Why are there multiple answers to Facebook maths puzzles?

Why are there multiple answers to Facebook maths puzzles?

Don't those Facebook maths challenges seem strange - a single equation and about 4 different responses.  Everybody is confused (except the smug few)! 

I mean, how the heck could they think it was zero when I got 56! And how did Larry Johnson get 50?! What is he stupid? And then it turns out Larry Johnson was right and you are stupid.  Hate that.

Time to change your luck with a maths secret. 

Those equations that seem so arbitrary actually tell a story. And they tell great stories!

For example, take this equation and try and answer it:

4 + 3 x 2

Is the answer 14 or 10?  

It is definitely one and not the other because the equation describes a very distinct series of events - a story. But the sequence is governed by BEDMAS not by the sequence the equation is written in. BEDMAS basically says to prioritise the multiplication and division sections before addition and subtraction. 

So put brackets around the multiplication and division sections and solve these first.

4 + (3 x 2)

So back to the story that the above equation tells.  

Once upon a time a young girl had 4 dollars given to her by her Dad.  Then the next day an amazing thing happened. Her Grandma gave her two dollars, her Grandad gave her two dollars, and then her Mum gave her two dollars!  Whooho!  How much money does she have?  (Do a quick count up). 10 of course!

You may be wondering why the story doesn't go like this:

4 + 3 x 2

Once upon a time a girl was given 4 dollars by her Dad, then another 3 dollars by her Mum. This happened two days in a row.  How much money does she have?

The reason the first story is 'correct' and the second 'incorrect' is that the little 'x' symbol actually means 'groups of'.  So 3 x 2 means 3 groups of 2.  It does not mean 2 groups of 3, even though this happens to be the same final amount. Make sure you understand this point. 2 x 3 means 2 groups of 3.

Think about how you would draw 2 x 3.

Here it is - two groups of three.

 But it is not three groups of two.

Also 3 x 2 = three tables (groups) with two people at each - but not two tables with three at each.

If I ask you to draw the following 2 x 4 you should draw two groups of four:   

If you change the amount of groups from that stated you change the whole story. For example:

4 + 3 x 2 

How many groups are there? There are three, not seven. By adding the 4 and the 3 first you change the story.  You change reality.  You change the meaning of life until you only live within the fragmented twisted world of your own imagination!

Stories

All maths is a story - those funny little shapes tell a story full of drama and intrigue!

For example:  3 x 4 - 7 is just crazy - dang! Check this story out.

Dana agreed to babysit her brother for three nights at $4 a night.  But blimmin heck, her little brother got his hands on her money and stole $7 dollars.  The tragedy of it all!

Or, how about this human drama:

2 + 2 x 4 - 3.  (13 or 7?)

It's a heart breaker. 

Donny had two toy soldiers that he loved and played with everyday. They weren't just those green plastic ones, but steel painted soldiers. 


An amazing thing happened to Donny. For Christmas his parents bought him two sets of soldiers, and each set had four soldiers in it! Gotta love Christmas. BUT... while he was outside playing on the driveway, Dad came home and ran over three soldiers with the car. Donny's heart was broken - he loved those soldiers.    How many soldiers did Donny have now? (Do a quick count up).

Okay okay so it's a dumb story. So make up your own darn story.  

Ideas for lessons

1. Turn all equations into stories - this will help the maths make sense and actually be meaningful. Get the story right. 'Bedmas' provides the rules. Google it.

2. Tell your students stories, and get them to write down the equation as you tell the story.  This working backward process is fun, and hugely educational. 

Here is one for you.

Damon drinks lots of coffee - he does this just to function like a human being. Everyday, Monday through Friday, he drinks eight cups of coffee.  But, on workout days, Monday, Wednesday and Friday he drinks an extra two coffees.  As a final insult to his body, Damon buys a coffee from a shop on the way home Friday night. Will Damon die of renal failure?

Five groups of eight, plus three groups of two, plus one. And yes - he probably will.

5 x 8 + 3 x 2 + 1 = 

It could be written like this  1 + 5 x 8 + 3 x 2 = but the answer is the same because the story is the same. Just because the 1 moved doesn't matter.  There are 5 groups of 8, and 3 groups of 2, and then one. 


Stories, stories, stories - not numbers numbers, numbers.

To finish, here is 'a' story for the opening problem.

Dale had 7 apples. Someone gave him seven more but he had to share these with six of his friends. Unfortunately, they only got one each. But at least he still had his original apple plus one more. Then someone gave him 7 packets of apples, each with 7 apples in it! The madness of it all. Because of his good fortune he gave seven of those apples to his mum. Because he loves her.

