Mathematical and cultural messages from the period between the two world wars : Elin Pelin ’ s story problems

In the mid-thirties of XX c. the renowned Bulgarian writer Elin Pelin published the children’s newspaper Path (in Bulgarian: Пътека) regularly including mathematical story problems. Th eir unusual imagery and plots were quite diff erent from textbook math problems. Th e leading character Old-hand Stanyo loved to tell his neighbours’ children stories in which he intertwined elements of mathematics or logic. Too lengthy sometimes, the story problems revealed various sides of culture to children and taught them to pay attention to detail, think logically, and not to neglect common sense. Whatever life situations were described, the moral was always the same – the knowledge of mathematics helped overcome obstacles. Elin Pelin’s newspaper was circulated all around Bulgaria inspiring children from towns big and small to learn counting, divisibility of numbers, systems of linear equations, binary number system, etc. Th ese topics continue to be in the scope of interests of contemporary students, teachers, and mathematics educators. Story problems of Elin Pelin’s series, exemplary pieces of which are analyzed in the article, attract modern readers with the unique mathematical and cultural values of their time.


Introduction
Mathematics curriculum for Bulgarian primary schools is based on the decimal number system. Th e pupils perform elementary arithmetic operations while solving numerical examples, simple linear equations or inequalities, and story (or word) problems. In mathematical story problems, numerical data and logical relations are interwoven in real 1 gorcheva@math.bas.bg life situations described through a narrative. To deal with them successfully, the primary school children need to acquire comprehensive reading skills. Th ey help them separate what is given from what is sought, recognize mathematical concepts and decide what operations to perform. In essence, these activities are mathematical modeling, which makes solving story problems crucial for building the little students' modern mathematical literacy.
For years the researchers in mathematics education like Pollak (Pollak, 2007), Niss (Niss, 2012), Stillman et al. (Stillman et al., 2013), Spandaw and Zwaneveld (Spandaw and Zwaneveld, 2010) and many others have been emphasizing the importance of teaching and learning of mathematical modeling in school. Th e authors of mathematical textbooks make signifi cant eff orts to prepare the pupils for the challenges of problem solving: aft er each unit they off er numerous story problems, which diff er by the level of diffi culty and allow the teachers to meet their students' needs. Math competitions have had a signifi cant share in raising the students' mathematical literacy as well. At the beginning of each school year the Bulgarian Ministry of Education and Sciences issues an offi cial schedule of all upcoming competitions and olympiads. 2 To train their students, the mathematics teachers organize circles where the children not only solve problems and learn extracurricular mathematical topics, but also make new acquaintances. In such a way mathematics turns into a subject that creates social networks among teachers, students, and parents.
Th e socializing eff ect of mathematical modeling for elementary schoolchildren was noticed and reported by Pollak (Pollak, 2007). Long before him, the Bulgarian man of letters Dimitar Stoyanov known by his pseudonym Elin Pelin also realized that very effect of mathematical problem solving. Although with no professional training in mathematics, he turned the spreading of mathematical literacy through a children's newspaper into his mission.

Writer Elin Pelin and his mathematical storyproblems
According to Rothschild, Elin Pelin is Bulgaria's "most popular interwar author", who wrote "superb children's stories and poems" (Rothschild, 1974: 395). Elin Pelin (in Bulgarian: Елин Пелин, 1877-1949 dreamed to be an artist, but became one of the most renowned Bulgarian writers who painted his images in words. Since the age of 17, for several years he worked as a teacher in his native village where he watched the children play, study and interact with their peers. Th is fi rst hand experience inspired many of the fascinating plots and characters in his later stories. Elin Pelin was also a publisher. In the mid-1930s he issued the children's newspaper Path (in Bulgarian: Пътека) where he regularly included mathematical story problems. Th ey are now collected in a small book (Elin Pelin and Podvarzachov, 1992) published posthumously by his son. Th e main character in these stories is Old-hand Stanyo who asks his neighbors' children to solve problems that test their logical thinking, wit, and attention to detail. Th ey oft en exceed in length and totally differ from ordinary textbook problems. Most of them are fairly easy to solve; others challenge even professional mathematicians and educators. What unites them is their reverence to mathematical literacy, common sense and nuances in speech. Th e stories are evidence of the type of mathematical knowledge the common people of the time regarded helpful to overcome the obstacles of life. I implemented many of them in my seminars with the prospective primary school teachers from the Department of Preschool and Primary school education at Sofi a University "St. Kliment Ohridski".

