The Mathematical Wonderland

INTRODUCING LUMI

Anyone who works or lives with a child knows all too well just how demanding it can be to help them grow up! Accompanying them through their evolutionary challenges means finding out just whose hand we are holding every day. It is a stilted path: each step forward followed by another step back. We have to follow our hearts and our instincts, and trust our own common sense if we want to understand how to act and what to do. Thanks to studies in neuroscience, today we have a much greater understanding of how a child’s brain works and how the human mind develops, and this information is of invaluable importance in the world of education and pedagogy.

LUMI is the fruitful combination of neuroscience, pedagogy, and evolutionary psychology. This young and innovative publishing range focuses on the needs of families today, sensitively and competently creating products designed to promote children’s well-being, in a global, efficient, and scientific way.

LUMI products are based on studies by various authors and researchers to give parents and teachers easy-to-use, enjoyable, and scientifically based tools to help them guide children along their evolutionary journey in this digital, interconnected, and intercultural world in which we live.

Through enjoyable activities and games, our little explorers can fully express their potential in cognitive, emotional, and social development, accompanied by knowledgeable and encouraging educational guides. The key words are curiosity, reflection, emotion, and autonomy

Educating

LUMI

LUMI is the name of a conceptual archipelago of islands, named after an old red and white striped lighthouse that served to light the way for all seafaring adventurers and guide them to the islands, where they could acquire a wealth of new knowledge, experiences, and skills.

LUMI CHARACTERISTICS

LUMI materials are mainly aimed at children but do contain a part for parents or teachers. In this way, the books or games can become an opportunity for adults to acquire new information on children’s development and how to fulfill their needs and respond to their behavior. Our products are designed for children who are independent, curious, contemplative, and creative; they are not passive learners, but choose to be the protagonists of their own learning. Third millennium children are the so-called digital generation; constantly connected to electronic devices and technology, they think and learn in environments that are fast, multimodal, and interactive.

With all this in mind, the traditional divides between the various fields of knowledge and disciplines are starting to crumble in favor of a more project- and module-based teaching, in which the various disciplines integrate and intersect, creating a kind of conceptual map to explain a certain subject.

THE ARCHIPELAGO AND . . .

These are the foundations of the conceptual structure of LUMI. Graphically portrayed by a small archipelago of seven islands, each representing a different area of development, the islands are all connected to each other, as growth is a global process involving various cognitive, emotional, and social dominions, all of which are interconnected.

The products in the LUMI range belong to seven distinct—but interconnected—areas to stimulate integrated and global learning. The contents of the activity books and games have been created based on the specific goals indicated and can be summarized in seven large categories that are transversal to the age range and represented graphically by the seven islands in the archipelago:

FEELING

CREATING

REASONING

... ITS INHABITANTS

You will also find some friendly companions among the animals living on the island who have taken on the role of accompanying our explorers, based on their age group.

0–3 years 3–5 years

Logi Uniq

This hermit crab is a sweet little animal. His strong shell means he can hide away whenever there’s a predator around. Logi is the perfect companion for our youngest adventurers, protecting them as they gradually explore the world around them, before they can eventually shake off their shells and wander off to new horizons, confident in their newfound skills.

Another resident of the LUMI archipelago is the sea turtle. As soon as they can move, these brave creatures head for the sea, never looking back. Uniq is the perfect companion for our young but intrepid explorers as they enthusiastically head off on a voyage of discovery of basic skills, encouraged by their conquests in physical, linguistic, and social skills.

5–7 years 7–9 years

Mega Ido

Our third friendly animal guide is the whale. Since the beginning of time, whales have always fascinated humankind, representing a symbol of humanity itself and of a search for knowledge. Mega’s fellow adventurers are strong and determined, ready to leave dry land behind and sail the seas to the very abysses of knowledge, on a discovery of their own individuality.

And finally, everyone in the LUMI archipelago has an invaluable friend in the seagull. Free, independent, determined, curious, and loyal to the group, Ido the seagull accompanies our more mature adventurers as they leave the safety of the beach to explore the more complex areas of the world of knowledge, transforming themselves into young but skilled leaders.

AN EDUCATIONAL VISION

Mathematics is a rigorous logical thinking tool that allows us to look at the real world, or any imaginary world, in a creative, curious, and fun way. If you’ve never experienced this, now is the time to try it!

