Extended Discussion and Reciprocal Maths

Why I chose this focus:

I have been working on developing numeracy for a couple of years now. I had a hunch that improving oral and read literacy would have a positive impact on numeracy, especially with comprehending word problems.

Where I am currently:

I have taught Extended Discussion through Reading. I have aligned my Reciprocal Maths roles with my Extended Discussion roles. I use both regularly in lessons.

Changes I have made to my practise:

I have built extended discussion in to my reading lessons. I have a specific lesson for working on word problems and allow students to discuss through a problem and teach each other.

Problems:

• Some students are not buying into helping others through conversation. They prefer to just solve problems. I may need to make specific groupings.

Next steps:

1. I have an idea for a more simplified Extended Discussion cue card. Hopefully this will promote more natural discussion.
2. I still need to move from Reciprocal Maths to the more open discussion format of Extended discussion.

Basic Facts Boxes v2

Basic Facts Boxes has been updated, four years after I first developed it. This version has several new features to improve the experience all coming under an altered basic facts focus and a different physical layout.

Basic Facts Boxes v2 focuses on the addition/subtraction facts to 20 and times tables, i.e. the first 100 multiplications. I work with students who have major gaps in knowledge and are often far behind expectations, so I need to figure out ways of helping them learn. The Numeracy Development Project takes children on a meandering path through number learning, which in my opinion makes children run sections of the path before they can walk. Focusing on the facts to 20 and times tables covers the main facts we English speaking people need as core maths skills.

To differentiate for those less skilled, there is now the option of choosing ordered or random. Ordered being the easier version... That's assumed, not tested. I would hope that students would be able to see a pattern forming so they can solve anything they don't know yet. Anyone who can't is likely to need a different level of education programme anyway.

The basic facts boxes that students fill in are now collated into two 10x10 grids (excluding frames). This change imparts several improvements:

• Increased practise frequency within a single "box".
• Smoother and faster number entry. Users now only need to use the tab and enter keys to move between cells.
• Improved screenshot captures.
• The ability to print to an A4 page.

Using a new format forced me to make improvements on the layout and programming. Everything except the title, grid, and buttons at the top are generated through the script. This brings everything one step closer to a "fool proof" spreadsheet where if anything (such as the grid) was deleted somehow, the whole grid would be reset/ regenerated by clicking a button.

Reciprocal Maths: first lesson for the year ... or not?

Week One was the intended first week of teaching Reciprocal Maths and applying it to word problems...that didn't happen, for many reasons. One such reason was my underestimation of the effect of covid based learning gaps, and the lack of formal maths education using maths exercise books.

Week Two ended up being a very brief introduction to Reciprocal Maths, while Week One ended up being a lesson on how to rule up a maths exercise book. It seemed that students took to the steps and process of Reciprocal Maths easily enough. However few groups applied the strategy taught prior to exercising Reciprocal Maths. More instruction and closer observation are needed.

This week, perhaps a choreographed demonstration is required to give an idea of how to run through the procedure.

Side note: it seems many students need more instruction on which way to begin opening an exercise book and how to find the "first" unused page.

2022 Teacher Inquiry...not

*Edit and disclaimer* The following post doesn't detail my confusion as to what the Inquiry for 2022 actually was. In subsequent posts you will see that my Inquiry was into Extended Discussion. This post was written before that decision was made. The principal talked about improving children's wellbeing and motivation to come to school in the new post lockdown era (seen in the post prior to this). That is what I got fixated on.

Post-ish Covid Teacher Inquiry...what to do? My Teacher Inquiry has already been chosen for me, but frankly, I don't see it as something I need to put too much thought into. More on that later. 

