The Implications

I started with the goal of investigating will alternative approaches to the 4 operations lead to better learning and the results are in; its a “Definitely maybe”.

From the experiential perspective, I think the new methods created greater motivation for learners who hadn’t experienced success with the standard methods ( a fresh start). The quantitative data would suggest no harm done with the possibility of a positive effect. I will use these new methods as an intervention strategy in the future with similar learners.

I have discussed these new methods with the department and we are in agreement that the traditional methods have slightly more flexibility but these methods are a good alternative. It’s definitely time to talk about methods.

What has happened ?

At the start of my enquiry, I wanted to investigate the introduction of new teaching methods of the four operations, their impact on fluency and understanding for the lowest attaining s1 class.

Many of these new techniques were inspired by a series of visual teaching strategies from Professor Arthur T Benjamin and Dr James Tanton.

I introduced each of the new strategies at first using a physical resource ( Dienes blocks),  atomising each of the components, before bringing the information together in the formal method.

The student’s reaction to these methods at first was a greater engagement than in previous years (subjective judgement), a novelty effect but an important one when trying to reteach something the students have failed to learn on many previous occasions.

For each of the four operations, I gave students an opening and closing test ( with delay), re-teaching any areas highlighted as insecure. This resulted in improved performance in each of these tests for students. I will monitor next year for the fade-out effect of this teaching. Performance or learning?

A key goal was to make sure the students in the intervention group made more progress in the departmental model that any other class. The results are positive in this area with students in the intervention group making the most progress, a feat not realised in previous years. This would suggest the new method at least did not have a negative effect.

Throughout the process, I shared with the students the vision of what we wanted to achieve by the end of the year and I’m disappointed that we were unable to conclude our journey. The lack of the final parental and students survey is a big miss in the data due to the Covid-19 outbreak, this is an area I will rectify when schools return.

I have shared some of the new teaching strategies with colleagues in the department, the general view was scepticism, this is understandable. The current methods have been commonplace in Scottish Education for 30 years. My colleagues are justified in stating this is evidence for the effectiveness of current methods but I feel this has created a lack of reflection or understanding of alternative methods that may be superior for some young people.




Why I did it ?


I have been thinking for years why a significant number of students (approximately 20 students,14% of our cohort ) enter high school without basic level 1 Cfe skills.

These students can’t number bond within 100, some can’t do it within 10. They are highly inaccurate in terms of any written algorithm. This is in the context of high performing catchment, about 50% of our cohort gain National 5 in s4(much higher than the national average).

I have worked with a similar group for the last 3 years and have made steady improvements, using physical resources, lots of spacing, interweaving and re-teaching on areas of weakness. It was in this vein I wanted to trial new methods for the algorithmic version of the four operations linking with mental agility and manipulatives.

The National progression framework highlights the importance of strong foundations of place value and mental addition before progressing to algorithmic methods( I agree). I disagree with the frameworks late addition of negative numbers. Students should meet this as soon they begin subtraction i.e 5+-3 should come before   5-3, others share this belive such as director of La Salle Education     Mark McCourt. I spent a significant amount of time teaching addition and subtraction of negatives before the new algorithms.

There has been very little research into which methods promote best understanding and fluency. The Lindy effect would suggest a type of cumulative professional evidence for the current methods.

Jo Morgon published a very timely book “a compendium of mathematical methods“, this gave me a much greater understanding of current and alternate techniques for teaching the four operations but Jo draws no conclusions as to which methods are superior.

Most of the students have made positive progress this year and greater progress than their peers from the in house models. This shows the new methods have at least not had a negative effect on learning but I do worry about transitions to other classes if this was not a consistent departmental method.



What I did

My enquiry was to improve the learning of the four main operations for the learners furthest behind their piers from Primary school. These learners are given an extra 2 periods in S1 maths, they are all level 1 in terms of Cfe and have a variety of ASN.                                 These pupils are in danger of not gaining basic numerical skills to cope with everyday situations.

This was to be achieved by teaching new methods of addition subtraction, multiplication and division.

The purpose of these methods was to link in with the manipulatives I use to teach and with mental methods.

I was looking to measure this impact by seeing improvements in pre and post-test, a greater rate of progress with the intervention group and an improvement in student confidence through a pupil survey.

I did not manage to complete the second pupil survey due to the covid-19 impact but will monitor this next year.

I would have also liked to share more of my experience with colleagues in the department and will aim to do this when I am back live in the flesh.



Update 3

One of the measurements of my intervention was a closing of the gap between the intervention group and the rest of the year. As a school, we have created a model to monitor this gap.

I will briefly explain this model before sharing the results.

We plot a coordinate for each student x coordinate( score on the current test – mean score from year group) and y coordinate ( actual score – predicted score). The predicted score is calculated using linear regression from 1000’s of students who have previously undertaken the test in other year groups, This gives a plot like the one below.

