Do You “See” Math?

Are you one of those people who can “see” math? Or are you a person who self-identifies as “not a math person”?

I’m an optimist. I actually believe that everyone can do math. But I do wonder sometimes, what is it that sets apart those of us who “get it” from those who struggle to see what is happening? In my classroom I spend a lot of time asking questions of my students, to understand what math they do see, so that I can help them to connect that understanding to new ideas.

this student had strong visual skills, creating a proof without words (or algebra)

this student had strong visual skills, creating a proof without words (or algebra)

In the last year as I have moved toward more inquiry-based math learning in my high school classroom, I have more insight than ever before into how my students think in the subject. And I have realized that we all see so differently. I have students who are incredibly strong with visual patterns but weak in algebra, and other students who can follow all the algebraic rules but can’t visually identify a 90° angle. I have noticed that some students struggle to see the 3D shapes in 2D drawings of solids, and I have wondered what I can do to help all of them.

I have read the research on the importance of block play in early childhood, and its effects on performance in secondary mathematics (this is so fascinating!). It’s one reason why the club members of GirlTech are engaged as leaders in our kindergarten: they lead block play with our youngest students.

Last month I had our school librarian order a copy of a recent publication: Spatial Reasoning in the Early Years. This book has been the best reading I have done as a math educator. What follows are the notes I took while reading: mostly thoughts, quotes, connections, and potential links to my classroom.

(@ddoucet, @aianovskaia, @timrollwagen, @gnichols, @lmcbeth, @danielleganley please read on, there is something for everyone here!)

Ch 1: Spatial reasoning is learned, it is malleable, and even if your kids didn’t play with blocks, there is hope for them to still be successful in math.

Ch 2: This chapter provided a breakdown of types of spatial reasoning. I found this helpful, as I often can identify a deficit but am not sure how to categorize it. Also, it connected to this article that I came across on twitter that weekend. Also, my mind exploded a bit when I realized the connection between spatial reasoning and multilingualism. From this Economist article in 2015, I knew that multilingual children perform better on these feats of spatial reasoning. That these skills transcend STEM subjects might be obvious, but this connection made me giddy. Language teachers: we build the same skills.

<note: at this point in reading, I wish I had bought the book myself, as I wanted to highlight and make notes in the margins. I have 26 stickies instead.>

Ch 3: This chapter focused on developing spatial thinking, and I wondered if this is what it means to “see” mathematics. From p.35:

“The strong link between spatial reasoning and mathematics raises the possibility that improving children’s spatial skills might serve as a way to strengthen mathematics learning.”

students building objects in order to observe patterns

students building objects in order to observe patterns

I was reminded of the brilliant work being done by the team at Westfield State University, on Discovering the Art of Mathematics. If you are a math nerd and haven’t seen their materials, go there now. I visited last November, and was impressed with the quality of math thinking in their undergraduate classes. The use of manipulatives was constant. Why don’t we use more manipulatives in high school?

Also in this chapter, I discovered more studies on the effects of block play. I loved the results from one study showing that free block play was good, guided block play was better, and narrative-driven block play was most effective. Narrative-driven. Did you hear that, English teachers? We need each other.

Intervention programs would be possible, given the malleability of the skills involved in spatial reasoning. This chapter inspired me to want to start a club/program for grade 9-10 students focused on spatial reasoning instead of more algebra/calculation/rule-following.

Ch 4: Here I read the history of math education. I was happy to discover that the transition from elementary to secondary mathematics has never been smooth. A quote on page 54 described the very real struggle of socializing as a math teacher:

“For we math educators, this discontinuity between elementary and secondary is often encountered as the cocktail-party confession, ‘I was good at math until grade 6.’ Such remarks, for us, are not unlike the statement, ‘I was a confident swimmer until my feet stopped touching the bottom.'”

Page 60 argues that math is not equal to calculation. AGREED. I think calculation is to math as spelling is to Literature.

Ch 5: Whoa. This chapter was deep, on Theory of Embodiment. I was grateful for my

teams work on logic challenges as they move around the game board

teams work on logic challenges as they move around the game board

liberal arts education with the philosophy requirements. I had to dig deep to keep up with this chapter. It described the body of the individual, the body of the group, and the body of knowledge, and I particularly liked the sections describing how concepts and movement are connected (I was already using drama games to warm up students’ spatial awareness!). Dan Meyer recently pointed me to this stellar resource on dance and mathematics. (PE Teachers – you know all about spatial awareness already. Let’s work together!) I also recently adapted Think Fun’s CodeMaster game as a full-body group game for a coding workshop. It was fun!

