Week 4: Math and the Arts

I am travelling and away this next week so have completed the week 4 work early and with the materials that I have available to me. 

In the introduction for the week, Gerofsky discusses the history of the divide between science/math and arts/literature, there is a question: "is it worth asking ourselves why contemporary mainstream culture in places like Canada and the is quick to separate people into 'math people' and 'arts people', and to look askance at work that bridges these 'two cultures'" (Gerofsky, 2024). This question really stood out to me as I often have experiences where people call themselves either 'math people' or 'artsy people' and I have never understood why one could not be both. If I had to put myself into one of these categories I would say that I am more of a math person however I do enjoy art as well (although I am not great at it) and spend a lot of my free time knitting and crocheting. When I think of these artsy activities I can easily see the bridge between the math and the art, perhaps why me as a 'math person' enjoys them so much? I don't think there needs to be this divide between the two "cultures" as they are both beautiful in their own ways and should be open to all who want to engage with them whether they see themselves as that type of person or not! 

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Reading Reflection 

Thomas, D., & Schattschneider, D. (2011). Dylan Thomas: Coast salish artist. Journal of Mathematics and the Arts, 5(4), 199–211. https://doi.org/10.1080/17513472.2011.625346

This reading dives into the artwork and inspiration of Coast Salish Artist Dylan Thomas. The introduction begins with a background on Coast Salish art and the fact that it almost died out as  traditional cultures. Recently, artists have been working on a renaissance of this style of artwork, Dylan Thomas is one of the youngest artists working towards this rebirth of his cultural art. The artwork takes on a modern style using elements of cycle, crescent and trigon. At 19 years old, Thomas learned to make pendants, earrings, bracelets and rings from a master artist. He then went on to work with three other influential artists who taught him carvings, prints, jewelry connected to myth and legend, carving totem poles and how to revive Salish art. 

The remainder of the reading, Thomas discusses various pieces of his artwork and Doris Schattschneider explains the mathematical elements of the art. Dylan explains six artworks that he has done or in the process of completing. His first is a Escher style tessellation with traditional Salish design of a spinning whorl, the subject he chose was his mentor-apprentice relationship and he uses salmon as it is sacred to the Salish people. Other artworks include a connection to the depletion of salmon in the world; house posts as welcoming figures for homes; reflection symmetry using vertical and horizontal mirrors; a cross-cultural mandala; and his new work an infinity tessellation. 

In his explanation of his first tessellation artwork Thomas says that "the problem was that I had no further education in mathematics past grade 12 and had no idea how to create a tessellation. I spent days staring at Escher's symmetry drawings such as his butterfly and lizard tessellations and from these, soon figured out how I could make a tessellation" (Thomas& Schattschneider, 2011, p. 202). This stood out to me as Thomas expresses how he struggled from having no math experience beyond grade 12 so did not know how to create a tessellation. When I think about my university math experience, I was not learning how to create tessellations and can only think of these being used in elementary classrooms. While he didn't have the math skills for this, Thomas used what he could and learned how to create the tessellation. I think this really shows how we can bridge the gap between people being 'math people' or 'art people' just because someone may be better at one than the other doesn't mean that they cannot do anything related to the opposite. 

Thomas uses Salmon in a few of his pieces of art, in his explanation of Salmon Spirits he says that it is connected to the depletion of salmon in the world. "Due to climate change and other environmental factors, the number of wild salmon has declined considerably over the past few decades" (Thomas & Schattschneider, 2011, p. 204). In our last class on connecting math to social and ecological justice my group members and I completed our first project on the decline of salmon. Through this project I learned a lot about the factors affecting the salmon population in British Columbia and the world and this really opened my eyes to some of the issues that are occurring out of the eye of the media. Seeing how an artist took an issue such as this one and created pieces to demonstrate the significance of salmon to their culture, the art shows the crowding of salmon in the spirit world and how there is need for preservation and increase of salmon. So often in creating lessons we focus on the content but this made me think about how showing students these artists work in conjunction with issues in our world can help them to understand the issue both on another level and from a different perspective than that of the educator. 

Questions: 

• Do you think of yourself as more of a 'math person' or an 'art person'? In what ways do you see yourself as both? 
• Have you used artwork (done by other artists or the students themselves) to help students understand lessons or concepts better? 

