Creativity is traditionally supposed to attribute to art and literature, but nowadays doing meaningful science has also been considered as a creative act. Liljedahl and Sriraman (2006) proposed that at the school level mathematical creativity can be defined as:

– The process that results in unusual (novel) and/or insightful solution(s) to a given problem or analogous problems, and/or
– The formulation of new questions and/or possibilities that allow an old problem to be regarded from a new angle

Examples of manifestations of mathematical creativity in the process of solving word problems:

Mathematical creative problem solving	Mathematical creativity
Mathematical creative problem solving	Creative abilities	Openness	Independence
Understanding mathematical problems	The ability to define the mathematical problem illustrated in the task from multiple perspectives	Tolerance to information that is incomplete, poorly defined, or polysemous	Constructing one’s own internal language, where mathematical concepts indispensable for solving the problem are set out and explained
Understanding mathematical problems	The ability to clearly visualize the situation presented in the task as well as vividly capture dependency relationships across data	Recognition of the potential value resulting from becoming acquainted with ways other than one’s own of perceiving and describing the mathematical problem illustrated in the task at hand	Separating the meanings of mathematical concepts from the meanings of everyday language
Generating possible solutions	The ability to formulate multiple and frequently atypical hypotheses referring to the possible solutions to the mathematical problem illustrated in the task at hand	Cognitive curiosity that results in readiness to become acquainted with possible ways of solving the problem	Courage in questioning commonly accepted rules and principles in order to find new and/or atypical ways of solving the mathematical problem
Generating possible solutions	The ability to create original images that render it possible to break away from typical solutions to the mathematical problem and use analogies in order to find new ones	Ease in analyzing new information and ways of solving the problematic situation presented in the task at hand	Autonomy and perseverance in searching for possible solutions to the problematic situation
Planning for action	Flexibility in applying various strategies of solving the problem	Openness to the verification of all possible solutions to the problem	Strong belief in the success of the undertaken activities aimed at solving the problem
Planning for action	The ability to transform images of possible solutions to the problematic situation illustrated in the task at hand	The acceptance of variability in applying the various problem-solving strategies	The ability to critically assess attempts – one’s own and other people’s –to solve the problem

Most math-traumatized adults (like myself) have a hard time believing how creative and inventive mathematics can be. Creative mathematics doesn’t always make it into the classroom for elementary school-aged children. Currently, elementary school math often focus on basic math “skills” and “facts”. However, creative mathematics allow children to come up with math problems that they encounter in day to day life and attempt to solve the problems creatively. This type of math education comes more naturally and is more functional. Children are naturally mathematically curious, as many mathematical problems just seem like they are part of a game. The creative and curious problem can stick in a child’s mind long after the lights are turned out for the night.

References

Leikin, R. & Sriraman, B. (2017). Creativity and giftedness: Interdisciplinary perspectives from mathematics and beyond. New York: Springer. doi:10.1007/978-3-319-38840-3

Liljedahl, P., & Sriraman, B. (2006). Musings on mathematical creativity. For The Learning of Mathematics, 26(1), 17-19.

Nadjafikhah, M., Yaftian, N., & Bakhshalizadeh, S. (2012). Mathematical creativity: Some definitions and characteristics. Procedia – Social and Behavioral Sciences, 31, 285-291. doi:10.1016/j.sbspro.2011.12.056

March 28, 2019March 29, 2019

Ideational Code Switching for Mathematic Creativity!

The earlier post, “Ideational Code Switching” described the three- and four- levels of creativity proposed by Beghetto (2007), and Beghetto & Kaufmann (2012). Based on the book Teaching creatively and teaching creativity (Gregerson, Kaufman, & Snyder, 2013), Mini-c creativity and be facilitated to little-c creative expressions.

Mathematics is an abstract subject. In school we often learn about math from Big-C, higher level concepts of math creativity that are conceptualized by math geniuses like Einstein or Leibniz. In order to discover mathematics in everyday activities, I wonder if we can teach students to code-switch using Big-C concepts and applying such to little-C everyday mathematic creativities? In essence, can we facilitate creative code-switching in reverse?

