Mathematics History Inquiries

Once a week I dedicate a period of our math study to looking at famous stories, paradoxes, inquiries and problems that real mathematicians have gone through in the past.  I approach it through the following general outline;

  1.  Recreate the historical story, understand who the players were and define the problem

  2.  Have the students attempt to work through the problem and come to their own conclusion on the outcome

  3.  Show the students how the actual mathematician solved the puzzle, and compare it to their own solution

  4.  Connect the concept studied to the larger picture of the history of math, society and how it shaped life and knowledge

So far, we have looked at the following problems this year;

  • The Case of the Lady Drinking Tea - Dr Ronald Fisher is at a party and serves a lady a cup of tea, however she declines and says she prefers her tea with the milk added first, not second.  He does a scientific test to see if she can actually tell the difference. The results of that test are the foundations of statistical methods (1925).

  • Pascal and Fermat and the Coin Flipping Game (The Problem of Points) - the two mathematicians are playing a coin flipping game (flip a coin, winner gets a point, first to ten points wins) in a Paris cafe when Pascal gets called away for an emergency.  The score was 8-7 at the time.  Fermat works out a fair solution as to how they should divide the pot and who wins. The solution he comes up with are the foundations of modern probability (17th century).

  • Eratosthenes - measures the circumference of the Earth in ancient Greece, using sections of a circle, distance and angle.

  • Seven Bridges of Konigsberg - There are seven bridges in the city of Konigsberg, Prussia, and Eulid tried to work out to walk through the city and cross each bridge once and only once. The solution to this problem led to the foundations of graph theory and prefigured the idea of topology (1735)

This is what I have so far, and I have a few more in the pipeline, but am hoping that I can continue these lessons all year long (the kids love them and they are great mathematical problem solving inquiries), but I am afraid I will run out of ideas.  There are tons of these problems out there, but I need help finding them.

Any advice for more?

  • PS – I am in the process of creating PDF files for teachers to use, and once I get to it (its on my list of things to do) I will post them here for you to use.

  • PSS – Bear in mind that I teach Elementary grade 5-6!


Organizing your Math Work

During our presentations today on our most recent Fermi problem (blog post about that coming soon), one thing that emerged from our discussions is just how important organization is; not just when presenting your work, but also during the process of solving the problem.  Several students had too many calculations on their page and they got lost and confused and mixed numbers up.  As a result, one student suggested we make a poster to help us remember some good points for organizing our work.  As a class we brainstormed and then went onto Powerpoint and made a little digital poster.  I printed one off and stuck to the wall next to the carpet, but I also printed some smaller ones off and the students use them as bookmarks in their math journal.


Here is the PDF Version;

Organizing our Math Work



Meta-Cognition in the Classroom

Hello all. I have a request for resources. I am looking at incorporating more meta-cognition and reflection into my classroom, and was wondering if anybody could share any great resources or books? Also, I am working through my Masters degree, and this will be a major component of my dissertation, so if you know of any theory based work, and practical based work, that would be great.

Thanks in advance if you can help.



The Librarian Who Measured the Earth

This really is one of the best picture books I have ever read related to math and critical thinking.  It is the story of Eratosthenes, who was the Head Librarian of the great library of Alexandria.  He was also an incredibly curious, and multi-talented scholar who did a bit of everything.  However, his work that he is best known for is measuring the circumference of the Earth.  He did this over 2000 years ago, and his calculation was within 200 miles.  How did he do it?

That is where a great inquiry math lesson pops up, because the book doesn't give you the answer quickly, it builds it up, step by step, stone by stone.  And what I love about this book is how you can take your students along for the ride with Eratosthenes.

It also covers a broad range of other subject disciplines and can be stretched out and kneaded into so many different shapes.  History, language, science, social inequalities, law, government (it has a bit of everything).

How I use it (and how I describe it below) is as a Math Mystery Investigation, and a good chance to reflect on some Meta-Cognition strategies.  Eratosthenes was a very reflective, curious, talented and mentally organized person, and how can we shape our own thinking to be more like his?