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Activity 

I chose to explore two artworks this week but was only able to recreate art that I could use paper coloring as I have limited art supplies available to me. 

I was assigned to the 2019 bridges conference: https://gallery.bridgesmathart.org/exhibitions/2019-bridges-conference 

The first piece of art that I chose to replicate was from Flora Olivier, this fish piece was created using only a compass and a ruler. Olivier (2019) explains that "it was real fun to find creative ways to deal with the limited degrees of freedom afforded by the two basic construction tools." 

I didn't have the same colours as the artist did for this piece however I limited myself to using a compass and a ruler for the entire project aside from drawing the scales on the fish. I found it a challenge to create similar curves and attach the straight lines in a way that was proportionate to the artwork of the fish. This project allowed me to see the mathematics in the curves and lines of geometrical shapes, I also found myself exploring proportionality as I had to make my fish fit on the page, but also the curves I drew needed to work with one another that would make my replica look accurate and less wonky. 

My version of Flora Olivier's 'Pisces Geometricus' 

I enjoyed this activity so decided to choose a second artwork to replicate. This one is by Emanuela Ughi. The original piece of art is a wooden puzzle based on the seed of life. The colours used in the artwork are the colours of Mondrian who I have learned in a Dutch painter. In the art description Ughi (2019) states that "straight lines and circles are the only curves, in the plane, that have a continuous group of isometries." This artwork explores geometry using only circles to create patterns based on different combinations of shapes and colours. 

In replicating this piece of artwork I used one circular object to trace, the top left image was the easiest to replicate however as I worked on others I found that even though they look more confusing, they all use the same basis of organization with different circles integrated into it. I was exploring shapes and patterns while I replicated this piece. 

My version of Emanuela Ughi's 'The Stijlish Seed of Life' 

This weekly activity inspired me to see more connections between art and mathematics, both in purposeful connections and less intentional connections. As discussed in the introduction for this week, connecting art and mathematics bridges the gap that we often see between these two realms. When we connect these two "cultures" we are opening up so many possibilities for creativity and mathematical beauty, something that we shouldn't actively be trying to avoid! 

Week 3: Math Outdoors

 I was excited for this week as I strongly support outdoor education and what it has to offer students. I love spending my time outside and appreciate any opportunities for outdoor learning and playing. As the introduction for this week says, "we may be able to put ourselves in a mindset of wonder and attentiveness to beauty, pattern and changeability in the living world, and the delight of finding elegant ways to connect with these patterns" (Gerofsky, 2024). Thinking about finding wonder, beauty and elegance while learning sounds quite lovely. 

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Reading Reflection 

Williams, D. (2008). Sustainability education's gift: Learning patterns and relationships. Journal of education for sustainable development, 2(1), 41-49. 

The reading begins with an overview of sustainable education and how it provides a solution for the isolation and silo thinking of traditional education - what Orr (1992) calls the "crisis of sustainability" (p. 41). There are three understandings as requirements for sustainability education being that (1) the whole is more that the sum of its parts; (2) living systems at all levels are networks; (3) relationships among members of an ecological community are nonlinear. After the introduction, the reading includes a case study on learning patterns and relationships. In this program a Food-based Ecological Education Design (FEED) program was started, through this program students participated in developing learning gardens and creating an environmental and nutrition farm. With these gardens students are learning about medicinal plants, nutrition, cultural significance, healing properties of gardening and resource cycling. Students felt ownership in what they were learning and developed a sense of connectedness. These school garden programs are emerging to help enhance curriculum understanding and encourage outdoor learning. 

The program emphasizes seasonal learning and showcases how the program takes shape throughout the year. I was thinking about school gardens that exist in communities around me and think they only use the garden in the warmer months when it is possible to grow plants, however I liked being able to see how this project can be carried out in an indoor setting, with designing and building garden beds and investigating soil. This would make the students feel connected to the project as it is a full year endeavor and not only a month or two of the school year. 

The students in the project were developing a concrete understanding of sustainability while creating a sense of connectedness and relationships. Students at one middle school use their school garden in conjunction with a service project to support a homeless shelter. This connects to the teaching for social justice course we took and I found myself thinking about how this could a great project for all students to make a difference in their community and see the value of the math that they are doing. 

