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Sometimes I'll provide links to sites that I like. Sometimes I'll provide links to interesting articles. Sometimes I'll just provide my opinion...not that it's worth much. :)

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Thinking Blocks

Posted on 4 May, 2017 at 20:10 Comments comments (15)

If you are looking for a Model Drawing website, try Thinking Blocks.

Singapore-style teaching helps solve problem of maths failure, says new research

Posted on 12 July, 2016 at 15:30 Comments comments (4)


Singapore-style teaching helps solve problem of maths failure, says new research

The study shows that children who were taught through the Singaporean 'maths mastery' approach learn faster than their classmates


The first conclusive proof that Far Eastern teaching methods can improve UK pupils’ maths performance is revealed in research just published.

A study, by UCL Institute of Education and Cambridge University, shows that children who were taught through the Singaporean “maths mastery” approach learn faster than their classmates - making, on average, an extra month of progress in a calendar year.

Academics also forecast that the increase in their maths skills is likely to lead them to earning more after they leave school - by up to £200 a year.

Researchers looked at the impact of the Singapore method of teaching on five and six-year-olds in 90 primary schools and 11 and 12-year-olds in 50 secondary schools. Schools in the Ark academy chain have adopted the new method of teaching.

The researchers argue their findings “are the first hard evidence that introducing East Asian teaching methods into a western schooling system influences children’s maths performance”.

“Until now there has been little proof that such an approach can be successfully introduced into western education systems,” they add. “Cultural and historical differences have made it difficult to establish why Singaporean children are so far ahead of their English counterparts when it comes to maths,”

The improvement in standards was roughly the same as was achieved by high level curriculum innovations such as the Literacy Hour introduced in Tony Blair’s first Parliament - which had a marked effect in improving reading standards.

The research study adds it suggests that “even a small enhancement of maths skills at age 10 yields long-term economic benefits for individuals and the country”.

Under the Singapore system, teachers generally do not split their pupils into different ability groups.

Instead, they wait for academically weaker pupils to reach a basic standard in each topic before the class moves on to the next concept. The able pupils study the topic in greater depth while waiting for the weaker pupils to master it - thus gaining greater depth of knowledge than their UK counterparts.

The method cuts down on the cost of remedial one-to-one teaching for pupils who struggle to keep up in class.

In addition, the Singapore system concentrates more on developing problem solving skills rather than mental arithmetic.

“These findings have potentially important implications for education policy and practice,” says the study.

It acknowledges that “it is unlikely that widespread introduction of this particular East Asian teaching method would springboard Western countries like England to the top of the ... educational achievement rankings”.

However, it adds: “This does not mean that implementing this teaching method is not a worthwhile investment to make.

“Thus, though we advise policy makers that further evidence is still needed, Maths Mastery nevertheless shows signs of promise and should now be tested over a longer time horizon and in a greater number of schools.”

Dr John Jerrim, the study’s lead author, added: “Maths Mastery shouldn’t be seen as a ‘silver bullet’ ... Yet, given the low cost per pupil, it may nevertheless be a programme worth pursuing.”

A spokeswoman for the Department for Education said that 72 schools had been trialling new textbooks based on the Singapore approach to maths teaching “to see how they drive up standards”.

She added that the new maths curriculum had been developed to reflect the key principles of the Singapore approach.

*The casual effect of East Asian ‘mastery’ teaching methods on English children’s mathematics skills by John Jerrim and Anna Vignoles is the latest working paper to be published by the Department of Quantative Social Science at UCL Institute of Education.

How it works

Maths Mastery concentrates on problem solving skills - using “props” like building blocks to find answers. For instance, in answering the question “There are three consecutive numbers that add up to 42 - what are these numbers?” the pupils are given three wooden bars. They then subtract one for each bar getting 39 and divide this by three to get the answer 13. As the question says the numbers are consecutive, they immediately get the answer 13, 14 and 15.

Case study

Children at an award-winning primary school have begun to develop “a genuine enjoyment” of maths as a result of learning through the Singaporean Maths Mastery programme, according to their teachers.

Daniel Upfield, headteacher of Ark Attwood primary school in Westminster, London,, was surprised during an educational visit to Singapore “how much more adept and at ease pupils were at maths”.

