MathsConf18 (Part 4)

4. Changing Units (Jo Morgan)

So here it is the highly anticipated fourth blog post on MathsConf18! This blogging thing really requires you to not be swamped by work & in the mood to do it, phew! (PS sorry for lying about this being written two days ago)

So the fourth session I visited was hosted by the esteemed Jo Morgan of resourceaholic fame. I have followed Jo for quite some time but as of late Jo has been delving into the annals of history through the medium of the text book which has really piqued my interest.

As a bit of an opener Jo posed the following question:

How did you solve it?
I went with the tried & tested ratio table with a mix of multiplications & divisions as necessary.

So why should we worry about Unit Conversion? Well it appears to be one of those topics that we, as teachers, seem to expect our students to be able to complete quite well but if we look at past exam questions they don’t.

Is it because it’s a topic that’s not covered in depth early in the students’ progression? Is it because there is pre-requisite knowledge that isn’t taught effectively?

Students first come across metric units of measure in year 3

They are then introduced to the concept of converting between different units in year 4…

…and again in years 5 & 6

So why do students struggle to convert units when they get to GCSE? Do we, as secondary teachers, ‘unteach’ them? Do we try & rigorously impose a deliberate & abstract methodology that removes the students’ conceptual understanding? Perhaps we need to rethink the way we teach this particular topic.

It’s certainly a topic that needs to be covered if you intend for your students to fully access the new GCSE as it seems to be a popular choice for exam writers to assess. Looking at the last two years worth of exam questions Unit Conversion appeared 10 times on Edexcel papers & 8 times on AQA.

So what are the barriers to learning? (Steps to success)

  1. Multiplying & dividing by powers of 10
  2. Memorising the Unit Conversions
  3. Performing the conversions

With these as our basis we can ensure that our students are more confident with converting units. Jo didn’t cover multiplying & dividing by powers of 10 & neither will I as this would really be teaching you how to suck eggs. Instead we should concentrate on perhaps the thing students struggle the most, the memorisation of the Unit Conversions themselves.

All the conversions your students will need to know

So how can we embed these?

Well there is the tried & tested tabulated prefixes linked to the multipliers.

Will this be enough? We can take a leaf out of one of Jo’s old text books & discuss what the metric system is, the next example is from a book that predates the widespread use (& legal recognition of it as the UK’s official unit of measure) of the metric system!

Before every child grew up knowing of the metric system it had to be described & spoken about in great detail, perhaps if we went back to doing this it would help to embed the understanding of it. Interestingly, did you know that all the multiples of a unit have Greek prefixes & the Divisions of a unit have Latin prefixes?!

We can also bring it to life using video, Jo has suggested that we take a look at the following YouTube video.

Don’t forget we have some excellent concrete manipulatives available to us within the classroom!

So now we have explored the idea of memorisation of the key facts, we need to look at the actual performance of the conversion. There are many options available to us in this regard, it is purely a personal choice as to which way you use, but Jo wanted to show us a way that she was exploring, namely “Last Man Standing”.

So the Last Man Standing technique relies upon your students understanding the concept of ‘cancelling’ (cough cough, excuse me, did I just say that word?!)

We also need to be able to write the number one in different ways, namely using conversion factors, which are widely used in science

So this takes a little practice. Lets say we wanted to 128m into cm. We’d need to choose the factor that had both cm & m in it, we’d also need to ensure that they were arranged in a way that enabled the metres to be (ahem) cancelled.

This would lead us to multiply by 100 giving us 12 800cm.

Let’s try another:

This can also be used with compound units as well:

Now, I’m glad it took me so long to write this blog, as it has meant that I have been able to trial this technique on my students. They were a little confused the first time we tried it, but the more we looked into the different aspects of it the more comfortable they became with it. They have also given feedback on it compared to some of the other techniques they have used; because of the way in which the calculation is constructed they feel more comfortable with why we do what we do when converting.

I’m hoping to get my last blog on MathsConf18 out by the end of the week so pleae, keep watching this space.

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MathsConf18 (Part 3)

3. Arrays & The Grid Method (Mike Thain)

Wow! One blog a day was pretty ambitious of me! I wasn’t able to keep that up along with planning, data, marking oh & a little bit of sleep as well. So here it is after a short delay my third blog about MathsCon from last weekend, if you haven’t read my first two you can find them here & here.

So my third session in Bristol was on Grid Method & Arrays hosted by Mike Thain. If you’re a maths teacher you are probably quite familiar with the concept of multiplication known as The Grid Method, some of you may know it as The Area Model a point I mentioned to Mike at the beginning of the session.

As a bit of a starter (or Do Now!) Mike posed the following question:


So the main ‘go to’ methods you thought of were most probably:

1.The Column Method (Long Multiplication)

2.Grid Method

3.Napier’s Bones (Lattice Method)

4.Stick Method

5.Repeated Addition

WHY WOULD YOU?!

Why so many methods? Why do students use what could be considered a method that’s ripe for misconceptions & errors (long multiplication)? Especially when you consider the student’s journey through primary school using multiplication.