Right here I will bullet point the maths programme that I had started last year with Grant Ogilvie (Go for Teaching) but wasn't able to continue. The details are as follows:

• Number knowledge & algebra, other strands, and strategy are taught on separate days in the week.
• The above divisions are taught every week.
• Other strands are taught in detail throughout the term. This is where the blog posts come from.
• Number sequence and order, place value, fractions, basic facts, and algebra are taught in two week blocks. Currently the two weeks are consecutive weeks. Preferably the weeks would be separated to encourage retention. This will happen from T3, when my new co-teacher has developed some familiarity with the programme.
• strategy covers add/sub, mult/div, proportions/ratios, as Gloss (assessment) divides them. 
• proportions and ratios are taught more than the others due to it being the focus as decided by the Senior Management Team. Preferably I would make sure add/sub and mult/div are understood completely before teaching prop/rat, but that's not how the NZ maths curriculum works.

Reasoning from observation as follows:

• skills and knowledge across the board were low
• division of groups based on overall level was not targeting students precisely enough.
• retention was poor
• many students with capable skills were testing low, with our hunch being they didn't understand the problems.
• our subsequent decisions were: to look more closely at the data to target specific weak areas for each student, specifically in strand, to continually and repeatedly cover all knowledge, and to develop comprehension of maths problems and thinking skills to solve them.
• finding weak areas in strand and teaching it over protracted periods is an attempt to boost strand knowledge
• repeatedly covering all areas of knowledge is to build retention
• to improve maths problem comprehension, we developed our "Reciprocal Maths" programme, which we haven't had a good run at due to Covid.

Our Reciprocal Maths programme has been covered here.

So what is my Teacher Inquiry for 2022? What I can do to keep/increase student motivation post Covid... I understand the need for keeping an eye on student "wellbeing", and that children's motivation for school might be lowered... However isn't keeping students motivated part of a teacher's job anyway?

Reciprocal Maths UPDATE

 Follow this link for an update regarding the Reciprocal Maths idea. Spoiler alert, there's been an overhaul.

Reciprocal Maths

 Read about the Reciprocal Maths idea Go for teaching and I developed here.

Create Staff Meeting Workshop

I ran a workshop at the August 2020 Create Staff Meeting Workshop on using Pencil Code for art and maths. The Create Workshop ran for Manaiakalani staff is a cluster wide staff meeting where teachers hold workshops to pass on skills for teaching children. The focus of the skills is on the 'create' part of Learn Create, Share, which involves children creating digital learning objects (DLOs) as learning tasks. 

My Pencil Code workshop took people through some challenges to learn the basics of coding. The first challenge for example, had people drawing a straight line, then expanding on that skill to create a house.

After creating the house, people could continue with the basic shapes challenge to create a smiling sun, or move on to the Coding Angles challenge.

The Coding Angles challenge tasks people with coding the turtle to travel between locations by measuring the angle and estimating distance between locations. This task involved the use of both digital and physical resources through using a protractor.

The afternoon was a good introduction for everyone into Pencil Code and how it can be used in the classroom for maths and art with these challenges. Pencil Code is more versatile and I would like to try coding simple quizzes or pick a path stories with children.

Oral language and comprehension

Oral language and comprehension, check out this blog post.

Addressing the basic facts gap

How have Grant and I begun to address the basic facts gap?

Place Value rears its ugly head again.

See this blog post on me having to teach Place Value again, this time successfully.

A Gap in Knowledge

See the new post about a knowledge gap in my higher ability maths learners.

Teacher Inquiry 2019: Maths ability and Reading comprehension; what's the link?

Nearing the end of Term 1 for 2019, I finally have some time to blog about my teacher inquiry for 2019. This year, Grant and I have decided to do a joint inquiry into poor maths performance and reading comprehension; more on this below. Since we are doing a joint inquiry, most of the posts for this inquiry will be posted on a separate blog with a feed to those posts underneath the header of The Wong Teacher.

The inquiry Grant and I are looking into comes from noticing that some of our students with poor performance in maths have skills and knowledge that are not presented in the standardised tests to evaluate mathematical understanding. There are other factors that contributed to deciding to look into the effects of reading comprehension on maths ability as well. Read about these and further posts to do with the inquiry on the LS1 Teacher Inquiry blog.