What are some ways we can facilitate ideational code-switching from the abstract to the concrete, day-to day mathematics?

Let’s consider an example.

Kai and Nahla are grade 3 students learning about basic probabilities. They both learned that Christiaan Huygens likely published the first book on probability in 1657. In these instances, it may be argued both Kai and Nahla are learning Big-C concepts and applying such to little-C, everyday activities. How can we facilitate ideational code-switching for Kai and Nahla?

Using the concepts of basic probability, Kai camp up with the idea of calculating the likelihood that his baby sibling would either be a boy or girl. On the other hand, Nahla came up with the idea that she could calculate how likely it is she would get the joker from a deck of cards.

Big-c Genius Creativity	Little-c Everyday Creativity
Christiaan Huygens’s idea of basic probability	Rosie used the insight gained from basic probability to calculate the chances she could draw an ace from a half deck of cards
Picasso’s Cubism painting style	Sam using Cubism idea to construct a mathematical model that calculates the cube’s position in three of Picasso’s paintings
Merce Cunningham’s Contemporary Dance style	Jacob used to concept of dance to illustrate math concepts, like the Math Dance

What are your thoughts on facilitating ideational code-switching in reverse? What do you foresee may be the barriers and benefits of such? Please comment below and let me know what you think!

Reference

Gregerson, M. B., Kaufman, J. C., & Snyder, H. (2013). Teaching creatively and teaching creativity (2012; ed.). New York, NY: Springer.

Image source: https://www.pinterest.ca/pin/343399540311137112/

March 27, 2019April 3, 2019

Strategies to Support Creativity in Math Classes

So what things can we do as teachers to offer opportunities for creativity and how can we embed these opportunities into our everyday practice? Piggott (2007) mentioned 3 principles to teach creative mathematics:

how we present content
how we model good practice
how we encourage our students to be creative

I would argue that the above 3 principles involve ideational code switching from big-C math concepts to little-C math ideas!

Presenting content

One concern raised by many teachers, is the need to cover the content requirements of the national curriculum and exam board specifications. However, by developing problem-solving skills and using problems to explore aspects of mathematics, learners can feel empowered to “think for themselves” and, as a result, become more confident when tackling standard questions. The interactive “Tilted Squares”, published in September 2004, is based on the ability to create tilted squares on a coordinate grid and to use this to investigate the area of squares with different tilts as shown in the diagram.

Big-C concepts	Little-c ideas
Pythagora’s Theorem	– Calculating length of slide in playground

Some suggested questions include:

What areas are possible?
What areas are impossible?
Why?
What observations, thoughts and conclusions can you offer?

Such environments enable students to explore and work from their own level of understanding, building on this towards new understandings. For example, in the case of Tilted Squares, students have worked at a range of levels:

some have made progress in understanding that squares do not have to be constructed with sides parallel to the edges of the paper they are drawn on;
some have begun to identify relationships between the amount of tilt and the areas of squares;
others have been able to generalize and offer a justification of Pythagoras’ Theorem for right-angled triangles with two short sides of integer length.

Modelling

How often do math teachers model problem solving in the mathematics classroom? How can we share with students the fact that we can also struggle with mathematics and that this is the “normal” state of affairs when meeting something new? What is important is that at any point of being stuck we acknowledge that we are stuck, and share our thought processes as we start putting our creative juices to work (Mason & Burton, 1982). At these points we should not be afraid to experiment and try ideas out – this is a common strategy we can all use. Perhaps is a good way to try this out is to walk into the classroom with a problem we have found in an old text book or mathematical activity book, such as books by Martin Gardner (Gardner, 1965), and say “Let’s look at this together” -and then spend time thinking out loud. This may push teachers out of their comfort zones, but we need to show the students that this is a fairly normal state of affairs by sharing such an experience with them from time to time.

