Week 3 Reading + Activity

Williams, D. (2008). Sustainability education’s gift: Learning patterns and relationships. Journal of Education for Sustainable Development, 2(1), 41–49. https://doi.org/10.1177/097340820800200109

 

This paper introduced a case study to use an interdisciplinary approach to develop a garden learning program for students.  This program is supported by schools, teachers, and students from the Leadership in Ecology, Culture, and Learning Programme (PIIECL).  K-8 students participated in the program to learn about food growing, harvesting and nutritious seasonal produce preparation.  This food-based and garden-based program helps students to build a connection with the local community and learn about the multicultural traditions of the land.  

 

Main ideas in Sustainability Education from the paper: 

1.     The whole is more than the sum of its parts: The essential properties of a living system are properties of the whole, which none of the parts have. 

2.     Living systems at all levels are networks.

3.     Relationships among members of an ecological community are nonlinear.

 

 

It reminds me of one of my friends who works with kids in the garden for summer camps.  She often told me her students are really concentrated and engaged in learning, and there is rarely a “class management” issue.  Compared to me, who teaches in a very traditional math classroom, where I always find my students are low-energy and distracted in class. Then I started to realize that outdoor learning is a significant part of students’ lives, and they will enjoy the learning content more if they are present in the environment.  Outdoor learning is something I really want to do, but I have hardly started (mostly because of the weather concerns; it’s just so good to find good weather here in Vancouver).  Thus, I started to think about whether there are some short activities we can do within 30 min, so that I can have the flexibility to arrange the time into any classes if the weather allows. 

 

STOP 1: “This care is vital for investing in the long-term health and vitality of self, land and community. The links between ecological communities and human communities are presented. No living being is seen in isolation.” 

This statement reminds me The First Peoples Principles of Learning: “Learning ultimately supports the well-being of the self, the family, the community, the land, the spirits, and the ancestors” and “Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place)”.  When students study inside the classroom, it is hard for them to be aware of the surroundings and get a holistic view of the world.  However, if we can engage in some outdoor activities, students can have a better connection with nature and the land.  To link back to last week’s topic of applying different senses in learning math, sometimes our current class design focuses too much on solving problems on a piece of paper, which isolates kids from exploring the real world.  

 

STOP 2: “More than half the grades 6–8 students indicated that they felt that they had a ‘choice’ to participate in the garden project and that they did not feel ‘forced’ to participate. This is interesting, because students were not given a choice; they were required to participate as this was not an ‘elective’ subject.”

I was really surprised when I first read this part.  The garden-based project was introduced as a required activity for all the students, but the students themselves felt like it was more like a “fun elective” rather than the mandatory, stressful course they took at school.  I think one of the reasons is that students really enjoy the project, and they had a great time in gardening, so learning is no longer a stressful thing at this point.  This is a really important point that it is possible to learn with fun, and it is more effective if students can learn something based on their interests.  In this process, students develop self-motivation in learning.  

 

Questions 

 

1.     How can teachers in different subjects collaborate to design the garden learning to help students develop various skills and cover the course contents at the same time? 

2.     What are some potential challenges of the garden learning program? (i.e. space, weather, time, etc.) 

 

Activity of the Week 

 

Firstly, I observed the surroundings. Then I drew something I saw in this piece of paper and put them into three categories.  I know that in this activity, we only need to focus on living things and human-made things, but I did find it interesting that clouds and rocks belong to neither.  

 

Living beings: trees, grass, a bee  

Human-made things: road, bench, streetlights, building

Others: cloud, rocks. 

 

 

What kinds of lines and angles did you see in most living things? How about in most human-made things? Are there typical lines and patterns that show up in living things vs. human-made things? Any exceptions to this?

Most human-made things are perfectly patterned, and the same object always has the same pattern.  For example, there is a specific number of pieces of wood in one bench, and this is the same for all the other benches.  However, the patterns in living things are always slightly different from each other.  The growth rings in trees are similar, but you can never find two identical growth rings in trees.  

 

How might you use close observation and drawing or sketching to help your students learn about

lines and angles?

I think this is a really good activity to introduce cartesian system.  Students can draw the objects around them and use themselves as the reference point.  Then, students can find the angles and distances of each object to the origin.  

 

Are there ways to experience lines and angles through whole-body movement or large body

motions outdoors? In relationship to the living world?

One idea I have in my mind is “tracing the walking paths”.  We can let students choose a location as the destination and walk from school to it.  Then, during the walk, the student can trace their path.  They may notice how the path is straight vs. curved routes in hallways or playgrounds, etc.  In the end, they can reflect on how nature can change the path and why they chose the specific path over the other (i.e easier to walk? shorter distance? etc.). 