Elin Pelin's approach to problem formulation
All students I worked with knew a lot about Elin Pelin's literary works, but no one had heard about his story problems. Th e example I used as a warm-up activity was more of a story than a mathematical problem and vividly demonstrated the culture of farmers' life and the nature of everyday problems at the time:

Iordanka G. Gortcheva
Story problem 1. "It was last summer, " Oldhand Stanyo began his story. "I was walking down the Pirin mountain, heading for the plain. On a paddock I saw two horses grazing. Th e stud was tall and black, robust, with a broad saddle and sleek fur. Although his tail was missing, he looked like a pedigree animal. Th e mare was very beautiful too -of a local breed, a little short, immaculately white, you just could not help stroking her, with streamers intertwined to decorate her magnifi cent tail. A really gorgeous animal! A few steps away, three guys were bickering about something, waving their arms… It turned out they were having an argument. Two of them were travelers. Th ey had come to that place at noon in the sweltering heat and left their horses to graze on the paddock while they took a nap in the shadow. Meanwhile the owner of the paddock came from the nearby village, woke them up and demanded that they pay him for the grazing of their horses. Th e travelers agreed. One of them dug into his pocket and produced 10 levs 3 which the villager willingly took. Yet when the other traveler handed him his 10 levs, he refused to accept it. " "It is too little, " he said. "Ten levs for the stud is a fair deal, but not for the mare. For her, if not 20, you should pay me at least 15 levs. " No matter how hard they tried to persuade him, he would not agree. Th erefore they summoned me to settle the dispute.
"Well, and did you, uncle Stanyo?" "I said straight away that the villager was right beyond a shadow of a doubt. Why do you, buddies, think I supported his opinion?" (Problem 4, Elin Pelin and Podvarzachov, 1992: 15-16) Th is small literary masterpiece not only excited the future primary school teachers, but also puzzled them: they were not sure whether it can be classifi ed as a mathematical story problem. Th e following speculations by Novakova (Novakova, 2003) 3 Lev (in Bulgarian: лев) is the name of Bulgarian currency.
shed light on the case, allowing the students to form their own opinions: "In order to have a story problem, there has to be a requirement (a question) that defi nes what is sought. " (Novakova, 2003: 165). In Story problem 1 such a question existed. Th e answer was also provided, which was not quite usual for problem formulation. Elin Pelin did it for a purpose -to boost the logical thinking of his readers. My audience reasoned out loud that the mare had to eat up more grass than the stud since more money had been demanded for her grazing. One of the students noted that in the wild animals do not overeat. Th us the idea that the mare could have been pregnant came up and the prospective teachers, most of whom were female, adopted that explanation. Th eir inference was plausible and Old-hand Stanyo would have probably accepted it. However, his reasoning diff ered: "Th e stud's tail was missing while the mare's tail was long and even had streamers for decoration. Th us the mare was able to easily keep the fl ies away and graze undisturbed, while the stud, continuously being pestered by the insects was able to pasture less. Th erefore, the villager was right to want more money for the mare's grazing. " (Elin Pelin and Podvarzachov, 1992: 80).
Th e students did not expect such a turn of events. Th ey became aware how skillfully Elin Pelin had led them to this conclusion: describing the stud's missing and mare's magnifi cent tail, he was giving them a key to his point of view. Th ey realized that writing his story and formulating the problem, the great writer had in mind both the concept and the solution. Th us the future teachers received a lesson in paying attention to detail and context.
One of Elin Pelin's noble ambitions as a publisher was to bring the culture of the world to the poor children from the smallest villages in Bulgaria. He created a story in which he told them about the enchanting nature of the Far East. But the readers would soon realize that the landscape and animals were just a background: Story problem 2. Once upon a time in China, on the bank of the Great River Yangtze, a boatman named Ha-de lived with his 10-year-old son Pupi. Ha-de built himself a large and strong boat and made his living by carrying bricks to the opposite bank of the river. Th ere the mandarin Ah Ti-Vrag was having a new palace built.
