The market scales puzzle & connections with ancient Egyptian mathematics: Homework for Monday Sept 28

 Here's a puzzle that connects with the principles of ancient Egyptian multiplication in interesting ways. There's a story connected with this puzzle that I'll tell you in class -- but here's the puzzle:

A market vendor sells dried cooking herbs in whole-number amounts from 1 to 40 grams. The vendor has an old-fashioned two pan weigh scale, and has exactly four weights of different amounts that allows them to weigh out any of these amounts of herbs -- without using the herbs or any other object as an auxiliary weight.

• Make sure you understand how a two-pan scale works...the weights can be placed on either or both of the pans.
  
  
• What must the values of the four weights be? Why?
  
  
• What if it was a one-pan scale -- i.e., a scale where you could only put the weights on one of the pans, rather than both of them? What five weights would you need to weigh up to, say, 31grams?
  
  
• How could you extend this puzzle to help your students understand the mathematics more deeply?
  
  
• How does this puzzle connect with ideas about number theory and bases you are already familiar with?
  
  
  

Please post your blog on this problem next  Monday September 28 by 9 AM.

Enjoy!

Dave Wagner: two-pan market scales, Bhutan. Used with permission.

 

Ancient Egyptian mathematics: Numeration

  

Value	1	10	100	1,000	10,000	100,000	1 million, or many
Hieroglyph
Description	Single stroke	Cattle hobble	Coil of rope	Water lily (also called lotus)	Bent finger	Tadpole	Heh[3]

(From Wikimedia)

What differences do you notice between ancient Mesopotamian/ Babylonian numeration and ancient Egyptian numeration systems?

(And if you are familiar with Roman numerals -- which we will visit later -- what similarities and differences do you notice?)

What affordances and constraints do you notice for the Egyptian system? For the Babylonian system? 

The 'Russian Peasant Multiplication' -- closely related to Ancient Egyptian mathematics and binary number systems

  Here is a fascinating Numberphile video from 2020 that shows the 'Russian peasant method' for multiplication. How is this the same and different from Ancient Egyptian multiplication? Why are they equivalent?

Numberphile Video (approx 5:00) < https://youtu.be/HJ_PP5rqLg0>

 

Examples of Ancient Egyptian multiplication and division

Here are examples of the Ancient Egyptian method of multiplication and division, by doubling and halving. They depend on the fact (clearly known in that culture) that all the whole numbers can be constructed from sums the powers of 2 (including 2^0, which equals 1).

Ancient Egyptians also used only unit fractions in the form 1/x, with the exception, for some reason, of 2/3 — something we will explore further. 

Take a good look at the division example. Would this technique work with all whole number divisors, or only even ones? Could this technique be used with other unit fractions?

                                                             Multiplying 22 by 7




                                                       Dividing 43 by 8

**EDIT: Blog post due Wed Sept 23: Did Mesopotamian scribes have algebra? (And how can we recognize 'rhetorical' and 'syncopated' algebra?)

Here is the link to an explanation of Babylonia algebra to help understand Babylonian word problems. (Note that this is the sheet we were using in class as a handout!)

https://drive.google.com/file/d/0B00n89L6TX5gZ01BQndwczRQaEE/view?usp=drivesdk&resourcekey=0-arSdOZ88pkPwDfajhNbQMg

You'll see that the form of the problem is different from the algebra we are used to -- and there is no use of compact alphanumeric symbols for variables. Instead, concrete words and images are used.

 

(**Note that algebraic relationships stated in words are known as 'rhetorical algebra'; algebraic relationships stated with some abbreviations but not a systematic symbolism are known as 'syncopated algebra'. It's interesting to think how this relates to our students' initial learning of algebra...!)

Questions to consider:

• How could one state a general mathematical principle in a time before the development of algebra and algebraic notation?

• Is mathematics all about generalization and abstraction?

