Alexandra O Fradkin: Story Math in Kindergarten: Two of Everything

[Reposted from Alexandra O Fradkin’s blog Musings of a Mathematical Mom]

Friday is story day in our Kindergarten math class.  For our first book we read Two of Everything, a Chinese folktale.  We then had a wonderful discussion and the kids asked some very insightful questions.

Here is a brief synopsis of the story: A poor elderly couple find an old brass pot in their garden and it turns out to be magic.  Whenever you put something into the pot, two of that thing come out!  The couple started doubling everything and soon became very rich.  One day, the husband accidentally pushed his wife into the pot and then fell into it himself.  After some initial arguing, the two couples realized that they could become the best of friends and use the pot to create two of everything, one for each couple.

At the end of the story, one of the kids asked, “But would there also be two pots?”  What a great question!  I said that I thought there would be only one pot, but some kids disagreed.  They spent several minutes debating whether it was possible to put the pot inside of itself to create a second one.

The discussion then moved on to how one would make lots of something.  The kids suggested that you could just keep putting the same object into the pot over and over again, creating one more each time.  I then asked them what would happen if we put two of the same object into the pot at the same time.  They all immediately yelled out that you would get three of that object.  My next question was whether only one of the objects would be doubled or both.  This led one of the kids (and then the rest) to realize that in fact, four of that object would come out.

I wanted to ask them about putting three or more objects into the pot, but it was time to move on.  Perhaps that was for the best because they already had a lot to take in.  I hope to come back to this topic and can’t wait to read more stories with them.  I feel that stories engage this age group like nothing else does.  And I absolutely love the questions and thoughts that the kids come up with!

Alexandra O Fradkin: Playing Math Detectives: First Week of Second Grade Math

[Reposted from Alexandra O Fradkin’s blog Musings of a Mathematical Mom]

The first week in our second grade class we did lots of time traveling. We played the role of math detectives and helped people from different time periods solve problems. We also learned about some ancient number systems.

On the first day we went back to several million years ago. (The idea for our scenario was taken from the wonderful book by Julia Brodsky, Bright, Brave, Open Minds: Engaging Young Children in Math Inquiry.) During this time period, there lived ferocious saber-toothed tigers with sharp teeth, crocodiles with awful jaws, and the first cave people, who had no strong jaws, long teeth, or sharp claws. How could we help those early people survive in their unfriendly world of dangers? The kids came up with making weapons out of sticks and stones, building fires, hiding in caves, running away, and climbing trees. I think they would have had a good chance of survival!

On the second day, we went back just 10-20-30 thousand years, to a time before numbers were invented but people had a need for keeping track. The kids’ task was to help a farmer determine whether his shepherd was bringing back all of his sheep at the end of day or whether he was stealing or losing some along the way.

The kids were split up into groups and each group got a bag of coins (which stood for sheep). They were told that when they were ready, I would take the “sheep” on a walk and bring them back. They would have to determine whether any were missing. The main rule was that they were not allowed to count in any way!

Here is their solution:img_2955

They made holes/homes for each of the coins/sheep, and when I brought back two fewer coins than they gave me, they were easily able to detect that because they had two empty holes. I was very impressed with their inventiveness. We then discussed and looked at pictures of how people actually did use dots, tally marks, stones, and knots to keep track of animals, money, and anything else they needed to.


Finally, on the third day we went back only several thousand years, to several locations around the globe. We visited the Babylonians, the Mayans, and the Romans, and learned how they wrote the numerals 1 through 10 in their number systems. The detective work consisted of helping them decide how they should write 11.

The kids examined the patterns closely, made suggestions, and discussed the merits of each one. In the end, they came up with versions that I think the ancient people would have been happy with.

Here is a picture of what it looked like (the Babylonian version did later get modified to be a horizontal wedge followed by a vertical one).


Next week we will begin our in-depth exploration of the Hindu-Arabic number system. More specifically, we’ll focus on the usefulness and meaning of place value and the importance of zero!

