The idea of using

The idea of using

**mathematical story picture books (MSPBs)**to enrich mathematics learning is not a new idea. In fact, it has been around for almost three decades, particularly in the early years setting. What is less common is using MSPBs to enrich mathematics learning beyond the early years level. I have been arguing - and will continue to argue - that the approach could also benefit mathematics learning of older pupils. Specifically, I would argue that the use of MSPBs could:*foster pupils’ conceptual understanding through multi-representation of mathematical concepts, variation of mathematical situations and the use of**common misconceptions as a teaching point**; develop language skills; and foster engagement with mathematics learning*.

**Foster conceptual understanding through multi-representation**

We can all (hopefully) agree that we do not teach mathematics so that our pupils become a human calculator, that is someone who is good at churning out correct mathematical answers but without conceptually understanding the concept behind it.

As part of one of my research projects, when Jack (pseudonym), a 9-year-old pupil, was asked by me what 20 ÷ 5 equals to, he was able to give me the correct answer (4) almost instantly. Then, when he was asked to (contextually) represent 20 ÷ 5 using a word problem, this is what he came up with:

*“Spanish Yoda had a can of Coke and a bag of bananas and apples and paint. How much did it cost her? Coke: £1.00. Bag of bananas: £2.00. Apples: £8.00. Paint: £9.00. Total £20.00”.*How Jack’s word problem is related to 20 ÷ 5 remains a mystery.

What Jack demonstrates here is a classic example of pupils whose procedural fluency (i.e. the mechanic aspect of mathematical learning) in relation to division is good, but have yet to fully grasp what the concept means conceptually.

As many mathematics education scholars have argued, in order to demonstrate conceptual understanding in mathematics, pupils must be able to represent mathematical concepts in different ways using different representations (e.g. contextualisation, visualisation, etc.). Here, I would argue that key features of MSPBs, such as narrative and page illustrations, make learning mathematics conceptually effective as pupils get to learn mathematical concepts through these different representations.

Take

**‘Divide and Ride’**

**(Murphy, 1997), for example. This is a story about a group of eleven friends who want to go on carnival rides. Some of these rides have two-people seats, others have three- and four-people seats. As these seats have to be filled up before each ride can begin, the children constantly have to work out how to group themselves. Due to 11 being a prime number, there is always, at least, one person being left out (a remainder), and additional children are consequently invited to join their group to fill up the seats for each ride. Through the storyline, children can visually see how division works and what a remainder means in real life. This helps children to contextualize the concept. Additionally, not only do the illustrations depict division through images of children filling up the seats, they also include a mathematical model at the bottom of each page to represent the divisional situation in a different way as well as corresponding numerals to help children connect visual representation with symbolic representation.**

Theoretically speaking, the more children are able to make meaningful connections between different representations of mathematical concepts, the more conceptual understanding they are demonstrating. Thus, effective mathematical story picture books carefully look at how these different representations can be combined seamlessly throughout the story.

**Foster conceptual understanding through variation**

Another key strength of teaching mathematics using MSPBs is the development of pupils’ conceptual understanding in mathematics through what I refer to as

*the variation of mathematical situations*that are often found in well-written MSPBs. To explain this concept, take

**‘Bean Thirteen’**(McElligott, 2007) as an example. The story follows two crickets, Ralph and Flora, who have collected twelve beans to bring home for dinner. When Flora decides to pick one more bean (i.e. Bean Thirteen), Ralph is convinced it will bring bad luck. No matter how many friends they invite to try to share the 13 beans equally, it is always impossible.

*Situation 1:*13 beans to be shared between 2 crickets (Ralph and Flora) resulting in 1 remaining bean (6 beans each)

*Situation 2:*13 beans to be shared between 3 crickets (Ralph, Flora and 1 friend) resulting in 1 remaining bean (4 beans each)

*Situation 3:*13 beans to be shared between 4 crickets (Ralph, Flora and 2 friends) resulting in 1 remaining bean (3 beans each)

*Situation 4:*13 beans to be shared between 5 crickets (Ralph, Flora and 3 friends) resulting in 3 remaining beans (2 beans each)

*Situation 5:*13 beans to be shared between 6 crickets (Ralph, Flora and 4 friends) resulting in 1 remaining bean (2 beans each)

In this example, while the number of crickets varies, the number of beans is invariant (kept the same). Through this variation of mathematical situations, rich mathematical investigations are made possible. Pupils can be asked, for example, to continue the pattern to prove that 13 cannot be divided evenly by any other numbers except for 13 itself (and hence demonstrating the meaning of prime numbers in the process). I argue that such variation of mathematical situations is crucial to foster pupils’ conceptual understanding in mathematics.

Another example of variation of mathematical situations can be found in

**‘Fractions in Disguise’**(Einhorn, 2014). This story focuses on the concept of equivalent fractions and it is about how George Cornelius Factor (who happens to share the same acronym, GCF, with – wait for it – the greatest common factor!) invents a machine, called ‘Reducer’ to help him find a very sought-after fraction (5/9) that has been stolen from a fraction auction, and has been disguised as another fraction by the villainous Dr. Brok. While at Dr. Brok’s mansion, GCF uses his Reducer machine to reveal the true form of a range of fractions (e.g. 3/21 is really 1/7; 34/63 is already in its true form; 8/10 is really 4/5, and so on) before he comes across 35/63 which is later revealed as the 5/9 fraction he has been looking for.