NOTE:  This was done with grade 5/6, and is geared towards the Elementary and JHS level.

Part 1

Read the first couple sections of the book.  It basically sets up his early life and makes particular note of his curiosity and his ability to generate and create questions.  There are a lot of little nuggets of gold in here, but the book itself is quite lengthy, so in the interest of time I will leave all those threads loose, and you can pick them up with your students.  The story brings his life up the point where he now Head Librarian, and he finally has the time (and the financial means) to investigate questions that are interesting to him.

Part 2

The tale starts to get interesting on page 30, when Eratosthenes asks himself an interesting question; is it possible to stand in one spot and figure out the circumference of the earth?

Have your kids try and answer it.  I had them in groups brainstorming possible ways to do it on a large sheet of paper.  They had some novel ideas (nothing that was even remotely possible!) and they flexed their creative brains (which is the point of this part of the lesson.  If we are going to solve a problem like Eratosthenes, we are going to have to go way outside the box!

Part 3

Finally, on page 35, a clue!  There is a wonderful diagram of a grapefruit on one side, and the earth on the other.  But don't read the text.  Just show the diagram and see if they can make the connection between why a grapefruit is like the Earth.  Brainstorm ideas on their piece of paper.

Part 4

Back to the story, Eratosthenes figures out that is he can measure the circumference if he can find out what one slice of the pie is.  All he needs is the inside angle of one of the sections; then he divides that angle by 360 degrees, and presto, he knows how many pieces of the pie there are in the whole.

I gave them a pie shape and some math tools and asked them to use this method to figure out how many pieces of pie would make a circle.  Once they had that, I asked them how to figure out the circumference of the whole circle.  Some of them were able it together and measure the top part of the circle and then multiply it by the number of pieces of pie.  Others struggled, but during the ensuing conversations, we were able to determine that this way does work, and he had some great math talk.

Part 5

So, what did Erasthonese do?  Before I went back to the book, I asked them to brainstorm ways that he could possibly do it, keeping in mind he doesn't have the technology we do, and he is trying to measure a big piece of pie!  Again, they came up with some interesting ideas, and a couple of them were even thinking about light (they missed the shadows).  We recorded our ideas on our piece of paper (which is getting pretty full at this point!)

Back to the story, Eratosthenes uses the light from the sun shining into a well in Syene  (the sun is directly over the well leaving no shadows on the wall, this is our straight line that goes to the point of the pie) and the sets up a big stick in Alexandria and measures the angle of the shadow that the sun leaves (this is the other line that is giving us our angle).

Give the students the measurements; 7.2 degrees, and let them figure out how many pieces of pie the Earth is made up of.  Like Erasothenes, they should get 50 (360/7.2).  Have them draw the picture of the Earth cut into 50 pieces.

Part 6

So, all he needs to do now is measure the top part of the circle, and multiply it by 50.  Simple right?  Ahhh, how do you measure a large distance (put it into perspective, maybe from Sendai City to Iwate-ken worked for us) with no technology, or vehicles?  This was 2000 years ago!

Let them brainstorm more ideas on their sheet of paper.

One of my kids said that he should walk it step by step and measure the steps.  The rest of the class laughed.  However, that is exactly what Eratosthenes did.  Not him personally, but profession walker-measurers called bematists.  They determined that the distance was 5000 stades (a Greek stadium).  One stade was roughly 157 meters (or 515 feet if you like that weird way of measuring things).

Note:  If you look online you can find several different interpretations of the numbers and conversions (and many arguments to go with them).  To me, that is not the point of this exercise, and the exact number is unimportant.  The point is the thinking that goes into it.

I asked them what they can do with these numbers?  And how can we use that number to find the total circumference of the Earth?  Again, on their sheet of paper they converted the stades into meters, and then meters into KM (those guys walked 785 km!) and they easily found the circumference of the Earth.