The Humanistic Mathematics Network Newsletter

The Humanistic Mathematics Network Newsletter (HMNN) was founded by Alvin White in the summer of 1987. The Newsletter was later renamed The Humanistic Mathematics Network Journal (HMNJ). The last issue of the HMNJ was published in 2004. The open access digital archive of the full run of the HMNN/HMNJ (1987-2004) is now available at

This journal does not accept new content. A related current journal is the Journal of Humanistic Mathematics.

The 7 biggest problems facing science

The 7 biggest problems facing science, according to 270 scientists, by Julia Belluz, Brad Plumer, and Brian Resnick on September 7, 2016 in Vox.

I love this example of deductive resoning:

  •  Academia has a huge money problem
  • Too many studies are poorly designed
  • Replicating results is crucial — and rare
  • Peer review is broken
  • Too much science is locked behind paywalls
  • Science is poorly communicated
  • Life as a young academic is incredibly stressful


  • Science is not doomed

Read the whole text.

Jack Abramsky: MathsWorldUK

I am writing to you, friends and colleagues, in an appeal to boost the number of Friends and donor sponsors of MathsWorldUK.
For those of you who are unfamiliar with MWUK, we are a registered company and a registered charity set up with the long-term aim of establishing in the United Kingdom the first National Exploratorium devoted entirely to mathematics and its applications. We will not be in competition with the Science Museum in London, because our philosophy is radically different from that of the SM. The new Mathematics Gallery (to be named the Winton Gallery) at the Science Museum will open in December. It will be essentially static with about 90 exhibits from the permanent collection of artefacts owned by the museum. Each artefact will be accompanied by a short description of around 90 words about some mathematical idea that the exhibit might exemplify. Our approach will be for fully interactive exhibits designed to illustrate some mathematical idea or mathematical application, with the visitor doing his or own individual exploration of the ideas underpinning each exhibit. So one approach is static, the other is active. The two spaces will complement each other, rather than be in competition with each other. A further fundamental difference is that The Science Museum’s primary function is to present scientific achievement, with mathematics as a small subsidiary of that endeavour, whereas the Exploratorium (note the emphasis on exploration, and hence discovery) will  put mathematics in all its manifestations and applications at the very heart of its activity.
Schools Competition MATRIX:  The top two prizewinning schools

Schools Competition MATRIX: The top two prizewinning schools

You can find out more about MWUk on its website (which will shortly be updated)
We have just hosted an extraordinary and wonderful conference at the University of Leeds called MATRIX with over 100 delegates from 15 countries.  This was the second MATRIX conference; the first was held in Dresden two years ago. MATRIX stands for Mathematics, Awareness, Teaching, Resources and Information eXchange. The conference was for museum folk around the world and others interested in improving public awareness and understanding of mathematics. We co-hosted this conference with the National Museum of Mathematics (MoMath) in New York and the University of Leeds. Full details of the conference are on our website, together with the winning entries for a schools competition that was run for the conference. Attached is a photograph of Hanna Fry with the top two school teams; Hannah gave out the prizes.
There are now over 50 mathematics ‘museums’ around the world with about 10 more due to open in 2017. Germany alone has 10 museums, either dedicated to mathematics or with substantial mathematical galleries. The UK has no such mathematical space. The new Musee Henri Poincare is due to open in Paris in 2020. The Director, already appointed, is Cedric Villani; he  is also the Director of the Institut Henri Poincare in Paris. Cedric Villani is a Fields Medallist in Mathematics (the equivalent of being a Nobel Prize winner in maths), who also wrote the recent best-seller Birth of a Theorem: a mathematical adventure. He was at our MATRIX conference.
MWUK has been invited by the Chairman of the House of Commons Select Committee on Numeracy and Mathematics to participate at a Conference on the Northern Powerhouse, to be held in Manchester, on 15th September. We have also recently had a meeting with Sadiq Khan’s mayoral team at City Hall in London.
I am not one who partakes in sponsored walks, cycle rides, marathons, treks or whatever. So I cannot appeal to you to sponsor me to do some incredible feat of physical activity and then denote the proceeds of such sponsorship to a charity of my choosing. Instead I am appealing directly on behalf of our charity. We urgently need donations to support the appointment of a full time director of fund-raising and to purchase some much needed equipment.
On our website you will be able to make a donation directly to MWUK, and also to request a form to become a Friend of MathsWorldUK if you so wish. All donations, large or small, will be greatly appreciated. Also, please forward this message to any of your contacts whom you feel may be interested in the MWUK project.
Thank you, in advance, for your consideration.
Jack Abramsky

