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.

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).

https://ako.ac.nz/.../Learning-progressions-make-sense-of...

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.

 

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. 

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. 

What to do when you don't know what to do!

Back in the video game: Maths Educator at Large

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.

On school and learning identity

I have spent the last years analyzing the data from a series of interviews I did with adult learners.
The one overwhelming finding is: Self-worth is tied up with academic performance - especially mathematics. Moreover, some environments are far more prone than others to enable judgments to be made about your worth (both by you and others).

Societal beliefs about what mathematics is and what it means to be good or bad at mathematics has damaged some of us.  And may still be doing so.

I'm trying to put into words how maths classes were described.  This is a rough start but perhaps this tale will help generate some thought about the impact social-pressure can have on us.

The tale

John has just turned 13.  Today is his first day at High School.  He is excited to meet new people and take part in what others are doing.  He is really looking forward to taking part in classes.

John has a positive view of himself, he is capable, inventive, gets on well with others and loves to play and have fun.  John is excited about going to maths class, he loves solving problems, talking about maths and enjoys reading about famous mathematicians like Aristotle.  His mum bought him a shiny, tidy new work book and he received an exciting text book.  The new pencils, pens, rulers and calculator made it even more exciting. 

On the first day of maths class he noticed the teacher would ask the class very direct questions.  He quickly realised that they weren't real questions because the teacher already knew the answers - they were more like mini-tests, however, he had a go at them  - his hand usually went up first.  When he answered, the teacher just carried on, as though the answer had come from his own mouth -but it felt good to answer, and the teacher seemed to approve.

One time, he answered wrong, and the class laughed.  He noticed that they laughed 'at' him, not with him. They didn't laugh because he was funny or had made a joke.  In fact, he wasn't sure why they laughed, but it didn't feel good.  It made him feel 'alone', for a moment, different, a sense of 'outsidedness'.  This didn't happen at his old school. 

Another time he answered a question in what he thought was a conversation with the teacher.  He realised too late that the teachers' question had been rhetorical.  He also answered with the wrong numbers. Other learners laughed at him again. One of them called him a 'dummy', but in a funny sort of way, not mean.  No one else had tried to answer.  Maybe he was breaking the rules about when to speak?  Why did they laugh?  Perhaps because he was so wrong, so surprisingly wrong, that it was funny?

He didn't want to be laughed at anymore.  He wanted to laughed 'with'.  So when a different student answered incorrectly - he laugh 'with' the others.  He stopped answering questions himself - rather he waited until someone else answered first and then he checked if he was right.  Often he was wrong, but so long as he didn't share it, it didn't matter.

John found that two worlds developed -his own world of math, conducted in his head where he answered and asked questions, and the public math, the math that occurred around him as he spoke to other learners and engaged with the teacher.  Two domains to think about math, one in the privacy of his head, and one public. 

Often he got things wrong in his private world, and hid it.  Often if he was wrong, and no one knew it, he would subtly rewrite the answer in his book.  If he wrote it lightly in pencil he found he could erase it and leave no smear marks.  But other times, as much as he tried to hide it he couldn't.  The teacher would ask him for answers directly and he had to give his answer to the whole class.  Sometimes they had tests and everyone found out how everyone did.  He would try and hide his score but someone would always ask.  To not tell would show you cared.  But that 'dummy' word stuck with him.  He wasn't sure why, but it came up again. 

He stopped enjoying maths, the class felt like a test and at stake was his reputation, his very self.  But outside the class he was great, especially away from school.  It gnawed on him though.  He was good at sports, and enjoyed some other subjects.  English wasn't too bad. 

What he didn't understand was how he could be confident outside class but be a dummy inside the maths class.  Which was he? Confident or dumb.   

He concluded that he was only dumb in the maths class.  That when it came to maths, he just didn't have it. But being good at maths means you are smart, so being bad at it means..?  This was too painful, so he told himself that maths wasn't his 'thing'.  He told his mum, "I'm good at sport but not maths.  Maths is for geeks anyway'.

Summary

John is struggling to construct a single coherent identity.  He wants to be strong, smart, capable of exerting his strength and getting things done.  But in the maths class he is positioned as the opposite, he can't be the guy 'who gets things done' and be the guy who is a dummy.  He can't be the dummy and the doer.  And hence there is a conflict between two identities.

John emotionally disconnects from maths.  He stops caring - because he has to. 

Statistics and critical thinking

Some quick thoughts on statistics, critical thinking, and foundation-level learners. 

Type one statistical thinking – Understanding how data is collected

Mark Twain noted ‘There are three kinds of lies: Lies, damned lies, and statistics’. He seemed to be implying that statistics were a tool used to manipulate the truth. Perhaps he was suggesting that politicians, advertisers, and others use statistics to mislead the public? What a concept.     

