## 2 March 2016

### Knowledge Gaps

Last week I watched a student teacher explain two-way tables to a Year 7 class. She gathered data from the pupils and drew a table on the board. The immediate reaction from a number of students was to shout out, 'It's a Carroll diagram'. The teacher looked to me for confirmation - she'd never heard the term Carroll diagram before so she didn't know how to respond.

The same thing happened to me in my NQT year. I introduced two-way tables to Year 7 and a student said 'Miss, that's just a Carroll diagram. We learnt these at primary school'. I'd never heard of them, so I googled the definition in front of the class. I found that Carroll diagrams are a type of two-way table. They organise data in the same way, but they show the individual pieces of data, rather than frequencies.
Neither Venn nor Carroll diagrams are specifically mentioned in the current Primary National Curriculum but it's likely that most children will have seen them at Key Stage 2. Students are required to "interpret and present data using bar charts, pictograms and tables" and it seems that Carroll diagrams are commonly taught here. But after Key Stage 2 they are never seen again. I expect that students find it a bit odd when their maths teachers aren't familiar with maths they've learnt at primary school. I wondered if there's anything else that gets lost in the primary-secondary transition.

Arrays
Last week I delivered my first fortnightly 'More Able Maths' session at a local primary school. Initially I was expecting to be working with Year 6, but instead I was assigned to a group of Year 3 children, followed by a group of Year 4s. At first I wasn't sure what sort of thing to prepare, but I had some useful input from Twitter, plus my very helpful primary teacher friend Helen gave me loads of information, including her school's 'Pitch and Expectations' documents for Year 3 and 4.

During my first session I was leading a discussion about square and cube numbers when one child got stuck trying to remember 9 x 9. I asked if anyone could suggest a good way of working it out. One child suggested drawing an array. Ah, an array... Now I know that arrays are all the rage in the US because I see them mentioned on Twitter quite often, but suddenly I wished I'd paid more attention. I think he meant that drawing 9 rows of 9 dots and counting them would be one way of working out 9 x 9. We discussed more efficient alternatives, such as 9 x 10 - 9.

Back at school, I chatted to my colleagues about arrays. Arrays are used at primary school as a way of explaining the concepts of multiplication and division. Most of us don't use them at secondary school because that understanding is well established by the time students get to us, and more efficient methods have been introduced. Even though it's unlikely I will start using arrays in my teaching, I would benefit from finding out more about how they are used.
CPD Priorities
Realising that I have gaps in my knowledge of pre-secondary level mathematics confirmed what I already knew - that I would benefit from spending time at a primary school, observing and getting involved in the teaching of younger children. I'm sure this would be beneficial to most secondary teachers, but our teaching commitments don't allow us the opportunity for this sort of CPD. Our timetables are such that we rarely have time to develop our understanding of how children learn and what they learn. When we do get time for subject knowledge development, our priorities often lie elsewhere - for example I'm currently focusing on learning FP2 (which I'm teaching for the first time this week) and learning more about the new GCSE qualification. I'm lucky that my own daughter starts primary school in September - I'm looking forward to seeing how she learns maths, I think it will be fascinating.

1. I find arrays are excellent at helping children see different properties of multiplication (commutative, distributive) as well as building up towards an area model of multiplication. When working with low achievers in Year 5 last year, many of them became more confident with mental multiplication for questions like 17 x 6 because they could draw and partition a model to 10 x 6 and 7 x 6, and slowly began to do this through visualisation. The grid method of multiplication then followed very easily from this, just with boxes not in proportion.

As an aside, I tutored GCSE maths recently and used the grid method when factorising quadratics by filling in the details in the answers section, and then looking at relationships between them to derive the numbers which should go in the brackets (my secondary subject knowledge letting me down with the terminology there).

1. Thank you, that's really interesting re arrays.

I agree - grid methods work very well with both expanding and factorising quadratics.