As a result, he was eager to adopt a similar approach in his own school in order to “revolutionise maths teaching”.

The experiment worked - within a year his first year children were achieving well above the national average for maths. The following year the school received a national award for maths teaching from the TES.

Mr Upfield added that as well as the significant increase in academic achievement he had noticed how much enjoyment there was of maths lessons - traditionally a subject that has failed to inspire a number of pupils, “They began to see themselves as mathematicians,” he added.

Dr Helen Drury, programme director of Maths Mastery at the Ark academy chain - which has embraced the project, said: “Every student is capable of being successful if they are given the right opportunities - and those opportunities are what we provide.”

Written by: Ghazal Abbasi      Originally published in the Independent.Co.UK

Original URL

Generational Math

Posted on 12 July, 2016 at 15:15 Comments comments (167)

I am not sure where I got this from.  If you know the author, please comment below.

Last week I purchased a burger at Burger King for $1.58. The counter girl took my $ 2 and I was digging for my change when I pulled 8 cents from my pocket and gave it to her. She stood there, holding the nickel and 3 pennies, while looking at the screen on her register. I sensed her discomfort and tried to tell her to just give me two quarters, but she hailed the manager for help. While he tried to explain the transaction to her, she stood there and cried. Why do I tell you this? Because of the evolution in teaching math since the 1950s:

1. Teaching Math in the 1950s

A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price. What is his profit?

2. Teaching Math in the 1960s

A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price, or $80. What is his profit?

3. Teaching Math in the 1970s

A logger sells a truckload of lumber for $100. His cost of production is $80. Did he make a profit?


4. Teaching Math in the 1980s

A logger sells a truckload of lumber for $100. His cost of production is $80 and his profit is $20. Your assignment: Underline the number 20.

5. Teaching Math in the 1990s

A logger cuts down a beautiful forest because he is selfish and inconsiderate and cares nothing for the habitat of animals or the preservation of our woodlands. He does this so he can make a profit of $20. What do you think of this way of making a living? Topic for class participation after answering the question: How did the birds and squirrels feel as the logger cut down their homes? (There are no wrong answers, and if you feel like crying, it's ok.)

6. Teaching Math in 2000

Un hachero vende una carretada de madera para $100. El costo de la producciones es $80. Cuanto dinero ha hecho?

7. Teaching Math in 2015

Who cares, just steal the lumber from your rich neighbor's property.


Why Math Fact Mastery is Important

Posted on 25 February, 2016 at 14:40 Comments comments (3)

Written by mathmatician Emma Devitt

“It is a profoundly erroneous truism, reported by all copybooks and eminent people making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of operations we can perform without thinking about them. Operations of thought are like cavalry charges in battle – they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”

Alfred North Whitehead, An Introduction to Mathematics


Math facts have become the bête noire of certain educational theorists in recent years. Thus, for example, Boaler, Williams and Confer (2015) argue that what is important to a mathematical education is developing number sense – the sort of sense, for example, that allows a child to infer that if 20+6 =26, then 19+6=25 – in addition to math facts. They argue, further, that one of the best methods of teaching number sense and math facts is a strategy called number talks developed by Parker (1993) and Richardson (2001).

It is difficult to assess the arguments of Boaler, Williams and Confer (henceforth, Boaler et al.) because they frequently resort to attacking what seem to be straw persons. They argue, for example, that a complete mathematical education “includes learning of math facts along with deep understanding of numbers and the ways they relate to each other” (2015). It is, however, difficult to imagine any educational theorist disagreeing with such an anodyne claim. Of course it is important – indeed crucial – to develop this sort of “deep” understanding. No serious educational theorist will deny this. (The Common Core Curriculum, for example, calls for both conceptual understanding and fluency with math facts). If there is a substantive disagreement on this issue, it must concern the relation between the learning of math facts and the learning of number sense.

The considered opinion of Boaler et al. seems to be that number sense is more important developmentally, and therefore should be made chronologically prior to the rote learning of math facts. Indeed, at various points, they call into question the utility of rote learning of math facts at all. It is very doubtful, however, that their supporting arguments establish either of these claims. Commenting on the public failure of a British Politician to successfully calculate the product of 7x8 – a sort of Dan Quayle ‘potatoe’ moment – Boaler et al. comment: “When asked to solve 7x8 someone with number sense may have memorized 56 but they would also be able to work out that 7 x 7 is 49 and then add 7 to make 56”. But this assumes that the product of 7x7 is the sort of fact that will be learned through number sense rather than rote learning. And this is something that the advocate of rote learning will deny. Boaler et al.’s argument is, therefore, question begging (in the technical, rather than colloquial, sense, of assuming the truth of the conclusion they desire rather than arguing for it).