The green dots are SIMD 8 to 10

The amber dots are SIMD 4-7

The red dots are SIMD  1-3

Each Quadrant can be interpreted  as follows

We can turn these progress scores into a box plot to see relative progress of different class ( must be interpreted  with caution )

The intervention class (my class) are in light blue at the top of the box plot. This indicates from this model they have made the most progress and closed the gaps on others.

This indicated improvement may be due to a variety of factors (most likely the extra time is given to this class) but it shows that the new arithmetical methods being trialled don’t appear to have a negative impact at this stage.

Update 2

Better later than never. 

Thankfully before the recent act of God, I had gathered some initial data.

For each of the new methods, I had been giving pre and post-test. Most of these tests are set at level 1/2 Cfe level. Each pre-test was given at the start of a topic and each post-test was given + 3 weeks from the end of the teaching period, this was to distinguish between learning and performance ( was this learning in their short term of long term memory). This has its disadvantages in terms of reteaching areas of weakness but since I operate a strand curriculum with this course I believed the this was correct balance.

Subtraction Test Pre-Test Cfe Level 1/2


Subtraction Post-Test Cfe Level 1/2

These initial results show some positive benefit of the new subtraction method taught but the students are still not at a level I would consider fluency. Some reteaching will be required.

I still have some doubts in terms of subtraction using this method and decimals. I will share some of further data and reflections on future blogs.

Update 1

(My Enquiry Focus – alternative methods four operations)

I recently read a new publication by Jo Morgan

” A Compendium of mathematical methods “.

I found this book enlightening. I especially like the scripts Jo put in for explaining these new concepts to students ( I will adopt these in my enquiry).


Jo confirmed that there hasn’t been much research done in this area

” For most topics in school mathematics, there has been little research into the ‘best’ methods.

— In more recent years, alternatives have emerged, and we are now left with a plethora of wonderful methods to choose from.” pg 9


Jo presents the methods without bias but believes it’s important for teachers to understand a variety of methods

” For if you take just one thing from this book, let it be this: we need to talk about methods.” pg 341.

My initial experiments with new methods have been going well in some areas, the new addition method has given a significant boost in post-test of addition, which is promising.

I do have some concerns with boundary examples of the new methods especially in their application to decimals





Both of the new methods require a much deeper understanding of place value.


In not sure if this is good or bad.

The group I’m targeting won’t focus on decimals as they are at level 1 Cfe but it worries me that this method doesn’t have as broad applicability.



My Enquiry Question – First Draft

I have decided to focus  in on the area in alternative methods of teaching standard algorithms  to link in better with physical and mental models.

I think the research in this field is under explored, Most teachers , teach the same way they have  been taught.

This is a form of evidence that these methods are effective , the lindy effect is suggest that ideas last for a reason.

The group  I will focus are group of s1 learners , the reason this will be focus is that standard methods have failed with this group. After 7 years of primary education these 15 students cannot number bond within 10 and cannot complete the standard computations with any accuracy. This limits their chances in maths , across the curriculum and in future life. So a new method maybe worth a go.


My Enquiry Question

” How do alternative approaches to teaching the four operations   effect learners mathematical confidence and ability to perform said operations consistently and  accurately”

What’s already known ?

I have been looking into both aspects of my inquiry focus for some time so much so this latest blog is a month late ( sorry BigBrother maybe BigSister), it’s been busy.

In one sense my exploration was fruitless, after much toiling in the field to find the effectiveness of different approaches to teaching standard computations my harvest returned barren and villagers are in for a famine. Many of the different approaches I have been trying, come from 2 different sources, Professor Arthur Benjamin  who has written a series of books

and Professor James D.Tanton

In these books and video lectures, they offer a series of alternative teaching methodologies, making arguments from the theory of multiple representation and links with mental calculations but I have yet to find the paper with a comparative study.

Jo Morgon is soon to publish a book on the use of different approaches (

I will read this with great interest.

Memory and Spacing

In my other area of interest changing the balance of lessons to review, the harvest was bountiful.

” Our data, to stress the point, suggests  that three complete sets of complete information interpreted and integrated into working memory, are the minimum needed for the construction of a new belief or concept”

Graham Nuthall , 2007

In my experience, the word minimum is vital in this sentence. With students with the poorest prior attainment needing more in the region  5 or 6 exposures spaced over time.

Direct Instruction the most successful program in Project follow through follows this 20% new to 80% review allocation.

” In this structure, lessons are shaped around multiple skills or topics rather than around a single skill or topic and each of these skills/topics is attended to for a short segment of time (five to ten minutes) and revisited over several days until these discrete skills/topics are mastered”

Explicit and Direct Instruction, Adam Boxer, 2019.

I’m looking forward to further exploring my practice in these areas, they are potential benefits in both but also areas where I can see problems arising.







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