“The dream of a wholly abstract, idealized, disembodied mathematics is simply not achievable; mathematics is a system of human interpretation of the world and has human qualities inextricably woven into its very nature.” (Gerofsky, 2011, p. 14 – quote taken from Radakovic, Jagger & Jao, p. 5)
The article I am reflecting on this week, is Writing and Reading Multiplicity in the Uni-Verse: Engagements with Mathematics through Poetry by Radakovic, Jagger and Jao.
c) Radakovic, Jagger & Zhao: Writing and reading multiplicity in the uni-verse
The article was full of stops for me.
Jagger is reporting on an experience in which she was looking for “ways to engage her anxious teacher education students with mathematics” (p. 2). The authors took a poem by Sakaki called, “A Love Letter” (1996) and Jagger invited her ‘math anxious’ students to create a poem, based off the example of Sakaki’s poem. Sakaki’s poem included concentric circles and representation of scale, with an initial increase of the circles by a factor of 10 and the geometric increase extending to the leap to light years. The first two stanzas of the poem are,
Within a circle of one meter
You sit, pray and sing.
Within a shelter ten meters large
You sleep well, rains sounds like a lullaby.
(It is worth a read, for sure.) In Jagger’s class students were asked to write a poem about their place and connect it to an exploration of place value. The hope was that students would include specific content knowledge. The authors were looking for accurate representation of distance and scale. They were also hoping that students would make real connections between mathematical measurements and their lived experiences.
The authors go on to discuss their interpretation of the student’s poems and resulting considerations of mathematics and poetry. They suggest poetry is a safe way into mathematics.
I have mixed feelings about this. Poems are highly expressive and personal in a way that other genres of writing are not. Perhaps this is linked with the interpretation (and sometime judgement) of poetry. I think it is why poetry has always been a little scary to me. I feel like I am being more vulnerable when I present my thoughts in poetry form. Another scary aspect for me is that I feel like in order to include mathematics into poetry, my understanding needs to be deep so I don’t make mistakes with the mathematics I include in the poem. On the other hand, I enjoyed creating the Fib poems and the PH4 poem this week and did not feel that pressure – and the experience was only slightly scary as I wondered if I ‘did the poems right’ or if ‘I was missing something’.
What I appreciate about this conversation, is the interdisciplinary nature of mathematical poetry. For the young ones, working an understanding of syllables into an understanding of mathematics, as in the Fib poems, is a unique and authentic way to integrate two content areas. In the introduction, Dr. Gerofsky asks the question, “What ideas do you have about ways to integrate literature with mathematics?” I wonder the same thing. The activities we participated in this week is a way. I’d love to hear others’ ideas of different ways to integrate the two.
Another stop for me in the article is the author’s discussion regarding the ideas of Barthes (1977). “Barthes proposes that the reading and subsequent interpretation of any text is a writing of a new text. The interpretive possibilities are infinite and depended on the knowledge, experiences and beliefs of the reader” to construct meaning (p. 4). I wonder how this fits in with mathematical content of a poem. How does mathematics in poetry affect the interpretive process? The genre impacts the reading and interpretive process, weaving specific ideas related to the genre into the meaning- making process of the poem. Is the truth seen in the text, or is it open to a multiplicity of meanings based on interpretation? These are questions considered by Trifonas and Jagger (2015). The authors of this article state, “reading of poetry is influenced by the readers’ personal experiences, understandings, and beliefs… we believe the subjective space and the blurring of ‘author’ and ‘reader’ can make poems more authentic, that is meaningful, relatable, and relevant for students” (p. 4). Perhaps the difference lies in whether we consider the poetry to be ‘mathematical poetry’ or ‘poetic mathematics.’