A similar charge of begging the question can be leveled at their advocacy of Parker and Richardson’s number talks. I agree that this is a powerful tool, one that should be used in classrooms and ideally at home too. However – and here is the rub – it is very difficult to do use this tool without prior knowledge of math facts. Thus, they employ the example of posing the problem of 18x5, and then describe students’ different strategies for solving it. Every single strategy they describe presupposes the knowledge of math facts and the laws of mathematics .


None of this, I should emphasize, is to deny the importance of conceptual or “deep” understanding of numbers in a mathematical education. On the contrary, Bezuk & Cegelka (1995) may be correct when they write: “Prior to teaching for automaticity… it is best to develop the conceptual understanding of these math facts as procedural knowledge.” (1995, p. 365). Moreover, Bruner wrote extensively about the CPA (Concrete, Pictorial, Abstract) approach that is now used by the Singapore Math Curriculum. Thus, I agree with Boaler et al. that understanding is very important. Such understanding can be reached through work with manipulatives and through modeling. But none of this is incompatible with according a similar degree of importance to the learning of math facts. When moving to the abstract phase one will have deep conceptual understanding of number sense. However, to progress to higher order abstract math you will still require automaticity with math facts.

Boaler et al. make much of the anxiety involved in the timed testing of math facts: such anxiety is “a debilitating, often life-long, condition.” If we wish to avoid such anxiety then, they suggest, we should jettison such testing. As someone prone to vomiting before major exams (and even some not so major ones) I can sympathize. However, there are few psychologists or psychiatrists who now argue that the way to deal with stress is to avoid the conditions that engender it. Rather, stress-inducing conditions need to be faced, and successful strategies developed to accommodate them.

This is another point at which Boaler et al.’s arguments are question begging. The guiding assumption is that since timed testing is stressful, the correct strategy is to eliminate such testing. But the best strategy for dealing with this stress might, in fact, be the rigorous and, therefore, successful rote learning of math facts. To quote a wise old karate sensei I once knew: “You need to do the kata until the kata does you”. Performing a kata – a ritualized but meaningful series of movements – in front of a crowded room is very stressful. One way to avoid this stress would be to give up karate. The other is to perform the kata so many times that it becomes burned into your brain and body, and you can perform it no matter what the environment. The cure for stress inducing timed testing of math facts may be more, not less, rote learning of math facts. Thus, as Cavanagh (2007) has argued, regular timed testing might, in fact, benefit students by allowing them to become accustomed to working under pressure.

The research of Miller, Hall & Heward actually shows that time trials improve accuracy and fluency, and students enjoy being timed (1995). Students can be motivated when they see their scores rising. Timed math fact tests are one of the occasions in a student’s education when there is an unequivocal, objective correct answer to each question, and if he or she learns it they are able to get 100%. After all, as Dewck (2006) has argued, dedicated students who put in the hard work and have a growth mindset can succeed at anything. There is no obvious reason why this cannot be applied to the learning of math facts.

I began this paper with a quote from the great Alfred North Whitehead, the co-author (along with Bertrand Russell) of the monumental work, Principia Mathematica. In this work, they try to deduce the laws of arithmetic armed only with the apparatus of set theory: essentially, relations between numbers are reduced to set-theoretical functions. Using this apparatus, it took Whitehead and Russell eighty-six pages to prove what they called the “occasionally useful” claim that 1+1=2. So, if there is anyone who has a “deep understanding of numbers and the ways they relate to each other” it is Whitehead. And, yet, far from eschewing the role of rote memorization – or automaticity – Whitehead emphasizes it.

