Yes, the problem below is solvable. But, it requires your mathematical skill, your wits, cunning and a little persistence. Oh! That just happens to be my definition of numeracy.
The problem is stolen from Stylianides and Stylianides (2015) which they took, and slightly modified, from a teacher training text book (Philippou and Christou, 1995, p.
Remember - it is solvable. You can do it and your children can do it. It just takes 'numeracy'.
By the way - 'product' in maths speak, means what number you get after multiplying. 'Sum' means what you get when you add.
I'll post the answer and explanation on Saturday.
You got this.
I hope you managed to give it a go. For those who did, you will have noticed that the problem required deductive, 'common sense' reasoning. When used with teachers, many teachers note that it does not look like a maths problem.
I thought it was interesting that we used
non-numerical clues to find the answer. It was
frustrating to work it out at first but made sense
in the end.
It made me realise that my initial thoughts about
the problem were completely wrong and this
problem was possible. I now understand that
problems that seem impossible or seem to have
irrelevant parts might actually be able to be
solved. Before dismissing any problem, put some
real effort into it and think about it in numerous
ways. Solution here.
Today I am wondering about what 'authentic' numeracy gains look like and how often they occur. As I trawl through hour upon hour of classroom recordings, it becomes clear that very little learning is occurring.
I have over 72 hours of classroom recordings (yip - listened to it all!). It doesn't matter if it is hairdressing, agriculture, employment skills or sport and fitness, the focus is always on 'completing' work.
Our sector has made a naive causative link between work completed and learning. It permeates the sector from the top to the bottom. I know that vocational training has a 'doing' emphasis. The learners are learning to 'do' things. But the work they complete in classes is not the work they will be doing in the target vocation. In some cases its almost pure academic.
This suggests to me an under-trained sector. For example, in mathematics education you are beaten over the head by the need to focus on learner thinking. Yet the tertiary sector appears to spend its time ignoring this or naively believing it is a side-effect of completing work.
I was reading Plato last night (Protagoras & Meno) and interestingly they do not even address 'work'. They seek understanding and insight - and by heck they get it. The Socratic method is frowned on in vocational settings and critics may have some small valid points. There is a belief in some circles that a. this approach is too intellectual for some learners, and b. it is abstract and therefore not applicable to the real world. Yet, the current state of affairs keeps the weaknesses of the Socratic method with none of the benefits of other methods.
Perhaps this 'completed work = learning occurring' is THE problem in the tertiary sector.
A new report out last week finds that the six year investment and increased focus on literacy and numeracy skills within primary school has resulted in less than anticipated results. On average there has only been a 1% improvement in results per year. The literacy and numeracy standards have done little to improve literacy or numeracy.
The Herald notes:
Thousands of children begin secondary school each year without the reading, writing or maths skills needed to make it through. In a new series 'The Primary Issue' we look at what more can be done to raise achievement for all Kiwi kids.
Primary school pass rates have virtually flatlined despite an six-year government literacy and numeracy "crusade" costing more than $250 million.
Data shows a quarter of children entering high school are below the National Standards in reading, writing and maths.
Of the almost 60,000 students who began Year 9 last year, 17,900 were unable to meet writing requirements, 18,500 were behind in maths, and 12,700 could not read at the expected level, meaning they would have to be rapidly "caught up" to have any hope of passing a high school qualification.
The figures remained largely unchanged over three years, rising an average 1 per cent across all year levels since 2012.
Side note: If you think primary schools are not doing so well, wait until you see the statistics on high school!
The fatal conceit
I have much to say about this (as usual!). There are three reasons why top-down educational initiatives don't result in improved learning. I could reference tens of Government initiatives that received huge funding yet made little real impact. Think, 'No child left behind' and 'Closing the gaps'. I'll briefly outline the most controversial reason here and look at the other two later.
Friedrich Hayek penned one of the most thoughtful books ever written called 'The Fatal Conceit". In it he described what in his mind is the grandest of conceits - the notion that a top-down remote leadership approach will impact the conditions of an established culture to which they don't belong (think the classroom). The notion that a remote decision-maker, far away from the situations in which the interactions occur, will be able to control what the teacher does in-the-moment is a conceit. He of course was describing the economical conceit of socialism, in which an external decision-maker will decide what your habits, interests, and worth ought to be, however his argument is valid. In addition, the approach used in NZ education functions under exactly the same paradigm. External bodies attempt to dominate what occurs in remote cultures (the classroom).
Understanding why a remote body cannot change in-the-moment interactions requires accepting that a classroom has a culture. Culture can be described as 'knowing when to do something, and how to do it'. Classrooms are powerful culture machines. A large body of research finds that classroom cultures powerfully influence behaviour. Children, adults and teachers act very differently inside classrooms than they do outside.