Questioning and encouraging students to think for themselves and share their understandings are important aspects of any curriculum and a focus on problem solving and posing offers a way forward. So, what are the key features of a problem-solving curriculum? One where students and teachers:

engage in problem solving and problem posing;
have access to experimental opportunities (environments) to explore which have the potential to lead to particular mathematical ideas;
are actively applying math concepts to real life (identifying the mathematics in situations);
make connections with other mathematical experiences;
engage in and examine other people’s mathematics;
are not constrained by the content of the previous lessons but supported by them;
value individuality and multiple outcomes;
value creative representation of findings.

Examples

Rebecca Goulding (n.d.) offers some math activities to inspire inventive and creative thinking. Below I present the activities and adapted them for use in the classroom.

Arranging Utensils for Multiple Possibilities

Big-C concepts	Little-c ideas
Multiple possibilities	– Arranging cutlery into different combinations – Figuring out a “secret password” from a set of letters and numbers given

Ask the students if they know how to count. “Of course,” they will respond. But the real question we want to ask is: “what is it that we are counting?”

For example, if a student is given a fork, a knife, and a spoon, it’s only three objects. How many ways can they arrange the items in a row? Here’s where it gets interesting:

Savvy table-setters might say “Six ways”: fork, knife, spoon; knife, spoon, fork; spoon, fork, knife; spoon, knife, fork; fork, spoon, knife; and knife, fork, spoon.
What if you allow the possibility of flipping the utensils upside down, so the handles face away from your body? The answer is then 48.
And if you can also flip the utensils over, so they face either way? With all four possible orientations of each utensil, the answer is 196.
Should the student be able to figure this out entirely, go ahead and add a salad fork to increase the complexity of the problem. Or, generalize to n different utensils.
On the other hand, if it’s a little too challenging, try to solve the puzzles with only a fork and a spoon, saving the knife for another time.

Cooking and the Commutative Property

Big-C concepts	Little-c ideas
Commutative Property	– Cooking procedure – Baking muffin grids – Ice cube trays

To get more abstract, the commutative property is at play in cooking! The commutative property describes how an operation (such as addition or multiplication) is applied to numbers. The commutative property tells us that 5 x 3 = 3 x 5, and 2 + 7 = 7 + 2. In other words, the order in which the numbers appear doesn’t change the result of the operation. In contrast, subtraction is not commutative, because 5 – 3 is not the same as 3 – 5. If the concept is taught in school, it’s usually introduced in around fourth or fifth grade, but even little kids can understand it in the context of the operations of making pasta.

Would your pasta sauce be the same if you added oregano and then basil, compared to adding basil and then oregano? (For the most part, sure!)
Would it be the same to boil the water, then put the pasta in, compared to putting the pasta in and then boiling the water? (Definitely not!)
But why is every young mathematical thinker so sure that 5 + 3 is the same as 3 + 5? When children first learn about multiplication, they often find it surprising that 3 x 5 (3 copies of 5) matches 5 x 3 (5 copies of 3). Why is that?
All it takes is laying out a 3 x 5 grid of pieces of pasta to see it. Try turning the table and see that it’s also a 5 x 3 grid!

Most important, have fun with mathematics. Let your students invent crazy ideas that don’t make sense, think about questions that don’t seem so mathematical, and grapple with “basic” mathematical ideas that might seem obvious to you. Because if math is fun, then your student may actually want to think about math all the time!

References

Boaler, J. (1997). Experiencing School Mathematics . Buckingham, Open University Press.
Gardner, M. (1965). Mathematical Puzzles and Diversions , Penguin.
Goulding, R. (n.d.). Creative Play with Math. Retrieved from http://www.pbs.org/parents/education/math/math-tips-for-parents/creative-play-math/
Mason, J., L. Burton, et al. (1982). Thinking Mathematically , Prentice Hall.
Olkin, I. and A. Schoenfeld, H. (1994). A Discussion of Bruce Reznick’s Chapter [Some Thoughts on Writing for the Putnam]. Mathematical Thinking and Problem Solving Schoenfeld, A, H. Hillside NJ, Lawrence Erlbaum: 39-51.
Piggott, J. S. and E. M. Pumfrey (2005). Mathematics Trails – Generalising , CUP.
Watson, A. and J. Mason (1998). Questions and Prompts for Mathematical Thinking , Association of Teachers of Mathematics.