How many apples has Dale got?

If anyone has a better story -PLEASE put it in the comments.

Important research for those working with lower-skilled adult learners 

It is quite common for literacy and numeracy tutors to have big hearts. Many of us got into the tutoring business out of a desire to improve the lives of others. This sometimes orients us toward protecting learners. The post below is not meant to be mean-hearted, but comes from my years of doing exactly what I describe.    

The tendency to protect manifests in several unproductive ways (and here is where I lose friends). The first is parental positioning. This is often evident when tutors say things like, "My learners...". 

While it may feel like it, they are not your learners. They are adults, not children. Your job is to educate and prepare them. It is a question of roles. In all likeliness, no one else is being paid to do your job. If you decide to put your education role into second place while you adopt another role such as counselor, friend or parent, then who is doing your job? I am not saying not to adopt these roles, but be careful of your priorities. Your first priority is to educate. I have learned this the hard way.   

The urge to protect also manifests with maladaptive expectations for learners. This is evident when tutors say, "My learners would not be able to do that... they can only..." or "this won't work with my learners, they can only...". Essentially, tutors become arbiters of the learners' experiences. They attempt to protect them from failing and in doing so impose their own limited expectations on adults. Here is the truth - the individuals you are working with are capable of much more than you realise.  [See the research below for more]

Limited expectations are constructed because tutors occupy an echo chamber of sorts. The learning situation is constructed in reciprocity between the tutor and learner, and settles into an equillibrium of non-threatening social norms. Unless you take steps to combat this it will just happen. You are like a fish in water, you cannot sense the water because it is all around you. The environment is safe, but unchallenging. Learn to beat this, and you will be on your way to being a great educator.  If you want to explore this concept investigate 'didactic contracts' or read Brousseau.   

Why this matters

The problem with a safe protectionist environment is that growth only occurs as a result of struggle. A recent Facebook post (the font of all wisdom) talked about the growth of a lobster. Take a look.

"The stimulus for the lobster to grow is to feel uncomfortable." 

In numeracy we use the word 'purtubation' to describe a necessary state for the re-organisation of existing cognitive patterns - in other words 'learning'.

perturbation

ˌpəːtəˈbeɪʃ(ə)n/

noun

1. anxiety; mental uneasiness.

   "she sensed her friend's perturbation"

   • a cause of anxiety or uneasiness.

     plural noun: perturbations

Yup - anxiety, mental uneasiness. Now knowing how to regulate this so it does not lead the learner to disengage is vital. But without it, you are not doing a fair service to the adult learner who is putting their trust in you. The lobster must feel pressure.

Final point

Below is a cautionary study. It shows that attempting to protect students, not only limits their exposure to content, but also reinforces negative beliefs - making their life much much tougher. 

“It's ok — Not everyone can be good at math”: Instructors with an entity theory
comfort (and demotivate) students

Rattan, Good & Dweck (2012) 


Abstract

Can comforting struggling students demotivate them and potentially decrease the pool of students pursuing math related subjects? In Studies 1–3, instructors holding an entity (fixed) theory of math intelligence more readily judged students to have low ability than those holding an incremental (malleable) theory. Studies 2–3 further revealed that those holding an entity (versus incremental) theory were more likely to both comfort students for low math ability and use “kind” strategies unlikely to promote engagement with the field (e.g., assigning less homework). Next, we explored what this comfort-oriented feedback communicated to students, compared with strategy-oriented and control feedback (Study 4). Students responding to comfort-oriented feedback not only perceived the instructor's entity theory and low expectations, but also reported lowered motivation and lower expectations for their own performance. This research has implications for understanding how pedagogical practices can lock students into low achievement and deplete the math pipeline.

The key lesson to tutors is this. There is a world of difference between being nice, and being kind.  Be aware that you have a tendency to impose your expectations on learners.   

THE CHALLENGING NATURE OF RATIO FUEL MIXES

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

Ratio petrol-oil	Litres of fuel
	1L	2L	5L	10L	15L	50L
15:1	66	133	335	670	1000	3340
20:1	50	100	250	500	750	2500
25:1	40	80	200	400	600	2000
30:1	33	67	165	335	500	1700
40:1	25	50	125	250	375	1250
50:1	20	40	100	200	300	1000
	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

	Petrol	Oil
Mililitres	25	1
Litres	1	?

   

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.