•Thinking about various areas of mathematical knowledge -- number theory, geometries, calculus, graph theory, etc. etc. -- how could you imagine stating general or abstract relationships without algebra?

• Do our students go through similar stages of rhetorical and syncopated algebra as they move toward learning and doing  symbolic algebra?

**EDIT (note change of date): Reading for Monday 28: More on Babylonian word problems

  

Our reading comes from a chapter on the history of word problems from my book, A man left Albuquerque heading east: Word problems as genre in mathematics education. This section is mostly about Babylonian word problems -- we may read the rest of the chapter as we consider the history of mathematics teaching and learning alongside the history of mathematics in itself.

Please post your response by  **EDIT: Mon September 28 at 9 AM.

A few additional resources: timelines and Babylonian word problem

  Here are some resources from Dr. Irene Percival (from SFU) including a timeline of overlaps between different cultures we're looking at and their mathematics, and an example of a quadratic Babylonian word problem!

https://drive.google.com/file/d/1EArxj8x8ihpA16iJwrHVLQOKELIdfyYX/view?usp=drivesdk

A Babylonian word problem -- and a consideration of what word problems are more generally!

 

Question: Were word problems ever simply the application of mathematics to real life situations?

 

And another small bit of homework for Monday September 21:Create your own Babylonian-style base 60 multiplication table for the number forty-five

  

Fragments of a Babylonian star calendar

In class, we experimented with a table of pairs of numbers that multiplied to sixty (in base 60, with base 60 fractions).

For homework for our class on Wednesday Sept 17, please experiment with creating a similar table (in base 60, using base 60 fractions), but find five pairs of numbers that multiply to forty-five!

Rules:

• At least two of your five pairs should include base 60 fractions in one or both of the numbers.
• Don't use 1 as one of your numbers. Challenge yourself a bit!

Here is an example (and please find five more pairs that multiply to forty-five!)

2    22,30
(explanation:  two times twenty-two is forty-four, but two time twenty-two and a half (,30) is forty-five)

Homework for Monday Sept 21-- Read and respond : Some interesting resources on the (Babylonian-based) history of time calculations, base 60 and base 12

   

Here's an interesting piece from Scientific American on some of the origins of our ways of telling time.

And here is a good piece elaborating on the origins and uses of base 60 by Mesopotamian/ Babylonian mathematicians, from the excellent University of St. Andrews math history data base.

 YOUR HOMEWORK: Read and consider these two short articles on the ways we measure time, and discuss them in relation to your own personal experiences and geometries for thinking about a year (and a season, a month, an hour…) Think about any inconsistencies between the two articles, and any surprises that have you reconsidering how we measure time.

 

Another interesting Babylonian tablet

Some Babylonian tables (translated into Hindu Arabic numerals)

     

What might these tables mean?

Note that, in this notation, commas separate place values (for both whole numbers and fractions).

Can you figure out the common theme here?

Why are certain numbers missing from the left hand column? For example, there is no 7, 11, 13, etc.

How do fractions in the Babylonian style connect with our fractions? (Keep this in mind as we learn about ancient Egyptian fractions later on...)

Could you create a table of this kind with a different number as its focus — for example, 45? 

Finishing up our blog set-up: Here are the instructions on how to invite the rest of the class to be authors on your blog

 Hi all! Some people have run into some snags in inviting others to be authors on their blogs, so I've written up the steps below that I hope will be helpful.

Please check your blog dashboard to see whether you have all your fellow students plus  me signed up as authors on your blog! If not, please follow the steps outlined below to invite everyone -- or if there are just a few people who haven't yet responded, please try deleting their original invitations and then re-inviting them. 

We're almost there!

Cheers

Susan

******************

How to invite the rest of the class to be co-authors on your blog:


• Open your blog, and choose "design" from the menu on the upper right hand corner of the screen to get to your dashboard. (I've used the 342 class blog as the illustration here, but this applies to EDCP 442 and EDUC 450 as well). 