Misha Gavrilovich: Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups

Misha Gavrilovich’s paper Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups, available from

is a follow-up to his paper in The De Morgan Gazette,

M. Gavrilovich, Point-set topology as diagram chasing computations, The De Morgan Gazette, 5 no. 4 (2014), 23-32

The paper raises important questions about optimal approaches to exposition of elementary group theory: quite a number of group-theoretic concepts (for example, solvable, nilpotent group, p-group and prime-to-p group, abelian, perfect, subnormal subgroup, injective and surjective homomorphism) can be expressed as diagram chasing in the category theoretic language.

Hamid Naderi Yeganeh: Fractal Africa

New mathematics artwork from Hamid Naderi Yeganeh.


This fractal is constructed by an Africa-like octagon and its lateral inversion.:

If we count the number of octagons of different sizes we get the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, … (see this animation:

The height of the biggest octagon is φ times longer than the height of second octagon; where φ is the golden ratio. (see the following image) It is a necessary and sufficient condition to see the white triangles in this fractal.


Vacancy, Postdoc position, PhD positions

  • A new vacancy for a candidate with a PhD in mathematics and I am sending this email in the hope that you might have recommendations for potential applicants to this job. The job title is “Senior Mathematics Curriculum Designer” and the salary bracket is £36,000–48,000. Full details about the job and a link for interested applicants to apply is available at
  • A two year postdoc in Mathematics Education, with a focus on problem
    solving, reasoning and educational design, is announced in Umeå, Sweden. The full announcement:  Information about Umeå Mathematics Education Research Centre:  Information about Umeå University:

The Digital Turn in Epistemology (DigTEp)
Research Project
DigTEp is a collaboration between the faculties of Science and of Humanities of Utrecht University (UU), and the faculty of Philosophy of Erasmus University Rotterdam (EUR). The multidisciplinary project DigTEp lies on the cross roads of:

  1. philosophy of mathematics (PhD 1)
  2. logic (PhD 2)
  3. ICT development and mathematics education (PhD 3)

All three PhDs will share an office at the Freudenthal Institute<> in the faculty of Science of Utrecht University.

Embodied, Embedded and Extended Cognition (E3C) marks the recent Turn in Epistemology, the philosophy of knowledge, as well as in Cognitive Science. The three Es indicate that our knowledge and capacities are not located in our skulls, but extend to, and are distributed over:

  1. our bodies, which are always embedded in their environment they interact with;
  2. the artifacts we use, varying from paper notebooks to ICT.

The mediation of knowledge acquisition and application by ICT is becoming so dominant and ubiquitous (smartphone, tablet, laptop, computer, world-wide web) that Epistemology must take a further, Digital Turn. DigTEp concentrates on the following epistemological questions: How to make sense of mathematical knowledge after the Turn in epistemology to E3C, given its abstract character? Does the Digital Turn affect the genesis and the essence of mathematical knowledge? When practical knowledge is primary and propositional knowledge is derivative (E3C), how does this ‘derivation’ work in a digital environment? To answer these pressing philosophical questions, an empirical case study in epistemology will be performed of the acquisition of mathematical knowledge and skills in a controlled ICT-embedded environment, by secondary-school pupils. For a more detailed description of the contents of DigTEp, and of the individual PhD projects, please send an e-mail to the supervisor of the PhD project you wish to apply for.