Week 2: Activity

This week, I chose to do the candy activity.  I have some lollipops left from Halloween.  The first thing I did was to count the number of each flavour left in my candy box.  I shared the candy box with my class, and taking a look at the leftovers can help me to understand my students’ preferences.  Kids favour chocolate and strawberry cream flavour more than cola and strawberry.  

 

Thinking of the geometry part, I have the idea of creating a “lollipop bouquet”.  For example, in a lollipop bouquet, you must center with one flavour and use a different flavour to cover it (fill the outer part).  We can give students a ratio (for example, if the center is “n” lollipops, the outer must have “2n +1”, etc.).  Then the student can receive a random package of lollipops with multiple flavours and decide how many bouquets they make in this task.   

 

Another idea I have is calculate the lollipops’ melting rate in water.  We can let students measure the diameter (radius) of the lollipop to calculate the volume.  Then we put a lollipop in a cup of water, measure the time, and the diameter (radius) afterwards to determine the rate.  

 

What difference (if any) did it make to actually experiment hands-on with

mathematical activities that you watched on the videos? What difference might it make to kids to learn from real 3D living things and/or objects with shape, texture, smell, taste, etc., as opposed to 2D printed images? 

 

By doing hands-on activities, math is no longer an abstract subject but something concrete that students can touch and feel.  I think all hands-on activities come from real life so that students can see the use of math and the purpose of learning.  From my teaching experiences, many students have a hard time understanding word problems.  I believe one of the reasons is that they have never seen or tried the scenario from the problem in their everyday life.  Engaging the use of senses can make math tangible and reinforce students’ memories.  

 

What difference do you think it might make for students with sensory impairment (low vision, auditory impairment, etc.)? 

 

From Stylianidou, A., & Nardi, E. (2019)’s paper, we learned that universal design is significant for teaching and for including everyone in the classroom.  To engage different uses of senses in math teaching, students with sensory impairment will no longer feel isolated in the classroom, and other students can have a better understanding of their peers. 

Week 2 Reading: Multisensory mathematics

Option C: Stylianidou, A., & Nardi, E. (2019). Tactile construction of mathematical meaning: Benefits for visually impaired and sighted pupils. In Proceedings of the 43rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 343–350). International Group for the Psychology of Mathematics Education.

 

This paper uses a sociocultural theoretical framework to create a tactile perception activity. Both visually impaired (VI) students and sighted students are invited to participate in the activity to investigate whether tactile perception can benefit math learning.  Observations of 29 audio- or video-recorded mathematics lessons and interviews with teachers and support educators are analyzed. 

 

In the activity, all sighted students need to close their eyes and determine the difference between the following two shapes: Shape X and Circle.  All the shapes were constructed on the same white A4 paper and distributed to each student.  Zak (sighted student) indicated that there is a straight line segment on Shape X, but he couldn’t identify a straight line segment clearly with his eyes.  He then described this experience as “the hidden facts on the shapes”.  Luke (VI student) identified the difference between Shape X and a circle that he said the circle can roll more, but Shape X cannot.  As can be seen, other students focused on the actual characteristics of the shape, while Luke used “embodied imagination” on what the shape can do.  

 

STOP 1: “The latter [universal design] denotes the design of environments, services and tools that can be used by every person, to the biggest extent possible, without the necessity for adaptation or specialised design.”

This is my first time thinking about the difference between reasonable accommodation and universal design.  Reasonable accommodation ensures people with disabilities can be supported by modification and adjustments to achieve an equal basis with others, while universal design focuses on creating an environment and way that inclusively covers the needs of all people.  This paper starts from the idea of using tactile perception to teach visually impaired (VI) students to transfer the strategies to a border picture that can benefit all the students.  I really support this idea that teaching is never an isolated thing that a good teaching design should be beneficial to all the students. 

STOP 2: “It would make the VI pupil feel that he is no more the only child in class who accesses mathematics differently from his peers. It would increase the sighted pupils’ familiarity with a sense which is mostly associated with VI pupils and often under-used by sighted pupils.”

This is an excellent example of “universal design” that benefits both VI and sighted students. People use different senses to experience math.  Most sighted students can easily trap their thinking into views (i.e. the visible parts of math; a written equation, etc.).  I have to say sight is probably the number one sense that many students use in a math class. Reflecting on the book Data Feminism that we read in our previous course, many students may assume numbers and data are something written on a piece of paper, but they never feel the actual power of numbers and data that exist in their everyday lives.  Seeing something doesn’t mean you are experiencing it. 

