# Multiples of 9 and 3 with Cuisenaire rods

You may have learnt at some point that multiples of 9 and 3 can be easily identified by checking whether the sum of their digits is itself a multiple of 9 and 3, respectively. But have you learnt why it is so? I, personally, didn’t. But it is a nice thing to look at, and it can be shown first with Cuisenaire rods before using algebra to generalise the observed pattern.

Let’s start by looking at the 5 first multiples of 9 with Cuisenaire rods in a 100-square, which is very useful when we want to focus on place value. …

# How to Improve the Teacher Training

I have posted several articles in this blog where I analyse, with the outside perspective of someone who was not educated in England, different aspects of the PGCE training and the English education system. I want here to summarise my critics on the PGCE training in a constructive way, as I deeply believe that improving the teacher training is a first step towards improving our education system.

The first school placement during the PGCE starts at the end of September. During the first weeks, teacher trainees merely observe lessons and familiarise themselves with the school and their future classes. During my training, only the eager ones taught a part of one or several lessons before half-term at the end of October. Most of us started teaching full lessons after half-term. From that moment on, we spent one day per week at university and four days teaching at our school. When we all turned up at university after our first week of teaching, I think any external observer would have noticed that almost all of us looked tense and tired. We taught only a couple of hours in our first week, but it required a huge preparatory work due to our inexperience and the associated doubts about any choice we had to make. And, above all, we experienced the immense stress of not only having to face a class of 30 kids, but also to do it while at the back of the classroom a teacher scrutinises every single one of our actions. In a nutshell: we had just got a flavour of the infernal working rhythm of a teacher and of the stress of a PGCE student. The latter would last for what we felt were the longest seven months of our life — for those of us who would hold on. …

# What Is a Teacher? — The Big Confusions

At his valedictory speech, the Director of the PGCE programme said that a PGCE is extremely difficult and that we should be proud to have accomplished it. According to him, the reason why a PGCE is hard lies in the fact that it is made of several parts, each of them being difficult on its own: teaching, carrying out a micro-research project, and writing academic essays. This may be true. PGCE students have indeed to distribute their work on completely different tasks: teaching (which is incredibly multitask in itself) and academic studies. Both are hard jobs, and they are also so different in nature that to conciliate both is a real challenge. Preparing and teaching lessons is extremely intense when you are a novice teacher. And I found it highly disturbing — and somehow distracting me from the core of my training , even if I actually enjoyed it, as I come from a scientific research background — to have to read research articles and carry out my own reflection on some aspects of schools’ policies in parallel to this. Some of us clearly chose to do the bare minimum for their essays just to pass, not seeing the point in all them. I believe that the very different nature of the tasks a PGCE student has to carry out is actually symptomatic of the confusion on the very purpose of the PGCE training, namely: Is it a professional training where the focus is primarily on developing PGCE students’ teaching skills through theory and practice? Or does it aim to train future education scientists? And this confusion reflects another, broader confusion on the teacher function. …

# Why Are Exercise Books Staying in Schools?

I have already mentioned Ofsted inspections in a previous post. Another perverse effect of the importance of these inspections is the issue with (exercise) books. When I was myself at school — and already at primary school — , I had different exercise books, often one per subject. The notebooks were an obvious learning tool, providing concrete elements of what had been done during the lesson, allowing students, when doing their homework, to go through the lesson content again and look at the exercises done and corrected in class. Also, homework was done in our exercise books. I would have thought this exercise book use is universal, and I hadn’t expected to find something else in England. Instead, exercise books are still used in England during the lesson, but they often remain at school. …

# Do We Want Our Students to Understand What They Are Doing or Follow a “Math(s) Recipe”?

In a previous post, I explained some aspects of the exam obsession in English schools. I want to show here with a few examples how math(s) educators sometimes choose to stick to a “recipe” in whatever situation, thinking it’s the best method to prepare their students to the exam, but at the expense of stimulating thinking and understanding.

The standard algorithm to solve linear equations is called the balance method. It is based on the fact that the equation remains true (balanced) as long as we perform exactly the same operations on both sides of the equation. The idea is therefore to carry out a series of operations on both sides of the equations to end up with = something. So, for the equation x -5 = 0, we add 5 to both sides, which gives x -5 + 5 = 0 + 5, i.e., x = 5. However, there is no need to apply the standard algorithm here since the question “What is the number that gives 0 when we take away 5 from it?” is easily answered by young children long before they can use algebra. But many teachers will stick to the ‘solving procedure’ whatever comes, to be sure students know how to solve an equation, even if, by doing so, they prevent them from thinking about what the equation actually says and linking it with their basic number sense. …

# The Lesson’s Structure in English Schools

When I chose to train as a teacher, I had in mind what my teacher friends in France told me, “Once you’re in your classroom, you can do it your way.” This autonomy was worth, in my eyes, earning less money than by working for a company where one is often closely monitored by a manager who can’t help but micromanaging their employees. The reality of English schools was however far from the image I had taken with me from France.

Quite at the start of my PGCE training, we were told about the lesson’s structure. Teachers are supposed to start every single lesson by presenting the differentiated learning objectives to the whole class (often in the form “At the end of this lesson, all of you should be able to do ‘ABC’, most of you ‘DEF’, and some of you ‘GHI’). The actual lesson should then begin with a starter, a short activity designed to engage students, and consist of student-led activities. The lesson finishes with a plenary where all the class are drawn together and learning is assessed and/or students are prompted to reflect on their own learning. …

# Family Maths Games with Cards

## A. Completing Number Sequences (from Reception/Kindergarten onwards)

From a standard 52-card deck, take only the 1–10 cards. Shuffle them and deal 10 cards to each player (2–3 players) or 6 cards (4–5 players). If the youngest player has an ace, they can start with laying this ace down. Otherwise, the player with an ace starts. If no player has an ace, the youngest player starts by taking a card from the pack. If it’s not an ace, then the next player takes a card from the pack, and so on.

Once there is an ace laid on the table, the next player can either put a 2 from the same suit next to the ace or put another ace. If none of these are possible, then they player takes a card from the pack. If they get a suitable card, they can lay it down. In the following, each player, at their turn, try to complete the sequence 1–10 in each suit. …

# Gender in STEM

It has been estimated that women make up around one third of the scientists’ population, all subjects taken together[1]. However, the proportion of women within each field varies greatly, with physics, mathematics and engineering being the subjects that attract the least of them.

When I was doing my Ph.D. in Physics in Germany, some people my age would look very surprised hearing I was a physicist, and they would ask me whether I really enrolled at university “as a woman”. What possible answers did they expect? …

# Student Assessment

Assessment is so central to education that it would seem impossible to imagine a school without any form of assessment. Here, we need to make the distinction between formative and summative assessments since their purposes are completely different, as can their effect on students’ motivation be as well. While summative assessment is an evaluation of students’ learning at the end of a teaching unit against some standard, formative assessment serves the purpose of promoting pupils’ learning. …

# Revisiting the Cosine Rule

The cosine rule (called law of cosines in the USA) is taught in England for the GCSE (at the end of year 11, which is the equivalent to grade 10 in the US) while it is taught in France only in year 13. This realisation led me to analyse further how it is taught in England and also to explore further the cosine rule itself.

First, I noticed that the most commonly used formula for the cosine rule in England is

in the triangle ABC where AB = c, BC = a, and AC = b, and the vertices A, B, and C are confused with the angles CAB, ABC, and BCA. [Sometimes, the three formula are given, starting with this one and then having b²= + – 2 ca cos B and = + – 2 ab cos C.] …