Article Summary and Stops!
This week I read the article titled “Seeing the graph vs. being the graph” by Susan Gerofsky (2011). In this article, the author describes the first two years of a study conducted around the use of gestures within graphing, and their relation to one’s overall understanding of the concept. Historically, gestures have been studied alongside oral language as a form of communication when observing the process of concept formation. The math concepts that are typically represented in those studies include patterns and relationships , size, space, quantity, and time. This study differs in that it focuses on the gestures used to describe graphs as a whole, and that the gestures were elicited through the use of pictures.
*STOP - This part of the article caused me to stop and wonder what other areas of math could be included in a study about gestures and communication. I started trying to formulate a list of math topics not listed above by going through the different concepts that I teach in grade 8 & 9. It struck me how many of the topics that are typically included in these types of studies are measurement based, or easily depicted with visuals. Are gestures helpful in learning, communicating and understanding math concepts that are not inherently visual, spatial or measurable? What would those topics be?
Participants (students) were asked to describe the graph in each picture using gestures and non-mathematical words. Their responses were coded and categorized for the types of gestures/movements that were used and the descriptive language or metaphors that were used. These categorizations were then compared with how the students performed in math class over the course of the year. It was found that the students who used whole body movements and rich metaphors to describe what they saw in the graphs, were consistently identified as “top” students within their class. In other words they were “being the graph”.
*STOP - In the video, Roger Antonsen states that the use of metaphors, imagination and storytelling indicates true understanding of a concept. This seems to support Gerofsky’s (2011) finding that the “top” students were able to make use of multiple rich metaphors to describe what they saw in the graph. This made me stop and wonder if there was a cross-curricular opportunity here between mathematics and humanities courses? For example, both teachers could work on storytelling, descriptive and figurative language, and creativity within their classes as a way of helping students make connections between the 2 courses.
Those who used small, controlled gestures that often involved eye level tracing of the line shape were consistently rated as “average” students. They “see the graph” but haven’t yet achieved a deep understanding of the graph as a whole. Finally, the third group struggled to describe the graph as a whole, and often used inaccurate language or gestures to represent it. These students (except 1) were consistently identified as “struggling learners” and were unaware of the features of graphs and other concepts such as symmetry. The author concludes that “being the graph in a fully embodied way fosters engagement and attentiveness far more than merely seeing the graph” (p.254)
Final Thoughts/Questions - I wonder why the “struggling learners” are not able to see the graph? Is this a learning disability? Do they have poor attendance and therefore major gaps? Do they also struggle in other classes such as English that require the use of similar skills? How do we support these learners in a different way?
Response to Introduction, Video, and Activity
First of all, let me just say that I feel very vindicated with this week’s activity! I am well known within my house for using anything but a measuring tape when buying furniture, hanging pictures, gardening etc. My husband is often exasperated with my insistence that I don’t need to use a measuring tape, and he’s convinced that when I do use a measuring tape, I usually end up with the wrong measurements. He very much subscribes to the idea that math equals precision and accuracy, and that there is one right way to do everything. That using my body to measure is childish and somehow lesser than his way of measuring. His way of thinking is reflective of the current methodologies often found in modern math classes, in which only young children should need to rely on such “simple” strategies.
My daughter Hannah (age 7) volunteered to help me with this week’s activity. She’s had some prior experience with using her body to measure objects at school, but had not taken the extra step of using a ruler to measure herself first. Below is a picture of the measurements that she recorded!
Initially, we were planning on measuring my car and the garage (it has oddly sized doors as opposed to the standard garage door), as a means of finally answering the debate on whether I could be parking in the garage or not. However, the -40⁰C wind chill forced us to stay inside and bake cookies instead. So she decided that she would measure the size of a cookie using finger widths, which you can see in the picture below. She determined that the perfect cookie size was 4 finger widths or approximately 2 inches. Anything smaller was too crisp, and anything bigger was too soft. Perhaps not practical to most people, but in the world of a 7 year old who’s learning how to bake independently, it was a pretty big deal to get the right sized cookies.
Overall she had a lot of fun and really enjoyed having creative control of how and what she measured. As a child with ADHD, she also found that being able to move while learning was helpful in sustaining her focus for a longer period of time. Ultimately, she was very proud of herself for solving the perfect cookie problem!
Hey team! I'm looking for feedback on the placement of my "stops". Inserting them into the article summary felt natural to me, but I can also see that it might be distracting for others. I'm curious to see how others approach the structure of their discussion posts!
ReplyDeleteThanks for your post, Courtney. I appreciate how you summarised the article and as well the placement of "stops". It made the reading of your blog post very smooth.
ReplyDeleteYour stop 1 made me wonder and think about the various topics of Math. I went through my list of topics that I cover in Math 8 & 9. Going through the list, it made me realise that all these topics including algebra and trigonometry, banking, graphing, fractions- they can all be represented in a visual form. For example - in algebra when we are learning about polynomials and combining like terms- a simple gesture of just pointing or making a v-shaped around two like terms helps students see which two terms to combine. And the same applies to any other topic. Gestures are a universal means of communication and can help increase the understanding of any given topic.
Secondly, you brought up a very important part about collaborations between Math and humanities. That's actually the area I am exploring in my own teaching practice about the possibilities of collaborating with the humanities department especially in terms of teaching social justice topics in my teaching of Mathematics.
What a beautiful example you have shared of baking cookies and having your daughter measure cookie width with her fingers. I can so much relate to this as I learnt from my mother to use a handful of rice or lentil when I am cooking. My husband on the other hand likes to be precise and use measuring cups to make 1 cup of rice. I just feel that it saves me so much time if I am able to measure something using my body rather than looking for measuring tapes and cups. This type of measurement helps in increasing our estimation ability too.
Hi Kanwaljit,
DeleteI often collaborate with my humanities teaching partner! I'd love to find more social justice connections between the 2 disciplines, but we're not quite there yet. Things that have worked really well so far are: using stories (both reading and writing) to explain math concepts, breaking down the vocabulary of math into prefixes/root words etc., and exploring the statistics within social studies textbooks.
I think putting your 'stops' in the article summary instead of following it was a good idea. I chose to do the same thing - like you mention it just felt more natural.
ReplyDeleteI too wonder why the struggling learners weren't able to 'be the graph' while the top students could. I'm curious about what the graphs in the study looked like exactly - were they smooth and symmetrical like a parabola, or more jagged and chaotic like a stock price over time?
I've noticed that being able to see a shape/curve/graph on paper and then also visualize it in your mind is pretty essential for success in mathematics, and I suppose if you can't visualize it you then can't take the next step to act it out in gestures. When teaching surface area and volume of compound shapes (tetris block type shapes but 3D) I've noticed some students REALLY struggle with tasks like matching nets to 3D shapes, or visualizing what the back view of the shape would be if it's printed on paper and not an actual model/manipulative. It makes me wonder what education research says about how visualization and gestures are related... if only there was some way to ask the author of the study about her thoughts on this:)
And as someone who enjoys a good cookie, I think your daughter's made some excellent observations about the relationship between cookie size and texture! Some people would turn to using a digital scale or a melon baller type tool to solve this problem, but using a body measurement like finger width is a simple yet effective way to deal with the issue.
Woodworking is a hobby of mine, and just like with cooking/baking, there are many traditional techniques that have been used for centuries to avoid having to pull out a measuring tape or do math calculations even when accuracy needs to be very exact. In fact with practice you can be MORE exact when not using numbers and calculations.