You’ll realize that mistakes are a precious opportunity to discover things, that difficulties provide stimulating and compelling challenges, and that there can be many paths to follow, just as there can be many possible results. Above all, you’ll discover how everyone, by their own means and at their own pace, can become an excellent mathematical apprentice

We hope that this book offers children (and parents) the opportunity to have a mathematical experience that encompasses all of the above. “The

only way to learn mathematics is to do mathematics.”

—Paul Richard Halmos (A Hilbert Space Problem Book, 1982)

A FEW TIPS FOR PARENTS

When you check your child’s math exercises, the best way to help and encourage them is to remember that:

With any learning process, it’s important to start with and use the knowledge the child already has on the subject.

It’s quite normal for learning to begin with misunderstandings. You don’t learn things that you already know; you only learn things that you don’t know yet.

Every “mistake” is useful. Misunderstandings reveal the way children are seeing things; they are useful clues to helping us understand the thought process of those who made the “mistake.”

If you think your child has made a mistake, before intervening on the basis of your way of dealing with a problem, you must first try to understand what their reasoning was.

Try to be accepting and inclusive; there are many different ways to participate in an activity, and just as many ways to represent and manipulate mathematical objects. It’s important to embrace them all. Not only does each child have their own way of learning, but learning is also flexible and evolves over time.

Everyone has the right to be given the time they need to do, try, experiment, or write something.

Just because they don’t understand something today doesn’t mean they won’t ever understand it.

Sometimes activities that don’t appear to be mathematical at all are actually very mathematical; mathematics can be applied in many different contexts and can be done in many different ways.

TOPICS COVERED IN THIS BOOK

In math, problem solving is an extremely important transversal skill. Many scholars of mathematics education have focused their research on teaching and learning this subject. One of them was the Hungarian mathematician George Polya, who in 1945 published How to Solve It.

The book presents four steps that, according to Polya, characterize the process needed to solve all problems, both mathematical and nonmathematical.

HOW TO SOLVE IT

STEP 1

Understanding the problem

STEP 2

Devising a plan

STEP 3

Carrying out the plan

STEP 4

Looking back

WHAT’S IN THIS BOOK:

INSTRUCTIONS ON HOW TO USE IT

This book is aimed at children ages 7 to 9. The story is about Adele, Saul, and their dog Babo, who find themselves in Strangeworld, a place where everything is a problem to be worked out and where the brain is always being stimulated.

The story contains various topics, such as classifications, algorithms, and relationships between numbers. It’s divided into six chapters, each of which contains George Polya’s four steps to problem solving:

1. Step one: Understanding the problem. Trying to understand the problem triggers the desire to solve it.

2. Step two: Devising a plan. Coming up with a plan is the key step, and the most difficult, to finding the solution.

3. Step three: Carrying out the plan. Patience, accuracy, and precision play a key role in carrying out the plan to solve the problem.

4. Step four: Looking back. In this step, you need to check everything you’ve done and the final result. Looking back at the process used to solve the problem helps you understand the problem even better, and also gives rise to new questions.

Entering Strangeworld

SAUL, ADELE, and their dog BABO approach the door to STRANGEWORLD , where there are several strange beings waiting in line to enter. “Why are you waiting outside the door?” Saul asks the last in line. “Aren’t we allowed to go in?” “Of course we are,” replies the strange little creature, but we must respect the RULE that’s ENGRAVED ON THE PLAQUE next to the door and wait for the sound of the bell BEFORE GOING IN .”

“THOSE WITH 3 OR LESS LEGS MAY ONLY ENTER 2 AT A TIME, WHILE THOSE WITH MORE THAN 3 CAN ONLY ENTER ONE AT A TIME.”

Pay attention! Don’t count your arms! HOW MANY TURNS will BABO, SAUL , and ADELE have to wait before they can enter Strangeworld?

MAGIC SQUARES, RELATIONSHIPS, AND OPERATIONS

AMONG NUMBERS

The Entrance to the Magic Park

The little monster BULLYBOB has decided to invite BABO , SAUL, AND ADELE to STRANGEWORLD ’s magic park—a magical place filled with numbers, symbols, and mathematical shapes! When the three friends arrive at the entrance gate, they find themselves standing on a square checkerboard on which there are a few numbers. To open the gate, they have to solve a “Sudoku” puzzle.

To complete the Sudoku, the three friends must WRITE NUMBERS RANGING

FROM 1 TO 4 in the empty boxes, respecting the following rules:

–Each of the 4 large squares on the grid must contain the numbers 1, 2, 3, 4.

–Each number may appear only once in each row, column, and square.