 

STOP 3: Furthermore, being aware of the characteristics of vision and touch – vision is wholistic and touch is gradual, allowing the exploration of an object from its individual parts to its whole. 

I like the idea of exploring the characteristics of the object gradually from individual parts to the whole.  I have to say, in our current time, most students are attracted to phones, videos and social media, so that one’s vision dominates most of the thinking.  It is really important to discover different senses that humans can use to connect with the world.  Sometimes seeing is far away,  but touching is close and tangible (I would argue that, for example, you may watch some travel videos online that show a lot of amazing places, but this is completely different from you actually going to that place and experiencing everything around you).  

 

Question: 

1.     What other senses can we apply to math teaching (i.e. listening, smelling, etc.)?

2.     How can we create an inclusive environment where all students are welcome to explore mathematics in different ways? 

Week 1: Body Measurement Activity

Hand Length: 16.3 cm

I use my hand to measure the length of the desk.  

The measurement of my current desk is 7 and a half hand lengths.  I did the calculation 7.5 x 16.3 cm = 122.25 cm.  

Then I use the meter tape to measure the length of the desk, which is 120 cm long! Not a surprise, my hand measure is pretty accurate! 

 

Next time if I want to get something from the furniture store, I don’t need to bring a meter tape with me. I can use my hand to measure the approximate length to determine if it can fit in my room.  

Week 1 Readings: Mathematics and the Body

Summary: Seeing the graph vs. being the graph: Gesture, engagement and awareness in school mathematics (Gerofsky, 2011). 

The article investigates how body movements and gestures can engage students’ understanding of mathematical graphs.  This idea came from a daily class task: when the teacher asks students to show y =4, all students use different gestures to represent the graph. 

In the research, students were selected from different grades in secondary schools and participated in the activity of using gestures to explain the given graphs to their partners.  The process was video recorded to qualitatively analyze how students use their gestures and bodies to communicate the graph to others.  

The results show three different ways of representing the graph: 

1)    Students use a small movement (fingers, hands or arms) to show the graph.

2)    Students use the whole-body movement (including hands and arms).

3)    Students have difficulty using gestures to accurately show the graphs. 

 

Dr. Gerofsky argues that using larger gestures and whole-body movements can be more effective in engaging students in mathematical learning.  “Being the graph” can let students make a deeper connection in learning.  

 

STOP 1: “variations in the placement of the x- axis in relation to the gesturer’s body, and potential cognitive, cultural and semiotic interpretations of this placement,” 

I like this statement that the gesture used can be influenced by one’s potential cognitive and cultural factors. This reminds me of how different cultures use fingers to represent numbers.  Growing up in China, we use the “telephone gesture” to represent the number 6, while I realized many of my students from other countries used two hands to represent 6.  Sometimes this discrepancy can be confusing.  i.e. I can unconsciously use this gesture to refer to the number 6 in front of my students, but some of them would be confused and ask me what that means.  In this research, purposeful sampling is used to select students from different grades with diversity in terms of gender, ethnicity, and math achievement and enthusiasm to make sure multiple perspectives are involved in the process.  

 

STOP 2: “In other words, being the graph in a fully-embodied way fosters engagement and attentiveness far more than merely seeing the graph.”

This reminds me of another article written by Dr. Susan Gerofsky that I read from my previous conference course, The Aesthetics of Scale: weaving mathematical understandings (Knoll et al., 2015).  It is a math workshop where three making components were provided for participants to work on, from small hand-held, tabletop, to large collaborative scales to make waves.  When the participants transferred the skills that they earned from the small and medium scales to the collaborative large scale, their senses started from “seeing the graph” to “being the graph”.  This idea resonates with Dr. Gerofsky argument that if students’ whole bodies are fully engaged in the activity, they can experience more attentiveness to mathematics learning (Gerofsky, 2011). 

 

 

Questions: 

1.     Besides using the gesture, what are some other ways you can let students make and experience graphs? 

2.     Can the gesture activity also apply to other math concepts? 

 

 

References:

Gerofsky, S. (2011). Seeing the graph vs. being the graph: Gesture, engagement and awareness in school mathematics. In G. Stam & M. Ishino (Eds.), Integrating gestures: The interdisciplinary nature of gesture (pp. 245–256). John Benjamins Publishing Company. https://doi.org/10.1075/gs.4.22ger

 

Knoll, E., Landry, W., Taylor, T., Carreiro, P., & Gerofsky, S. (2015). The Aesthetics of Scale: weaving mathematical understandings. Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture, 533–540. http://archive.bridgesmathart.org/2015/bridges2015-533.pdf

Hello Math World :)

First post for EDCP 553.