At the end of the reading Williams (2008) is discussing the challenges for establishing school gardens, he states that "the challenges for establishing these gardens have more to do with whether these projects will become mainstream or they will stay at the margins embraced only by those who believe that environmental sustainability must be our priority" (p. 49). This resonated with me as I have seen lots of teachers avoid any kind of outdoor education either because they do not believe in sustainable education as a priority, or because they don't like going outside themselves. As someone who believes in sustainable education and thinks taking learning outside is important for all ages, I find these opportunities exciting but I know that some teachers may find the idea of a school garden project or outdoor learning a daunting feat. 

Questions

• A school garden is a big project: What are some other examples of sustainable education you have done or seen in schools? 
• How can we encourage teachers to see the importance of sustainable education and make it seem like less of a challenge to those who may be hesitant? 

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Activity 

I decided to do this activity twice: once at the ski hill where there are a lot of human-made things and then again as I was walking my dog one day on a trail in the forest where there are less obvious human made things. I have made a list of things I saw but did not add all of them to my picture. 

Ski hill 

living beings: trees, mountains, people, birds 

human made: chair lift, skis, boundary sign, ski run signs, cars, buildings 

Walking Trail 

living beings: trees, squirrel, dogs, people, birds, plants growing through the snow 

human made: trail sign, mittens left on tree, barrier to block off one of the trails, cars, barn 

This picture looks more summery than it did outside, we have a lot of snow right now!

Reflection 

Most of the living things I drew did not have straight lines, or they were a mix between straight and curved lines. The human made things seemed to have a more straight line to them and similar angles in each shape (aside from a few). The human-made objects seem to have more rigid shapes whereas the living things include more flow to them. There are some exceptions to this and I think it depends on how one draws the images as well. I did notice that clothing (the mitten on the tree) was more round in shape than other human made things such as the chair lift and the signs. I feel like these patterns exist as humans make things that are easier, take less time or are more efficient. When I think of the signs both at the ski hill and on the trail I imagine that they came from a larger piece of the same material, more signs could be cut from the material if the lines are straight and align with one another than if they were circular. This reminded me of the bees from my reading last week! 

Using observation and drawing to help students learn about lines and angles allows them to develop a deeper understanding and connect what they are learning to concrete examples, this will allow them to bring meaning to what they are learning in the classroom. These activities also allow students to enrich their relationship with the world they are apart of (Gerofsky, 2024). Like in this activity, exploring the differences between the living and human-made objects, students could see how the human-made are a part of the world and are different than those that exist naturally without manipulation. 

I think there are a lot of ways to experience lines and angles with whole-body movement. Going out into a field and using students to create shapes, either with their bodies or using footprints in the snow would be one that could be done on school grounds. I grew up figure skating so an activity that I think of is to use an outdoor rink and explore the lines and angles that the blades mark on the ice or if students can skate well they could use a similar idea to the dancing on the beach as seen in the weekly viewing. 

Circling back to the beginning of my post, through this activity I was able to sit in nature with wonder on my mind and explore the beauty and patterns that exist in the world around me. It was nice to pause and take time to sit in tranquility as I explored these spaces that I visit often. 

Week 2: Multisensory Math

This week found me connecting to things I had done before, learning new things, and being a little (lot?) bit confused while I was excited about what I was learning. This is a long post, sorry! 

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I found I had the most stops when reading the Week 2 introduction particularly in the paragraphs discussing rethinking ideas about disabilities. Gerofsky (2024) states that "in order to minimize disability, the society would need to stop disabling people!" This really stood out to me as I was thinking about the society we live in where those with a disability often are in an "other" category and how we can change this simply by changing the narrative we use towards those with impairments. The introduction says that disabilities result from a restriction of activity that disadvantages and oppresses the person. It seems like a fairly easy fix to stop disabling people by providing everyone with the same opportunities that will support those with impairments. Obviously this is easier said than done and may take a lot to support everyone's impairments but I think it is doable especially if we focus on one classroom at a time. I really liked when Gerofsky (2024) said that "everyone experiences impairments to some degree at different points in their life" thus there is not a clear divide between who is disabled and who is not. We need to work towards moving the barrier between those who are 'able-bodied' and 'disabled' so that everyone feels supported. 

This reminded me of the Universal Design for Learning model where multiple means of engagement, representation, and action & expression. Through this model teachers are providing all the students in the class with the same opportunities to optimize their learning. 