A question I have is, “can students develop their own conceptual understanding through the exploration of mathematical concepts in the context of poetry?” I think the answer is yes, because I would say that in the process of creating poems this week, my understanding or at least my experiences of Fibonacci deepened. I have also experienced this in my own teaching. I have been experimenting with students using story to show mathematical understanding. By asking students to include mathematics in a story they create, I was able to see conceptual understanding of students; understanding that had been hidden to me when asking students to complete traditional assessment activities. I am aligned with Davis and Renert’s (2014) view of mathematics as described in the article. Mathematics is a “collective, connected, and context dependent enterprise in which the focus is on knowing (something dynamic) rather than knowledge (something static)” (p.6). I resonate with the ideas that mathematics is dynamic…it brings mathematics alive and it becomes playful, rather than static and something to be left on the shelf. When we allow students to play, we are able to access a deeper knowledge of their understanding.
Hi Joy,
ReplyDeleteI would like to comment about one of your stops:
“Barthes proposes that the reading and subsequent interpretation of any text is a writing of a new text. The interpretive possibilities are infinite and depended on the knowledge, experiences and beliefs of the reader” to construct meaning (p. 4).
I like to read the blogs of other classmates who had the same article that I have summarized. The may idea of the article is the same for all the blogs, however, each person has different interpretation of it. Depending on each classmate's interests the focus on the article is different. For this reason I agree with Barthes.
Also, you are right, writing poems is very personal. Even if the topic is the same, each person will write the poem differently.
It's so true that even though we read the same article, the things that stand out for us often differ. That's why I really connect with the idea of writing about our stops... and the idea that a stop can be about something we agree with, struggle with or need to work a little harder to understand. Looking for the stops has helped me to form connections in a different way than I had before. I like the idea that our 'stops' demand our attention and are an invitation for us to engage more deeply with the ideas.
DeleteHi Joy,
ReplyDeleteI found it interesting that the Radakovic et al article was about using poetry as a way to quell teacher candidates’ math anxieties. This is something I am quite interested in and it comes up from time to time in our Lower Mainland Math Contacts meetings – around how teacher candidates are engaging with mathematics, including how some still come out of their B.Eds quite anxious about teaching mathematics. To connect to Lida’s article, it seems the authors were using a humanistic mathematics approach in considering their students as living, breathing, FEELING beings. If the authors are connecting with their students this way around mathematics, hopefully this serves as a model for the future teachers to draw upon in their own teaching.
Two of my stops in your post are connected to one another. I stopped and thought when you wrote, “I feel like in order to include mathematics into poetry, my understanding needs to be deep so I don’t make mistakes with the mathematics I include in the poem.” My connection with this was I felt this as a reader/listener of the Bridges 2017 poetry readings. For example, I listened to (and viewed because she displayed a written version of the poems) Sarah Glaz’s poems Death of Euclid and Archimedes. I enjoyed the content of the poems that was about the mathematicians themselves. However, I wasn’t sure about how to examine the poems mathematically. I’m in awe of the creativity of the poems in the collection as a whole but also intimidated that there is some advanced math hidden in the writing that I won’t understand. Nevertheless, I enjoyed the poems. I’m glad for the two types of poems Susan suggested we try because they were a great low-floor place to start with mathematical poetry with the possibility of stretching ourselves within the structure (high ceiling) if we wanted to keep practicing. This experience made me realize that I don’t need to jump into some fancy, clever application of an advanced mathematical concept to engage with creating mathematical poetry.
This connects to another stop for me in your writing, which was one of your stops in the article – you quoted Radakovic et al, “reading of poetry is influenced by the readers’ personal experiences, understandings, and beliefs.” To me, thinking about level of mathematical understanding and engaging with mathematical poetry, the readers’ mathematical knowledge influences the interpretation of the poetry. So, how I read and enjoy and interpret some of the Bridges poetry will be different than how Maria, who teaches upper level high school math, experiences it which will also be different than how one of the other Bridges poets reads a poem by another author. None of it is wrong and I don’t know that one reader would get more enjoyment out of it than another; we could get equal enjoyment from different experiences.
