Kindness in Mathematics

Deeds not words. (eng) Facta non verba (latin)

This story aimed at 5-7 year olds. It introduces addition and subtraction. But also can be told at 9-10 years old when studying fractions.

One rich Khan from Persia has accumulated ten thousand gold coins. From this sum he took one hundred coins to feed all hungry people in his kingdom. His cooks begun to prepare a tasty dish from rice and meat, while the khan’s servants went to invite poor men for this lunch. On the way these servants met in the forest one poor woodcutter and ordered him to come for the khan’s feast.

“The generosity of our Khan is boundless,” servants have told.

“Thanks, but I used to live by my own work,” the poor man answered.

Servants become angry, seized the woodcutter and have led him to their khan.

“Today I spent ten gold coins and with this money I managed to feed more
than one thousand persons, and what great deeds did you do?” the Khan

“Well, today I have earned only four copper coins and on this money I was
able to feed only four persons, this is what I’ve done. I fed my wife,
myself and two sick old men, my neighbours,” modestly answered the
woodcutter.

At these words the Khan’s face changed. “Who is more generous: me or this
woodcutter?”  thought the Khan to himself.

Questions and tasks to the fairy tale :

• What part of the money was spent on the people by the Khan, and what part – by the woodcutter? Try to compare it.
• What do you think, who is more generous, the Khan or the woodcutter?
• Imagine that you have six sandwiches and split sandwiches in equal shares between you and your hungry : dog , brother and sister, five hungry friend. What part of the whole amount will you give in first, second and third case?

Prepared by Ada Mehmed and Danilo Borovnica from book “Magic Mathematics. Kind plus and Wise minus” by

Math Catcher

Math Catcher are stories/movies that you can use in your classroom to promote math:

• Small Number Counts to 100. Boring task instead will be like this. When you divide number 100 with number 7 which is remainder? The story can be shown to elementary school students as a counting practice/puzzle or as a pattern recognition problem. For high school students it can be a way to introduce arithmetic progressions, modular addition, or an idea of number systems with a base different than 10.
• Small Number and Old Canoe. The idea behind this approach is to give the moderator a few openings to introduce or emphasize various mathematical objects, concepts and terminology.  The aim of the question is to lead to an introduction at an intuitive level of the concept of a function and the essence of the principle of inclusion-exclusion as a counting technique. Is is also opportunity to appreciate that in order to understand a math question, one often needs to read (or watch) a problem more than once.
• Small Number and Basketball Tournament. Small Number demonstrates how a basic understanding of combinatorics can help in all aspects of life, even basketball!
• Small Number and Scateboard Park. Geometry problem solved by Full Angle.
• Small Number and Salmon Harvest.
• Small Number and the Big Tree.
• Small Number and the Old Totem Pole. There is a similar story song and poem about huge sugar beet whom trying to pull out grandfather, grandmother, granddaughter, doggie and kitty. Unsuccessfully. Finally mouse help them to pull out beet.

Aboriginal peoples in Canada are the indigenous peoples within the boundaries of present-day Canada. They comprise the First Nations, Inuit and Métis. Although “Indian” is a term still commonly used in legal documents, the descriptors “Indian” and “Eskimo” have somewhat fallen into disuse in Canada and are sometimes considered pejorative.

The performance of Aboriginal students in British Columbia (Canada) for the last five years has been significantly lower than the performance of non-Aboriginal students :

• As early as grade 4, Aboriginal students lag behind their non-Aboriginal classmates by about 20% in their performance on the Foundation Skills Assessment in numeracy.
• By grade 10, the gap widens and only 47% of Aboriginal students fulfill the expectations in numeracy, compared to 77% of non-Aboriginal students
• 2% of BC’s Aboriginal population completes Principles of Mathematics 12 compared to a completion rate of 25% for the whole BC population

The project Math Catcher: Mathematics Through Aboriginal Storytelling is an outcome of the BIRS supported First Nations Math Education Workshop from 2009.

During the workshop, Veselin Jungic (former Yugoslavia) and Mark MacLean co-wrote a story, Small Number Counts to 100, which served as the cornerstone project.

To promote mathematics among Aboriginal learners they have created a series of stories with mathematical themes. These stories are based on the storytelling tradition of Aboriginal peoples. The fact that all of stories have been translated into several Aboriginal languages is probably the biggest recognition.

The Program is based on the belief that it is crucial to engage Aboriginal students in mathematics and science at the early age.

Program aims are to promote mathematics and scholarship in general by encouraging elementary and high school students to recognize how math is used in everyday life and how it forms the basis for many of our daily decisions and life-long choices. The storytelling, pictures, models, and hands-on activities encourage young people to enjoy math and help dispel myths that math is boring and abstract.

Again, can you find similarity with your country?

Prepared by Danilo Borovnica.