The reason is relatively obvious. Holding things in working memory is hard work. There is much erroneous talk in Boaler et al.’s (2015) article about mathematical facts being stored in working memory – based, it seems to me, on a misguided interpretation of empirical work of Beilock (2011) and Ramirez et al. (2013). In fact, nothing is stored in working memory. Working memory is not a store: it is a (functionally defined) place where information is temporarily placed in order to be manipulated. This is hard work! There are severe limits on how much can be brought into working memory and how much can be done with it once there. Rote learning of mathematical facts acts much as an external store of information whose function is to reduce the complexity of operations performed in working memory.

Imagine how difficult it would be to do long division – say 1,000,000,870,050,705 divided by 353 – in the head. Instead we avail ourselves of an external representation. The function of this external representation is to break down the overall process into a series of smaller processes. Each stage in the process can be achieved by way of a pattern mapping operation – learned by rote memorization. We fill in the result of this pattern mapping-operation, write it down, and then use this result to direct us to the next stage of the operation. In this way, the complexity of task is offloaded – or distributed – onto the environment. The external representation, written on paper or screen, is an external resource we can use to reduce the complexity of the task we would otherwise have to perform in the head. This idea forms the basis of the embodied turn in cognitive science that has become prominent in the last fifteen years or so (Rowlands 2010), and which is, I think, deeply consonant with the work of Bruner. My point is that with respect to working memory, the results of rote memorization are akin to an external resource. The more we can avail ourselves of this resource, the less we have to do in working memory. Rote memorization is not the opponent of deep understanding: rather, it is its best friend.

Pace the claims of Boaler et al., this idea is, I think, now widely accepted by educators and cognitive psychologists. For example, Whitehurst (2003): the “ability to recall basic math facts fluently is necessary for students to attain higher-order math skills”. Delazer et al. (2003) argue: “Recent research in cognitive science, using MRI’s, has revealed the actual shift in brain activation patterns as untrained math facts are learned”. And, crucially, as Dehaene (2003) has shown: “Instruction and practice cause math fact processing to move from a quantitative area of the brain to one related to automatic retrieval” (Dehaene, 2003). These are all processes whose function, or result, is clear. As Geary (1994) argues: when you have automaticity of your math facts you free up your working memory to work on higher order tasks, problem solving and learning new concepts and skills.

Nowhere in any of these processes is rote learning the enemy of conceptual understanding. On the contrary, they complement each other very nicely. The reason for this stems from certain features of working memory and, in particular, its relation to both short- and long-term memory. As Willingham (2004) has demonstrated, transferring a fact from short- to long-term memory requires “over learning”—this involves not merely being correct in fact recall on one occasion, but doing so repeatedly. Moreover, as Rohrer, Taylor, Pashler, Wixted, & Cepeda (2005) have shown, retaining the memory for a significant period of time requires additional, periodic, practice subsequent to initial mastery—emphasizing the importance of regular review of learned material. Hill and Schneider (2006) have shown that repetition of this sort produces significant changes in the brain, the thickening of neuronal myelin sheath resulting in faster retrieval.

Elementary mathematics deals with many different concepts like money, telling time, using the four basic operations on large numbers, measurement, estimation and rounding. Working with all these disparate concepts requires the use of math fact fluency. Fluency with math facts means that students will be confused less, find calculations easier, and will be faster and are more accurate in performing such calculations. With fast automatic recall of math facts students have time to concentrate on the higher order math problems they are being asked to compute or navigate. They will also complete tests within the allotted time. In short, rote memorization of math facts is not the enemy of deep understanding. It is its best friend. Bryan and Harper (1899 - quoted in Bloom 1986) put it better: “Automaticity is not genius, but it is the hands and feet of genius”.


Beilock, S.L. and Carr, T.H. (2005) ‘When High-Powered People Fail: Working Memory and ‘Choking Under Pressure’ in Math.’

Bezuk, N. S., & Cegelka, P. T. (1995). Effective mathematics instruction for all students. In P. T. Cegelka & W. H. Berdine (Eds.), Effective instruction for students with learning difficulties Needham Heights, MA: Allyn and Bacon.

Bloom, B. S. (1986) ‘Automaticity: The hands and feet of genius’, Educational Leadership, Vol. 43, No. 5, pp. 70-7.

Boaler, J. Williams and Confer (2015), ‘Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts’, Stanford University. Accessed December 31st 2015.

Bruner, J. S. (1963).The process of education. Cambridge: Harvard University Press.

Bruner, J. S. (1966). Toward a theory of instruction. Cambridge: Belknap Press of Harvard University Press.

