​​​(October 24)​

Many educators recognize the importance of teaching for creativity, yet few successfully integrate it into their daily teaching practices (Beghetto, 2010). However, creativity and problem-solving are vital components of teaching to meet academic needs of gifted learners (Maker & Schiever, 2010). Some teachers attribute this problem to the lack of dedicated time in the curriculum, while others point to the binding nature of standards-based teaching. Acknowledging these factors, researchers also point to another fact, which is that teachers are not quite sure what exactly creativity is, how they can teach and foster it without overwhelming themself (Beghetto et al., 2015). Regardless of the source of the problem, teachers need a systematic understanding of what mathematical creativity looks like in the classroom to promote creative problem solving (Bahar et al., 2021).

Unfortunately, creativity has been largely underestimated in most mathematics curriculum as well (Bahar & Maker, 2011). According to Mann “Teaching mathematics without providing for creativity denies all students, especially exceptionally talented students, the opportunity to appreciate the beauty of mathematics and fails to provide the gifted student an opportunity to fully develop his or her talents.” (p. 236). Regarding fostering creativity, teachers play an essential role; when the teacher’s will in fostering creativity is high, the motivation of students to generate creative outcomes will also be more likely. In a similar vein, when creative behaviors are supported and valued, the creative achievement of students will be higher (Egbert, 2017). Specifically having students engage with relevant tasks, which can tap creative thinking skills, is one of the essential ways for fostering mathematical creativity in the classroom (Bahar & Maker, 2015). In this paper, I will discuss three research-based strategies that were found to support teachers design promising activities/problems to foster creativity in mathematics classrooms: (a) Designing Open-ended Tasks, (b) Modeling, and (c) Problem Posing.

Designing Open-ended Tasks

Much of academic content in mathematics curricula could be identified as an aspect of a problem-solving process (Bahar & Maker, 2015). Researchers (Maker & Schiever, 2010). suggest that a problem can fall into one of the three types: (a) closed, (b) semi-open, and (c) open-ended. Dunbar (1998) defines an open-ended problem (ill-defined) as one in which the solver does not know the operators, the goal, or even the current state, whereas all these attributes are known to the solver during solving a closed-ended problem. Similarly, Bahar and Maker (2015) suggested that problems were classified as either closed or open based on the number of alternatives available to the problem solver. For example, a problem was defined as closed ended if it could be solved in only one way and open-ended if it could be solved in a multiple (sometimes infinite) number of ways. 

Creativity is applicable to most situations and always relevant to ill-defined or open-ended problems (Bahar & Maker, 2011); however, most of the problems and tasks in our math textbooks or curricula have been in a closed-ended form. To me, trying to foster students’ mathematical creativity through teaching closed ended problems is like eating soup with a fork. Of course, learning how to solve closed-ended problems through use of basic facts is valuable and much needed practice but not enough to nourish creative thinking skills. Having said that, it is a good practice for teachers to know to make a closed-ended problem (or a task) an open-ended one. Let’s have an example: 

Example 1: Determine the unknown number that makes the equation true in each of the equations 8 + ? = 11. (This problem is taken from Grade 1 U.S. Common Core Standards (Grade 1 » Operations & Algebraic Thinking: Work with addition and subtraction equations - CCSS.MATH.CONTENT.1.OA.D.8, which can be reached through http://www.corestandards.org/Math/Content/1/OA/)

As seen in the example, this problem is a closed-ended problem because it has only one correct answer. We can easily convert this problem into a semi-open or open form. For example, “Use these numbers to write as many correct equations as possible (8, 11, 3).” If you place a limit on the amount and use of operations (e.g., use either addition or subtraction only once), the problem could have four basic solutions for a first grader [8+3=11, 3+8=11, 11- 3=8, 11-8=3], qualifying this problem as a semi-open problem. If you do not limit use of any operations at any amount, this problem could even qualify for an open-ended problem and can have an unlimited number of solutions. In this format the problem begins to address students in higher grade levels too. For example, (3 + ((8 - 3!)3)) can be a solution to the problem from a high schooler. Discussing possible solutions in the classroom will allow students to see the possibility of using different operations and number orders to arrive at the maximum quantity of answers in novel ways. By doing so teachers can encourage students to look for original solution methods.

Example 2: Find the volume of a rectangular prism with dimensions 1/2 cm x 3/4 cm x 8/3 cm. 

This problem is taken from Grade 6 Common Core Standards (Grade 6 » Geometry: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. (CCSS.MATH.CONTENT.6.G.A.2): Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. 