Place Value...a teaser

EDCURRIC 714 Exploring Mathematical Thinking taught by Gail Ledger at the University of Auckland is a great paper for learning about how humanity has thought about how number systems work and how the place value system came about. The practical values come in understanding where to start developing the concept of "one" or a single unit and building that into the concept of place value among other mathematical concepts.

Follow this link to the University of Auckland calendar website.

As it happens, the next weak spot in my groups' understanding to work on is place value. I used the idea of teaching the Ancient Egyptian numbers and numeral system last year to teach place value. The kids enjoyed it. Whether it helped them further develop an understanding of place value, I don't know. This coming term I'll be doing something similar, but a bit more 'hands on' and a bit more localised...

Ancient Egyptian numbers. Source: Grauberg, Eva (see below).

Now, time for the obligatory declaring of evidence that I have done some reading on the topic and not just pulled these ideas out of thin air. There are many texts on place value and ancient number systems, but here are a few (no, I'm not using any particular format e.g. APA).

Place Value:

• Elementary mathematics and language difficulties; a book for teachers, therapists and parents (1998). Grauberg, Eva.
• Children's Developing Understanding of Place Value: Semiotic Aspects. Maria Varelas and Joe Becker. Cognition and Instruction, Vol. 15, No. 2 (1997), pp. 265-286

Number history:

• Numbers through the ages (1989), Written Numbers pp. 76--130. Flegg, Graham.
• The Beginnings of Arithmetic, The Mathematical Gazette, Vol. 12, No. 177 (Jul., 1925), pp. 401-414. L. N. G. Filon

Ancient number systems:

• Ancient Egyptian - see above sources
• Chinese -
• Joseph, G. (2011). The crest of the peacock: non-European roots of mathematics (3rd ed.). Princeton: Princeton University Press
• Pellatt, V. (2007). Numbers and numeracy in Chinese culture, language, and education : the social substratum of the development of mathematical thinking.

IKAN results... a small ray of hope

My pupils were tested on their knowledge earlier this term. Generally in New Zealand, this involves the IKAN test. My pupils however often need to be retested on the JAM (junior assessment for maths) to get a clearer idea of what they know as opposed to what they can hear, read, parse, recall, then write down within 3 seconds.

After seeing the results, I had a sense of a general increase in certain domains of maths, especially fractions. I have put it to the test and made a comparison.

Colour coded IKAN shift for 2018

With green being the code for an increase in level, this screenshot of my spreadsheet shows that there has indeed been a general improvement in the Overall Level out of the pupils present for both tests (there is one coding error for pupil #28, which made no overall change). IKAN's Overall Level though, is taken from the lowest score within the domain scores, so it doesn't necessarily show improvement. The large number of increases in fractions demonstrates this phenomenon.

I have taken the small win in fractions and have moved on to improving basic facts for the remainder of this term (since about Week 7), and next term I will focus on place value.

Basic Facts Boxes

Basic Facts... a gap in knowledge that I never had when I was growing up. A gap in knowledge that prevents or hinders improvement in most areas of mathematics. How can Basic Facts knowledge in children, who are in some cases quite far behind expected ability, be brought back to average and boosted further?

Here's something I developed to give it a go.


This Google Sheet populates boxes with random addition and subtraction problems to solve. On entering an answer, the sheet checks the answer and gives immediate feedback. Another sheet in this Spreadsheet does the same for times tables and the corresponding division facts.

I tested the first version of this on my own pupils to limited success. The positive is that many of the kids wanted to keep going until they got everything correct. The negative was that they didn't take screenshots of the boards that had errors, so there was no record of their improvement.

A colleague, Robyn Anderson tested version 2 (without the times tables) with her class. You can read about her success and outcomes I never expected from this tool on her blog post.

The inspiration for this tool came from two places. The first being a Japanese tool for memorising basic facts and the second being Monty Jones's (Tāmaki College) spreadsheets for practising maths problems at secondary school level.

Feel free to copy it, use it, attribute it, and tell me about the successes and challenges you have had.