Cavanagh, S. (2007) ‘Understanding 'Math Anxiety’, Education Week

Dehaene, Stanislas, et al. (2003) ‘Three parietal circuits for number processing." Cognitive Neuropsychology 20.3-6

Delazer, Margaret, et al. (2003) “Learning complex arithmetic—an fMRI study." Cognitive Brain Research 18.1

Geary, D. C. (1996). ‘Children’s mathematical development: Research and practical applications’. Washington, DC: American Psychological Association. (Originally published 1994).

Miller, A.D., Hall, S. W. & Heward, W.L. (1995) ‘Effects of sequential 1-minute time trials with and without inter-trial feedback and self-correction on general and special education students' fluency with math facts’ Journal of Behavioral Education Volume 5 Issue 3

Parker, Ruth. 1993. Mathematical Power: Lessons from a Classroom. Heinemann Press.

Parrish, Sherry. 2010. ‘Number Talks, Grades K–5: Helping Children Build

Mental Math and Computation Strategies’. Math Solutions: Sausalito.

Richardson, Kathy. 2011. ‘What Is the Distinction between a Lesson and a

Number Talk?’

distinction.pdf. Accessed December 27th 2015.

Rohrer, Doug, et al. (2005) ’The effect of overlearning on long-term retention.’ Applied Cognitive Psychology 19.3 (2005).

Rowlands, Mark (2010) The New Science of the Mind; From Extended Mind to Embodied Phenomenology, Cambridge; MIT Press.

Whitehurst, G. (2003, February 6). IES Director's presentation at the Mathematics Summit, Washington, DC.

Willingham, D. T.: ‘Practice makes perfect—But only if you practice beyond the point of perfection.’ American Educator, (2004).



Would you like to be a guest Blogger

Posted on 22 November, 2015 at 0:35 Comments comments (2)

I am currently looking for guest bloggers.  Please email me if you are interested.

Great new book!

Posted on 25 March, 2015 at 19:30 Comments comments (2)

If you are looking to read a great summer math book I recommend; Uncomplicating Fractions to Meet Common Core Standards in Math K-7.  You don't need to be a CCSS state to get powerful information from this new publication.  The ISBN# is 978-0-8077-5485-6.  It can be purchased at the NCTM online bookstore. 

Free Singapore Math Webinar 11/21/14

Posted on 10 September, 2014 at 15:15 Comments comments (3)

Howdy.  I am fortunate to offer you an invitation to connect via webinar to other educational professionals throughout the world, INCLUDING Singapore to discuss how we are using Singapore Math ® in our classrooms. You will need to register at the link I have provided below as currently there are only 85 participants allowed. Please also send me a response ([email protected]) so that I know you will be on the live webinar On Friday, November 21st  from 7-9 pm (EST).



  • A webinar for Math teachers, educators, authors and parents in and outside Singapore to exchange experiences and discuss about Singapore Math.

    Below is the tentative agenda. In an Un-conference, the agenda will be updated based on participants' interests and suggestions. If you have a topic to share, materials you would like to present or would like to facilitate a session, write to [email protected] after registering.

    Time and dates take into account the end of daylight savings.

    Session A (Friday 7pm to 8.15pm EST => to 9.15am Singapore Time Saturday)

  • How is Singapore Math the same and different from Standard Common Core Math. A comparison of content, pedagogy and assessment methods. Is Singapore Math any different?

  • How teachers outside Singapore integrate Singapore Math into their classrooms. What are the pros and cons?

    Session B (Friday 8.15pm to 9.30pm EST => 9.15am to 10.30am Singapore Time Saturday)

  • The stack method to problem solving in Singapore Math.



    Sherri Adler

  • [email protected]

Social Media

Posted on 8 September, 2014 at 16:40 Comments comments (2)

If you have not been on my  Singapore Math Linkedin site, you should take a look...Singapore Math (duh).  There is a free webinar that is being offered.  Also, there is now a Facebook page SIngaporeMathSupport and a Twitter handle Singamathexpert that I manage as well.

Singapore Math Pentagon

Posted on 8 August, 2014 at 1:10 Comments comments (0)

Love this on Pinterest

Posted on 30 July, 2014 at 20:55 Comments comments (1)