You wrote about having mixed feelings about Radakovic et al claiming that poetry is a safe way to experience mathematics and about your own feelings about writing poetry as scary because of it requiring, to you, a level of vulnerability. This made me think of our students. I don’t think every experience will be THE experience to connect every student with mathematics in a way that engages them in developing understanding. That is why we offer variety. If a student is reluctant to write a mathematical poem, maybe they are the one that loves creating complicated dance patterns. In this course I have definitely gone deeper into the activities some weeks than other weeks because of curiosity and comfort. Each week I learned something through the experiences but had inclination to dig deeper some weeks. So, if poetry writing isn’t your thing, or a student’s favoured activity, but you still try, you get an experience and learning. I think you discovered this as evidenced in your discussion of writing the Fibs.
Hi Sandra,
DeleteI had a similar experience with reading/listening/experiencing the Bridges Poetry readings. I loved hearing the authors read their own poetry and especially enjoyed Sarah Glaz's poems, as you mentioned. I also was unsure in some of the poems how involved the mathematics was. I wasn't sure if I was missing something or if I was understanding, 'correctly'. I also agree that the types of poetry we wrote this week was an easy entry into mathematical poetry. It is something I would like to try, even with my grade one students. I'm going to be spending some time in a grade one class, so I'd like to try it out, even if we write one together as a class - since we'll also be working out the idea of syllables at the same time. I just love the thought that young students can start their education career, seeing, experiencing and considering mathematics in a different way. Many already come into grade one with pretty well developed ideas of what math is.
Which lead me to your thoughts on variety. This course, is like a smorgasbord of ways of thinking about and incorporating math through the arts and outdoors - non-traditional places. This is one of the things I am wrestling with is how to include this smorgasbord for students in a productive, meaning-making, skill building way. I'm starting to grasp different ways of including these ideas. I am currently in a book club on a book called "Unlearning" by Katie Novak, which is about UDL and allowing students different way to represent their learning is a big part of that conversation. It ties in with assessment as well. How do we open up opportunities for students to actually show us what they know, rather than asking us to show us very specific targeted skills that don't give us a window into their deep conceptual understanding. A statement in the Radakovic et al. article that I highlighted said, "the idea is not for educators to unidirectionally gain insight into students' knowledge, but for students to develop their own conceptual understanding through the exploration of mathematical concepts in the context of poetry" (p. 5). Beside that statement, I wrote, "show what you know." How often do our assessments and activities place a ceiling on students' abilities?
Evening Joy,
ReplyDeleteI think as a secondary math teacher, one of my reservations to using poetry in a math class is how subjective it can be when it comes to marking. Math has always been to me more right or wrong, even when I follow a student's written work, I know where the thought and error in their ideas are. With poetry, at times, I do worry, what if I misinterpret what a student wrote? But if poetry is used as a creative outlet, where it is not being used as a form of marks, and really as a form to journal their thoughts, then I may be comfortable in using that as a method of checking for students' understanding.
I wonder if we need to stray a little from assessment as always being for marks... or rethinking what counts as assessment. I know this is the tricky part especially for secondary teachers (and students, and parents). I am not a part of that world but I sense a shift (and a good one) is coming. Often getting the "right" answer only shows a small part of the understanding. If the students understand 80% of what they are doing, but get the wrong answer, how do we assess that? How can we know what they understand, if it is not all based on a correct answer? Could poetry or other forms of writing play a role? I don't know the answer but they are questions I am asking.
DeleteThank you, Joy, your origami and poetry play encourage a healthy attitude to life. I find the group discussion you initiated helpful with respect to ways in which shifting thinking can support each of us to be hopeful and I feel energised by this.
ReplyDeleteI struggle at times to switch perspectives, especially in my relationship with the Welsh culture, where I’m from, as this has often been incomprehensible and emotionally charged. I’ve found listening to and creating stories and poems that speak to the cries of the earth, the passionate dreams of revolutionary artists, and the age-old restraints of mathematicians really supportive.
Remembering that mathematics helps, as shared in your conversations with Lida, Sandra, and April, could well be the difference that makes a difference!