manifest as them failing other content that relies on multiplicative thinking. For example, Jenny may not grasp the concept of return on investment (ROI) and fail to understand further content.

Developing multiplicative thinking

So, how do we develop multiplicative thinking? Below are a couple of ideas, these are explored further in the numeracy toolbox series.

1. Embedding multiplicative descriptions into your instruction

One way of supporting learners such as Jenny is to embed an activity that helps adults recognise how multiplication differs from additive thinking before introducing the investment scenario in the task. The key is to stress that multiplicative thinking is about how many groups we add or take away, not single numbers.

For example, we might introduce the idea that money gained on investments is worked out multiplicatively and often represented as a percentage. This is because we don’t really care about the amount of money invested (the primary group), but rather how many groups we can grow from it.

Using grid paper (or perhaps on a Power Point) you can shade in 2 squares, then 4 squares, 6 squares, and finally 8 squares (see attached picture).

Then show that Frank, Tane and Gill, each invested two dollars (a group of two), represented by the 2 squares.

Write Franks name above the 4, Tane’s name above the 6, and Gill’s name above the 8.

Explain that these are the returns each got from the investment.

Ask learners how many times the 2 goes into the 4. Jenny can count and answer ‘2’. Then say something like, “That means Frank got 2 times his investment. Because his original investment fits 2 times into his return.

Let’s look at Tane’s return. How many times does 2 go into 6?”

Let learners work out how many ‘twos’ go into 6. When they find that 2 goes 3 times into 6, say “that means that Tane made 3 times his investment.”

Then let learners talk together in small groups to work out how many times 2 goes into 8. Learners count how many groups of 2 are in 8 and find that Gill made 4 times her investment.

You might then show that an investment of $1000 that grows to $3000 over time, represents 3 times the original.

If you wanted, you could cut out the ‘2’ and physically measure how many times it fits into other amounts. A conceptually oriented activity like this takes about 10 minutes and can seriously help most learners grasp the new situation. If you wanted to you could then link percentages to this easily in a further activity.

The idea of embedding is that you front foot difficult literacy or numeracy content before learners get stuck on difficult content. Of course, the more time you put into such work, the better the results.


2. Develop your own numeracy skills.

The approach above will certainly help most learners, but many will require concerted and targeted support to develop multiplicative skills. In such cases the research suggests that the first thing is to make sure you are not only multiplicatively competent but that you can also identify the difference between additive and multiplicative situations. The research recommends that tutors spend time reasoning through rich quantitative tasks to become familiar with the type of content. Here is an example of a rich task (note: you are not asked to solve the task, just reason through it).

My brother and I walk the same route to work every day. He takes 40 minutes to get to work, and I take 30 minutes. Today, my brother left 5 minutes before I did. How long will it take me to catch up?

Take some time with this task. The goal is not to solve it, but rather to use it as an opportunity to think. Think about all the things you know first such as how fast the brother walks, the difference between their speeds and so on. Draw pictures, make diagrams, charts etc. Exercise those numeracy skills. Doing this will make you a better tutor. It will force your brain to make connections – chew on the problem over a whole week if you need to.

Or… you could go to the research link at the end of the article (Sowder et al.) for information on the problem and how people solved it. I don’t recommend giving these to learners – they are for you to stress out over, not learners 😉


3. Use different problem structures to cultivate multiplicative thinking.

Now, remember that we want to develop conceptual understanding for learners who are seriously struggling, around step 2 (perhaps a high 1 or low 3) on the Learning Progressions. Word problems can be useful if used as thinking tools. The trick to using word problems with adult learners is to avoid using them like work sheets or mini tests.

Our goal is for learners to recognise multiplicative structures. One method is to break learners into groups and ask them to explain how they solved various problems using drawings or diagrams (only one or two problems). Then get them to show their diagrams/drawings to the whole class. Another way to use word problems is to ask learners to rewrite them as a division problem (restructure the problem in some way). This helps them conceptualise how multiplicative thinking works.

Example 1

There are 2 tables in the classroom and 4 people seated at each table. How many people are there altogether?

There are 8 people sitting equally between 2 tables in the classroom. How many people sitting at each table?

Example 2

Sue has $3000 in her Kiwisaver and Tane has 4 times as much money in his. How much money does Tane have in his Kiwisaver?

Sue has $3000 in her Kiwisaver and Tane has $12000. How many times greater is Tane’s Kiwisaver than Sue’s?


4. Go back to the Learning Progression activities

Teaching Adults to Make Sense of Number has 8 specific activities sequenced in order designed to develop learners' multiplicative thinking (see page 17 at the link).


There are many interesting ways to embed multiplicative thinking into your practice and lots of fun activities that learners love and learn from. Please add them to the comments if you have time.

Finally, the article below is old, but personally still really relevant to the learner’s error in this case.

Thanks for reading this far! All the best!

Related research

Sowder et al., (1998). Educating teachers to teach multiplicative structures in the middle grades:

The link is in the comments below.


Thursday 2 July 2020

Numeracy Resources for Entry Level Engineering

Many years ago I was asked to put together some resources for entry-level engineering students who struggled with numeracy. The content is good, but the resources were never presented as a coherent whole. 

Below is an image that includes them all. It has both video and resources attached. Certainly nothing flash, but worth while if you are teaching numeracy concepts to lower-skilled adults. 

Friday 19 June 2020

A new project

I've been working on various areas of numeracy for over 20 years and find I have quite an original and dare I say 'state of the art' approach and content. 

The plan is to capture some of this content and put it into a 'Master Course' in adult numeracy. The idea is that you sign up to the course and work through the modules and complete a small task at the end of each session. A session consists of a video clip, an information sheet with tasks, and an online space to interact with other people. There are about 10 sessions. 

For a limited time here is the second video clip. The challenge has been getting the language right. The audience is primarily trades tutors. Trades tutors are highly intelligent but not used to education language that teachers are familiar with. So I've chosen a down to earth tone that I would have appreciated when I was just starting out. 

It's not perfect but as a first draft its a step in the right direction. 

Thursday 14 November 2019

Saturday 24 August 2019

Wednesday 27 February 2019

Whiteboard Problem no.7

Below is one of the best puzzles I have ever come across.  The clip below embeds it within a communication context, but you can just copy it straight to the whiteboard.  My advice is to try and solve the puzzle before watching the entire clip.

It's a little long winded I know. Endure.

Good luck.