The Decimal point and Place Value

 My daily twitter browsing showed me the recurring argument about "moving the decimal point" in relation to multiplication or division by 10, 100 , 1000 etc:

The thread is worth a read (you can get to it if you click the image). I read it, and some of the quote tweets around it, and it got me thinking about what I think are two possible misconceptions around the decimal point and place value that seem to exist in the minds of teachers of maths. I am sure some will disagree with one or both of these, and may think that these are misconceptions in my mind instead. But anyway, here goes...

Misconception 1: That the decimal point is a fixed immutable point between the ones and the tenths

There are a number of people saying in that thread that the decimal point cannot move, that it must stay between the ones and the tenths. This for me seems false. The job of the decimal point is to separate or mark the transition between whole values of our unit, and values that represent part of that unit. Of course, in most cases our unit of counting would be ones and the decimal point therefore marks the transition from ones to part of one. But that isn't always the case. Consider for example:

2.6 million

In this, the value to the left of the decimal point does not represent two ones, it represents two millions. The unit we have chosen to count in is millions, and so the decimal point separates the whole millions from the parts of a million. This has obvious parallels to something most mathematicians will be familiar with converting between standard form and our "ordinary" decimal number system:

320000 = 3.2 × 10⁵

In converting to standard form we are literally changing our counting unit from the ones to which column has the highest value in the number we are working with. This means that the decimal point does move, it moves to separate whole values of our new counting unit (in the above example 10⁵) from parts of this counting unit. In converting from standard form we do the reverse; we change our counting unit from the largest valued column back to the ones column, and the decimal point moves concurrently. One could even make the argument that converting units of measure could be viewed in the same way:

3.25 metres = 325 cm

Have we multiplied by 100 to go from left to right here? The physical distance hasn't changed? Would it make more sense to consider that the decimal point has moved due to our change of unit choice from metres to centimetres, and so what was separating whole metres from part metres, is now separating whole centimetres from part centimetres.

Misconception 2: That the decimal point moves when we multiply or divide by a power of 10

Despite what I have written above, and some compelling arguments within the thread, I still come down on the side that it is wrong to teach pupils that the decimal point moves when you multiply or divide by a power of 10. This is not just because of its tendency to be used a trick for teaching without understanding, but more because conceptually it actually doesn't fit with what is happening when we multiply or divide. If we accept that the decimal point moves when we decide to change our unit (perhaps a big if for some), then the decimal point cannot move when we multiply or divide by a power of 10. Consider:

3.2 × 100 = 320

There is no change of counting unit in this situation. In all three numbers, the counting unit is ones, and the decimal point separates the ones for the parts of one. What has changed is the physical size of the numbers that each of the digits represents in the number 3.2. The 3 has become 100 times bigger to become 300, and the 0.2 has become 100 times bigger to become 20. This is synonymous with moving the digits up the place value columns, and definitely not the decimal point down - or even translating the column headings down. Although I can see the argument that says "we can consider 3.2 × 100 as having 3.2 hundreds, and what we are doing is converting that back to a number of ones" I can't make that fit in my own mind with the importance of making sure pupils recognise and appreciate the multiplicative relationships between the place value columns. Even if our learners are completely secure with this, I can't see why we would then use a unitising approach to multiplication to model multiplicative calculations with different powers of 10 - except maybe if we had reached the point where pupils were so secure in this that we were opening them up to another way of making sense of such calculations and perhaps attempting to highlight that moving the decimal point is akin to the equivalent multiplication or division.

In summary, the decimal point definitely can move, but probably shouldn't if we are teaching multiplication or division by powers of 10 unless we are taking a unitising approach to this sort of calculation.

What is a valid mock?