Th e mandarin was very mean. One day he called Ha-de and told him: "Listen to my order! Tomorrow you must tell me how much my favorite elephant Sambo weighs. Otherwise you'll be in trouble. " Th e boatman panicked: he did not have such big scales to weigh an elephant on them. Desperate, Ha-de told his son about this impossible task. Pu-pi just laughed; he was sure that his father could deal with it. And here is what happened: the next day Hade stepped before the mandarin and told him exactly how much his favorite elephant Sambo weighed.
How do you think the boatman and his son managed to fi nd the weight of the elephant? (Problem 1, Elin Pelin and Podvarzachov, 1992: 8-9) Th e situation described was again unexpected for the undergraduate students and for a while they pondered how to approach the problem. Another reading of the text and the practice with the previous problem led them to carefully interpret the information which the expressions "he built a large and string boat" and "he did not have such big scales" carried. Th ey concluded that the boatman had to have some scales, although not big enough to weigh an elephant on them, but pretty good to identify the weight of a brick, for example. Th us they almost reached the solution Elin Pelin had in mind: "Pu-pi, the boatman's son, hauled the boat into the water, lured Sambo inside it and drew a line on the boat's outer side to mark the level of the water. Th en Pu-pi took Sambo back to the bank and began to load the boat with bricks until the water reached the mark; then he weighed one brick on the scales, counted the bricks in the boat and calculated the elephant's weight." (Elin Pelin and Podvarzachov, 1992: 80).
I asked the students to collectively formulate Elin Pelin's problem as it could be written for a modern primary school textbook, and make a comparison with the original version. Th e text unanimously suggested was: Students' version of Story problem 2. A single brick weighs 3 kg. How many kilograms do 1,200 such bricks weigh?
Th e future teachers' comments on their own version were that it was "mathematically precise, doable in a minute, free of emotion and boring", because "no insight is needed for this problem to be solved. " One student expressed an interesting opinion: "Th rough this story problem Elin Pelin sends a message to the children that mathematics is not an isolated subject. Th at is what is missing in our version." Th e literary works which Elin Pelin left document various aspects of common people's life in the period between the two world wars, including their fi nancial hardships. As the text of Story problem 1 reveals, they count every penny and negotiate each bargain. Understanding the importance of interpersonal relations which occur at the markets, the writer uses this environment as a setting for several of his story problems. From Old-hand Stanyo's next story contemporary readers learn that buying chocolate to a child was a rare event: a reward or maybe a bribe for proper behavior. Th e humorous plot aims not only to provide mathematical knowledge to the children, but to instill good manners in them and teach them respect for parents and grandparents: Story problem 3. My grandson Mitko is a very spoilt child. Th is morning my wife and I headed for the market. As loving grandparents do, we took Mitko with us to spend the day together, but he became a pain in the neck: whichever store we stopped by, Mitko always wanted us to buy him a present.
In a grocery store I accidentally saw big scales and asked the seller to check my weight. I was just about to step on the scales, when Mitko grabbed my pants: "Grandpa, I want to be with you on the scales!" "Let me fi rst weigh myself, buddy. Th en I'll weigh you as well. " No matter how hard I tried to persuade my grandson, even by buying him chocolate -he did not agree to get off the scales. I was forced to give in. I allowed Mitko to weigh with me and the scales showed 111 kg. Aft er we got off them, I called my wife to quickly weigh herself while I was entertaining the troublemaker Mitko. But he caught her skirt and did not drop it until he stepped on the scales again with her. Th us both Mitko and grandma's weight together came exactly to 82 kg.
"So you and your wife did not weigh separately, uncle Stanyo?" "No, we didn't. We barely managed to get together on the scales. Th us both grandma and grandpa, without Mitko, weighed 139 kg. Tell me now, how many kilograms each and every one of the three of us weighs. " (Problem 16, Elin Pelin and Podvarzachov, 1992: 45-46) Probably the most straightforward way to approach this problem is to use a system of linear equations, but the topic is not included in Bulgarian primary school mathematics curriculum. Th erefore the undergraduate students were to step into children's shoes.