Maths Vlog

In another attempt to encourage learners into speaking about maths and develop their understanding of maths, I am trialling a task I'm calling Maths Vlog...again until I can think of a better name. This task involves learners creating a stack of objects in front of them and talking about the objects in relation to the concept being learned. This monologue is recorded then embedded into a Google Slide as if creating a vlog.

As can be seen in the example, the task is reusable (unlike worksheets), realia based (unlike many apps), and has some basic scaffolding to aid in learning.

The idea of Maths Vlog came from the success of SSR Selfie for promoting reading and FlipGrid for talking about maths. Hopefully this will be as successful as SSR Selfie in engaging learners with their basic maths concepts and provided the much needed mileage in using maths verbally.

Less than? Before? Minus one? What does that mean?

Observation of learners over the past years have shown that children in the tail end often have severe limitations in their vocabulary and understanding of basic concepts in size, order, and direction among others. Concepts need to be taught and repeated before internalisation occurs and application to problems can happen. How can I do this while meeting the demands of a full schedule? I know about apps, but frankly many are naff or incomplete for my needs. The many resources available from teachers around the world are often too babyish for my year 4/5/6 learners because these concepts should be developed in kindergarten or the first couple of years at primary school.

My solution is a series of semi open-ended tasks dubbed "Knowledge Cards"...until I can think of a cooler name. These are used with maths materials or other realia around the classroom to reinforce concepts in the learner's mind. Learners are also supposed to describe what they have done aloud in a Maths Vlog to aid in both reinforcing concepts and learning how to use the vocabulary verbally.

So far there has been a positive response to the cards I have developed. There are still some cards that I would like to make for both simple and more advanced maths concepts such as dividing/sharing items and part-whole number relationships.

Talking about capacity using Flipgrid

Capacity. What is it? How can it be measured? How is it usually measured? These are some questions for a maths concept that can develop some much needed language skills. But this is maths, how does one get children talking about maths? FlipGrid to the rescue.

In these lessons about capacity, children learned about containers and different ways of filling them up to find out how much they held. Children then learned how to talk about the comparative sizes of containers and how to measure exact capacities using a measuring cylinder and millilitres.

The capacity and verbal components to this lesson were very successful. The learners enjoyed using water and scientific instruments to find out about capacity. They also enjoyed sharing their learning by talking about it and using a simple video response recording system.

Problems arose with the word problems I had developed for them based on the learning done in the lesson. There were two main problems, both of which could be foreseen with knowledge of my learners. These problems were the inability to comprehend the problem despite being written in a simple and consistent format, and a lack of basic facts knowledge.

In subsequent lessons I structured and walked through how to understand, respond, and answer these problems with some success. I will need to continue scaffolding responses till learners become reasonably proficient before dropping the scaffold.

 

Maths Synonyms - reflection

The synonyms task I tried earlier this week was not as successful as I'd hoped. I had not anticipated two things: the learners misunderstanding the task and the learners not looking up the meanings of words they were unfamiliar with.

I still feel the vocabulary task would have been a valuable task for the learners to engage. Learners would be exposed to both the connections between everyday and mathematical vocabulary, and the nuances between synonyms such as trim and cut.

Ultimately, a positive is that the learners gained a stronger understanding of the meaning of the words addition, subtraction, multiplication, and division. However these synonyms will most likely need to be revisited someday.

Maths Synonyms

Bolstered by my mild success from the basic vocabulary lesson, I decided to try connecting the language of mathematical operations to everyday vocabulary they may already know. The idea was to also expand their vocabulary with related words they may not have already encountered.

In the lesson, learners focused on the core mathematical operations add, subtract, multiply, and divide. In groups, they used beans to demonstrate and explain the actions made by the operations.

The follow up task required learners to make videos demonstrating the meanings of the following words: sum, take, split, count up, remove, withdraw, total, cut, reduce, group, trim, share, tally, decrease. The words and videos were to be organised under the headings of the main operations.