When teaching statistics to foundation-level learners it is useful to think of statistical critical thinking skills as two skills. The first is to understand ‘how data was collected’ and the second ‘how the statistics are represented’ to the reader. The first is closely related to typical critical skills, and the second more mathematically oriented. Foundation-level learners often enjoy the first and are not so familiar with the second.   

Understanding ‘how the data was collected’ is useful when presented with information such as this old advertisement for smoking.

Critical questions here might include:

Did they ask every single Doctor whether they smoked? How many did they ask, where were they from, how many answered, and how didn’t smoke and how many did? Of those who did smoke how many smoked Camel? Why did they smoke Camel? Were the Doctors given any incentives to promote Camel?

Question: Based on the evidence in the advertisement above is this scenario possible.

The researchers asked 50 Doctors whether they smoked. Five said they smoked. Three of these smoked Camels.

Could this be true based on the advertisements claims?

An old ‘good boy’ dog food scam.

A large company sold a popular brand of dogfood called ‘good boy’. The company was putting together a new advertisement and wanted some statistics to use in the advertising.

Here is how they produced the statistics.

An employee rang ten people and asked them what dog food they used to feed their dog. They recorded the amount of people out of ten who named the company.

They then rang another group of ten people. Then another. Then another. After doing this they simply selected the group of ten in which the highest number had selected their dog food. 

The statistic used was ‘six out of ten dog owners use good boy’. It was true, one of the groups of ten people had six members who used Good Boy. What wasn’t mentioned was that this was the highest of the ten groups. Some of the groups included only one or two people who used the dog food, others three, four or five. If the groups were combined the statistics would have read ’36 out of 100 dog owners use Good Boy’, or ’66 out of 100 dog owners do not use Good Boy’. Quite different. Yet the statistics are accurate – it’s the method needs to be critiqued.

However, be careful...

Despite these dubious methods of generating data, don’t allow learners to become cynical of all data. Most data collection methods are very good; understanding the strengths and limitations of the methods is the goal. Critical thinking when taken to the extreme means you can never know anything.  Don’t teach learners to be cynics, teach them to be critical evaluators.

Type two statistical thinking: Understand how statistics are represented to support certain ideas.     

Adults also need to know what statistical data analysis methods have been used and how the approach produces and represent findings. For example, if the term ‘average’ is used, does it refer to the mean, median or mode? Is the average even relevant? The ‘average NZ house price’ is frequently used in the NZ Herald. Would it not be better to know how many houses were sold in various price brackets? If two houses are for sale in a neighbourhood, one for 100,000 and one for 1,000,000, does the average of $550,000 fairly represent the situation? I don’t think so because an average of $550,000 suggests that people with one fifth of this amount cannot buy a house – and it isn’t true based on the situation.    

For example, the classic statistical scenario: There are two buckets, one full of boiling water and one full of freezing water. Stand with a foot in each. Stop screaming, because on average you feel warm.

If you only know the average, you don’t know enough to form an opinion.  

In sum, adults are bombarded with statistical information through politics, advertising, workplace statistics, dietary recommendations, safety courses, sports and health advice. In order to make sense of the information, adults require some knowledge of how data can be collected and how it is represented.

Petrol Maths

Stolen from David Farrar. 

"So when petrol prices are around $2.50 a litre, how much money does the Government get from you to fill up a 50 litre car?

• National Land Transport Fund $31.51

• GST $16.31 (on a $2.50 retail price)

• Regional Fuel Tax (in Auckland) $5.00

• ACC Levy $3.00

• Local Authorities Fuel Tax $0.33

• Engine Fuel Monitoring Levy $0.15

But that isn’t all. To pay for the $125 petrol bill, you need to (if top tax rate) earn $186.57 which means there was also income tax of $61.57.

So every-time you spend $125 filling up the car with petrol, the Government will have got $117.86 as you will have had to have earned almost $190 to pay for it!"

Note from Damon: This might not be the content to use in a financial literacy class as it may just make the students angry.

Good teachers/tutors make the difference

Note: The quality of teacher/tutors is directly related to outcomes.

Chauvel (2014) notes:

New Zealand research conducted over the last 10 years has consistently highlighted that the quality of teaching and the effectiveness of the learning environments facilitated by teaching staff is crucial to Pasifika and Māori learner engagement in tertiary education. As stated by Alton-Lee (2003): “…high achievement for diverse groups of learners is an outcome of the skilled and cumulative pedagogical actions of teaching in creating and optimising an effective learning environment… Quality teaching influences the quality of student participation, involvement and achievement (including social outcomes)” (p.1-2).

Supporting teachers and tutors is pivotal to improving learning outcomes. Not sure why this is argued?

Yes - this blog is ALIVE!!!!

And we're back online... The PhD is done and dusted, and all systems are go. 

This blog was designed to present the process and findings of the PhD in a usable way to the sector. I started uploading lesson plans, activities, ideas, strategies and so on and then it sort of puttered out as life got busy.