Ready for the Grand Tour

The three friends have entered the park. There are three caterpillar trains waiting for them: one for the playground, one for the infirmary, and one for the Grand Tour of the Magic Park. The conductor calls out to the passengers: “Caterpillar train departing for the MAGIC PARK TOUR !”

The three decide to go on the tour and ask him which caterpillar train they should get on. “The one with the MAGIC SQUARE drawn on its side.”

WHAT’S A MAGIC SQUARE?

A magic square is a square containing NUMBERS that in each VERTICAL, HORIZONTAL, AND DIAGONAL ROW ADD UP to the SAME VALUE .

This value is called the MAGIC CONSTANT , and in Strangeworld this is the price of a ticket. Here’s an example of a MAGIC SQUARE

What’s the magic constant in this square? It is 34 .

Help ADELE, SAUL , and BABO catch the correct caterpillar train by working out which of the three squares is a MAGIC SQUARE .

The caterpillar train for the Magic Park tour is number ................... And how much does a ticket cost? ...................

Magic Games

At Station 15, our friends are greeted by ZUC , the Scorpion Wizard: “Pleased to meet you. My name’s ZUC , and my specialty is magic games! Who wants to join me?”

ADELE bravely steps forward. “Think OF A NUMBER BETWEEN 1 AND 20 ,” ZUC tells her, “then write it on this piece of paper, and don’t let me see it for anything in the world!”

WRITE the NUMBER YOU THOUGHT OF in the blue box.

“Now,” continues Zuc, “PERFORM the OPERATIONS I tell you to.” Adele gets ready to write. Use your number and pay attention to the colors.

1. MULTIPLY THE NUMBER you chose BY TWO

2. ADD SEVEN

3. Now, SUBTRACT DOUBLE THE NUMBER YOU THOUGHT OF

4. LASTLY, SUBTRACT TWO.” What

“I bet the number you wrote down is the NUMBER OF BUTTONS ON MY JACKET ,” says ZUC . “Incredible!” cries ADELE . “How did you do that?”

“THANKS TO THE INVERSE OPERATIONS TRICK !”

Let’s take a look at the OPERATIONS I asked you TO DO :

“That’s cheating!” says SAUL . “If you don’t believe me, try it with ANY OTHER NUMBER !

The answer will always be the number. of buttons on my jacket.”

“And it’s all thanks to math!” exclaims ZUC , enthusiastically.

Stamps for the Islands

The three friends land on POSTAL SERVICE ISLAND

From here, all lost items are sent back to their rightful owners on the various islands in the archipelago.

To identify which islands to send the lost items to, they use STAMPS made up of 4 SQUARES in a row, in which the FIRST SQUARE is always YELLOW, AND THE OTHER THREE are always YELLOW, RED, and GREEN. THE ORDER of these LAST 3 SQUARES makes it possible to IDENTIFY THE ISLAND the package must be sent to.

For example, the stamp for the ISLAND OF ODD SOCKS is as follows:

ISLAND OF ODD SOCKS

Whereas the stamp for the ISLAND OF DROPPED COINS is:

ISLAND OF COINSDROPPED

If EACH STAMP CORRESPONDS TO A DIFFERENT ISLAND , how many ISLANDS are there in the Archipelago of Lost Things?

Try and draw all the stamps, and then guess the number of islands in the archipelago.

a) 3

b) 4

c) 6

d) 12

Cutting the Cake

In room 5, the three friends meet THREE TRIPLETS: PLATELICKER, TRUSTY , and GREEDYGUTS

PLATELICKER and GREEDYGUTS are arguing about how to DIVIDE the birthday CAKE , without leaving EVEN A LITTLE PIECE FOR TRUSTY , who’s fast asleep.

“I don’t trust you to divide it fairly! I want to cut the cake,” says GREEDYGUTS .

“Well, I don’t see why you should choose your piece first!” replies PLATELICKER .

“How about WHOEVER CUTS THE CAKE CAN’T CHOOSE THEIR PIECE FIRST ,” suggests ADELE . “That seems like the best plan if you want the cake to be cut into two equal pieces.”

Try DIVIDING THE CAKE so that GREEDYGUTS AND PLATELICKER will have TWO EQUAL PIECES . There are lots of different ways to do it.

TRUSTY suddenly wakes up. “Don’t forget about me!” he exclaims. “THE CAKE HAS TO BE DIVIDED INTO 3 !”

After a long discussion, they find a way to divide the cake that all three agree on. Which of the following do you think divides the cake EQUALLY , SO THAT ALL THREE PIECES ARE THE SAME SIZE ?

Circle all the solutions that seem equal to you IN RED .