• Choose "Settings" from the menu on the left hand side of the dashboard.

• Scroll down in Settings to "Invite more authors"

• In a separate tab, open our class spreadsheet of emails and URLs.


• Highlight and then copy the whole column of our emails


• Go back to your blog Settings --> Invite more authors and paste the column of emails there. Check to make sure you are not inviting yourself! You can delete just your own email from the list.


• Press "send", and you should have invited the whole class to be authors on your blog.


Once everyone has finished these steps, and we have everyone accepting one another's invitations, we will take the next step to set the privacy setting to Reader Access --> Private to authors. (But please don't do this till we get everyone signed on!)

Some very helpful books on the history of mathematics for teachers

 Hi everyone! Here are the books I brought to show you in our EDCP 442 class yesterday. I can recommend all of these as excellent resources for you as a teacher, and to have in your classroom math teaching library! I'll bring along some more books to upcoming classes, and post them on the blog as well.

Assignment 1: Doing ancient problems in ancient and modern ways

  

Our first assignment of three assignments for this course involves choosing an ancient puzzle and solving it in ancient -- and modern-- ways. The aim is to begin to understand what it might have been like to do these problems with the mathematical/ conceptual tools and approaches of a historical culture, and to understand the equivalences to our contemporary mathematical understandings. This should deepen our understanding of the mathematical ideas involved, and help us appreciate the ingenuity of human cultures over the millennia and around the world.

You should find two partners to work with, and each group will choose a different problem to work on. Each group of three will present:

• their problem and the background to it
• their solutions to it, using both ancient and modern mathematical techniques
• an extension to the problem

...in an interactive 15-20 minute presentation with the class. Presentations will be tentatively scheduled for Oct 5 and 7 . Groups should post their solutions and extensions to your individual blogs by Monday October 5 at 9 AM.

•You are also required to write a short personal reflection after your presentation on what you learned and took away from the process of doing this project.

Here are the problems you can choose from:

• #1-7 From 5000 Years of Geometry (Scriba & Schreiber): Choose from Problems 1.2.1 to 1.2.7 on pp. 25-26. Background information that you need comes from the preceding pages, pp. 15-24. [Note that in problem 1.2.6, there is a typo in formula 1.2.11, which should read "s=1/2(d-(sqrt d^2-c^2)).]

• #8 From Episodes in the Early History of Mathematics (Aaboe):  Problem 1.5, with a preamble that explains a bit about the problem.

• #9 Also from Aaboe: A preamble and a problem on Babylonian calculation of Pythagorean triples.

• #10 Ahmes' loaf sharing problem. Your task, if you choose this one, is to work through the problem as explained in this (translation of) the original papyrus, and to be able to explain how the Ancient Egyptian solution worked, in both ancient and modern terms.

• #11 As Ahmes was going to St. Ives. Your task, if you choose this one, is to work out how these powers of 7 could have been arrived at using Ancient Egyptian calculation methods we have learned, and to be able to show these methods. You should also research how these calculations would have been made in Fibonacci's time (around 1100 AD in Italy), before the introduction of Hindu-Arabic numerals and the algorithms we use today, and to show how these medieval European arithmetic methods worked as well.

Schedule for presentations: Each group will have 15-20 minutes to present, including a brief interactive activity.

Monday Oct 6 

1. 

2. 

3. 

4.

Wednesday Oct 8

1. 
   
   
2. 
   
   
3.  
   
   
4. 
   

Homework reading and response for Wednesday: Crest of the Peacock introduction

   

Your reading for Wednesday’s class is from Crest of the Peacock -- an introduction to non-Eurocentric math in history. https://drive.google.com/file/d/0B00n89L6TX5ga0p6QUdFVndrN3M/view?usp=drivesdk&resourcekey=0-8OhOE_L_BmuYoZn-PZBjBQ

Please read this and write a blog post on it by Wednesday, September 16 at 9 AM. Your blog post should address three things that surprised you (and why)!