As Figure 1 shows, the solution process was not smooth for everybody. Pretending to have only a little pupil's mathematical knowledge was tense for the particular prospective teacher: she obtained the same answer twice (Mitko's weight), but unintentionally omitted to point out the third answer required (the grandfather's weight).
Elin Pelin loved to be in line with the latest developments in science and technology. In an unpublished telephone interview from 2006 with Elin Pelin's son -Mr. Boyan Dimitrov, -I learned a lot about the great writer's interest in science and technology. He subscribed to the journal "Science" and took many of his ideas from it; he assembled three radio sets all by himself; he regularly went to the market and as he was a keen observer and an attentive listener, he probably borrowed ideas and characters to later implement in his story problems. In the last narrative discussed here the knowledgeable reader will recognize the characters from the fairy tale "One-eye, Two-eyes, and Th ree-eyes" published in 1812 by Brothers Jacob and Wilhelm Grimm (Grimm, 1909(Grimm, -1914. What makes Elin Pelin's story unique is its mathematical content that cannot be found in the original Brother Grimm's tale.
Story problem 4. Once upon a time there lived an old and cruel witch. She had three daughters. Th e fi rst had only one eye placed in the middle of her forehead; the second had three eyes and the third two eyes, just like other people do. Th e old witch loved the One-Eyed and the Th ree-Eyed daughters, but hated the Two-Eyed one: maybe because she was the most beautiful and her two eyes were just in their right places.
One day the witch called her three girls and gave them apples: 50 wonderful apples to the One-Eyed, 30 wonderful apples to the Th ree-Eyed and only 10 unripe and wrinkled apples to the Two-Eyed.
"Now, girls, go immediately to the market and sell all your apples, " she ordered them. "However, make sure that each of your brings back the same amount of money!" Th e Two-eyed daughter was scared: "O Mom! I cannot earn as much money for my wrinkled 10 apples as my sisters for their pretty 50 and 30 apples. " "Shut up!" chided the old hag. "If you bring even a penny less than your sisters, you will be in trouble!" Th e Two-Eyed girl began to cry: "Th en let me sell my apples at a higher price than them. " Th e old woman scolded again: "Listen! If you dare sell at a price even a penny higher than your sisters' prices, I will punish you. Go!" Th e three girls headed for the market: Th e One-Eyed and the Th ree-Eyed sisters, dressed up and dolled up, ran ahead, laughing. Th eir Two-Eyed sister, wearing her only faded cotton dress, dragged behind, crying miserably. Th us the wretch was left a whole kilometer behind her sisters. Meanwhile, her beloved golden-horned goat caught up with her and turned the ten wrinkled apples into ten irresistibly fresh and juicy ones.
What happened later? Soon the Two-Eyed sister reached the market and stood there next to her sisters. Whatever price they had asked for their apples, the same price she wanted for hers. At the end of the day, although Th e One-Eyed sister sold 50 apples, Th e Th ree-Eyed sold 30 apples, and Th e Two-Eyed one sold only 10 apples, the three girls earned an equal amount of money and brought it back to their mother.
At what price, do you think, the three sisters sold their apples to fulfi l the demands of their witch mother? (Problem 9, Elin Pelin and Podvarzachov, 1992: 29-30) When the prospective primary school teachers read the problem from their handouts, they were puzzled saying it was unsolvable. Several female students suggested reading it again, in parts, changing their voices appropriately. Th e careful analysis of the text revealed that a clue to model the situation mathematically still existed: it was to decipher not only what was told to the readers, but also to guess what was not told. Th e only thing the mother required was that all the sisters demanded equal prices for their apples at the market. For example, they were not forbidden to change the prices. Th us a two-stage sale of a diff erent number of apples could bring the same revenue to the girls.
Elin Pelin provided such a solution which I showed to the audience. We discussed it for a long time because the undergraduate students had difficulty believing it was possible. In the end they realized the problem had a solution (Figure 2).
Th e students' surprise was even bigger when I demonstrated a few more solutions obtained by the method of linear programming (Kelevedjiev and Gortcheva, 2010). Th e students were fascinated by the existence, as well as by the potential of this method. Tables 1 and 2 show two such solutions.
