Monday 7 November 2016


I've been thinking about the types of numeracy tasks adults are required to do and learn for various vocations. The current model of ‘mapping’ the demands using the Learning Progressions is useful up to a point. However, my hope is to turn this into actionable information. For example, Yes, I want to map out the demand on the learners, and the demands on the tutor. But, I also want to identify learners’ prior knowledge and analyse how this might enhance or constrain their understanding. Following this I want to develop effective pedagogical approaches. This means going deep.  It means understanding the challenges of the task, the interference of prior knowledge, and the interference of tools.  It’s a little heavy, but below is some initial work on fuel mixes for chainsaws and other tools. Below is what I think of when I think of analysing a task.

The need to mix fuel for two-stroke engines 
Many tools used in the agriculture and horticulture sectors are powered by two-stroke engines which require a fuel mixture different from that of four-stroke engines. Depending on the situation this requires users themselves to combine petrol and oil in a pre-specified ratio. Chainsaws, widely used in industry, typically require either a 25:1 or 50:1 mix. While this is a reasonably common task, anecdotes abound of workers interpreting the 25:1 ratio as 25 milliliters of oil to one litre of petrol, rather than 25 parts of petrol to 1 equal part of oil. If adding oil to 20 litres of petrol to create a 25:1 mixture, the user operating under the former understanding of the ratio would add 500ml of oil, instead of the correct 800ml. This error would result in a chainsaw seizure, requiring either an expensive reconditioning or complete replacement. Equally damaging however, might be the damage to the individual’s reputation among his or her peers, as errors are often interpreted as indicative of undesirable traits such as low intelligence (Van Dyck, Frese, Baer and Sonnentag, 2005). The failure to reason proportionally, and apply rates and ratios has potentially dire effects on workers’ economic and social wellbeing.  
Proportional reasoning is considered a difficult yet essential skill for students and adults to attain. In regard to importance, it has been described as the capstone of elementary school and the cornerstone of high school (Lesh, Post and Behr, 1988). It is also considered essential for a wide range of everyday contexts (Dole, 2010). In regard to difficulty the New Zealand Learning Progressions for Adult Numeracy framework situates the ability to use multiplicative and division strategies to solve problems that involve proportions, ratios and rates at the highest level, step six (TEC, 2008). Assessment results taken from 203,000 learners attending embedded literacy and numeracy vocational programmes showed approximately only 20% of learners meet this criteria (Earle, 2015).  The ability to reason proportionally is so difficult that Lamon (2005) suggested up to 90% of adults were likely struggling to do so. Given that proportional reasoning is difficult to develop, it stands to reason that adult learners with historic difficulties with mathematics are likely to be amongst those with lesser skills, and that this content will present a considerable challenge. 

Theory of proportional thinking
According to the NCTM (Lobato, Ellis, Charles and Zbiek, 2010) proportional reasoning requires ‘one big idea’, and 10 essential understandings. The ‘big idea’ consists of the recognition and understanding that two quantities are related proportionally when they possess an invariant relationship. Recognition of the invariant relationship is inherently difficult because it is not explicitly indicated, but rather is deduced by the user. 

How individuals come to recognize these relationships is largely a result of the learner's own mental actions. For example, Piaget posited three kinds of knowledge; physical, social, and logicomathematical (Kamii and Warrington, 1999; Piaget, 1954). The first, physical, is knowledge gained from empirical observations of external reality. Kami and Warrington used the example of knowing that counters do not roll like marbles as this type of knowledge. The second, social knowledge, related to social conventions such as one-third being written as 1/3, or the knowledge of an algorithm. In contrast, however, logicomathematical knowledge is developed from the learners’ own mental actions. Piaget argued that learners were required to construct this knowledge themselves. While an educator can provide the stimulus, they cannot not ‘transmit’ this knowledge. Logicomathematical knowledge is argued to be necessary for learners to recognise invariant relationships. This provides a rationale for developing adults logicomathematical understanding of ratio, not only physical or social. It also requires learners to take a proactive role in classrooms, rather than passive observers.     

Mixing petrol and oil to prepare fuel mixes for two-stroke engines is a particularly challenging aspect of proportional reasoning. Proportional instruction should build on learners’ intuitive understanding (Diez-Palomar et al., 2006), yet prior experiences may interfere with learners understanding of fuel mixes. The ratio of petrol and oil mixes for chainsaws are ‘commensurate’, in that they are expressed as the same unit, such that 25:1 refers to 25 mililitres of petrol to one mililitre of oil.  However, adult interactions with rates and ratios are often in contexts in which the units differ (Chelst, Özgün-Kocaet and Edwards, 2014; Lesh et al., 1988). For example, a cars’ fuel efficiency is measured as kilometers travelled per litre of petrol. Even more problematic are pesticides and herbicides in which different units are used to measure the same attribute, liquid. For example, 10ml of Weedkiller to 1L of water (Lesh et al., 1988). Chelst et al., noted this confusion and posited commensurate ratios (equivalent units) as the most complex. 

Adding to the complexity, fuel mix ratio charts tend to express petrol in litres and oil in mililitres (see Table 1 for the chart used in the classroom). While the charts have utility in the workplace, from a teaching point of view they may obscure or interfere with key concepts of ratio. Furthermore, because the ratio is represented within the charts as non-commensurate, conversions between milliliters to litres is necessary (See Table 2). Knowledge of the metric system and multiplicative fluency support this but given the lower skills of learners in entry-level vocational programmes, this may be limited. 

Table 1. Chart used to teach two-stroke oil mixing ratios
Litres of fuel
Mililitres of oil to add to fuel

The complexity of fuel mix ratios, and the utility of the ratio charts is likely to contribute to petrol and oil mixes being taught procedurally with the chart. If so, one litre of petrol is likely used as the starting quantity for instruction. A small mercy for educators is that typically fuel mixes begin with petrol poured in whole litres, followed by the addition of oil. For example, 1 litre of petrol will have 40ml of oil added rather a total one litre mix comprised of 962ml of petrol and 38ml of oil.  Thus, the construct is part-part whole, rather than the more complex part-whole that might have been the case. However, it is safe to say that the concept is challenging, and requires learners to actively engage in order to make sense of the content. 

Table 2. Unit conversions implicit within fuel mix chart

A summary might be 'fuel mixes - more complicated than they seem'. 

My plan is to develop these types of analyses into a package that includes a series of lesson plans, lesson resources and tutor support material, using a mixture of paper based and video content. Let me know if anyone is interested in an efficient approach to developing these skills within a programme. I have a system that would improve the learners' achievement, while decreasing the time.

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