Maths is Visual

In this reflection I will discuss Lesson 3 in my learning journey on mathematical mindsets from Jo Boaler at Stanford University. Find my reflections from lesson 2 here. 

In Lesson 3, Jo introduced the idea that our brains want to think visually about maths. Building students' mathematical understanding doesn't just mean strengthening one area of the brain involved with abstract numbers. It means strengthening connections between areas of the brain and the visual pathways.

Visual Representation and The Brain

Jo Boaler, Stanford University

As children get older - between the ages of 8 and 14 - they develop the ventral visual pathway (shown in orange). The brain becomes more sensitive and specialized in representing visual number forms. This showed an important and increased interaction between the two visual pathways - VTOC/pFG and IPS/SPL. This indicates that the brain becomes more interactive as children learn and develop, connecting the visual processing of symbolic number forms, such as the number 10, with visuospatial knowledge.

Throughout this lesson Jo shared many examples of how we can use visuals in math. She discussed finger perception and the importance of finger visualisation, such as counting with our fingers and visualising our fingers when they are hidden. She focused on a common misconception many teachers have, which is to move on from finger counting and discredit its value. Jo shared an article and a Tedx that gives more detail about the science behind using our fingers in math. 

The Painted Cube

A task that was posed to us

Jo Boaler, Stanford University Imagine that we paint a 4 x 4 x 4 cube blue on every side. How many of the small cubes have 3 blue faces? How many have 2 blue faces? How many have 1 blue face? How many have not been painted at all?

My attempt By taking the 4 center cubes from each face we would have 24 cubes with only 1 blue face (4 cubes x 6 faces). Then we would take the side cubes from each face (2 cubes x 12 edges) giving us 24 cubes with 2 blue faces. This leaves us with the 8 vertices having 3 blue faces painted and lastly 8 cubes from the center of the cube that have no faces painted blue. If I am correct this should equal 64 because 4³ is 64 so 24+24+8+8=64. I think I am right, thoughts?

Student Instruction

Jo shared with us this same painted cube task as she had delivered it to students. First, she outlined the parameters and expectations for how she presented the lesson to the students. From there we watched the students work on the task. They had sugar cubes in front of them and coloured the cubes. There were clear frustrations and confrontations throughout the task. The students tried to prove their point through mathematical thinking. Students learned from each other, a few aha moments were provoked by their peers.

Jo Boaler, Stanford University

From the painted cube activity Jo and her team collated the student responses in a table. They gathered information like, how many single side blue faces, how many cubes with 2 blue faces etc. Jo then asked a question, when going from a 4x4x4 cube to a 5x5x5 what do you notice? She then sat in silence for a while and let it linger, which is something I am trying to do in my own practice. Students came up with a range of conjectures and theorised the validity of the statement. Jo used visualisations, added on to students' ideas and left their thinking as a question. I really liked how the students took the time to build on their own ideas and tackled their own misconceptions in the process. Jo really emphasizes the importance of visuals - they are not merely a tool to get to more abstract ideas. Something she said really stuck with me - "Thomas West, who's written a really interesting book called Thinking Like Einstein, says that it's masochism for a mathematician to do without pictures". Definitely on my must read list.

Considerations for Teachers

Jo asked us to consider an area we are currently teaching. I am in the middle of a unit on algebra, we have recently been looking at factors and multiples. I did a bit of searching and found this cool blog post from Anthony Persico and MashUp Math. Jo reiterated that visual thinking is really important. Visual ideas can inspire students and teachers to see mathematics differently, to see the creativity and beauty in maths, and to understand mathematical ideas.

So what can you do to help all students, including visual creative thinkers, be valued instead of just the memorizers and calculators? What changes would help this? Personally I think the problems we choose to pose to students area big way to help students visualise ideas. Using resources such as Youcubed and NZmaths will really support this approach by using open-ended rich tasks for students to explore.

My Takeaways

Awesome lesson on visualisation. I really liked the cube activity, it showed the concept that Jo was trying to portray clearly. I enjoyed the ‘realness’ of her class. However, watching her class did raise some questions for me. When we allow students to explore their own learning and misconceptions, how are we ensuring all students are engaged? In her group discussions I noticed dominant personalities taking over. In the class discussion there were students exploring ideas and learning as they went, many were engaged and learning from each other - but was the whole group? Overall I think this is a great approach but something I am aware of for my own practice. 

Jo also brought up this point -

I know that many of you are stuck with low-quality curriculum that districts have chosen for you that may show drawings, but really don't engage students in visualizing or drawing or building or modelling. We can't wait for publishers or districts to actually read research. Students need us to offer this high quality work now.

Great statement, I think we definitely need strong voices in all parts of education, ministries, school districts and boards, administrators within their schools and teachers. However throughout my career I have seen a lot of teachers doing directives they don’t believe in, why? A multitude of reasons, my thoughts though is that it is hard to ask a teacher as an individual to stand up against their administrations and further beyond. Maybe we need a new structure for change?

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