Find guidance on how to address the issues and challenges in addressing connections between numeracy and mathematics in schools.
So how do we go about addressing this challenge?
Whilst there are moves to highlight the importance of numeracy across the curriculum, as with the numeracy general capability in the Australian Curriculum, and in related work and research about the training of all teachers in numeracy (e.g., see Forgasz, 2019; Goos et al, 2014; Goos 2016), there is the need for more work and to extend this approach.
To date, schools and curriculum authorities have not generally acknowledged the range of challenges associated with implementing the General Capability of numeracy. Without this, the cross-curricular development of numeracy skills is unlikely to succeed. (Carter et al 2015, p. 609)
There are a number of issues and challenges to face in order to move forward. These challenges include, amongst others:
- the disconnect between school maths and the real world, including the world of work,
- a lack of awareness and knowledge about the way to teach numeracy, and
- the related issue of word problems in maths.
Following is a short introduction to each of these issues. The final section has some suggested professional learning activities for schools and teachers to undertake to discuss and address some of these issues and challenges.
The disconnect between secondary school maths and the real world
Despite calls from many reports, research, and mathematics organizations to contextualize mathematics, many schools continue to present mathematics as abstract, decontextualized problems. (Gravemeijer et al 2017, p. 109)
As the above research is indicating, Australian schools generally do not prepare students particularly well for mathematics in the real world.
If students have little experience grappling with messy real-world situations and problems, and if they can only apply mathematical procedures when problems are packaged in very familiar ways like in a classroom, then why would we expect them to be numerate and be able to see, use and apply maths in the world outside the classroom? The AAMT research project mentioned earlier observed this, with one of the teachers involved commenting on this disconnect:
This is one of the most interesting aspects/concepts of this project. The relationship between workplace mathematical skills and school mathematics could be described as 'distant' at best – Teacher observation (AAMT & AiGroup, 2014)
So a challenge is to use ways and approaches whereby we can engage our students in real world and authentic numeracy tasks and activities. A key to being numerate is to be able to start from a real world problem or situation and excavate the maths and formulate it as a mathematical problem to solve, as highlighted above. This implies we need to take the students out into the real world to do some mathematical investigations, or alternatively bring the real world into the maths classroom.
How this can be done is addressed in a number of the activities in the next section.
Lack of awareness and knowledge about the way to teach numeracy
Related to the above, because of this disconnect between school maths and the real world, including the world of work, the challenge for school teachers of maths is about how to see and incorporate numeracy as an integral part of their teaching.
A current common approach is to teach some maths content, and then maybe find an application where that maths can be used and applied. Unfortunately there are a couple of issues with that approach:
- the maths content is already specified—so the student does not need to formulate the problem or decide on what maths to use, not like in the real world,
- often this approach uses an unauthentic context or word problem, which can have a negative impact (see below).
As in the models above, and in the AAMT research, the ability to recognise, access and identify how and when mathematics is used in the workplace and the real world is critical.
Numeracy needs to be explicitly taught—leaving it to providence will not guarantee success. We need:
- to use problems set in real world contexts,
- an explicit focus on using different mathematical processes (such as communication, modelling, devising strategies, representation, and reasoning),
- an explicit focus on all the stages of mathematical modelling (formulating, employing, interpreting/evaluating) and how to see and excavate the maths out of a context.
Teachers need to teach students how to "read" and interpret the context in which the maths is embedded, often in a physical object or in a printed or digital format. Moreover, in teaching numeracy you need to explicitly cover a range of different cognitive processes and mathematical content areas, the use of appropriate tools and representations, and the skills required to reason, argue, reflect and evaluate and then communicate the results as in the Goos et al model of numeracy described earlier.
Challenges regarding the value of "word" problems in maths
Four horses cost as much as three cows, four sheep as much as two horses, and three lambs as much as one sheep. How many cows could I exchange for 40 lambs?
A farmer has cows and chickens. He only sees 50 legs and 18 heads. How many are cows and how many are chickens?
Many traditional school-based mathematical word problems often disregard and challenge students' sense making and only help serve to distance students from the real world, and the usefulness and value of mathematics. It is critical therefore in the teaching of numeracy and mathematics that a key value in mathematics is about its relationship with real world things—whereas word problems often do the opposite.
Using authentic situations in numeracy activities and tasks helps encourage a more positive disposition towards solving relevant and engaging maths problems, not irrelevant, word problems as can sometimes be met in school maths classrooms. This issue of authenticity and word problems in mathematics education has been researched and documented (e.g., see Palm 2006, 2008; Verschaffel et al., 2009; Stacey, 2015).
Instead of using traditional word problems of the kind shown above, we encourage greater use of short realistic mathematics tasks more like those used in PISA. See the activity below, Engagement activity 3: the challenge of word problems, for more information.
The approach recommended is to integrate the actions below into your classroom teaching and learning:
- use a problem solving, investigative, open-ended approach—use real texts and real situations—make connections between maths and the real world,
- start from the real world—teach students how to identify and extract the maths from the messy, real-life situations that they are likely to face (what can be called "excavating the maths"),
- and a key part of this is to make the maths explicit and then, when the need arises, or gaps in knowledge appear, teach the maths that is required: "just in time, not just in case".
For the engaged and the disengaged student, and the competent and less competent student, the context can provide the challenge, the motivation and the purpose for understanding and using mathematics. However, in order for this to work in the classroom, we need to explicitly integrate the Goos et al or PISA cycle of skills into our teaching and learning.
Two activities in the next section attempt to demonstrate how to do this:
- Engagement activity 3: the challenge of word problems
- Engagement activity 4: connecting to the real world – how to do it?