This blog post was written by OU student Jim Darby.

We have republished this (with permission) from Jim’s personal blog. Jim has encapsulated the learning outcomes of reflecting on his own thinking and appreciating the range of learners thinking. Jim has described a rich working relationship with the school he works with. Our students vary from those who have such opportunities to those who work with one or two learners perhaps from their own family.

After sending in my final EMA (End of Module Assessment) for ME625 I find myself reflecting on it all.

I ought to begin by introducing myself. My name is Jim Darby and I work full-time in the computing industry specialising in security of the computing infrastructure of large commercial enterprisies. I started with the Open University (OU) doing an Open Degree with Spanish as the first option because the local college had stopped all adult education. After completing Spanish I began to look for other things. Astronomy sprang to mind but I wanted to revise maths first. Maths (MST124, MST125 and MST224) followed then I moved onto Astronomy (S282) and Planetary Science (S283).

[**Brief aside:** For non-OU students you’ll see a lot of references to letters followed by numbers. These are the identifiers for specific courses, more accurately called *modules. *Also “student” refers to an Open University student and “learner” refers to someone learning mathematics that the OU student is working with.]

About this time I had begun volunteering at a local school, initially to work with mentoring learners (pupils). A new trust had recently taken over the school and were very keen to improve learner outcomes, with mathematics identified as a key area. The school had identified several learners in need of additional support but I felt woefully inadequate for the task, then I saw *Developing Mathematical Thinking in Schools* (ME620)…

Reading the course details it seemed pretty much ideal because it was the underlying *thinking* that I wanted to address and develop. I made the choice and took it.

**It utterly changed my views on learning and doing mathematics.**

I have to be very clear here. This is *not* a teaching course. I’ve found a fair amount of confusion about this: firstly myself and then from the teachers I’m lucky to work with.

What the course is about is the *study* and *development* of mathematical *thinking *both by the OU student taking the course and by the learners they’re working with. The modules (ME620 and the ME625) are focused on investigating and developing how we think about mathematics and how we learn it. The modules are based on highly reflective work where the student considers how they work to solve specific tasks and later on how the learners go about the same task. This is reflected in the assessments where questions are often in pairs to allow students to compare their processes with those of their (typically) younger learners when faced with a similar task.

There are major differences between these courses and teaching courses. These differences are very important. It would clearly be unacceptable to spend an hour long maths lesson focusing on a tiny proportion of a class and ignoring the rest. With the ME-series (Maths Education) modules we work with small groups or (most commonly) one-to-one to conduct an in-depth investigation of their learning. The emphasis is strongly on encouraging them to solve the problems *their* way with as little scaffolding (support) as is possible. In fact, revealing where their processes *differ *to that of us, the OU student, is an essential part of developing understanding of how *everyone *learns.

I am *extremely* lucky in having a great and highly-cooperative school to work with. Without their support I would not have been able to complete the courses. They lent me some *amazing* learners with whom it has been a pleasure to work. To be able to work well on the course you will need access to learners of mathematics (of any age) but they will need to be in a small group (often one-to-one) to allow the “deep dive” of what’s happening: a class of thirty just isn’t suitable.

Some of these learners had difficulty in accessing mathematics and presented with widely divergent levels of achievement, motivation and engagement. I was able to investigate their approaches to mathematical thinking and this helped me with the modules and (more importantly) the learners with their understanding of mathematics. Being able to “deep dive” their mathematical *thinking* using the ideas, concepts and models from both modules over the course of a year gave me a wide range of strategies to help them overcome some or their barriers.

It’s certainly possible to use just a single learner on the courses, but personally I found having varied learners in the school beneficial in contrasting mathematical thinking: an essential core of the modules. The point is to investigate *how* the learners’ approach solving mathematical problems and *why* they make the choices they do.

Having a basic understanding of the learner’s current achievements is essential to session preparation. They need to be challenged, but not *too* much.Getting the level right is often difficult, especially if you have a group you haven’t worked with before. Set it too easy and they’ll just march right through it revealing little about their problem solving processes. Set it too hard and you may find, as I did, that you’ll end up with a student sitting under the table glaring at you! If that happens you may need to reduce the task level…

However, once you’ve established a good working relationship with your learners then the ME courses are immensely rewarding. I found that working one-to-one with those learners having problems accessing classroom mathematics often helped them overcome the issues they had with learning mathematics and allowed them to make additional progress. I used many (if not all) of the modules’ concepts to analyse these barriers and assist the learners with breaking them down.

In analysing how effective various strategies were, I was able to gain substantial insight into how *others* access mathematics and the obstacles they face. The differences to my own learning processes were a great surprise and to me this was by far the most important end result of the modules.

Additionally, in a few cases the learner’s issues surfaced as behavioural issues, often borne of frustration. However once the learning issues were reduced their behaviour improved. Similarly for those becoming bored in classes and wanting a greater challenge I was able to provide tasks that deepened and broadened their understanding. Both of these are of great benefit to the learners, myself and the school.

I found one of the major parts of the modules is the one-to-one time to analyse the learner’s thinking *in great depth*. This would be substantially harder (if not impossible) in a class of 30-plus but it is an essential component of the ME courses. The initial analysis occurs during a session with a small (ideally one) number of learners. Later a more reflective account plays a central role in the course assessments. It is very much expected that this reflection will enhance the student’s understanding of how we *think* and *learn* mathematics in general (for ME620) and Algebra specifically (ME625). There are many “module ideas” and their use in developing this thinking, both in terms of practice and in terms of the reflection by the OU student. These ideas and techniques are useful both as approaches to support learning and to describe what actually happened.

I was able to have hour-long sessions with my learners and I would suggest that this is more-or-less the ideal length. Shorter and you don’t get enough time to go through a task, longer and your learner’s attention is going to fade. I believe that these small group sessions were well worth the time and effort because they enabled the less confident learners to better long term participate in mainstream education. With the more confident ones it allowed the exploration of topics in greater depth. Ultimately it is worth investing in for both the course student (you) and the learners.

Full-time teachers or TAs (Teaching Assistants) will find it difficult to make the time for these small-class sessions. You should be aware of this before beginning the course.

If you’re considering working (volunteering) with a school then it is *essential* to have a good working relationship with them. You should be familiar with how the school works in terms of lesson planning, timetabling and general ethos. It is a privilege to be able to work with learners so you’ll need to ensure that it all goes smoothly. You will almost certainly need to obtain records from the Disclosure and Barring Service (DBS) as well as being familiar with the school’s safeguarding process and principles. It is critical to be able to work well with the school’s Mathematics Department as well as its senior leadership team.

Returning to the theme of understanding how *others *learn and think about mathematics I would like to highlight an example of how my views were so radically changed. A few of the learners were finding fractions hard to work with. Before undertaking the modules I would have thought that this was “obvious” and have given a perfunctory (and ineffective) description. However, by employing skills learnt on the modules I was able to provide far more useful advice by first determining what they already knew and then working with them to expand that into a deeper and broader understanding. This was a very interactive approach often starting with physical models used to ensure that the core *thinking* was a sound foundation before building on that. At each step I would ensure that they were not repeating what I had just said but instead had grasped the underlying concepts. We would then use these new concepts and build upon them to the next stage. I really enjoyed the time that we had to explore how they thought about the concept of fractions and how they work.

I must end with a young learner’s comment made during my final EMA. After working with her for about three quarters of an hour she appeared *most* upset. However, she smiled as she said “You tricked me into learning *ALGEBRA!” *

To me, that’s the ultimate aim of the ME modules.