Through such variation of mathematical situations, both ‘Bean Thirteen’ and ‘Fractions in Disguise’ make it possible for their readers to take their time to digest the new mathematical concept they are learning by providing them with several mathematical situations or examples to show them what is and what is not prime numbers and equivalent fractions, for example. The goal is that once they have seen enough examples of what is and what is not prime numbers and equivalent fractions, they will arrive at their own definition (and hence understanding) of these concepts themselves. Authors of well-written mathematical stories think carefully about what kind of variation their story needs that could help

*scaffold*students’ learning of a mathematical concept in question.

**Foster deeper understanding using**

**common misconceptions as a teaching point**

Effective mathematical stories incorporate readers’ common misconceptions about a particular mathematical topic in the stories as a teaching point. A good example is

**‘Sir Cumference and the Fracton Faire’**(Neuschwander, 2017) which follows Sir Cumference and his wife, Lady Di of Ameter, to a local Fracton Faire where local goods are sold and where different shopkeepers show how numerators and denominators can be useful for customers to indicate how much of each product they want to buy (e.g. one-fourth of a roll of fabric, four-eights of a cheese wheel). The story addresses a common misconception that the bigger the denominators, the larger the parts. Specifically, in the story, Sir Cumference is surprised to learn that four-eights of a cheese wheel that he wants is the same size as two-fourths of the same cheese wheel that Lady Di has chosen. Authors of effective mathematical stories do research and consult with experienced mathematics educators to identify such common mathematical misconceptions and weave them in their story.

**Develop language skills**

From my earlier research (Trakulphadetkrai, Courtney, Clenton, Treffers-Daller, & Tsakalaki, 2017) and those of others, it has been found that children’ mathematical abilities are linked to their language abilities. What is exciting is how recent research (e.g. Hassinger-Das, Jordan, & Dyson, 2015; Purpura, Napoli, Wehrspann, & Gold, 2017) have also found the positive impact of using stories when teaching mathematics concepts to young children on the development of their language abilities particularly their vocabulary knowledge. Why not kill two birds with one stone? Why not teach mathematics using MSPBs to develop both pupils’ mathematical and language development at the same time?

**Engagement through emotional investment**

Another key advantage of teaching mathematics using MSPBs is that pupils arguably do not see MSPBs in the same way that they see, for example, mathematics textbooks or worksheets with word problems after word problems to be solved. They are more likely to view MSPBs as something that they can be emotionally invested in, and something that they can enjoy interacting with over and over again either together with the whole class or in their own time at their own pace. Research (e.g. McAndrew, Morris, & Fennell, 2017) has recently found that the use of stories in mathematics teaching can help to foster children’s positive attitude towards the subject.

**Final words**

Teaching mathematics using MSPBs should not only be found in Nursery and Reception classes. This creative, cross-curricular and research-informed mathematics teaching and learning approach should too be utilised by teachers teaching at the primary school level and beyond.

If you want to explore our 500+ recommendations for MSPBs that can be used to cover 40+ mathematical concepts, please head to our Recommendations

**here**, and if you want to see examples of how MSPBs can be integrated as part of your mathematics teaching, see some of our examples

**here**.

**References**

Einhorn, E. (2014).

*Fractions in Disguise*. Watertown, MA: Charlesbridge.

Hassinger-Das, B., Jordan, N. C., & Dyson, N. (2015). Reading stories to learn math: Mathematics vocabulary instruction for children with early numeracy difficulties.

*Elementary School Journal, 116*(2), 242–264.

McAndrew, E. M., Morris, W. L., & Fennell, F. (2017). Geometry-related children’s literature improves the geometry achievement and attitudes of second-grade students.

*School Science and Mathematics, 117*(1-2), 34-51.

McElligott, M. (2007).

*Bean Thirteen.*New York, NY: Penguin's Putnam Publishing Group.

Murphy, S. J. (1997).

*Divide and Ride*. New York, NY: HarperCollins.

Purpura, D. J., Napoli, A. R., Wehrspann, E. A., & Gold, Z. S. (2017). Causal connections between mathematical language and mathematical knowledge: A dialogic reading Intervention.

*Journal of Research on Educational Effectiveness, 10*(1), 116-137.

Trakulphadetkrai, N. V., Courtney, L., Clenton, J., Treffers-Daller, J., & Tsakalaki, A. (2017). The contribution of general language ability, reading comprehension and working memory to mathematics achievement among children with English as additional language (EAL): An exploratory study.

*International Journal of Bilingual Education and Bilingualism*. DOI: https://doi.org/10.1080/13670050.2017.1373742

About the authorDr. Natthapoj Vincent Trakulphadetkrai is a Lecturer in Primary Mathematics Education at the University of Reading’s Institute of Education (UK), and founder of MathsThroughStories.org. His website is Natthapoj.org, and he tweets at @NatthapojVinceT. |