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Reading Reflection 

Kepler, J. 1611/2010. The six-cornered snowflake: A New Year's gift. Paul Dry Books. 

While I was excited about what this book included, I found it hard to read and ended up re-reading many sections to fully try to grasp what was being said. The book explores why shapes are the way they are, why snowflakes always have perfect symmetry with six points. In the section I read for this week, Kepler bounces back and forth exploring the shape of beehives and pomegranate seeds. He also discusses shapes, mainly a rhombus, and how they can create other regular solids and Archimedean solids. Kepler explains that a beehive is composed of a hexagonal plane where each cell is surrounded by six others and each bee has nine neighbors. When explaining pomegranates, the seeds are round at the start but as the pomegranate grows the rind stiffens and the seeds are forced to crowd, forming a rhombic shape. 

In explaining the shapes of pomegranate seeds, Kepler explains how this connects to beads of equal size and material being contained in a vessel. The beads are squeezed into a rhombic shape. There are two arrangements of spheres when placed in a container - either they will align in a triangular or a square arrangement. While playing with my dog and her tennis ball, I decided to try this out myself so that I could connect to the reading a little more (please ignore the hair, I know it's gross but these are the only round objects I have.) 

This is a recreation of Kepler's explanation to show what is occurring with the tennis balls

*Paisley guest appearance because I had her toys* 

While I was reading, I was thinking about how easy it is to make these changes and implement multisensory experiences into elementary schools but am struggling to think of ways that they could be used in an academic high school math class. I can see how they work in science classes with the hands on activities that arise but am stumped thinking of how they may work, for example in a calculus class, with math students who are academically driven and don't necessarily have the time for multisensory experiences. I can see how many of the students I work with would be frustrated that they are doing these activities as opposed to doing math the way that they will be doing it in University. 

Questions: 

• Have you consciously tried to remove the barrier between 'disabled' and 'able-bodied' before? Can you think of any instances where this sort of approach will not work? 
• How can we implement these multisensory experiences into high school math classes that traditionally focus on lectures, assignments and tests in a way that supports all students learning?

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Activity

I completed the activity prior to the reading however after reading for this week I connected what I read to the activity I completed. 

Part 1: The beginning of the activity suggest finding mathematical foods such as a pomegranate, apple, persimmon, zucchini, squash, tomatoes and honeycomb. I struggled with this part of the activity as the less common foods are not available in my grocery store and I did not want to purchase foods that I would likely not use or were quite expensive. I tried to use what I had at home already and supplemented with pictures off the internet. 

Apple: When I cut the apple horizontally I saw the clear "flower" shape in the center of the apple, this had five petals like the flower blossom below. The apple seeds fall into the cavities of this flower pattern but do not appear to be evenly distributed between them (maybe if they all fell to one half they would be even). I have used this cut of apples before with young kids as a paint stamp as the pattern sometimes shows up in the artwork. I have to say, I don't know if all apples are like this but I can't think of any types of apples that do not start with a flower blossom, therefore I assume that most apples start like this and follow the fivefold pattern of the flower. 

Apple blossom: https://commons.wikimedia.org/wiki/File:Apple_blossom_01.jpg

Pomegranate: I couldn't buy a pomegranate at either grocery store in my town so this is an image from the internet. most of the online images are cut from the top down (vertically). The seeds of the pomegranate appear to be enclosed into patterns, the second image shows how they have a similar "flower" like shape to the apple, with each one having two rows of evenly distributed seeds. The first image does not appear to have much of a pattern except for the fact that the seeds exist in between the pomegranate 'membrane'. I saw that the apple followed the shape of the flower blossom but I realized I didn't actually know how a pomegranate grew - they also start with a flower however they vary from three to five flowers.  

https://pixabay.com/photos/pomegranate-fruit-fruit-kernels-2018968/
https://commons.wikimedia.org/wiki/File:Inside_of_a_pomegranate.jpg

Other: The other fruits / veggies that I looked at were a zucchini and a tomato. The zucchini I had didn't have much of a clear pattern it just looked like a few different colors however I have noticed before that a cucumber does have seeds enclosed in the middle. I cut my tomato the wrong way so had to redo it with a horizontal cut and forgot to take a photo of it this time. I can see in the tomato that they follow a similar pattern of the seeds being enclosed by the outside, harder, edge. 