Remote decision makers cannot influence what really matters, which is the in-the-moment interaction between the teacher and the learner. This in the moment interaction is partly what Giroux called 'the hidden curriculum'. The hidden curriculum refers to what is really learned inclassrooms. It can be argued that it is within the interactions that learning occurs (think symbolic interactionism and negotiated meaning). Not even close decision-makers can influence this - think about the effect of learning advisers, PD professionals, and think of Nationals new super teachers that mentor other teachers. These are the interventions that people who are removed from the situation think will work.
Because decision makers cannot directly influence the classroom culture, they have to create secondary levers, which according to Hayek, they actually believe will make a difference. Thus, they use the following tools: policy, curriculum, resources, classroom design, assessment practice and targeted funding. They do so under the false belief that these make a significant difference to an existing culture. Unfortunately the lesson of history (AGAIN) reveals that these have a negligible impact on actual learning. I want to scream this from the rooftops, if only someone would listen.
Policy, assessment, funding etc, are the areas that the decision-makers have control over, so this is what they attempt to implement. These are all top down approaches. An apt metaphor might be: trying to fix a motor using two long sticks held in either hand. Always at a distance, never accurate, and lacking the dexterity to enact real change.
If only the Ministry of Education had learned from the Tertiary Education Commission's strategic attempts to improve adult literacy and numeracy. They have adopted almost identical methodology, and the results (although currently obscured) are not at all impressive. An estimated 160 million dollars has been invested since 2008 into adult literacy. An estimated 260 million dollars has been invested into the schools national standards. We have new resources, assessments, frameworks, policy, new qualifications and personnel - but learning gains? Almost nil.
I'm sure most educators who do not work directly with learners day in and day out will disagree with this post. However, for those who are in the classroom, most will agree. The top-down initiatives simply do not penetrate the culture enough to make a difference to learning.
So, let me clarify where I stand.
First, we need good policy, we need effective frameworks, we need appropriate useful assessments. But these are support structures, not effects in themselves.
What we desperately need is:
Highly, highly, highly trained teachers and tutors. The research constantly shows that teachers make the difference. Highly trained, experienced, passionate teachers win every time. They are the best people to make decisions - train them so they make good ones.
If that money had gone into training, retaining high achieving tutors, holding tutors accountable and rewarding them for real learning gains for all learners, we would not still be at the starting line after spending half a billion dollars.
A way forward
The Content Knowledge for Teaching Mathematics (CKT-M) tool measures how well teachers and tutors know the content they are teaching. This tool is used internationally and predicts learner outcomes. I.e. the higher a teacher scores on the tool the higher their learners achieve. In other words - wouldn't you love to know the score of the teacher of your children?
We could develop a similar tool, or use the existing tool, and quickly begin to measure and improve teachers and tutors skill levels. Perhaps it is simply too easy.
Several people have asked me lately for a list of activities that can be done in adult classrooms. I thought I'd re-post this, because many in the adult sector didn't see it. Although the clip is designed for parents, it's equally useful for adult tutors.
All the activities in the clip can be adapted to the classroom and 'gamified' (turning them into games). The clip is a little long, if you skip the first three minutes you'll be straight into activities.
Also, I will be posting more problem-solving content very soon.
The first problem at the three minute mark is the 'golden handshake'. If you want more information on this problem go to the clip below and skip to 2:35.
Continued professional development a better indicator of a good numeracy teacher than a qualification
The title says it all. A qualification is great... but it does not compare with ongoing professional development when it comes to improving tutor performance. This finding is consistent - from Li Ping Ma to Diana Coben to Bass, Ball and Hill.
For example, note this comment from Sheila Macrae, taken from Diana Coben's (2003) review of Adult Numeracy provision:
The findings also raised questions about the sort of mathematical knowledge needed by teachers in order to be effective. Contrary perhaps to expectations, being a highly effective teacher was not positively correlated with high levels of mathematical qualifications, a finding supported, as noted above, by Begle’s research in the USA (Begle, 1979). Instead, the amount of continuing professional development in mathematics education undertaken by teachers was a better predictor of their effectiveness.
Here's my thought. Educators are in the business of educating. Therefore, an educator must be a master of learning, and that means commiting themselves to ongoing learning. Imagine being taught to drive by someone who didn't drive. There are things you can only learn about mathematics by actually learning mathematics.
Do note however, that the evidence does NOT diminish qualifications. A person with a qualification still outperforms one without (in general).
If you are teaching maths, the best thing you can do is to start to study. It will absolutely reinvigorate your teaching. It'll make you better, and it'll make your learners better.