Professional Development

Sharing the Joy of Math: Making Deep Connections When Teaching Mathematics

  • Sophie Murphy

Before we can get students to love math, we need to find ways to grow their confidence in math and avoid 'math trauma' by moving beyond the surface level to deeper levels of learning. 

In July of 2018, I met Professor Jo Boaler from Stanford University after we both did keynote presentations at NCSM in Washington D.C. On the same day, I interviewed Jo for a podcast that I was co-hosting at the University of Melbourne (Talking Teaching) where she provided an insight into her mathematics research. She spoke about the power mathematics teachers have to empower learners with confidence in math to avoid what she called ‘math trauma’. She detailed the things students described as giving them math trauma’ that has followed them throughout life and into adulthood. Jo said that math lessons in elementary school that timed students on how quickly they could answer math equations were hugely problematic. In addition, these quick response test that relied on working memory alone and not make connections or allowing time to make connections didn't give learners the confidence to believe in themselves and destroyed their self-efficacy in mathematics from very early on. In fact, it had nothing to do with their ability to do math, it relied on how quickly they could respond.  


I could relate to what Jo had described. I grew up thinking that I was no good at math. I was not fast in my responses. I always hoped that the teacher wouldn’t single me out and ask me a math question in front of my peers. I couldn’t remember the formulas and would often give up. Sadly, I think this occurred from very early in my schooling life as she had described. I remember thinking and telling myself (again from very early on) I'm not good at math, but I'm good at reading and writing. I just accepted that. Funnily enough, I loved learning math in some classes, yet in other math classes, I would often talk to my friends for the entire lesson as I just believed that I couldn't do it, so failed to even try.  


My mum would say the same, “Sophie, I'm no good at Math, but I'm good at reading and writing. I can help with those subject areas, just not math.” It affirmed my own math beliefs into always thinking that it was a family thing – I was destined to be no good at math! As I moved into secondary school, I would often ask myself and my teacher, When am I going to use this? or Why do we need to learn this? I can't ever remember getting the answers that I needed to hear. When I was questioning the value of trigonometry or algebra as a teenager, I couldn’t understand the reason that I needed to learn it. Then, when I couldn't remember the formulas and failed the exams, it just felt that it was all too hard.  I never understood the ‘why’ and I was being assessed on the end product, not the process. I always saw mathematics as black and white, with no shades of grey. Just right or wrong. We know that this is certainly not the case and we value the process, as well as the many shades of grey when problem solving.  


Fast forward a few decades to 2013 when I began a M.Ed with Professor John Hattie. We were researching effective classroom discourse from a student perspective. As a quantitative study, we developed a questionnaire, which required statistical analysis. I recall being petrified about doing the analysis and making mistakes. I wanted to be confident about doing it and couldn’t wait to work out the results. However, the math trauma had come back. I watched many YouTube clips on statistical analysis. I would make notes and teach myself. I was so surprised that I could do it, but even more surprised that I enjoyed it.  


Now in 2020, I near the end of my PhD with Professor Hattie and those math skills come into play again. I no longer ask how or why. I can see the connections; I know the relevance and importance of these math skills. I wish I’d known them back in elementary school and believed in my mathematical ability. I wish I’d understood the connections or knew there even were connections to be made, to go deeper than the surface level. 


Over the past few years, at conferences such as NCSM, NCTM, ASSM and CAMT, I have been overwhelmed by seeing the LOVE of math from the thousands of math educators attending these inspiring and significant conferences. I have learned so much from the motivated math lovers and teachers all wanting to find the very best ways to teach math. I would see thousands of math teachers wearing an "I Love Math" t-shirt or a funny math t-shirt and see first-hand an excitement about the math learning that I’d never experienced before. I would hear stories of the joy of patterns from Fibonacci, theories, ratios, zero, and the excitement of different problems that could be solved mathematically. The math went far beyond the facts, definitions, formulas, and basic terminology.  I would hear about math theories with math educators creating dialogue about the day and whatever we were doing, how this related to math.  


My two children have traveled with me and have also been a part of these robust conversations with passionate math educators. I would watch their eyes light up as they were challenged yet excited to be doing such fun problem-solving tasks. I have seen their confidence in math grow as they see the excitement in everyday math and continue to make connections. They ask where the connections are. Much like the video that I shared last week with my children, they made deep connections with Minecraft and mathematics and the many mathematical concepts associated with Minecraft. They love math! 


Math teachers can inspire this excitement and love of mathematics in their lessons, in the classroom and remotely. For teachers that can see their students online regularly, keeping them actively engaged in the learning process is essential. Ensuring that students know why they are learning a math concept and how it connects to the real world or can be transferred from one context to another is crucial. How can we develop a love and confidence in math? From my perspective, we need to find ways to grow confidence in math and avoid math trauma – moving beyond the surface level to deeper levels of learning. 


Last week, Big Ideas author, Laurie Boswell and I discussed what teachers should prioritize as the school year nears closely to the end. I suggested that teachers use a framework to ensure that not all lessons are 'surface' and rely on working—memory, that they provide opportunities for deep level learning. It should be noted that surface level is just as important as deep and to ensure that you are not going too deep too quickly. 


As I suggested in the video, rather than overloading students with many concepts that are taught quickly on a surface level, choosing ideas that provide students with the opportunity to go deep in their learning, moving from surface to deep in a sequential way. The SOLO Taxonomy as a framework to support the planning and instructions of each math concept. I use the SOLO Taxonomy to ensure that an understanding is created with a sequential lesson that moves from surface to deep and then ultimately to transfer. When moving through the learning sequence, using a visible success criterion that includes two surface-level "I can" statements and two deeper-level "I can" statements shows students what this success looks like upfront with visible goals that move from surface to deep.  


Using SOLO Taxonomy framework, teachers can ensure that lessons move past the surface-level lessons that usually have one answer and rely on rote, recitation, and working memory. This structure also provides that there is the right amount of scaffolding required to move into a deeper level of learning that requires connection and understanding. If students move too deep too quickly, they feel confused and lack confidence. This is something, particularly in math, that we want to avoid to ensure that students are in the 'Goldilocks Zone', not too easy and not too hard. Just right.  


The SOLO taxonomy 

 Biggs and Collis developed the SOLO Taxonomy. In this sequence, or cycle, the following stages occur: 


  • Prestructural. There is preliminary preparation, but the task itself is not taught yet.
  • Unistructural. One aspect of a task is picked up or understood serially, and there is no relationship of facts or ideas.
  • Multistructural. Two or more aspects of a task are picked up or understood serially but are not interrelated.
  • Relational. Several aspects are integrated so that the whole has a coherent structure and meaning is made and connections are made.
  • Extended abstract. That coherent whole is generalised to a higher level of abstraction and concepts can be transferred to new or different concepts. 


You can find examples of learning intentions and success criteria that I have written, using the SOLO Taxonomy at the front of the Big Ideas Math textbooks. If you would like to know more about using the SOLO Taxonomy, I will be talking about it in my next video and am happy to share more examples about using SOLO in the coming weeks. 


If you have a topic that you are interested in and would like to hear more about, please share with me on Twitter @sophmurphy23 


Keep be amazing and inspiring your students and all those around you with a love and passion for math. And, most importantly, stay safe. 


Kind regards,  


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