This is another good example for a closed-ended problem from the sixth-grade curriculum. Let’s rewrite this problem in an open-ended form so that it allows students to show their creativity. For example: “Draw as many rectangular prisms as possible with a volume of 100 cm3” or “Draw a unique (nobody else in the classroom can think of) rectangular prism with a volume of 100 cm3”. These open-ended problems can encourage students to find original solutions by using unique methods. 

Tasks, activities, and problems are influential for learning. Specifically in mathematics, “benefitting from a model that encourages use of closed, semi-open, and open tasks/problems in a variety of combinations can empower teachers to make conscious decisions about fostering creativity while promoting rigor” (Bahar et al., 2021; Bahar et al., 2024). Although the predesigned textbooks and very structured curricula are sometimes major barriers for teachers for teaching for creativity, using practical strategies, teachers can continue to cultivate creative minds.

Modelling

Modeling is an effective way of cultivating creativity in mathematics classrooms. It is best defined as a process in which students investigate and apply mathematics to make sense of a problem arising in everyday life. Modeling entails posing a mathematical inquiry about a real-world situation and the development of new methods to address these problems using sound mathematical knowledge (Niss & Blum, 2020). Through modeling, students find opportunities to make assumptions about the problem and analyze their outcomes, and finally implement strategies to see if they work to solve the problem (Wickstrom & Aytes, 2018). 

In fact, formulating and representing such models, and analyzing them is appropriately a creative process, which depends on creativity and expertise (Standards for Mathematical Practice, 2021). Therefore, using them in mathematics classrooms can tap learners’ creativity and foster their creative thinking skills (Chamberlin & Moon, 2005; Lu & Kaiser, 2022). According to CCSSM, different modeling tasks and problems can be provided for students across grade levels. “In early grades, this might be as simple as writing an additional equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another.” (Standards for Mathematical Practice, 2021, Model with mathematics section)

Example: 393 / 25 = ?

This problem is taken from Grade 4 Common Core Standards - Grade 4 » Operations & Algebraic Thinking (CCSS.Math.Content.4.OA.A.3: Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.) 

This simple math equation can be converted into a modeling problem such as: 

We have picked 393 apples from our orchard today. A basket can hold 25 apples. How many baskets are needed to bring all the apples to the grocery store? What are alternate methods to bring the apples to the store?

Math modelling is more than a task, it is a process that involves representations, analyses, predictions, and insights (Neubert, 2020). For more information and guidelines, teachers can visit the Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME), which was drafted by the Society for Industrial and Applied Mathematics (SIAM) or visit MathWorks Math Modeling (M3) Challenge webpage. 

Problem Posing

The act of identifying and formulating a problem, which is separate from and possibly more significant than problem solving, is a major part of creative thinking and creative performance in various domains (Jay & Perkins, 1997). 

Example: Let’s refer to the same math problem we used in the ‘Designing Open-ended Tasks’ section; we do have a rectangular prism with dimensions 1/2 cm x 3/4 cm x 8/3 cm. The teacher can ask students: “What math questions can we ask about this rectangular prism?”. Initially, students will most probably come up with similar questions such as:

• What is volume of the rectangular prism?

• What is surface area of the rectangular prism?

When students are encouraged, it is not surprising to see students pose very original problems such as:

• How many cubes (with a side length of 1/12 cm) can get into this rectangular prism?

• What is the maximum length of a stick that can get into this rectangular prism?

Given that problem posing process requires seeking out a problem, transforming of a given situation into a new version, and finally creation of a new problem (Silver, 1994), it also naturally taps students’ creative thinking skills. When teachers use this strategy continuously in their daily teaching activities, posing/creating a problem turns out to be a habit/skill of mind for learners in the classroom. 

Although problem posing is a great strategy to foster mathematical creativity and problem-solving skills, it is never an easy task for teachers as well as students. Problem posing needs more than a simple modification of an existing problem or situation; becoming a good problem poser requires structured coaching and guidance. Teachers will notice shortly how their students will develop into better problem posers when the students instructed systematically for problem posing. As students advance in their skills in problem posing, their problems will shift from random variations to those with a distinct mathematical goal (Making Mathematics, 2002). 

Author Notes

Credits given to Dr. Bahar’s writings in his column ‘Imathination’ published in Teaching High Potential (THP), which is National Association for Gifted Children (NAGC)'s quarterly magazine written for educators of high-ability learners.

References

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