Part 7

The story concludes with the answer and by saying that Eratosthenes lived a long and happy life.  It also says that he never stopped asking questions, and he was continuously learning something new.  This is a great kick off into the last part of the project, which had to do with analyzing his mind, and our our minds.

First, I gave them a picture or Eratosthenes and had them write the big ideas from the story around the picture.  We have defined the term Meta-Learning in my class, so these terms are part of collective language.  See one example below;

After that, in our math journals, we made mind maps of the skills and traits that Eratosthenes has, and how he used them in the story (Resilience, Curiosity, Resourcefulness, etc.).  Next, we tried to think of times where we used those skills and did a bit of reflective writing.  Finally, we set a goal for the month, based on the skills we that Eratosthenes has illuminated for us.

I can see a lot of this book becoming part of the classes collective intelligence and culture.  After this activity, it is something that will be able to refer back to as a point of inquiry and thinking for the rest of the year.


Moving Learning into the Stone Age

This post is inspired by a wonderful post over on What Ed Said.  Thanks Edna for some inspiration!  Keep up the good work, I am a fan of your site.

We will be starting a UOI on Ancient Civilizations in the coming weeks.  I sat down with myself and I, and we had a bit of a brainstorming session.  We decided that we need to move the unit back to the Stone Age.  I looked at the old unit plan and re-tinkered it to meet our modern, past tense focus.  The only way to go forward is to go really far back.

What has changed?

They used to...... use Lego to make a stop motion video about what a day in the life of an early human was like.

Now, we will go outside for a day and live it.

They used to..... use timetoast to create a timeline retelling the history of their civilization.

Now, we will find a large rock, make our own paint, and tell the story through cave-painting art.

They used to..... use Google Sketch Up  to make a 3 dimensional model of an ancient dwelling.

Now, we go out into the woods, gather materials, and build it ourselves.

They used to.....  do online research into what ancient tools were, and then make a Prezi to send to University Anthropology professors for feedback.

Now, we go into the woods and use trail and error to find the best way to make tools.


Context is everything.  I use a lot of technology and I am willing to have a go with any new tech tool and see if it works for my group of kids.  We make a lot of films.  We do a ton of photography.  We Skype with classes from all over the world.

Sometimes, however, it just feels good to go outside and get dirty.




Translating our Thinking

I have this little boy in my class who is very, very good with math.  He gets everything, he can apply it, and he does it in his own way that makes sense to him.  The problem is, and his major goal for the year, is to be able to explain to others how he does it.  I am not to concerned with operations or calculations, but rather how he explains his thinking.  How he makes it visible for us to see.

Today, instead of getting him to explain his thinking in more detail, I got another student to do it for him.  One of my ultra-neat, super-organized girls took his scrawling mess of a piece of paper and sat down to try and translate how he had got to the solution.  With help from the little boy (it was a great mathematical discussion) she was finally able to organize his work into something coherent.

This was a valuable lesson for both students.  For the boy, he had to describe his thinking to another person on a one-on-one basis, without the pressure of the social group staring at him.  He knew that the girl was an organized, neat-thinking person, so he had to adapt his style to meet hers.

For the girl, this was a lesson in outside the box thinking.  She was able to take something that is completely alien to her (invented strategies in subtraction) and turn it into something that is neat, logical, and works for her.  She even commented after they finished that she liked this way of doing it, and she would try it with questions in the future.

This kind of translating thinking is something that I will do again in the future.  I think it would work well if you have a mixture of kids in your class who are Kumon-type math kids, and who are more of the invented strategies type.


The Noisy Math Class

Kids like to be noisy, and teachers like to make them quiet.  Line up in rows.  Sit down with your hands on your desk.  Five point check.  No talking in the halls.  Stay in a straight line.  Show me how you listen.  I could go on.