 So Ofqual have released (on a Saturday) the criteria for deciding what a "valid mock" is for the basis of appealing a pupil's A-Level, AS-Level or GCSE grades. If you haven't yet seen them, the criteria can be found here: https://www.gov.uk/government/news/appeals-based-on-mock-exams

One of the criteria in particular caught my eye:

I remember in 2015, when the new GCSE in Maths started, I made myself very familiar with the approach that exam boards take to set their boundaries. It is quite involved to say the least, using national data on prior attainment of the cohort, previous performances of other cohorts, all sorts of data. I doubt any individual school or trust could match those standard - I remember the laughable outcomes when PiXL first tried and the flack they got for the grade boundaries they came up with. My blog on our process at the time is still my most read blog: https://educatingmrmattock.blogspot.com/2016/11/new-gcse-grade-boundaries-my-thoughts.html

Which of course presents a problem. For a mock exam to be graded in line with the exam board's examination standard, I can't see any alternative but to have used the grade boundaries provided by the board, on the same paper(s) provided by the board. Anything else simply doesn't (in reality) live up to examination standards. No exam board would arbitrarily reduce boundaries to reflect that pupils haven't completed a course, which is a common practice in schools. No exam board would simply use the average of a set of grade boundaries, or apply one years grade boundaries to a different set of papers. Any school or Trust who has done this can't sign the form to say their exam was graded in line with the examination standard provided by their exam board. Nor can those schools that use papers they have created themselves, or those that use practice papers that don't have grade boundaries provided.

In my own department we use a practice paper that comes from the exam board, but does not come with boundaries. We set initial boundaries for that years ago, before the first ever exams on the new GCSE and so before their were any past papers in existence. We sat the exam at a similar time to some other local schools, pooled our results together, and applied a similar process that exam boards were applying for their first set of new GCSE papers. We then compared the results that pupils eventually got with their mock exam results, and adjusted the boundaries for the following year. We used the same papers for mocks every year since, continuously refining the grade boundaries as we got more real results to compare to. This is about the most robust system any school could implement to grading, but even this falls short of examination standards. 

Of course, the head of centre only needs to sign a form saying "I confirm this mock exam meets the criteria". No one is going to check. No one is going to ask how they know the boundaries meet the criteria. Unless this is going to be used as a "get-out" clause to deny large numbers of appeals, then I doubt this line will end up meaning much. But it should, and that is the shame of it all.

Maths as an option?

This tweet caused quite a discussion recently on Twitter.

My answer to this was quite to the point.

However, I was asked to elaborate as to why, and I thought it would make more sense to write it in a blog rather than as a series of tweets that might (would) turn into an overlong thread.

My reasons for opposing Hannah's proposition fall broadly into three camps:

1) There is so much more to maths than what you can do with it (its functionality).

2) The central place that mathematical knowledge serves in our society and history.

3) The inherent biases that exist within education and the minds of maths teachers.

Regular readers of my blog/listeners to my various presentations/interviews will know that I am a big and loud (in both senses) proponent of the view of maths as a collection of connected ideas, and that our job as maths teachers is to support pupils in reaching an understanding of these ideas and how they are connected. This doesn't necessarily mean a "guide at the side" approach to teaching - indeed a wise sage that knows exactly what things to say, questions to ask, examples and representations to show is often exactly what it needed to support in gaining wisdom. What this does mean is a need for teachers to really have a depth of knowledge about each mathematical idea, and its place in the larger concept.

Let us take a couple of examples, in fact the ones mentioned in the thread, namely constructions and histograms. Both are touted as superfluous to requirements in a "modern" mathematical education. However, constructions are intricately and inexorably tied to wider knowledge and understanding of 2-D shapes and their properties. This is often not made explicit because, as Dani Quinn (@danicquinn on Twitter) pointed out, constructions is very often approached as a series of "do this, do that" steps with no effort made to tie it to the larger concept. Getting specific, consider trying to create the perpendicular bisector of a line segment. But rather than thinking about the steps required, think instead about which 2-D shapes have the property that at least one of the diagonals perpendicularly bisects another. The list you might have come up with is:

1. Square
2. Rhombus
3. Kite
4. Arrowhead (if you class that separately to a kite).

Now think again about the standard image for a perpendicular bisector:

Can you see which shape we have created in order to create the perpendicular bisector? It is of course the rhombus. Do we make it clear to pupils that this is the case. Or that alternatives exist? Our list suggested 4 different shapes we might draw that could lead to the same result. Can you see how we would create the others using similar techniques? The images would look something like this:

Except....there is something strange about that last one....it can't be done! Not precisely anyway; you might luck out and get both sets of two arcs to cross on a line that is at 45^o to the line connecting the dots, but no way to make sure. So although there are four shapes that could lead to that perpendicular bisector, only three of them are constructable. Can you see why? What is different about the definition of a square compared to the others? Yes, it is because a square is defined both in terms of its sides and angles, whereas rhombi, and kites are defined just in terms of their sides. And although an arrowhead must have a reflex angle, it isn't particular about the size of that reflex angle, whereas the angles in a square must be 90^o.