March 25, 2019March 29, 2019

Resources

Here is a list of resources to encourage mathematics creativity!

Videos

https://www.youtube.com/watch?v=JQgFly5-QlA

Two years ago, when Ivan Zelich was a 17-year-old school student, he co-developed a theorem that took the global scientific community by storm. He believes that the way that maths is taught in school needs to adapt and change – that we need to think of it as a creative journey and not simply a list of formulas to memorise. At age 17, Ivan Zelich co-developed a groundbreaking mathematical theorem that works faster than a computer and has applications in better understanding geometric structures. The Liang-Zelich Theorem paved the possibility for anyone to deal with the complexity of isopivotal cubics having only high-school level knowledge of mathematics. A paper on the theorem was published in the peer-reviewed, International Journal of Geometry, making Zelich and his collaborator Xuming Liang, the youngest contributors ever to the journal. Aside from his passion for numbers, Ivan is a swimming state champion, speaks six languages, and has represented Queensland in chess. He is currently undertaking his fast-tracked, undergraduate degree at University of Queensland.

Ted Talk by Dan Meyer. Today’s math curriculum is teaching students to expect — and excel at — paint-by-numbers classwork, robbing kids of a skill more important than solving problems: formulating them. Dan Meyer shows classroom-tested math exercises that prompt students to stop and think.

Books

Creativity and giftedness: Interdisciplinary perspectives from mathematics and beyond

This book provides readers with a broad view on the variety of issues related to the educational research and practices in the field of Creativity in Mathematics and Mathematical Giftedness. The book explores (a) the relationship between creativity and giftedness; (b) empirical work with high ability (or gifted) students in the classroom and its implications for teaching mathematics; (c) interdisciplinary work which views creativity as a complex phenomena that cannot be understood from within the borders of disciplines, i.e., to present research and theorists from disciplines such as neuroscience and complexity theory; and (d) findings from psychology that pertain the creatively gifted students.

Leikin, R., Sriraman, B., & SpringerLink ebooks – Education. (2016;2017;). Creativity and giftedness: Interdisciplinary perspectives from mathematics and beyond. New York: Springer. doi:10.1007/978-3-319-38840-3

Mathematical creativity and mathematical giftedness: Enhancing creative capacities in mathematically promising students.

This book discusses the relationships between mathematical creativity and mathematical giftedness. It gathers the results of a literature review comprising all papers addressing mathematical creativity and giftedness presented at the International Congress on Mathematical Education (ICME) conferences since 2000. How can mathematical creativity contribute to children’s balanced development? What are the characteristics of mathematical giftedness in early ages? What about these characteristics at university level? What teaching strategies can enhance creative learning? How can young children’s mathematical promise be preserved and cultivated, preparing them for a variety of professions? These are some of the questions addressed by this book.

Singer, F. M., & SpringerLink ebooks – Education. (2018). Mathematical creativity and mathematical giftedness: Enhancing creative capacities in mathematically promising students. New York: Springer. doi:10.1007/978-3-319-73156-8

First & Second Grade Math Activities

These activities are appropriate for ages 6 to 9. Browse to find the first or second grade math activities that work best for your student.

Reflections on Teaching for Mathematical Creativity

Math Teachers reflect on how they can incorporate creative activities when teaching math.

Cultivating Creativity in Math Class

Creativity in the mathematics classroom is not just about what students do but also what we do as teachers. If we are thinking creatively about the mathematical experiences we offer our students we can open up opportunities for them to be creative. In this article, Jennifer Piggott shares some of her thoughts on creative teaching, and how it can encourage creative learners.

Journal Articles

Mathematics Creativity in Elementary Teacher Training

Creativity plays an important role in mathematics learning, so teachers must provide students with appropriate learning opportunities. This means using tasks, in particular those with multiple solutions and/or multiple resolutions, that usually require creative thinking and it could be a possible way to promote creativity in students. In this paper, we identify some traits of creativity in elementary pre-service teachers through tasks productions used during math classes.

Category: Foster Math Creativity

What is Mathematical Creativity? Why is it important?