I assigned the undergraduate students to create again their formulation of the problem. Collectively, they produced the following text: Students' version of Story problem 4. A retailer rents a store in three city locations and pays the same rent for each. Today he sells top fashion T-shirts, at the same prices in all three stores. In the fi rst store there are 50 T-shirts to sell, in the second -30, in the third -10. As the rents he pays are equal, he wants to have the same revenue from each store, keeping the prices equal at every moment of the sale. At what prices can the retailer sell the T-shirts in order to ensure the same revenue from the three stores?   Asked to compare the two versions of the problem, the undergraduate students, mostly young ladies with vast shopping experience, found the Tshirt version more realistic. Th ey eagerly discussed: "Even if we hadn't been able to reach such exact solutions, we would have invented at least a strategy to them. "

Discussion of the results
Th e selection of Elin Pelin's story problems which I present herein gives an idea about the writer's interest in mathematics and sciences, but their scope remains uncovered. Fortunately, I managed to introduce the majority of the story problems to the students at my seminars. Th ey all enjoyed his ingenuous ways to imbed concepts like divisibility of numbers, binary number representation, Diophantine equations, graph theory, permutations, or Cartesian coordinate system in fabulous stories about animals, rascals, buckets of water, navy ships… Although not so frequently, ideas from science and technology were also found there, as well as literature related tasks.
Initially I was concerned whether using such old-fashioned story-problems was justifi able in 21 st century teacher education and in its projectionprimary school education. Confi rmation came from the students. Here is how they refl ected on them: Student 1: "I never imagined that Elin Pelin could write math problems for children. " Student 2: "I changed my opinion about men of letters which only at fi rst glance did not understand maths. " Student 3: "From a literary point of view the writer did a perfect job: while Old-hand Stanyo talked, the children listened to him to realize in the end that it was not just a story, but a problem to be solved. As a reader I felt the same way. " Student 4: "If all word problems had been represented in such manner, mathematics would have been my favourite subject at school. " Th e fact that a group of undergraduate students started impulsively reading out loud Story problem 4 because it seemed unsolvable to them can be interpreted as a form of "expressive engagement" (Sipe, 2002): by immersing themselves in the plot the prospective teachers tried to "dominate the text" (Sipe, 2002), i.e. to solve the problem.
I observed the same motivating eff ect on the students of all Elin Pelin's story problems. In my opinion, to great extent it was due to their unique cultural messages which my audience with a background primarily in humanities highly valued.
As reported by Lindgren and Bleicher (2005), many future elementary school teachers fi nd science content courses "boring or diffi cult, with little relevance to their lives or to teaching children" (Lindgren and Bleicher, 2005). Th erefore, situations like the one described in Story problem 2 boost students' interest in science and make them search for more practical problems and connections among school subjects.
My experience shows that Elin Pelin's story problems can be successfully used even as a source for a drama project in primary school (Gortcheva, 2014). For such a purpose, it is not necessary to literally use the texts, but only the way mathematical ideas have been intertwined.

Concluding remarks
Th rough their specifi c contents Elin Pelin's story problems help contemporary readers touch the culture and lifestyle of common Bulgarian people from the period between the two world wars and learn about mathematical knowledge and skills they needed most. Properly balanced, these problems still have place in modern primary school education, as well as in core subject courses for prospective teachers. Th ey challenge even profession-al mathematicians to model the nonstandard situations described. Th e multiple solutions which some problems allow make them appropriate for implementing inquiry based learning in the classrooms.
Elin Pelin's story problems can serve teachers to eff ectively combine the traditional "3Rs" of primary school education: Reading, wRiting and aRithmetic. Th e reading skills developed by such texts are crucial for children's success in problem solving. By focusing both on the context and detail, they help pupils analyze situations, fi nd numerical and logical data in texts, get rid of unnecessary information, and answer questions. To communicate their mathematical knowledge, students should possess strong writing and oral skills. Only then will they be able to persuasively express their opinions and results on paper, in fi les, presentations, blogs, social networks, podcasts... And here come to help the messages of Elin Pelin's story problems to their young readers: to study hard in order to have things to say and write about . др Јорданка Гочева Институт за математику и информатику бугарске Академије наука, Одељење за образовање, Софија, Бугарска