Posted in Algebra | Tagged , | 1 Comment

Logic with the cow

My first exam at university was Logic and I still remember the joke told by proffesor. Now this very joke one can find everywhere on  internet. As well as in a Marc Hadon’s book The Curious Incident of the Dog in the Night-Time chapter 181.

There are three men on a train. One of them is an computer scientist and one of them is a engineer and one of them is a mathematician. And they have just crossed the border into Scotland (I don’t know why they are going to Scotland) and they see a brown (or black) cow standing in a field from the window of the train (and the cow is standing parallel to the train).

And the economist says, “Look, all the cows in Scotland are brown.”

And the logician says, “No. We can say there are cows in Scotland of which one, at least, is brown.”

And the mathematician says, “No. We can say there is at least one cow in Scotland, of which at least one side appears to be brown.”

There are different options for passengers but in my mind these three works best since a computer scientist just wants to get within an order, an engineer wants a realistic approximation within the permissible error, and a mathematician wants it precisely right. Story makes it clear to us the level of accuracy we are striving.

Story speaks of  the nature of generalization and needs for precision. On the edge of a cliff one millimeter makes a difference between life and death, while in the light of the evolution one millennium is just a blink in the eye of time.

At least one” is a mathematical term meaning one or more. It is commonly used in situations where existence can be established but it is not known how to determine the total number of solutions.

Is this the only possible conclusion one can come to here? Try to flip it around: is there a way to criticize the mathematician and praise the computer scientist?

The moral of the story is that we should be careful making generalizations. Examples of wrong generalization you can find everywhere. People make conclusions based on a single case.

Prepared by Danilo Borovnica.

How to hang a picture

Professor Knot was very proud of his collection of paintings that were hanging on all the walls in his apartment. Our story begins when a neighbor from a nearby apartment started  indoor reconstruction.  The walls were shaking because of  vibration. Professor was worried for safety of his paintings and decided to add one or even two nails.

And, he did it! First picture was hanging on two hooks.

Then he wanted to increase the safety and wrapped the rope several times, as shown in figure.

After that , he was quite satisfied.

Anyway, he decided to test security by pooling out one of the two hooks. He first pool out the right hook.

He was very surprised when the picture fell down on the floor! This means that the left hook does not hold a picture. Meaning: right hook keeps all the weight.  He wanted to be sure.

Than, he repeated the whole procedure and  picture was hanging on the wall, holding on two nails.

This time he pulled out the left hook because the right one holding a picture . He almost fainted when the picture fell down on the floor, again.

If neither the left nor the right hook  does not hold a picture, what holds the picture?

Prepared by Zorica Marinković based on article  Boromejski prstenovi ili Kako umetnički okačiti sliku, Rade T. Živaljević, Časopis Tangenta 59/3, 2009-2010

Literature:

The same fence, different area

Once upon a time, there was a king named Euclid who was very fond of mathematics so he built identical castles with beautiful gardens with fencing in the shape of a triangle, square, pentagon, hexagon, octagon and circle. He wanted to have a lot of children who would each have their own castle once they grow up. At the same time, he said to the architects that the fence of each castle had to be the same length.

He married the beautiful Hypatia, with whom he had (amazingly) six children who were called Dopey, Grumpy, Sleepy, Bashful, Sneezy and Smiley.

When the time came for them to learn to rule, the King gathered them and told them that he had decided that everyone would get their own castle to manage it. He also added that the one that chose best would inherit the throne.

Dopey chose the castle with the fence in the shape of a red triangle, Grumpy took the yellow square, Sleepy took the blue pentagon, Bashful took the purple hexagon, Sneezy chose the sky-blue octagon castle and Smiley chose the castle with the  fence in the shape of a green circle.

Help the king to choose his heir.

You can talk with your students about what it means to have the fences with the same length (same perimeter for regular polygons and circle). What is the best decision for the heirs (to choose the biggest figure with the biggest area).  Who will inherit the smallest castle?

What is the lie in this story? Snow White and the Seven Dwarfs. Who is missing?

Prepared by Danilo Borovnica based on Brilliant.org task.

Photo Store

This is a story wich can be intro story for commutative law multiplication as well as for symmetry and rotation.

One student is a Customer who does not know math very well and another student is a Seller in Photo Store. Seller standing behind the counter and Customer walks into a shop.

Customer: “Do you have any frames that fit a 7×5 photo?” (looking at the exposed frames)

Seller: “Yes. We have some.Here you go.” (and shows few frames with big 5×7 title)

Customer: “I like THIS one, but you only have it in 5×7.”

Seller: “Yes?” (Seller do not understand the question)

Customer: (whines disappointedly while holding one frame)

Seller: “Um…” (looking at the customer wondering)

Customer: “But I need one that’s 7×5, not 5×7!”

Seller: (slowly turns the frame on it’s other-wider side)

Customer: “Oh, wow! It is amazing. How do you do that?”

