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.