Why Base 60? (EDIT:**NOT for homework as we have already had some discussion about this!)

We have seen that Babylonian mathematics had a base 60 (sexagesimal) place value system -- that is,the same symbol could be used in the units column, the 60s column, the 3600s column,... with different meanings. 

We'll be doing some further readings on this in our upcoming classes!

For homework, please write a blog post about why you think the Babylonians chose base 60 rather than the base 10 system we are used to (even though they did have a special symbol for 10). 

To write your blog post for Wednesday, please engage in a "speculative phase" and a "research phase". Begin by thinking, wondering and speculating for yourself, and only after that, go to doing some research online or in the library:

Speculative phase:

(1) Think for yourself why 60 might be a convenient, significant or especially useful number to use as the base for a number notational system. What is special about the number 60? How is it different from 10?

(2) Then think for yourself how we still use 60s in our own daily lives, in Canada, and across cultures if you have knowledge of other systems (like the Chinese zodiac and time-telling system, for example.) Why is 60 significant in so many situations involving time and/or space?


Research phase:

(3) Finally, do a bit of research via the internet and/or the library to find out what others have learned about the significance of 60 in Babylonian numeration systems, in our contemporary world, and possibly across cultures.  

Spreadsheet with our blog urls & emails

Bakshali manuscript from India: first use of zero

 To invite everyone in the class as co-authors on our blogs (and close off the privacy settings to just our
class), we'll need to share our blog URLS and emails. Here's a link to a spreadsheet where you can do that!

Interpreting a Babylonian cuneiform clay tablet

 

Our detailed look at the history of mathematics will start in an area known as Mesopotamia (meso: 'between', potamia: 'rivers' -- between the Tigris and Euphrates Rivers), in what are now the countries of Iraq, Syria, parts of Turkey and Kuwait. The area is sometimes called the 'fertile crescent' because the river systems flowing down to the Persian Gulf made agriculture and cities viable.

In accounts of mathematics history, the mathematics of Mesopotamia circa 3100 BCE - 300 BCE is usually called Babylonian mathematics. But if you look more carefully at the history of Mesopotamia in this period, there were several different peoples and nations that governed this region, including the Sumerians, Akkadians, Assyrians and Babylonians. 

For our purposes, we will use the term 'Babylonian mathematics' to refer to the fairly unified mathematics traditions over this 3,000 year period (starting approximately 5,000 years ago). 

Babylonian writing was done with a wedge-shaped reed stylus in wet clay tablets the size of a person's palm, and then dried in the sun or in a kiln. Their ingenious writing system was known as cuneiform: 'wedge-shaped', and it was possible to write words, numbers and other symbols with just these wedge-shaped forms. Because these baked clay tablets are very durable, we have many of them in museum collections to this day -- and quite a few of these seem to be teaching tablets for scribes learning mathematics to take government jobs in the Mesopotamian cities!

Here is an example of one of the existing mathematical clay tablets from ancient Mesopotamia. Your job is to figure out what is written here, and how the writing system for this mathematical tablet works!

 

Hello! Here's some information on the Orchard Garden, where we'll be holding our classes for the next

weeks till Thanksgiving. 

As mentioned in class and on our tentative course outlines, we are planning to hold our classes in the UBC Orchard Garden outdoor classroom from now till Thanksgiving (Oct. 12), weather permitting. This upcoming Monday, Sept. 14, looks like good weather, and we'll check in on the Wednesday Sept. 16 weather as we get closer to the date.s

How to get to the UBC Orchard Garden:

Address: Totem Field Studios, 2613 West Mall (1 block south of Thunderbird)

Go through the wooden gate between the buildings at Totem Field (the pedestrian gate on the left side is always open -- just push!) Turn right and walk about 100 steps through the 'Xmas tree farm'/ conifer research project, and you will be at the Orchard Garden.

Walking: It's about a 10-15 minute walk south and slightly west of Scarfe.

