The joy of teaching and learning

This meme cracked me up for some reason.  I've been the teacher and the student.

The 'Consolidation of Responsibility' was the tip of an iceberg (Why adult mathematics classes fail)

Early research with adults in mathematics classes revealed a phenomenon called 'the consolidation of responsibility' (Howard & Baird, 2000; Karp & Yoels, 1976). This is when the members of the class cede personal agency to a few members of the class that they believe are more capable. Thus, a small number of the class members account for the majority of interactions between the class and the tutor, while the others remain silent.  If you are a tutor, you will be aware that you will have a learner or two in your class that you end up addressing more than others. It's bad, because we learn by engaging in discussions, not just listening to others. The problem is adult learners letting others take the responsibility for asking questions - however that is the very tip of a nasty iceberg.

My research elaborates on the previous research by allowing us into the private world of learners. This was achieved through the use of microphones that recorded all utterances, at all times.  I am able to recreate every moment of a lesson from various perspectives.  Imagine taking a 3D video of a class which you can freeze, fast-forward or rewind while being able to move around in any direction.  You could take the perspective of any one learner for the whole class, a group, or the tutor. I can do this in an audio sense, and thus fidelity is high. The results are fascinating.

I have been busy putting some theory around this using a symbolic interactionist (SI) framework. A key tenet of SI is that people are constantly interpreting others. They then modify their behaviours in order to achieve some need, usually a smooth interaction or collaboration, often in order to verify their identity in some way. In short, people read others and adapt. We 'read' others based on our beliefs about what their interactions and behaviours mean. Our subsequent adaptions are then 'read' by the others, changing the way they act - and on it goes! If this becomes maladaptive it can create interactions that are negative, which in a classroom will erode learning. This is particularly true for mathematics learning where we know that a particular pattern of behaviour results in learning and another does not.  

It looks a bit like this within a typical vocational embedded numeracy class. Group members assign status to members they believe have a high proficiency in mathematics. They then take somewhat passive or supportive roles and acquiesce to these learners. The learner with higher status is more able to assert their expectations for normative behaviour on the group. These behaviours are not consistent with effective learning behaviours - in fact they are completely negative.  The pattern of behaviour however, reproduces the conditions that produced status, and perpetuates a cycle of high/low proficiency members.

In other words - this pattern perpetuates the failures many learners experienced in school.

In conclusion, the research on the consolidation of responsibility ought to have attracted more attention from the adult mathematics community. It was the top of a very nasty iceberg that potentially negates all the learning potential in an adult class.

Developing Maths Skills at Home

Due to my role working with adults, I have a unique perspective on children's education.  My view is 20/20 because I get to look back on a persons' learning history and figure out what went wrong. A persons relationship with maths is ALWAYS a factor, for reasons I've discussed a lot on this blog.

The family culture possibly plays the BIGGEST part in determining whether a child will achieve mathematically or not. And it needs to start early.  The clip below gives ideas about how to cultivate a family culture that will contribute to a healthy mathematical outlook.  It also includes math problems you can use with your children or yourself.  If you are home-schooling this should be useful.

The YouTube version is here (recommended).  Or watch below.  

Developing mathematical prowess

A single sentence by Lakartos gets me thinking about what we are actually trying to achieve when we attempt to teach adults and children mathematics.

Speaking of the argument he is about to make in his book, Lakatos states: 

[The book's] ... modest aim is to elaborate the point that informal, quasi-empirical, mathematics does not grow through a monotonous increase of the number indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations. [Emphasis mine]

In other words, mathematical prowess does not primarily improve by adding the knowledge of algorithms and formula to our strategic repertoires.  Thus, if you decided as an adult to learn mathematics (and I hope you do!) you would not really benefit from attempting to memorize all the methods taught in your maths books.  Ah, okay, what should you do then?

You would improve your use of mathematics substantially by developing your ability to make guesses (speculating) and then criticizing the guess.  That means improving the quality of HOW you make guesses, and improving the quality of your criticisms.  This requires argument, proof and refutation - the building blocks of mathematics.

This is very different from how things are reflected in the NZQA numeracy standards and how content is taught in the adult sector.  We could argue about the differences between numeracy and mathematics, in particular that numeracy ought to be concerned only with giving people the skills to complete a job.  However, this argument generally hits the rocks when we realise that 'mathematical understanding' and 'agency' are still pivotal to improving numeracy.  

Anyway, if you want to improve your own, your learners', or your children's math skills then think about how you might improve their ability to guess and then to criticize their guess.  Because it is these two skills that help the learner to 'tack' in a zig zag way toward correct answers.

   

Invitation

Quite amazing that what we are doing and learning here in NZ is highly relevant to the rest of the world.

PhD tips

Your methodology needs to be your baby.  Love it and nurture it.  Look after it frequently.  Constantly work on it.  It is central to all that you do.

Hamburg - Here we come!

Well, it would appear we'll be heading to Hamburg Germany to attend and present a paper at the ICME 'Mathematics in and for Work' conference.  I have to say I'm pretty excited.  I've never been to Europe so it's a new experience for me.  

The article is fairly simple, it's looking at learner engagement patterns within a particular classroom. It doesn't address a whole lot of theory and hence does little in the way of answering either why learners engage in different ways, or what we can do about it.  I sometimes wonder if I'm making everything more difficult by making classroom interactions so much more complex.  But I quickly think of the quote attributed to Einstein: If I had one hour to solve a problem, I would spend the first 55 thinking about the problem and five about the solution.  

I think my research is part of the 55 minute problem-exploration component.  Without which, we may offer trite solutions to problems we don't really understand.  

A brief example can be found in the fact that many mathematics research articles on reform-or constructivist-based mathematics often end with a list of recommendations for teachers such as; value and promote discussion, value conjectures and justifications, draw on classroom knowledge and so on.  But HOW do you do this in a class of adults who do not want to make conjectures, but simply want you to show them how to solve the darn problem.

"Just show me how to do it!" he screamed at the tutor, who up to now had refused to do so.  The tutor ignoring the growing frustration kept repeating "Tell me how you have been trying to solve it?"  

Clang! - goes the clash of beliefs about how maths ought to be learned.

The issue of poor learner engagement is not purely a pedagogical issue, yet that is usually how it is framed.   It's complex baby!  And we'll be discussing it in Germany!