A little something for my classroom came from p. 76 “… in re-cognizing the class as a potential collective body engaged in mathematics, we can ask questions about the class, and how it works as a body in and of itself during the lesson with respect to spatial reasoning.” What that means for me: class discussions are important in learning, but the classroom atmosphere that makes the class a body is equally important.

Ch 6: This chapter was on early elementary tasks. I was happy to discover that complex 3D geometry tasks are as appropriate for young elementary children as they are for teenagers; we don’t usually associate young minds with this level of sophistication in thinking.

Also, p 96 stated, “Research shows that people who perform well on tasks of mental rotation also tend to perform well across a range of mathematical tasks.” I don’t want to confuse correlation with causation here, but I do think that it would be worth a try to improve the spatial skills of secondary students as a step in remediation.

Ch 7: DRAWING! This chapter had my mind racing. I have so many notes.

  1. Drawing of shapes & 3D images is not based in cognitive development, it is a learned skill based on convention & personal experience.
  2. Awareness of shape properties are supported by purposeful attention to drawing & visual representation (Art department, are you still reading?)
  3. I would like to encourage more drawing of spaces and motion in grades 6-12.
  4. Drawings help connect the experience of the world to symbolizing the world (p. 120). Art department, let’s talk!
  5. I have done some visible thinking tasks with my students. Weak math students often struggle with these tasks, as they don’t “see” how to draw a solution. From the examples in this chapter, I think I should adapt the tasks in the future, and first do a physical activity with the students that they have to draw afterward. Then in subsequent tasks, I can have them move straight from the problem to the drawing.

Ch 8: 2D to 3D. This is something I see my students struggle with and I have been trying to help them through the use of manipulatives like Zometool. This chapter is the first time I have seen acknowledgement that the 2D representation of a 3D object is assumed to be obvious, and the connection is glossed over in math books. It deserves more attention in class. The chapter also stresses that this type of representation is all cultural convention and should be explicitly taught. I would like lesson/activity ideas around this.

Ch 9: The final chapter is a discussion on revamping curriculum to include more spatial reasoning. There is a call for more research (sign me up!!). Seriously, how do I get involved?

Finally, a lovely quote from a 1979 study on p 148,

“Spatial reasoning plays a key role in narrative comprehension and memory across every domain of human engagement.”

That’s a strong closing argument.

I give the Spatial Reasoning Study Group a standing ovation for this impressive book!! I was aware of the connection between math understanding and spatial reasoning skills, and I find myself reassured by the fact that these skills are malleable: we can help students develop them. Along the way, we tie in English, Foreign Languages, Drama, Art, Music, all the Sciences, and certainly the Humanities. This book was such a joy for me to read, as it provided details of fascinating research supporting much of what I believe about math learning. More importantly, it gave a new dimension to the interdisciplinary nature of the subject. Math is often sold in school as being so widely “applicable”, as if students will often calculate the trajectory of a projectile in motion on the baseball field, or solve simultaneous equations at the grocery store. The focus on spatial reasoning makes math interdisciplinary on a much deeper level, and these connections are worth pursuing, for our students and for ourselves.

2 Replies to “Do You “See” Math?”

  1. Fascinating stuff. I was reading an article (http://www.math.cmu.edu/~wn0g/noll/mnb.pdf) recently that reminded me of the essential difference between arithmetic and mathematics. It is prudent to remember that distinct types of reasoning are reliant on learnings from diverse subjects. As you hint at throughout your notes, we need each other!

  2. Hi Ruth,

    This is a bit literal but I think that you might enjoy the read as an educator: Have you seen
    “Born on a Blue Day” about Daniel Tammet who sees numbers as shapes and colors? He is able to perform calculations by putting the shapes together and has recited tens of thousands of digits of Pi based on their shape and color.

    I’d love to hear what you think about it: https://www.amazon.com/Born-Blue-Day-Extraordinary-Autistic/dp/1416549013

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