Seller: “I just do the math”

Customer: “You kidding me? What math have to do with it?”

Now teacher can talk with students what math have to do with it and what is math behind this story.

Prepared by Danilo Borovnica based on story by Denise Gaskings.

Engaging the imagination

“Engaging the imagination is not a sugar-coated adjunct to learning; it is the very heart of learning” Kieran Egan

“You have to be a magician to keep a kid’s attention for more than five minutes these days!” Foghorn Leghorn

Mathematics is often perceived as a collection of facts and skills to be learned, and often these facts and skills are counterintuitive to the learner. When this happens a common student’s reaction is to seek refuge in the meaningless memorization of rules.

First act of wondering  can become a pivot for further mathematical instruction. To engage students in mathematics our goal is to ignite the fires of curiosity, to get them wondering why things are as they are. “I wonder why…” is the beginning of wonder and inquiry. A very direct way to get students to wonder about things in mathematics is to ask them ‘why’ something is as it is.

Imagination

Mathematical activity inherently related to the imagination. How to describe, perceive for example negative numbers, all variables, division by zero, infinite set, non-Euclidean geometry without imagination.

Understanding mathematical concepts often involves seeing things in new ways and venturing beyond our limited perspective.

Images created in traditional oral cultures have the crucial social role of aiding memorization. Some images, no doubt, are influenced by pictures in books, but it is common to find that the most vivid and evocative images are those we generate for ourselves while listening to stories.

If we are to make knowledge meaningful for students, then we must introduce it in the context of these human emotions—and the imagination is the best tool for accomplishing this task. Imagination accompanied by a charge of emotion.

Emotion

Emotional connection is important  for good storytelling. We need to convince our audience that what we are talking about matters. But equally important is to know which details should be omitted in order to understand the punchline. Sometimes you have to lie to tell the truth.

If young student do not know  what is the dodecahedron it will not destroy his chances to succeed in math. But if the young student hated and does not understand anything in math, it will destroy his chances to succeed.

Math Storytelling is a great opportunity to get children excited about math through stories wich can include logic, patterns, puzzles and numbers, story that asks a question. Stories are told to help people learn a lesson or understand a problem. This is a step to appreciate all the ways math enhances our daily lives.

Not all math stories are happy. Many grown-ups have their math agony stories, telling of giving up on mathematics. Share story with us if you have such math grief story.

You probably learned many fairy tales when you were young. We ask people who come from different parts of the world about the stories they learned when they were young. Are they the same kinds of stories?

Wonder

The task of stimulating interest in mathematics often involves locating mathematics in the wider context of wonder.

What is there about mathematics that is so awesome to wonder? There is no answer to such a challenge. It is same for beauty : wonder is in the eye of the viewer. We can only say that for many, ourselves included, there is wonder in everything in mathematics.

Make it human

Science and mathematics texts seem particularly ‘inhuman’. Text-books, in particular mathematics textbooks, have tended to disguise from us the simple truth that all knowledge is human knowledge. Do not forget this.

Engage

We tell stories in the mathematics classroom to achieve an environment of imagination, emotion, and thinking. We tell stories in the mathematics classroom to make mathematics more enjoyable and more memorable. We tell stories in the mathematics classroom to engage students in a mathematical activity, to make them think and explore, and to help them understand concepts and ideas.

What we are going to do

We share techniques for storytelling that makes telling more interactive and more appealing. We present a framework that may help potential storytellers create their own stories, as well as ideas as to how existing stories can be enriched and adapted for the needs of any particular audience. By such means we hope that more teachers and more colleagues will story-tell in their classrooms, and be patient enough to wait for long-term benefits.

• Stories can provide a frame or a background to mathematical problems, stories can deeply intertwine with the content, and stories can explain concepts or ideas.
• Spark interest, assist in memory, reduce anxiety are some of the advantages of storytelling in the classroom.
• Make that students acting like their heroes, create empathy, provide them something to hold to, make the lesson more relevant and more vivid, make break from the routine, creating a refuge to return to and to seek more stories.
• Whether in the form of oral, pictorial, written, or film media, stories help in the exchange of experiences from one to another.
• Mechanisms such as art, drama, music and movement combined with visual images help strengthen the ability to transfer knowledge.
• Introducing stories in mathematics classrooms will change the stories about t he mathematical experiences of
• Creating interest with a story is an important initial step. Describing a chain of events may engage students, create excitement, mystery or suspense, and motivate thinking about a particular problem. Stories may convey passion and enthusiasm.
• Attention is a delicate thing. Although it is easily Sparking students’ initial interest with a story it is hard to sustain this interest, to sustain students’ engagement and not let it evaporate as the story ends. That is why some of our stories never end. Constant stimulus, the change of the rhythm in the classroom is necessary.
• Variation on a story can help in solving a problem or gaining a better understanding of a solution
• Make that storytelling is frequent and regular activity in a mathematics classroom.

Literature: