I have officially finished with the Discovering Mathematics module! Reflecting back on my time I think that I have greatly expanded my mathematical knowledge, thought about ideas I hadn’t considered before and most importantly had fun!

I think that having fun should be the foundation of any mathematical class, concept or learning. This, I believe, will give mathematics the much needed boost that it needs to become just as important to us as other subjects. Everyone looks forward to an art class, why can’t we look forward to a mathematics class?

The biggest thing I have learnt during this module is just how much mathematics is in the wider world, so much more than I could have ever possibly imagined.

Moving forward I plan to embrace mathematics, to allow it back into my life. When I am carrying out specific tasks I will look for the mathematics within and recognise it.

I have already found myself saying to others “Do you know there’s mathematics in what you are doing right now?”

Moving forward mathematics is not going to be something I learned in the classroom, or a module I took in university, it is going to be a part of me in my life, and this module has helped me to recognise this.

A little advice to future students partaking in this module, don’t get too hung up on the actual sums behind the inputs, enjoy and embrace them instead, because the essay and the blog-posts aren’t assessed on your mathematical ability that you developed in school, it’s about your engagement!

In a lecture with Elizabeth Lakin we discussed the ways in which teaching science and mathematics may be problematic.

We discussed how sometimes for a child the application of a skill set or knowledge from one subject to another can often be difficult, this is widely applied to mathematics and science. Again I am going to mention the mathematical myth which limits the application of mathematics outside of the classroom. A pupil who engages with this myth will find it hard to use their mathematical knowledge out with the mathematics lesson. It is important as teachers that we try to diminish this myth but also if we are planning a science lesson which uses the same concept being taught in a mathematics lesson to teach the concept in a scientific context in order that bridges are built between the two subjects.

We also discussed the difficult with the stage at which mathematical concepts are taught. Quite often one of the first ideas in science is the interpretation of information from graphs. However in mathematics the introduction of graphs and data analysis is in the upper stages of primary. The issue is then we are expectant of children in the science classroom to be able to interpret information from a complex source which is entirely new to them.

The idea of staging relates to Liping Ma’s idea of longitudinal coherence. If the teacher of the class had achieved PUFM they would not be tied to the idea that graphs is taught in the upper stages but would tailor the learning to the needs of the children. They would plan in order that the learning took place at the most beneficial time.

How likely are we to win big when gambling? Gambling is entirely centered around probability, the likeliness of a win or result. We bet on the likeliest result and cross our fingers for a result. Once every now and again, we will win, so how can it then be said that the bookies always win, because quite often a person wins their bet.

It is because the bookies decide the odds that they give out. They calculate the odds to ensure that they will make a profit on the bets they receive.

For example, in a game of Dundee F.C. versus Dundee United F.C. the odds of Dundee winning are 3/10, which means for every £10 bet only £3 is paid out, plus the initial bet.

For a draw the odds are 4/1, a £4 pay out, plus the initial bet. And for an away win 9/1, £9 payout plus the initial bet.

No matter what the result, there is always a return for the bookie. The mathematics behind this includes fractions, chance and probability, multiplication etc.

This relates to Liping Ma’s idea of multiple perspectives, however not mathematically. Here we are looking at the different aspects of gambling to gain a greater understanding of the foundational principles behind it. The effects for the person making the bet and the effects for the bookmakers.

In conclusion, the bookies do always win, and this is because they use their mathematical knowledge to ensure their odds are in favour of them. From this I have learned that I am studying the wrong profession, the bookmakers is evidently the way forward!

Cake again!

When baking a cake, you use an incredible amount of mathematical processes, but you don’t think of the mathematics you are using as you are doing it.

When measuring the ingredients for a cake, we use the basic knowledge of measurement taught to us in primary school.

You need 500g of sugar for the recipe, so half of the bag – FRACTIONS!

The recipe in in ‘kg’ but the measurement on your ingredients are in ‘g’ you have to use your BASIC KNOWLEDGE OF MEASUREMENT!

You only require a quarter of what the recipe makes, to work this out we use RATIOS!

To set the oven timer to cook the cake for 20 minutes we need to understand the basic concept of TIME!

Who knew there would be so much maths in baking, especially ratios!

But why don’t we immediately think of mathematics when we are doing these processes?

It is perhaps due to the mathematical myth discussed in a previous post that ‘We do not use mathematics outside of the classroom’, quite often if we say something to ourselves often enough it becomes truth. If we believe that maths is not applicable in the wider world then perhaps we do not look for it as we do not expect to find it.