When discussions like this are taking place around these mathematical ideas, then constructions becomes a rich, deep vein in which to explore all manner of aspects of properties of 2-d shapes. Similar discussions can be had around angle bisection, and construction of other shapes. A nice link to make is between the perpendicular and angle bisectors. I will leave these three images as a prompt and say no more...

So what about histograms then? No one uses them in real life! No one creates unequal groups! Whilst this may be true, my thoughts are simply:

1. They should!
2. They are still rich links between these and other areas of maths and the social sciences to exploit.

There are plenty of situations where classes naturally arise that aren't equal in size. Many companies wage structures have jobs in salary bands that aren't equal in width. Boxing weight classes aren't equal in width. Months of the year aren't all equally sized. Any situation where frequency is being collected and analysed in these sorts of situations should not be represented in a simple frequency diagram, and kids need to know why. What is it about frequency diagrams that doesn't work if classes are unequal? What can we do if we are in a situation where it makes sense for classes to be unequal, without arbitrarily forcing them to be equal. Take for example this frequency table of salary bands (made up!) for a company:

 

If you were to draw this as a standard frequency diagram, it might look something like this:

There are several problems with this. The greatest problem is that the 30 to 40 group appears far larger than the 22-25 group, where in fact it only represents one more person. This is of course because the area of the 30-40 bar is far larger than the area of the 22-25 bar, and this is almost entirely down to its width. A similar problem exists in comparing the 18-20 and 20-22 bar with the 22-25 bar. It appears that both of the first two bars would fit almost perfectly inside the third, where in fact there are 4 more people in the first two bars combined compared to the third - a quarter again of the frequency it is supposed to represent. All of these problems arise because we are immediately drawn to the area of the bar as representative of its size rather than its height, particular when bars are different widths.

If we now see the same data plotted using frequency density:

What we have now is a fundamentally different distribution, with the earlier wages much more pronounced. Looking at the first diagram one might thing that there is an abundance of people being paid in the range of 25-40k. No one looking at the second diagram would make the same claim.

This of course comes back to the idea of unit measure. Unit measure is a fundamental concept in geometry and measures. What histograms allow for is a reexamination of unit measure in a new context, and the skilled teacher can use this as a vehicle to re-highlight this concept and show its importance even outside the realms of measuring quantities. The links to the idea of density as being per unit are also important - I always make links to population density and materials density when teaching histograms, and what they have in common around the idea of defining a unit to allow proper comparison. Histograms are also a great place to look at units other than "1"; in the example above the unit is per £1000. Properly taught, histograms allow us to impress upon students that a unit can be whatever we want it to be, as long as it is the same for all applicable circumstances. One of my fondest memories of my teaching pre-lockdown was with my Year 7 class, and defining the unit of a "Mattock" as my height, and then having other pupils estimate how many "Mattocks" tall they were (of course they are all shorter than me so it was a nice way of looking at estimating decimals as well). The joy and understanding that came from that activity were palpable, and if I am teaching those kids in Year 10/11 I will reference it back again when we come to histograms.

Every aspect of the curriculum up to 16 (maybe even 18!) has these connections between and within concepts. Part of the job of a teacher (in my opinion) is to understand these links and use them to create that connected learning experience for every pupil. No pupil should be denied the opportunity to see where and how these ideas resurface and develop - this is at the heart of what it means to teach maths.

This leads me nicely onto my second point - to me maths is universal (in two senses). Mathematics is at the very heart of our society on several levels. Advances in society are inherently tied to advances in mathematics. Our ever increasing technological development is down to advances in mathematics. Mathematics is the language humanity uses to describe the physical processes of our universe. As we develop mathematically, we develop a better language to describe these processes, and this language is what allows us to manipulate these processes to our own betterment. The recent advances in areas such as quantum computing are due in no small part to an increasing sophistication by which some people are able to use mathematics to explore what is possible.

