'When students learn through visual approaches, mathematics changes for them, and they are given access to deep and new understandings' (Jo Boaler, 2018).

 Cooperative work between mathematics educators and neuroscientists, such as that of Jo Boaler’s  You Cubed project at Stanford University, has generated new teaching strategies that depart from the abstract, formulaic approach to teaching which is common in many high schools and elementary schools throughout Europe and the English-speaking countries. The new approaches rely heavily on visualisation as a teaching strategy for mathematics.

In this session I will report on a mathematics program which used only visual strategies -  but no formulas - to teach mathematics to first year nursing students at an Australian University from 2016-2018. Much of the mathematics needed by the nurses was at upper-primary to lower-high school level, so the visual teaching strategies used would be appropriate for school students as well as beginning university students who need mathematics, such as pharmacy and veterinary science. 

Mathematics topics included place value, multiplication and division, fractions, decimals, percentages, proportion, rates and measurement conversions. One particular visualisation, the 'region model', was used in different ways to support most of this learning. In this paper, I show how the region model was adapted in various ways to convey conceptual understanding of these mathematics topics. Some of these adaptations were taken from the mathematics literature; others were developed by the presenter as she taught the course. Presenting one visual model as primary, and adpating it to different contexts, proved a highly efficient way to teach the above mathematical concepts. The use of one visual model, albeit with adaptations, lowered the cognitive load for students, and also encouraged  them to conceptually link the mathematics topics to each other. This produced a high level of success for students; much higher than had ever been the case for first year nursing students at this university. 

In 2017, results from the 9-hour tutoring program were very good. The lowest achieving students (n-37) scored 52% in a 25-question pre-test, 88% in a post -test after 9 weeks of tutoring for 1 hour per week, and  88% in a second post test that was taken 21 weeks after the pre-test, and 12 weeks after the tutoring stopped, showing they had retained their mathematical understanding 3 montsh after the course ended. 

The program will be repeated in March and April 2018, during which the use of visualisations will be further refined, and student feedback on this teaching/learning approach will be gathered. Accordingly  I will be able to present the 2018 results by the time the conference occurs. 

Visual mathematics teaching strategies such as those used in elementary school (see Reimer, 2005) and middle school, high school and college (Sowell, 1989) encourage higher achievement for students. Youcubed (a Stanford center dedicated to giving research based mathematics resources to teachers and parents) provided visual teaching strategies world wide in 2017. They found that eighty-eight per cent of teachers said they would like more of the activities, and 83% of students reported that the visual activities enhanced their learning of mathematics. The Stanford approach is based on the understanding from neuroscience that the neurobiological basis of mathematics cognition involves complicated and dynamic communication between the brain systems for memory, control and detection and the visual processing regions of the brain (Boaler 2018). A number line representation of number quantity has been shown in cognitive studies to be particularly important for the development of numerical knowledge and a precursor of children’s academic success (Kucian et al., 2011; Hubbard et al., 2005; Schneider et al., 2009). A visualisation called the 'region model' was used at Stanford University recently to teach algebra (see Boaler 2018, page 8) ; this same model was used in the project reported in this paper to teach multiplication and division, fractions, decimals, percentages, proportion and rates to first-year nursing students.

The findings focus on the efficacy of using visualisation to teach multiplication and division, fractions, decimals, percentages, proportion and rates to primary, secondary and beginning tertiary students of mathematics. Using the 'region model' as a visualisation was a very successful approach to teaching these subjects in 2017 and 2016; results from 2018 will be presented, along with an exploration of the ways in which the region model encourages students to inter-link their understandings of these maths topics, thus avoiding the disjointed, formulaic learning of 'traditional' mathematics teaching in Europe and the English-speaking countries.

Boaler, J., Chen, L., Williams, C., Cordero, M., (2018). SEEING AS UNDERSTANDING: The Importance of Visual Mathematics for our Brain and Learning. https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/2017/03/Visual-Math-Paper-vF.pdf. Accessed 31 Jan 2018. Hubbard, E. M., Piazza, M., Pinel, P., & Dehaene, S. (2005). Interactions between number and space in parietal cortex. Nature Reviews Neuroscience,6(6), 435-448. doi:10.1038/nrn1684 Kucian, K., Grond, U., Rotzer, S., Henzi, B., Schönmann, C., Plangger, F., . . . Aster, M. V. (2011). Mental number line training in children with developmental dyscalculia. NeuroImage, 57(3), 782-795. doi:10.1016/j.neuroimage.2011.01.070 Reimer, K. & Moyer, P.S. (2005). Third-Graders Learn About Fractions Using Virtual Manipulatives: A Classroom Study. Journal of Computers in Mathematics and Science Teaching, 24(1), 5-25. Norfolk, VA: Association for the Advancement of Computing in Education (AACE). Retrieved January 31, 2018 from https://www.learntechlib.org/p/18889/. Schneider, M., Grabner, R. H., & Paetsch, J. (2009). Mental number line, number line estimation, and mathematical achievement: Their interrelations in grades 5 and 6. Journal of Educational Psychology, 101(2), 359. Sowell, E. J. (1989). Effects of Manipulative Materials in Mathematics Instruction. Journal for Research in Mathematics Education, 20(5), 498. doi:10.2307/749423