Cycling or scootering: Faster -- about 5 minutes from Scarfe

Bus: There is one of the smaller community buses (#68) from the UBC Bus Loop that I have seen has a drop-off point right outside Totem Field. However, it is scheduled only every 20 minutes on school days (for example, at 12:20, 12:40, 1:00,...) Google maps say to get off after 5 stops at eastbound Thunderbird @ Eagles Drive, but it's still a 450m (4.5 block) walk from there (I've done that...) I'm not quite sure how to access that bus stop right in front of Totem Field, but maybe you can figure it out and share this?

Car: There's no street parking on West Mall and they do ticket. You are probably ok parking inside the Totem Field gates -- or at the UBC Botanical Gardens lot across Marine Drive? (I don't have a car).

What to bring/ wear:

• A sun/ rain hat -- very important!
• A water bottle -- remember to hydrate frequently
• Any medication, epipen, etc. that you need. (Please let me know about any relevant allergies or medical conditions for safety precautions)
• Appropriate layers of clothes for the changing weather
• Appropriate close-toed shoes or boots for walking on uneven or muddy ground
• Light jacket or windbreaker
• Sunscreen and lip balm with sunscreen (I've had sunburned lips, and it's not fun!)
• Laptop, tablet and/or phone for accessing the internet and writing
• Notebook and pen may be helpful as well
• Snacks

Looking forward to seeing you at the Orchard Garden for our classes this week and onward!




Cheers

Susan

An exciting, relatively new book about the role of the history of math in teaching and learning!

    Here's an exciting and very relevant new book published in 2023 by Springer and CIEAEM, on the role of the history of mathematics in teaching and learning. We may be using some of the essays here as part of our class discussions and readings -- and you can download the ebook for free from the UBC Library website! Highly recommended!

Our first reading: Why teach math history?

  Here is a link to this thought-provoking article!

Please read this and post a blog response by next Monday, September 14 at 9 AM on your personal blog.

Your response (2-3 concise paragraphs) should address:

1) Your pre-reading ideas about whether, why and how math history should or could be incorporated into you own math teaching.

2) 2 -3 things in the article that made you 'stop' and wonder, question, connect, agree, disagree or be surprised by what you read.

3) Any changes in your ideas after having read this piece.

Welcome to our math history for teachers course!

  Hello everyone, and welcome to this very interesting and exciting course. We will be exploring aspects of the history of mathematics from a wide range of eras and cultures, through problem-solving, working with mathematics in historical ways, and through connections with our current BC curriculum and the arts.  

It is truly amazing and eye-opening to learn about the ways that our ancestors and people of every culture and historical have worked with sophisticated concepts of number, shape, and patterned relationships. We will explore many of the roots of ideas we teach in school and tend to take for granted nowadays -- and I hope this will just be the beginning of your ongoing interest and understanding of the history of mathematics and the sciences.

Here is a link to our course outline (which may be revised as we go along). Happy historical learning!

** Note that blog posts will be assigned a mark from zero to 3 each week:  3: extraordinarily thoughtful & insightful  2: very good, showing understanding, engagement and thought  1: adequate, showing that the reading has been completed although the response is not strong.  0: assignment is not complete/ reading has not been completed.  Most of your blog posts will get a mark of 2, and if you have all 2's, you will receive full marks for your blog! A mark of 3 is just to let you know that you did an exceptionally impressive piece of writing and thinking from our points of view, and a 3 counts as full marks as well.

If you receive a 0, 1 or 1.5, you will need to write or revise that blog post before the next marking date (in about a month) or by the end of the course. The comments you receive should let you know what needs to be added or revised. Please don't delete the original post, but add an **EDIT to let us know what you have added that is new! Most of your smaller blog posts and responses will be assigned from September to early November -- after that, you can concentrate more fully on your culminating assignments.

Cardano's anomalous version of the 'planetary' magic squares