I'm going to be developing more content and this blog will be where I test ideas in a risk free environment.  I need an outlet that is low stakes to try things out. So if you are interested, I'll be dumping all sorts of stuff here to do with adult education. It'll be pretty raw, but fun, and I hope highly interesting and useful.

Last few days of the PhD

Well, this is it - five years of work nearly over. Expect lots of content soon!

Cheap and nasty?  No way! 

It's all about engagement!

Some people may wonder why I do not use better materials than paper and dice to develop numeracy skills.  Well, I do.  But only when they improve learner thinking. 

My philosophy to teaching mathematics and numeracy is to focus entirely on student thinking.  Student thinking (which I call 'engagement') is a moment in time when the student stops thinking about everything except the task at hand (some call it 'flow')- they are absorbed fully in the moment as they struggle to make sense or meaning out of the information at hand.  It is in these rare moments that the learner 'constructs understanding'. And hence these are the most precious moments in a teacher or tutors day.  

Everything else is superfluous.  Everything else is killing time.  Everything else distracts.  

So, because I believe that it is only 'what happens in the head' that results in learning I am very utilitarian about my gear.  

My only question when using material is:  does this facilitate learner thinking?

To sum up the wise HULK below: Fancy resources look good but you do what you must do to get the job done. 

Self-regulated learning

The single biggest factor in learner success is a learners ability to self-regulate their learning.  It is the exact opposite to 'learned helplessness' that you should read about here.  Learn about it, understand it, and you will become a better educator.

Of course, self-regulated learning (SRL) is not easy to develop in learners particularly learners who have been beaten around by the educational sector.  Also SRL can be broken down into many sub-skills and dispositions that are generally developed separately but used cohesively.

I used to think that SRL was the secret to developing the potential of adult learners with problematic learning histories.  I used to think it was the answer to the literacy and numeracy problem.  No more embedded ELN, no more literacy or numeracy provision - just SRL.

I now realise that sitting below SRL are belief systems that dictate SRL - beliefs are the mainspring of downstream effects.

As such the form below is no good on its own, even with an extensive PD package that was almost developed and ready to go.

Moving on.

The problem of engagement

There is a line of thought in the numeracy discourse that goes like this: A learner's motivation in a certain area can be used to piggyback numeracy content.

An example is a learner who loves cars, and is on an automotive engineering course.  The theory states that they will engage to a greater degree with numeracy if it is presented in the context of car specs, and if the content is relevant to their needs.

It sounds true.  And there is some self-report data to support the idea.  Learners who believe that numeracy is useful tend to achieve better than those who don't.

But, I'm not so sure about all this.

Engagement in 'numeracy tasks' at a micro level is how numeracy skills are developed. Does the relevance of the situation context improve micro engagement?  Not from what I am seeing.

Here are some questions for numeracy tutors:

• Do your learners engage more in numeracy tasks that are fun or important?
• Do your learners engage more in numeracy tasks that have clear performance goals (how many can I get right) or conceptual understanding goals (a new way of thinking about an idea)?
• Do your learners engage in different ways depending on the time of day, or the day of the week?
• Are your learners motivated by passing assessments, or developing meaningful understandings of numeracy concepts that will be useful in their lives?

Finally, do your learners actually believe that the numeracy content they are learning on the programme is relevant to their needs?  And how would you know?

The joy of teaching and learning

This meme cracked me up for some reason.  I've been the teacher and the student.

Adult Numeracy Back in the House

I've been away for a while but back in business (PhD thesis submitted!). The clip below is useful for adult educators teaching numeracy. Someone else edited it, so some key points have been cut that would have rounded it out - and made it epic. So its not epic, but it does make some good points I think.

  

In anticipation of some kickback - note that OECD study that explored strategy use, utilised a methodology to ask learners, not teachers what strategies they were taught. Of course all the teachers said they promoted elaboration strategies, but the learners are a better indicator in this case. 

Echazarra,A., et al. (2016), "How teachers teach and students learn: Successful strategies for school", OECD Education Working Papers, No. 130, OECD Publishing, Paris.

Stolen from Tim Ferris

“Inspiration is for amateurs — the rest of us just show up and get to work. And the belief that things will grow out of the activity itself and that you will, through work, bump into other possibilities and kick open other doors that you would never have dreamt of if you were just sitting around looking for a great ‘art idea’.”-Chuck Close, American artist who achieved fame as a photorealist through massive-scale portraits

How to get the most out of mathematical problems

I had the privilege of speaking at the He Taunga Waka Wananga/Fono last week. It was a fantastically run conference attended by a great group of people and included plenty of thought provoking speakers. Boy, there are a lot of talented people out there. 

Well done to the team who put it together. Thanks for your hard work.

A few people have asked for a run down on a problem I used as an example in the presentation. It is covered pretty thoroughly in the clip below which I've edited. The problem starts at 2:15. 

Additionally, all the conference presentations will be made available on the He Taunga Waka website. 

They are not up yet but will be soon. The clip is below.