When trying to work out how to only make a quarter of the cake, the baker would more than likely spend a great deal of time trying to work out how much of each ingredient they would need, instead of quite simply using a ratio formula.

This relates to the idea of connectedness by Liping Ma. Looking for mathematical links in the wider world, in order that the learning is supported by these connections.

If we as teachers continually make real life applications then perhaps the children will make these applications in their everyday lives. The aim is to not let mathematics die in the classroom, but to revive it in everyday use, where we recognise it, not just simply use it, unknowingly.

One of life’s’ burning questions, what came first, the chicken or the egg? In a lecture with Richard we discussed statistical information and data analysis, and the same premise from the question above became applied to our discussion.

Do you record the data first, or do you interpret the data first?

Similarly, this seemed to be a question which had no answer, much like the idea of the chicken and the egg, but does it have an answer?

Some may argue that you have to have data recorded in order to be able to clearly interpret that data. For example, if you were measuring how many brown haired people were in a room in comparison to blond, it would be hard to interpret this data without the exact numbers presented in ‘black and white’. The interpretation of this data would not be accurate without the information being recorded.

Others would argue the case that you have to have an understanding of the data before you record it. There has to be an interpretation of the data to determine its worthiness of being recorded.

This question is a conundrum, similarly to the chicken and the egg, there is no definitive answer. Whether recording or interpreting happens first is different in any situation. It also depends what is meant by the word interpret, does it refer to an understanding of the concept which you are investigating or an interpretation of the data presented by the concept.

This relates to Liping Ma’s idea of basic concepts. We are not simply looking at recording and interpreting data but the reasons behind why we use these processes. We are looking into the structure behind data analysis, and gaining a greater understanding, not simply the use of it as a process.

I remember learning fractions in school, and every teacher, without fail, related the idea to cake!

As you can imagine this was very engaging as a child, and now in my later life I often will say “I will have a third of that cake please”. That’s if I decide to share it! Most of my understanding of fractions relates to food and portioning of food.

Until recently, when we had our input on mathematics and music, I had no idea that fractions feature in music!

When reading music, the note determines how long the sound is played for. A whole note would last one measure and half a note would last for half a measure. Each of the notes in music relates to a fraction of the measure of time the note should be played.

It is important for a musician to have an understanding of fractions in order to interpret a piece of music.

We can use music to teach children about fractions in the same way that we use food. Using another context for learning could emphasise to children that there is not only one application of fractions in the wider world, but many.

I feel this relates to Liping Ma’s idea of connectedness. Not just connectedness between the mathematical concept itself but connections between the concept and ideas out with the subject area. This results in the different subject areas not being fragmented but the curriculum being recognised as a whole.

When I first started this module and we were introduced to the idea of ‘profound understanding of fundamental mathematics’ (PUFM), I was slightly terrified. It sounds terribly confusing but actually it boils down to very simple concepts. “By profound understanding I mean an understanding of the terrain of fundamental mathematics that is deep, broad and thorough” (Ma, 2010).

Connectedness – “A teacher with PUFM has a general intention to make connections among mathematical concepts and procedures.” (Ma, 2010). This simply means being able to identify ways in which mathematical concepts procedures connect to one another, and highlighting this when we teach so that children can then identify these links. In practice this would mean the learning of a child was not fragmented.

Multiple Perspectives – “Those who have achieved PUFM appreciate different facets of an idea and various approaches to a solution, as well as their advantages and disadvantages.” (Ma, 2010). This means that the teacher respects the different aspects of problems and solutions and allows children to explore these different aspects in order that they have a flexible understanding of the subject.

Basic Ideas – “Teachers with PUFM display mathematical attitudes and are particularly aware of the “Simple but powerful basic concepts and principles of mathematics” (e.g. the idea of an equation)” (Ma, 2010). This means that teachers encourage children to explore the ideas in relation to a problem as opposed to simply calculating the solution. This will mean their learning and understanding of the subject will be more in depth.

Longitudinal Coherence – “Teachers with PUFM are not limited to the knowledge that should be taught in a certain grade; rather they have achieved a fundamental understanding of the whole elementary mathematics curriculum.” (Ma, 2010). This means that teachers are willing to revisit learning done in previous years, but also to plan accordingly with the flow of the classrooms curriculum and meet the present needs of the child within their studies.

PUFM is more than simply understanding the subject area that you are teaching. It is embracing the subject as a whole and appreciating the foundations of the subject. Having a profound understanding is what we aspire to as teachers. If we are expectant of a child to have great depth of a subject, then so must we.

Liping Ma, 2010, Knowing and Teaching Elementary Mathematics, New York, Routledge, p.104