Is Problem Posing a Tool for Identifying and Developing Mathematical Creativity?

Abstract

The mathematical creativity of fourth to sixth graders, high achievers in mathematics, is studied in relation to their problem-posing abilities. The study reveals that in problem-posing situations, mathematically high achievers develop cognitive frames that make them cautious in changing the parameters of their posed problems, even when they make interesting generalizations. These students display a kind of cognitive flexibility that seems mathematically specialized, which emerges from gradual and controlled changes in cognitive framing. More precisely, in a problem-posing context, students’ mathematical creativity manifests itself through a process of abstraction-generalization based on small, incremental changes of parameters, in order to achieve synthesis and simplification. This approach results from a tension between the students’ tendency to maintain a built-in cognitive frame, and the possibility to overcome it, which is constrained by their need to devise mathematical problems that are coherent and consistent.

Appendix

Problem 1 (posed by Diana, grade 4): On the planet Zingo live several types of aliens: with two or three eyes, with two or three ears, and with five or six hands. They are green or red. How many aliens should shake hands with Mimo to be sure that he shook hands with at least two of the same type?

Problem 2 (posed by Emilia, grade 4): In the Infinite king’s castle there are 43 corridors with 18 rooms each. Each room has 52 windows. At every window, there are three princesses. How many princesses are in the Infinite king’s castle?

Problem 3 (posed by Malina, grade 4): In Princess Rose’s jewelry boxes there are sapphires, emeralds, and rubies. 27 are not rubies, 31 are not emeralds, and 32 are not sapphires. In total, there are 45 jewels. How many jewels of each kind does Princess Rose have?

Problem 4 (posed by Paul, grade 4): If a group of students sits by two at their desks, seven students remain standing, and if they are placed by three at the same desks, seven desks remain free. How many students and how many desks are there in the classroom?

Problem 5 (posed by Sabina, grade 4): In the Fairies’ Glade, live 60 unicorns and fairies. They have 160 legs in total. How many beings of each kind are there?

Problem 6 (posed by Sergiu, grade 4): One day, the plane leaves Cluj [a city in Romania] to go to Japan. It departs at sharp hour in the morning, when the hour and the minute hands of a clock form a right angle. The hour hand points to a number bigger than 4. This plane travels at 60 km per hour, and the distance Cluj-Japan is 540 km. In the plane climbed three times more men than women, who are 24,484 people. The cost for a man’s tickets is the first odd number greater than 7. Women’s tickets cost as double of 3 added with 4 and the result divided by 2. (A) When did the plane leave Cluj? (B) When did the plane arrive in Japan? (C) How many men boarded the plane? (D) How many women boarded the plane? How much money did the pilot receive, if he received all the money, without 3,000 of total?

Problem 7 (posed by Tudor, grade 4): On a farm, there are two cows, some geese and horses, a total of 86 heads and 328 feet. How many horses are there at the farm?

Problem 8 (posed by Victor, grade 4): In the world of letters, each letter represents a number. M is two times greater than N, and the difference between these two letters is equal to A. A is less than B by seven, and the sum of A and B is neither bigger nor smaller than X. If we add two to X, we get Y. The sum of X and Y equals Z. Knowing that Z − (A: O + P: P + Q: Q + A: R) = 30, find M × N.

Problem 9 (posed by Alin, grade 5): (A poem!) If one places three cakes in each box/There’ll be three cakes left/If one places five cakes in each box/There’ll be an empty box left. (…) How many boxes and how many cakes/Do I put on the shelves?

Problem 10 (posed by Cosma, grade 5): Two boys need £67 to buy a game. The price of the game decreases by 50%. If the first boy pays three times more than the second does, how much money should each pay?

Problem 11 (posed by Andrei, grade 6): On the planet Uranus in the T316B2 city, there are less than 101 and more than 49 aliens. 1/2 of them are red, 2/7 are green, 1/14 are yellow, and the rest are blue. How many aliens live in E943S4 city, the capital of the planet, if their number is 149 times greater than the count of blue aliens from T316B2?

Problem 12 (posed by Cosmin, grade 6): P⋅R⋅I⋅C⋅E⋅P⋅I⋅P⋅R⋅O⋅B⋅L⋅E⋅M⋅A = x. Knowing that different letters represent different digits, find the last digit of the number x. (He multiplies the letters meaning YOU UNDERSTAND THE PROBLEM.)

Problem 13 (posed by Cristiana, grade 6): Maria has to solve the following problem: “4, 14, 1114, 3114, 132114, 1113122114, … What is the next term of the sequence?” The mathematics teacher gave her some advice: “You must empty your mind of all other mathematical information.” Can you help Maria to solve the problem?

Problem 14 (posed by Cristiana, grade 6): Maya the puppy has six bones. She wants to make four equilateral triangles out of these six bones, but she forgot one essential rule: one has to think out of the box. Can you help her?

Problem 15 (posed by Maria, grade 6): A number is “special” if it can be written as both a sum of two consecutive integers and a sum of three consecutive integers. Prove that: (a) 2,001 is special, and 3,001 is not special, (b) the product of two special numbers is special, (c) if the product of two numbers is special, then at least one of them is special.

Problem 16 (posed by Mihai, grade 6): Because the sixth-grade students were the best, they received a prize consisting in 1 h free on paintball field. The field has the dimensions 80 m × 120 m, and two people are able (and allowed) to shoot one another if they are at no more than 29 m distance. Prove that howsoever 26 students place themselves on the ground, at least 3 get shot.

Problem 17 (posed by Nandor, grade 6): Dan has a 24 hour display digital clock that is broken: the first digit of the hours’ counter and the last digit of the minutes’ one get switched every 5 hours. Example: if switch occurs at 17:42, the clock will show 24:71. The clock continues to run correctly after that and stops at 99:99, when it gives an error (1 hour is transformed in 100 minutes). If the clock breaks when the correct time of the day is 10:10, what will be the time before giving the error?

Problem 18 (posed by Radu, grade 6): Prove that any parallelogram can be divided into 16,384 congruent parallelograms.

Problem 19 (posed by Teofil, grade 6): In 2011, 300 students went on the field trip. Knowing that the percentage of girls was 45%, find the number of boys who participated.

Problem 20 (posed by Vlad, grade 6): A new quarter was built near a forest. The residents put their garbage in waste containers with a capacity of 750 kg each. At every 10 kg of garbage throw away by the residents, 4 kg disappear, being consumed by bears leaving in the forest. The residents produce 20 kg of garbage per hour. (a) Find out how long it take to fill a waste container; (b) Knowing that in the neighborhood live 500 families that fill 86 containers per month, that each family should pay 7.8 euros garbage fee, but only 400 families are fair and pay, calculate how much money is collected as garbage fees in a month.