2. I read your post with interest as I'm in exactly the same position as you! My son is in reception and I am KS3 coordinator and deliver fortnightly outreach lessons! It has always amazed me that we are not trained in KS1 & KS2 and have developed so much having had to consider what lies below the old level 4 (because of course all Y7's come to us with a level 4 ;). The arrays you mention are very interesting as of course grid multiplication has it's roots here and we all know how useful that is!...one thing that I've been thinking a lot about recently is the importance of what number comes first in the multiplication and is 3 x 4 the same as 4 x 3 and what does it look like....algebraically we all know that 3 x a is a + a + a so 3 x 4 is 4 + 4 + 4 and I remember recently on fb something that showed an american exam question something along the lines of show 5 x 3 is 15 that when the pupil wrote 5 + 5 + 5 had been marked incorrectly. I also have just bought my son a times table book that lists each times table like this; 7 x 1, 7 x 2, 7 x 3, etc..which really upsets me....am I wrong? Thanks for the great posts.

1. I learnt my times tables parrot fashion at school/home, and I remember chanting 'one seven is seven, two sevens are fourteen, three sevens are twenty-one...' etc, so to me it seems right to write 1 x 7, 2 x 7 etc. But in a conversation with Tilly Warden on Twitter today, she explained: "actually tables should be 2x3, 2x4, 2x5 etc, where 2nd number is multiplier which helps to move away from repeated addition (lots of) to 'times bigger' or 'times as much' which is much more useful".

So instead of thinking as 2 x 7 as two lots of seven, we should see it as 7 x 2 ie seven times bigger than two. This is all new to me, I need to have a good think about it! Interesting stuff.

2. mmmmm, that's really interesting I can see that but we do also want to think of 2 as the multiplier in 2 x 7 or 2 x a as well later on....

3. Scotland teaches the 7X1, 7X2, 7X3 notation while England teaches 1X7, 2X7, 3X7

3. I know this is US and not UK, but as a HS teacher, I found the following helpful in knowing how our students are learning fractions: https://www.illustrativemathematics.org/progressions

1. Thank you! I'll have a look.

4. I completely agree - my sons are in Y3 and Reception and I've learnt a lot from their learning. My 8yo is currently learning fractions using arrays and my 4yo is learning how to construct number sentences. Absolutely fascinating, and it's making me a more sympathetic y7 teacher.

1. That does sound very interesting. I have all this to come. I can't imagine my four year old constructing number sentences!

5. Hi Jo,
Another brilliant post. Last year I decided to join a school where I get the opportunity to teach from KS2 to KS5. The experience has been amazing. It is fascinating seeing the progression from KS2 to KS3 especially at this time where Mastery is all the rage.
Have you heard of Number Talks? I heard of it through reading Jo Boaler's 'Elephant in the room'. It is fantastic for building number sense. A quick 5 mins in the morning daily does wonders for a class.

Will it be possible to share the 'Pitch and Expectations' document - i.e. if your friend Helen is happy to. I am very intrigued by the sound of it.

Cheers again for all you do to inform us.
Hilda

6. Thanks Hilda, I can email you that document.

That must be fascinating teaching KS2 - KS5, what a wonderful opportunity.

1. That will be great Jo, I will inbox you my email on Twitter. You are more than welcome to come and teach my children anytime :)

7. When I did my primary specialist maths course, part of it involved working more closely with KS3 and so I went to see KS3 children being taught and the secondary school maths leader also came in to observe me. I gained a lot from this sharing of ideas and teaching methods but sadly it was a one-off because time is always so precious. It is this sort of time that has been squeezed out unfortunately.

1. I agree Deb, it's such a shame that this becomes low priority.

8. Ciara Murphy left this comment but it has disappeared for some reason! Ciara, I am happy to help - please email resourceaholic@gmail.com

"Really enjoyed this post, as a primary teacher I would admit my lack of knowledge is the exact opposite to yours! Primary and secondary need to collaborate more! Would it also be possible to have a copy of that pitch and expectation doc? Does she have a year 5/6 document too?"

9. Informative post! I really like and appreciate your work, thank you for sharing such a useful information about knowledge management system strategies, keep updating the information, hear i prefer some more information about jobs for your career hr jobs in hyderabad .