***Aside Rant: Where else in society is it an expectation to start at attention in rows and wait to be given orders?    I can think of one.  So, why do we make it an expectation in school?*** 

I like a noisy classroom.  I like incessant chatter like the drone of a million cicadas (I love the sound of Cicadas in Japan, get outside the cities and it is an amazing wonder).  I have been thinking a lot about listening and sound (great TED Talk), and the impact it has on our lives.  With that in mind, I came up with this lesson today;


We did a mental math activity today.  I had the students walking around the room practicing how to show their thinking.  Instead of doing it in a notebook (a messy one!) we did with words.

Teacher:  Question one; what is three hundred and twenty five plus one hundred and twelve?

Response: Three hundred plus one hundred equals four hundred and twenty five plus ten equals thirty five plus two equals thirty seven, so therefore the answer is four hundred plus thirty seven, which equals four hundred and thirty seven!

The students would all yell out their thinking at the same time.  Each one would be doing it differently, but that is the point.  After we have our cacophony of math yelling, we would pair off and share our strategy with somebody else.  Next, the two would have a math debate about which strategy was more efficient.  Then, we get the next question and start the whole process again, but this time pairing off with a new partner.

I took a quick recording with my iPad, and here is what it sounded like;

Cacophony of Math

Sounds like music to me.



Being messy with Math

We were working on Mental Math strategies today, and I had the students attempting to show their thinking on paper.  First, we try to think of all the steps we do in each problem we try mentally.  Instead of doing this in our heads though, we write down what we see in our heads as we see it (not always an easy task with grade 5/6!).  For example;

134 + 143

In my math notebook I would write

134 + 140 = 274

274 + 3 = 277

These are the two steps in which I followed to do this problem in my head.  We have been working on writing our thoughts on paper as they come into our heads.  During the exercise, I noticed something funny going on, so I called an emergency math congress on the carpet.

Me:  Unicorns (I call them unicorns), I want you to look closely at the following two notebooks, and tell me which one you think is better.

Notebook One was immaculate and neat, everything was lined up in a row and written in perfect numerals, however, under the perfect script were remnants of erased numbers, barely visible.  Notebook Two was covered in scratch marks and lines and X's and it was sloppy and messy.

Kids:  Notebook one!  (in unison and immediate).

Me:  Why?

A: Because it is neat and easy to follow and she got the answer right.

Me:  Interesting. What about this one?

B:  It is messy and there are lots of black marks on the page.

Me:  Did he get the answer right?

A:  Well yes, he did, but he made lots of mistakes along the way.

Me:  Did Notebook One make mistakes?

A:  Yes, but she erased the work and wrote it neatly.

Me: Ok, imagine you are me, and you have to grade this.  I need to know what you were thinking when you did this problem.  Which notebook is easier to understand what the person WAS thinking at the time?

B:  The messy one.

Me:  Why?

A:  Because you can see his mistakes, and you know when he changed his mind.

Me:  Good.  So, from now on, lets try and be messier when we work out problems, make sure you write down everything, even the mistakes you make.


I love a messy math notebook.  I love scribbles and doodles and X's, and big heavy black pencil marks that are smeared, and maybe some paper that is crumpled because of frustration, and random numbers in the margin, and shapes and pictures.

It is like a snapshot of the brain at work.  A recording of thinking in action.

Go on kids, be messy.



Henry Climbs a Mountain

I found this picture book in a second hand store in Canada for about 0.50 cents.  It is the story of Henry David Thoreau, and how he spent a night in jail in protest over the governments acceptance of slavery (he didn't pay his taxes).  The artwork is brilliant.  It is very cool to see how the cell transforms from a plain white space into an area of natural splendor.  I personally love Walden and Civil Disobedience, and it is great to see the general philosophy from those books brought to a level that young children can understand.  There are a lot of big ideas to chew on in this book as well;

Is it right to break a law that you feel is unjust?  Does everybody have the same perception on what unjust means?  What do you think unjust means?  Did he accomplish anything by going to jail?

Henry Thoreau is a colorful and interesting character, and we will certainly come back at some point this year and make some more connections to him.

If you are working in a PYP school, and have a UOI on laws, society, fairness, equality; this is a great conversation starter.