Mathematics for teachers of mathematics

A new paper at The De Morgan Gazette:

A. Borovik, Mathematics for teachers of mathematics, The De Morgan Gazette 10 no. 2 (2018), 11-25. bit.ly/2NWECtn

Abstract: 

The paper contains a sketch of a BSc Hons degree programme Mathematics (for
Mathematics Education). It can be seen as a comment on Gardiner (2018) where
he suggests that the current dire state of mathematics education in England cannot
be improved without an improved structure for the preparation and training of
mathematics teachers:

Effective preparation and training requires a limited number of national institutional units, linked as part of a national effort, and subject to central guidance. For recruitment and provision to be efficient and effective, each unit should deal with a significant number of students in each area of specialism (say 20–100). In most systems the initial period of preparation tends to be either

•  a “degree programme” of 4–5 years (e.g. for primary teachers), with substantial subject-specific elements, or
• an initial specialist, subject-based degree (of 3+ years), followed by (usually 2 years) of pedagogical and didactical training, with some school experience.

This paper suggests possible content, and didactic principles, of

a new kind of “initial specialist, subject-based degree” designed for intending teachers.

This text is only a proof of concept; most details are omitted; those that are given
demonstrate, I hope, that a new degree would provide a fresh and vibrant approach
to education of future teachers of mathematics.

Advertisement

Alexandra O Fradkin: Exploring Rectangles

Friday is a special day in our math classes at the Main Line Classical Academy.  We read and discuss mathematical stories and we do exploration projects.  Here is the project that we did with the 2nd-4th grades last Friday.

It began with one of my favorite questions to discuss with kids: What is a rectangle?  Some of the kids in each class had participated in previous discussions with me on this topic, but this was close to 2 years ago and so probably had very little effect on the outcome.

Here is what the boards looked like after the 2nd grade and the 3rd/4th grade discussions respectively:

The kids used a lot of hand motions in their initial descriptions, but I told them to pretend that we were talking on the phone and I couldn’t see them.  They would also sometimes come up with very long and convoluted explanations, which I also refused to write on the board.  After each initial set of properties, I’d try to draw a shape on the board that fit them all but was not a rectangle or did not fit some of them and was a rectangle (some of the shapes unfortunately did not make it into the pictures).  The kids had a lot of laughs when I would draw a silly shape and ask them “is this a rectangle?”  In the end though, I believe that we settled on a set of properties that succinctly characterized rectangles.

The second part of the class consisted of making all possible rectangles out of a given number of squares.  The kids had to make them out of snap cubes and then draw them on graph paper.  The second graders all got 12 snap cubes while the 3rd/4th graders initially got 12 and then each their own different number between 18 and 32.

I was very surprised that no one tried to draw the same rectangle in different orientations.  Some kids did, however, try to make and draw rectangles with holes in them.  A few of the second graders initially had trouble because the squares on the graph paper were smaller than the snap cubes, so tracing the structure did not work.  However, after a brief discussion, they were all able to make the one-to-one correspondence between the cubes and the squares.

Here are some pictures of the process:

In the end, we discussed with both groups how to make sure that we have made all the possible rectangles.  One of the older kids pointed out the connection with factors/divisors of a number.  None of the kids had formally studied area or multiplication (although most know what those are to various degrees), but those will both be big topics in the 3rd/4th grade class this year.  I think that this served as a good indirect introduction to them.

Olivier Gerard: Learning mathematics as a Russian interpreter

You might be interested in reading How I Rewired My Brain to Become Fluent in Math, by Barbara Oakley, in Nautilus, October 2, 2014.

A quote:

“Time after time, professors in mathematics and the sciences have told me that building well-ingrained chunks of expertise through practice and repetition was absolutely vital to their success. Understanding doesn’t build fluency; instead, fluency builds understanding. In fact, I believe that true understanding of a complex subject comes only from fluency.

In other words, in science and math education in particular, it’s easy to slip into teaching methods that emphasize understanding and that avoid the sometimes painful repetition and practice that underlie fluency. “

How to raise a genius: lessons from a 45-year study of super-smart children

How to raise a genius: lessons from a 45-year study of super-smart children, by Tom Clynes, 07 September 2016, in Nature | News Feature.

On a summer day in 1968, professor Julian Stanley met a brilliant but bored 12-year-old named Joseph Bates. The Baltimore student was so far ahead of his classmates in mathematics that his parents had arranged for him to take a computer-science course at Johns Hopkins University, where Stanley taught. Even that wasn’t enough. Having leapfrogged ahead of the adults in the class, the child kept himself busy by teaching the FORTRAN programming language to graduate students.

Unsure of what to do with Bates, his computer instructor introduced him to Stanley, a researcher well known for his work in psychometrics — the study of cognitive performance. To discover more about the young prodigy’s talent, Stanley gave Bates a battery of tests that included the SAT college-admissions exam, normally taken by university-bound 16- to 18-year-olds in the United States.

Read the rest of the story

Alexandra O Fradkin: Who’s the Oldest: Conversation with Kindergartners

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

Yesterday, I overheard a wonderful conversation between our Kindergarten teacher and the Kindergartners.  The kids needed to line up to exit the classroom and the teacher told them to line up by age, oldest to youngest.  Immediately, one of the kids (K1 from now on) had a question.  “But how can we do it?  I’m five, K2 is five, and K3 is 5, so that means we’re all the same age!”

Teacher: Are you all the exact same age?
K1: Yes.
Teacher: So you were all born on the exact same day?
K1: Noooo. (giggling from the other kids)
Teacher: Ah, so some of you were born before others.  When are your birthdays?
K1: July.
K2: May.
K3: May.
Teacher: When in May?
K2: May 5.
K3: May 17.
Teacher: So who is older, who was born first?
K1: K2 is older.
Teacher: Why?
K1: Because she is taller!
Teacher: So taller people are always older than shorter ones?
All kids: Noooo.
Teacher: So in order to figure out who is older we need to determine what comes first, May 5 or May 17?
Silence.
Teacher: Well when you count, do you say 5 or 17 first?
K1: 17.
Teacher: So we count 1, 2, 3, 4, 17, and then five comes at some point later?
K1 (after much giggling): Noooo, it’s 1,2,3,4,5.
Teacher: So who’s older?
All kids: K2!

After that conversation it still took them a moment to get into the correct order, but they did it, and off they went! I love hearing kids of this age group reason because they are, for the most part, still not afraid of being wrong and they will say whatever comes to mind. This allows you to analyze how they think and is just plain lots of fun.

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:

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 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

Conclusion:

• 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

You can find out more about MWUk on its website (which will shortly be updated)

http://www.mathsworlduk.com

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

Maths GCSE: repeats are of low values

As this table from Education Data Lab shows, show, retakes are of low value.

Read the whole article: Repeat After ‘E’: the treadmill of post-16 GCSE maths and English retakes, by Rebecca Allen