Honeycomb: the honeycomb has a clear pattern. There is a 6 sided shape (hexagon) connected to another on each side. All of the shapes are the same and share their sides with the neighboring hexagon. I think bees make their hives this way for simplicity reasons and less work, it is easy to create a hive where each wall is shared with another. This appears to take up less space to allow for more honey in the hexagons.  

As I was reading Kepler, I confirmed that the bees create this hexagonal plane because hexagons have a lot of space inside and therefore store more honey than other shapes and because the hexagons share walls it requires less work from the bees. 

https://www.pickpik.com/honeycomb-bees-hexagons-comb-honeycombed-insect-148106

Part 2: It snowed many days this week but was wet snow so I could not find any snowflakes that were photo worthy. I will keep looking as it snows more and see if I can update with my own images. 

Every snowflake has six sides, and resembles a hexagon. When I look quickly on google it states that snowflakes create a hexagon shape because the hydrogen and oxygen (h20) molecules of water combine together more efficiently to form a hexagon. 

Part 3: I made the following shapes using the cut and tape templates and tried to make some of the origami ones but found myself frustrated as they were not working easily for me (I have good origami paper but couldn't find it anywhere) so ended up giving up on those ones for now. 

Paper shapes made with my nephew. We used tape to hold them together but found the larger shapes tricky to make.  

I have also made a lot of these shapes with sonobe unit origami before. I first learned how to make these in my undergrad math education class and have since used them in many classes that I have taught. Here are some images of the sonobe units we learned to make, then the cubic shape that the students have made as well as some larger shapes. 

This site shows how you can make sonobe units and the modular shapes: 

https://mathcraft.wonderhowto.com/how-to/modular-origami-make-cube-octahedron-icosahedron-from-sonobe-units-0131460/ 

Sonobe Units bundled into 6 each for a cube
Students working to make their cubes. 

The students cubes they made on their own! 
Sonobe units, hexahedron, triangular hexahedron, and octahedron

Reflection questions: Cutting up the fruit, looking at images and building the shapes helped me to understand what the text was saying in multiple ways. I found that this helped me to develop a deeper understanding especially since I found the text somewhat confusing to understand. As in our introduction for the week, it says that "multisensory, bodily experiences of mathematical patterns and relationships have potential to serve as meaningful, symbolic or representational resources as students develop the ability to work with generalization and abstraction in mathematics" (Gerofsky, 2024). Through using these multisensory experiences with the fruit and shapes I was able to use these objects as representational resources to help me develop this understanding.

Using 3D objects with shape, texture, smell, taste etc. invites opportunities for all the senses to be used rather than relying only on sight and hearing. These additional opportunities to use the remaining senses can help all students thrive in an educational setting. Not all students are capable of simply sitting, listening and doing work and even if they are they likely would benefit from additional activities. By including multiple modalities to teach students more complex or abstract concepts educators are allowing them to understand the concept in a way that makes sense to them without feeling like they are less capable than others. This also applies to students experiencing sensory impairments, "innovations supporting learning for students with sensory impairments will support learning for all" (Gerofsky, 2024). By implementing multisensory activities all students are supported without needing to feel different than the rest of the students, activities such as this one create opportunities for all senses to be used. 

Week 1: Mathematics and the Body

Hello! 

As I was measuring my body to complete the first part of the activity, I went to visit my Mom who was helping my nephew do a very similar activity. He is in Kindergarten but in a blended program where he does work at home half the week, his math activity for the day was to measure things using his body parts. It was exciting to see what I was doing for this course being used and working with him to see what different things we could measure together, discussing why my measurements were different than his (he needed to use more hands than I did to measure the book size) and how we could still get an accurate measurement. He was so happy to do math together and show me what he was learning! 

Working with him to use his body as a measurement tool showcased the significance of embodied experiences, my nephew was using his body as a resource to connect what he was learning about measurement and connecting these ideas to his next lesson on proportion. While he does the majority of his math work at home, his learning experiences are engineered in a way that supports his natural ways of thinking. 

Activity 

I started the activity by completing the chart with my body measurements, I found through this activity that I was getting caught up on accuracy in my measurements and had to remind myself that the width of my finger doesn't have to be super exact. I measured some things using my measuring scale inside my house (flower pots, desk, couch) and some things outside (width of a chairlift, fence, car). 