Key Stage 1

Mike began by outlining how a student first sees the idea of multiples & multiplication. Arrays seem to be the order of business in the early stages. Students look at counting on in numbers larger than 1, they link these ideas to arrays & multiplications, using Concrete manipulatives.

In year 2 this concept is extended into grouping & sharing, again the concepts are explored through the notion of arrays & the use of manipulatives.

The use of arrays is further explored through the times tables & fact families in year 2.

Key Stage 2

In year 3 the use of arrays is EVEN FURTHER explored & linked to times tables, multiples again using concrete tools to manipulate them.

In years 4 & 5 a more formalised written method is introduced to students & we begin to see a move away from arrays toward the column method (which is essentially an abstract concept with little ability to be represented by concrete manipulatives), a few weeks ago Mike posed the following question to KS1 & KS2 teachers:

Notice the shift from KS1 (33% using arrays all the time) to KS2 (16% using arrays all the time). Does this mean that the natural progression of areas represented in arrays toward areas represented in grids is subverted by the KS2 curriculum or is it taught along side the formal column method? Well, a bit of both really, since students are also taught to use the ‘bus stop method’ in order to develop a written method for dividing two & three digit numbers by a one digit number.

One problem there though. The ‘bus stop method’ is actually a reverse Grid Method! So this natural progression is explored to an extent but there is a necessity to teach the abstract column method.

Year 5 is also the year where students first explore the distributative law which is wonderfully illustrated by the grid method.

So students have a greater understanding of grid method than most KS3 teachers might think, in years 7, 8 & 9 when secondary teachers ask students to take on board the grid there is no real reason why they shouldn’t be able to adapt to the concept of the grid method. The formal methods required to be taught in year 5 could potentially undo a lot very valuable work completed in earlier years. So we should definitely push students toward fully embracing it as they have all the pre-knowledge & concepts buried in their memory. It is a very adaptable method & should definitely forge ahead with teaching it as it can be used to complete:

  1. multiplication/division of large numbers
  2. linear algebraic expansion/factorisation
  3. polynomial expansion/factorisation
  4. surds
  5. matrix multiplications (apparently)

It would appear that most KS3/4/5 teachers do, in fact, embrace the grid/area model as Mike also posed the same question to KS3/4/5 teachers. Perhaps if we make the links to arrays in our teaching we could rekindle our students’ love for multiplication again.

At the beginning of the blog I mentioned that grid can be seen as The Area Model, Mike touched upon this during his presentation.

Well that’s your lot for tonight, tomorrow, Jo Morgan’s session on converting units.

MathsConf18 (Part 2)

2. Atomisation (Naveen Rizvi)

As detailed in my previous blog I recently went to #MathsConf18. If you haven’t read my first blog you can find it here. In this blog I want to detail the second session that I went to.

For those of you who haven’t heard of Naveen I would highly recommend following her on twitter. Although we have been in teaching for the same amount of time she has a much deeper insight into the methodology of teaching mathematics than I do, to which end Naveen is currently working for a MAT developing year 7 & 8 resources. In fact (& I happened to mention this to her tonight at a completely different CPD event) she has the job that I would love to have!

So back to the session…

This session was aimed at atomising a topic into its individual sub-topics, the idea being that when you spend some time thinking about all the different elements that make up a topic you won’t flit around between the sub-topics & (most likely) miss some out.

What are the positives of Atomisation?

  1. When you take the time to really think about all the sub-topics of an element of mathematics you will identify 100% of the domain that needs teaching
  2. Once you have all the sub-topics identified, you can order them in a way that builds up the knowledge required in an accessible manner
  3. If you don’t consider each element you might end up teaching to the assessment as you try to identify what’s needed
  4. By considering each individual sub-topic you end up differentiating for every ability group (the lower ability groups can access the first few elements & progress whilst the higher ability groups gain a greater insight as well as being challenged by the later elements)
  5. If you have ordered your sub-topics correctly then the next element is building on the previously taught content, it also means that each step introduces new knowledge in smaller manageable chunks

What are the negatives of Atomisation?

  1. When you first start using atomisation it takes a while to really consider what every element is in a topic (this improves with practice)
  2. It also takes longer to teach the earlier elements as, again, you are focusing on EVERY element of a topic. This obviously means that the earlier elements are covered in as much detail as the later elements (the pay back is massive as you have built the knowledge up slowly)

What’s the best practice to implement it?

Naveen heavily recommends that this isn’t a process to be undertaken by a lone teacher, it needs to be a team effort. The recommended approach is to take a topic every week & as a faculty discuss what sub-topics it contains. Once you have the list you should decide on which order to teach them in. The whole process should take around 15-20 minutes. After this it is a case of creating resources & pedagogy to best deliver each element, you should meet again & QA them so that they can be used again (or improved upon) the next time you teach it.

Having now attended two separate sessions by Naveen covering this I am building up a pretty decent picture of how best to create a scheme of work based upon this concept.

