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 https://www.amazon.co.uk/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401/ref=sr_1_1?keywords=arthur+benjamin&qid=1573195097&sr=8-1


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 (https://www.amazon.co.uk/dp/1912906600/?coliid=I2C8H7H393BK8Q&colid=13VVNRXI8UQ9Y&psc=1&ref_=lv_ov_lig_dp_it

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.







Exploring My Itch

After watching the lecture series ” The Power of Mathematical Visulasisatioin ” by Professor  James Tanton

( side note the great course has loads of brilliant maths courses).


I have been inspired to alter the way I teach the four operations, linking  with a visual model of teaching.


As with all new methods, I am working left to right. This links well to how you would use dienes blocks and mental methods. It has the added benefit of making adding  in alternative bases simpler.


Again left to right and gives a better feel in terms of number awareness, the students are constantly reinforcing place value.



Links well to physical resources and links well with multiplying brackets (5x + 6)(4x+3)  = 20x^2 + 39x  +18. Here the students can start to spot the relationships.



I have made the choice to leave the final answer in mixed fraction form,  again to link with a manipulative and further algebraic division. Later on, I will explore the link between fractions and decimals.


I hope to test these methods with my 9th set s1 class. These students have had exposure to standard methods and failed. I will hope to show that these methods will allow the students to acquire the knowledge to complete basic calculations and develop a greater conceptual understanding.


The second drive I wish to explore is the link between memory and forgetting.

In his recent book “Teaching for Mastery” Mark McCourt recommend an 80%-20% split in the review of each lesson and the teaching of novel content, this point was also made by Siegfried Engleman in his book ” Successful and Confident Students with Direct Instruction”.

Both authors make similar justifications that content has to be embedded throughout time, revisited after forgetting and moved in with a phased approach from direct teacher instruction to independent practice /behaviour. This is an idea I think has the power to improve understanding and retention for all pupils.

I will adjust the ratio of my lessons to have 80% review, 20% or so independent practice/ behaviour of material spiralled in the curriculum, guided practice through the use of mini-whiteboards of recently learned material (interleaved) 60% and then 20% novel idea teaching with a heavy focus on direct instruction. I hope this change will improve retention and far transfer.


Reforming Scotland : “What future for Scottish Education” – Reflection

I recently reviewed “Reforming  Scotland Future: What future for Scottish Education”.  The main suggested reforms from the author where

  1. Increased Autonomy: His conclusion was increased Automony would improve results, this, of course, is the current policy of the Uk government in relation to English schools but as Dylan Willaim points out in his book “Creating the Schools our Children Need” increased autonomy has little impact on increasing a nations educational standard. If given complete freedom some schools will perform exceptionally well ( Michaela in England being an example) and some will perform extremely poorly ( Al-Madinah school in England). The overall effect is a net-zero or negligible improvement in the overall educational attainment of a nation.
  2.  Learning from experience: This improvement strikes with a sense of irony since the introduction of the curriculum for excellence educational experience in every international metric has fallen in Scotland and in internal metrics such as SSLN. His conclusion from the piece is it’s just not been implemented correctly this strikes me as a classic case of “new true Scotsman fallacy”, what evidence would be acceptable to show CfE has been a failure? We have respected international educationalist posting articles such as
    Learning lessons from the failure of Scotland’s “Curriculum for Excellence”   https://gregashman.wordpress.com/2017/03/22/learning-lessons-from-the-failure-of-scotlands-curriculum-for-excellence/

    3. Technology is transforming education argument, this argument can be found in books of 19th-century philosophers. I have yet to read a study in which the use of technology in the classroom has significantly improved leaning.   The mere presence of a mobile phone in the classroom led to 20% reduction in attention, concentration and performance ( Thorton 2014 Social Psychology)   Schools that banned mobile phones received a 6.4% increase in their exam results going up to 14% for the previous lowest-achieving ( Beland and Murphy 2015).   “laptops use can negatively affect performance on educational assessments , even or perhaps especially – when the computer is used for its intended function of easier note-taking” (Muller and Oppenheimer, 2014).

To transform Scottish education and close the achievement gap we need to focus on the areas that have been shown to work time and time again. Teaching for long term memory, spacing, interleaving retrieval, a strong knowledge-based coherent curriculum with a focus on mastery at each step, firm and loving discipline.

We should invest in teachers and in CPD, scrap failing skills-based curriculums and invite our poorest children to share in rich knowledge-based culture, giving them opportunities and life chances currently reserved only for they’re well off piers.




Why are so many Students performing poorly in maths ? Part 1 Data

Students performance in maths ( and all other measures) have been steadily declining in Scotland.

Below are PISA scores from the Scottish Gov Website.

A steady decline .

(We will see in December 2019 if this trends reverses , don’t hold your breath.)

Another graph based on country comparisons. We can see a massive decline since 2003, the worst in the UK

The now scraped  SSLN data

What has happened ? Is it a problem  with our general society ? Why then have we  declined in comparison to England.

The problem must surely lie with our school system. I will expand on this  conclusion in future posts.


(I have focused on maths in this post but the same results are replicated for literacy levels. )


Elements of Teaching

I stumbled upon Will Emeny’s website a few years ago and have always been impressed with his clarity of thought . It was Will that started me on my path into the research of memory. Thanks Will for this and for  Num Ninja’s , still  having  a high impact on my classes for years ( completely free of charge).

He recently had another great idea a periodic table of teaching , I have been following weekly for updates but instead I thought I would use his idea to help create my own version. Not as good as Will’s will eventually be but a start.

Follow the link below to see my fledgling attempt.





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