Indoor Measurement: I had a new roommate move in with me at the beginning of the month, there is not a ton of storage space so we were trying to figure out if his hockey bag would fit in the tiny storage space we have open. I measured that the space we had was 4.5 of my feet x 5 of my feet and then measured his hockey bag to be 4 of my feet x 3 of my feet. Without finding a measuring tape and trying to make it fit we figured out that the bag will fit in the limited space. 

Extension: We had a lot of visitors at my parents' house over the break and lots of cars in the driveway. Working with my nephew we decided to try and figure out how many cars could fit into the driveway if we parked them all straight ahead. We measured how big the driveway was using paces and then how much a vehicle was using feet. I helped him with the organizing and converting measurements. Using his counting skills he worked with his grandma to figure out how many cars we could fit. (I forgot to take any pictures while we were working on this activity). 

In the TedTalk, Antonsen (2015) explains that understanding has to do with the ability to change your perspective and use your imagination. In changing your perspective you are learning something new. I found that this was true for me as I worked through this activity, I had to remind myself to change my perspective and not measure things super precisely. This activity also changed my perspective on measurement - I think its easy to think of measuring as using a ruler or a 'standard' measurement tool but using your body or other less conventional measurement tools provides the same mathematical skills in a way that embraces the embodiment of math. Antonsen also explains mathematics to be finding patterns, representing patterns, making assumptions and doing cool stuff. I really connected this activity to the 'doing cool stuff' aspect. Being able to use my imagination to measure and think of things to measure allowed me to see how my nephew thinks about math as a five year old and how I can create new math experiences. 

Reading Reflection 

The reading I read this week was an excerpt from Foundations of Embodied Learning (Nathan, 2021). There were a lot of stops for me throughout reading this, I found myself thinking about new terms I haven't heard of before, connecting to my personal experience and nodding my head as I agreed with the author. This reading discusses embodied learning, misguided educational systems and grounding metaphors for math education. 

Nathan (2021) discusses early algebra education and the importance of giving students time to think before teaching them how to solve the problems. The students intuit how to work towards solving the problem and apply a guess-and-test method to see what works, guiding them closer to a correct answer. When students were asked to describe how they used intuition to solve algebra problems many students will say "I cheated" as they view this method not acceptable for school. This section really stood out to me as students are feeling like not using a set method to solve a problem but instead using what they know with trail and error is cheating. In so many ways society has taught students that cheating is wrong so I am curious as to how these scenarios have encouraged these students to continue this method of solving problems. I am interested in knowing more about how this method has worked in classrooms to support students understanding of concepts but also how it has worked with students who struggle to try unless they are given explicit steps. I can see how this method would positively foster a growth mindset towards math but at the same time could be frustrating to students who may take more time to solve problems. 

Grounding metaphors provide the basis for meaning in math and help to link one branch of mathematics to another. These grounding metaphors are good for simple foundational concepts however linking metaphors are used when cognitive offloading is needed to understand a more complex concept. There are examples that outline how these grounding concepts apply to different mathematical concepts. While some of these metaphors may seem intuitive to us as adults who have an understanding of mathematics, they need to be learned. Nathan explains that often these concepts are taught at home or in the community, there are instances where students do not learn these ideas, "when it is

not learned culturally, it must be taught explicitly." (p. 148). Students who are not experiencing these learning at home are missing out when they come to school - this is often seen in underserved communities. I connected to this idea as I work in communities where there is a large mix between privileged and underprivileged students. There is often a clear divide among students who have learned foundational concepts at home and those who haven't. This made me think about how some students who struggle in math at an early age may just need to learn these foundational metaphors and it may be beneficial for early education teachers to learn more about the activities suggested to help teach these concepts. 

Questions

Have you ever given your students time to solve a problem without any prompting or prior instruction? how did it go? If not, do you see this being successful? 

Have you heard of the grounding metaphors before or have you learned ways how to develop these in students who may not have learned them before school? 

Hello World!

 Hello everyone! 

Here is my first blog post for EDCP 553! 

We've got a lot of snow this week and it is supposed to get very cold here in a few days. Here is a picture of my dog Paisley enjoying the snow (taken before we got snow this week)!