Here is the example that Naveen delivered to us at LIME Oldham

Perimeter

  1. Perimeter of irregular polygons
  2. Perimeter of rectangles
  3. Perimeter of parallelograms
  4. Perimeter of regular polygons
    • include polygons of different sides but with equal perimeter
  5. Manipulating perimeter of regular polygons
    • what’s the side length given the perimeter?
    • if different polygons have equal perimeters but different number of sides, which has the longest sides?
  6. Manipulating the perimeter of a rectangle
    • what’s the missing side length when you’re given the perimeter
    • write the different side lengths allowed given a perimeter
  7. Manipulating isosceles triangles & isosceles trapeziums
  8. Perimeter of compound shapes
    • give non examples of a compound shape as well as examples to build the understanding of what a compound shape is
  9. Manipulating the perimeter of a compound shape
    • Finding the short length (subtraction)
    • Finding the long length (addition)
  10. Combine regular polygons to find the perimeter of the new compound shape
  11. Perimeter: given a shape does another shape have greater, less or equal perimeter
  12. Can you draw all the possible rectangles on a grid for a given perimeter
    • Match various rectangles on a grid to given perimeters

My first Blog!

So I’ve been meaning to write a blog about my musings for quite a while but never really had that much to say! This weekend I attended #MathsConf18 hosted by the ever knowledgeable Mark McCourt.

For those of you who haven’t attended (you’re missing out!) or those of you who don’t know what MathsConf is (where have you been hiding?!) it’s a regular event put on by the aforementioned Mark & La Salle Education. At the conferences there are many sessions delivered by maths teachers who are keen to share best practice with & get to know other maths teachers from across the country & settings.

MathsConf18 is the fourth (or is it the fifth?) one that I have attended. I have always left with either a lot of books (there are lots of stalls selling really great material as well as stalls representing the major examination boards), loads of inspiration, new techniques to try, food for thought and, most importantly, new friends.

There are six sessions throughout the day that you can attend (which you select in advance), I was very kindly asked if I could write a quick blog to describe the sessions I attended this time around. So (as the old saying goes) Here goes nothing:

1. Feedback NOT Marking (Cat Ashby)

So the first thing to mention here is that the school I work in has a marking policy that places the onus on the students & not on the teachers, the students mark their own work in class with purple pen & we give them verbal guidance as to what they should write as their own feedback. We produce a whole class feedback sheet every half term that the students personalise. I was very keen to attend this one to see if my school is unusual in this approach & it would appear we are. I was also very eager to see how I could improve my own practice with feedback as I have tried to automatise as much of it as possible.

The cycle most schools probably use

Cat pointed out that there are many schools that operate a policy of detailed & regular marking with feedback for every student. Think to your own school policy, how much time do you dedicate to this practice, how much time do your students dedicate to it AND how much does it actually help your students to improve?

The suggested alternative is that, instead of you using lots of your valuable time to mark & feedback only for your students to give a one word response to it, you get your students to put more time into the feedback cycle & ultimately make better progress for it.

Cat’s idea to ensure that this happens is to give your students a list of topics that are going to be covered in a two weekly cycle & link them to a Hegarty Maths Clip (this could easily be a MathsWatch video or one of CorbettMaths‘ vidoes). The students are told that they need to complete 30 minutes of work based around these topics, which they chose from. At the end of the two week cycle the content is assessed, this includes a literacy element & also content form previous years or cycles (Ebbinghaus’ forgetting curve anyone?). These assessment are peer marked in class & the teachers’ job is merely to look through them & identify students who are a cause for concern.

Now the best thing about this is that from there on in the feedback is only guided by the teacher, the majority of the work is completed by the students themselves. This gives them more ownership of it, ensures that they engage better with it, recognise their weaknesses & better yet learn how best to correct it.

The really detailed Feedback/Feedforward sheet you see here is (almost) entirely filled out by the student. They select a question they got wrong & watch a Hegarty clip attached to it, make some notes, copy an example question & create a similar question to it. They then correct the selected question from the assessment. Finally they make some notes as to what they have done about the topic(s) they are less confident with, why they think they might have struggled with them & what they can do next time to ensure they don’t struggle with it again.

Cat has noted that she is in a lucky position with regard to SLT buying into this idea & that there were some compromises made in implementing this form of feedback. But the students are engaging better with it, they are themselves saying that they are finding it more useful & it gives them a more immediate fix to their weaknesses.

So if you are being weighed down by endless marking & feedback perhaps you should consider something like this yourself? Personally I operate in a similar fashion, again I am lucky that my SLT don’t have a stringent marking policy & the expectation is that the students put more into it than the teachers. This session has given me some things to think about:

  1. Give my students the list of topics in advance, to get them to be more proactive in their study
  2. Create assessments with older topics to create that need for continual revision & recall
  3. Make my assessments so they are split into OA1, OA2 & OA3 sections so that I can better see where my students’ struggles are

Wow! I actually do have a lot to say! I won’t make these blogs too long as you won’t read them all the way through. So, I will do a blog a day this week covering the rest of the sessions I attended.