Not only in technology does maths wield its power. The more recent advent and development of statistics has given us a window into human life that simply wasn't possible a few hundred years ago (along with our technology allowing us to connect and share in ways unthought of until recently). We know more about human life as it is now than we ever have done at any point in history, and mathematics goes to the very core of what allows us these insights. Developments in all areas of mathematics are what allows society to develop, and although mathematics has developed to such a point that fewer and fewer people are now capable of advancing the field further, this doesn't mean that we should only be sharing that story with those that might be capable of taking up that mantle. Mathematics has at different points played a central role in our history, our culture and our development as a species. I hope very much that I am not alive in the generation that decides that this birthright should not be passed on to every human being possible, even if relatively few will make use of it in any meaningful sense.

Mathematics is also universal in a different way, in that it is something that literally every human with a functioning mind can advance at. Some people are born without the use of arms and legs that might proclude them from many opportunities that others take for granted. Some are born with medical conditions so severe that they will never walk, never run, never build, never take apart or dissect. Mathematics, as primarily a discipline of the mind, is open to all. Every human with the capacity to think can advance their own understanding of and ability within mathematics, and as such I strongly believe that mathematics has to stay a subject open to all.

This leads me to my third point, the inherent biases of our education system and of maths teachers. Let us take as fact that mathematical attainment correlates well with future earnings (for those that can't take that as fact, this article and this report should be enough to get started with). Let us also imagine that maths, beyond basic functional numeracy, was optional for pupil, post-14 say. Who do we think would be opting for it? And more importantly, who do we think would be "discouraged" from opting for it because schools/teachers are worried about how their success rates in maths might look? I would suggest that those from disadvantaged backgrounds, those from certain BAME backgrounds, those with SEN and other learning difficulties, they would make up a large proportion of those pupils that didn't "opt" to continue their mathematical education. They would be the ones where teachers are having conversations like "maybe functional numeracy would be better for them to support their future aspirations", where of course one of the big things education should be trying to achieve is to broaden kids horizons so that they can continue to re-examine those aspirations in light of what they learn is possible. I can easily imagine the situation where a white middle class family is pushing their child through a mathematical education, supporting with tutors and being ever more demanding of their child's school, where huge swathes of disadvantaged pupils who might actually do well from a mathematical education don't continue it. We all know those pupils that despite everything have come good in maths, and we also know those pupils who haven't. But to deny that opportunity from anyone goes against everything I stand for as a teacher and a human being. Maths as an option is a pretty sure fire way to ensure that the gap widens between the haves and have nots, between the mostly white middle class and those of BAME backgrounds. In a time where, as white people, our relationship with and treatment of other races is under increased scrutiny (and rightly so), the changing of maths to an option would be a huge backwards step in making sure that everyone has the best possible opportunities throughout the rest of their lives. Of course, pupils from disadvantaged and some BAME backgrounds already typically underperform compared to their white middle class peers, and we really do need to look hard at that and take every step to deal with it quickly, but I can't see that part of the solution would be to create a system where large numbers of these pupils can simply be pushed out of it altogether.

The final thing I would say on this matter is this: as a sector we really need to look at what we class as "success" in a mathematical education. If we are talking about anything up to 30% of pupils who reach the age of 14 and are still functionally innumerate, enough that we need a whole separate qualification for them because GCSE isn't suitable, then that is a real problem. I know at this point people will start saying "but the exam system is set up like that" and I understand that (although there is flexibility in it if large numbers of pupils start performing significantly better), but regardless of the grade, I am talking about what these kids actually know and understand. If we are saying that 30% of pupils cant reach half way through a Foundation GCSE paper, after 11 years of maths eduction (bearing in mind that much of the same content also appears on a SATs paper that Year 6s are given) and that it is wrong to even try to support them through that, then I think the problem is not the qualification so much as the previous 11 years. The fact that only 1 in 5 pupils can get over half way through a higher paper is a damning indictment of our education system, and offering an "easier" qualification will not solve that problem. I think our attention needs to be directed much more to making sure that no pupil reaches 14 without already being functionally numerate rather than what we might do for the final two years when they do. On this I have no answers, as my results as a teacher and head of department are no better or worse than many others out there, but as a head of department that is what I want to spend my time thinking about, not which kids I should allow to continue studying maths, and which are going to be denied entry to all of the opportunities that studying maths allows.