How people learn is difficult to measure and define because we are all so different and have unique backgrounds and cultures. When I think about explaining my theory for how people learn it is difficult to explain how it functions for everyone. However, through my experiences and reading about different learning theories, I have come to the conclusion that the way we learn is a mixture of sociocultural learning and constructivism.
In typical Marissa Misura fashion, I wanted to show my theory for learning in a mathematical and methodical way. When I encounter the word "theory" I think of theorems. In math, theorems are statements thought to be true and can often be proven true with different methods of proof. As a teacher, the three main methods of proof that I show my students are: two-column proofs, flow-chart proofs, and proofs by construction. In a proof you take what is given to you and through a series of linking true statements you come to a conclusion which is what you are trying to prove true. I have completed a two column proof of my theory for how people learn below. Theory of Learning By: Marissa Misura Given: A sociocultural and constructivist environment. Prove: People will learn.
Despite how much this proof shows my thinking, sometimes my students (and I) struggle with two-column proofs. So I also show my students how to prove things using flow-chart proofs. I like flow-chart proofs because you can start with what you are trying to prove true and work backward until you connect all the ideas to what is given to be true. In the short gif below I completed my theory for learning proof in the flow-chart style, the complete proof in the picture following that, in addition to my previous two-column proof.
Finally - and sometimes the most fun way to prove something true - is proof through construction. A proof by construction often uses mathematical tools, such as a ruler or compass, to help show something is true. In reflecting on my two weeks of in-person learning during my hybrid program at MSU, I realized this was exactly a proof by construction! During our two weeks, I was taking in all of the knowledge presented to me by my professors and peers more knowledgeable others - and then constructing new knowledge that was uniquely my own. I was collaborating in small groups in our classes and during our many field trips. During Ed Camp when I suggested a session on Interactive Student Notebooks (ISN). In the session, I shared my ISN knowledge and experience with small group of educators. I am hopeful that they learned from me, as I surely gained new ideas when talking with them. By taking all these experiences together I am constructing the proof that we learn by being guided by those around us who are more skilled (sociocultural learning) as well as building that knowledge off of previous knowledge and experiences that we have had (constructivism).
Just like some theories in math cannot be proven true, I cannot truly prove my theory for learning is infallible. But, based on my experiences and observations of my students I have found teaching successes when using this theory of learning when creating opportunities for my students.
Resources:
Cherry, K. (2022, June 3). Sociocultural theory of development. Verywell Mind. https://www.verywellmind.com/what-is-sociocultural-theory-2795088 Kurt, S. (2021, February 21). Constructivist Learning Theory. Educational Technology. https://educationaltechnology.net/constructivist-learning-theory/ Misura, M. (2022, August 12). Theory of Learning [Image]. Misura, M. (2022, August 14). Theory of Learning, Flowchart Proof [Video]. https://youtube.com/shorts/NTEsLMHX_xs?feature=share Misura, M. (2022, August 14). Theory of Learning, Flowchart Proof [Image].
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These past two weeks of in person learning during my graduate course work at MSU have me really considering how people learn. While my own theory of learning is a combination of sociocultural and constructivism, I have to wonder how and if my students learn differently?
Because I knew my face to face learning experience during my graduate coursework at MSU was coming to an end, each day felt like “the last” of something. On Monday and Tuesday it was the last time my classmates and I were going to pick an interesting place to eat lunch in downtown Lansing. Wednesday was our last field trip on campus to visit the Planetarium and the last day in our classroom. And Friday, being the actual last day, came and went so quickly I almost didn’t have time to soak in all of the parting moments. Because of these lasts I tried to make the most of all my learning opportunities - with my professor, with my classmates, and within these creative learning spaces. Am I getting more out of these experiences because I have more of a motivation to learn as is an assumption of Knowles's Theory of Andragogy (or adult learners)? Or do my students have the experiential background to also appreciate these important moments in their learning experiences as well? (Kurt, 2020)
One of the culminating activities of our hybrid learning experience was to attend an Ed Camp . I wanted to sit back and watch what my peers would bring to the table, but soon realized that I needed to be an active participant in my learning experience. I suggested a topic that I am passionate about, creating an Interactive Student Notebook, or ISN, with my students, and it ended up being one of the sessions of the day. Starting the conversation about ISNs during the session was a bit scary, but as other teachers began sharing their knowledge, questions, and ideas on ISNs I began thinking of new things to incorporate into my ISN, it sparked my creativity to consider how to make my ISN better, and overall had me excited to start the school year learning and creating with a new group of students.
Attending this Ed Camp further strengthened my constructivism/sociocultural learning theory as being a powerful combination of ways to learn - as an adult or child. My only hope is that I am able to create these powerful learning experiences for my students within the confines of my context. Creating genuine collaboration between my students is something I can help them learn by setting up group work norms during the first few days of school and giving them reminders throughout the school year. Sparking creativity through different activities is something that I can add to my curriculum by relating activities to math, or not, to inspire my students to think differently. The most challenging aspect is motivating my students by learning in different spaces because of the limitations of time, my district, and because of safety concerns. However in small ways I can do this - have students change groups, work in the hallway, go to the library or cafeteria to change up their perspective or outlook I hope will spark creativity in them as it did for me these past two weeks.
Resources:
Kurt, S. (2020, July 11). Andragogy Theory - Malcolm Knowles. Educational Technology.
This past week was an excellent reminder that immersing yourself in different situations, collaborating with different people, engaging in traditional “creative” type spaces are all excellent ways to bring out, or reinvigorate your own creativity. During my graduate program coursework my driving thought has always been: How can I incorporate this into my teaching? During this past week we engaged with a lot of different activities and tasks, which on the surface felt as if they would not have a place in a high school mathematics classroom and initially I thought this as well. Some mornings we would collaborate with another branch of our hybrid course where students come from vastly different backgrounds, some not even in the traditional classroom setting. We took field trips on MSU’s campus to the STEM building, the stadium, Broad Museum, and we also traveled off campus to Michigan’s Capitol building, Michigan’s House of Representatives offices, and local well known eateries. We also engaged with in class creativity creations from this past week. The infographic below is a collection of experiences in these different spaces.
However, reflecting on all of these experiences so far I have come to the conclusion that they don’t need to be incorporated one to one to my math classroom for me to get something out of it. Again, being surrounded by different types of creative outputs encourages me to be more creative, or has my brain think in different ways. So, when I think about redoing a unit for my Algebra 1 class I am inspired to think up new tasks or new ways to approach, some may argue, bland topics. Being able to collaborate with my classmates and professor, or even just hearing about their ideas for their own unit designs, I am inspired and motivated to push my unit to be even better. Visiting museums, architecturally interesting buildings, a panel discussion with educational leaders, MSU staple restaurants and landmarks gives me new perspectives. This has encouraged my brain to work in different ways that what it normally would, which in turn allows me to think up and create differently than I traditionally would.
All of this speaks to my own theory for how we learn. As I mentioned in my previous blog post, my theory for learning is a combination of sociocultural learning and constructivism. By surrounding myself with more knowledgeable others, whether that is my professor or classmates I am learning more about topics, them, and myself. By experiencing new places and people I begin to construct new ideas. And by physically creating new artifacts during our creativity tasks I push my knowledge and understanding to different or new levels.
When it comes down to it, how to best teach our students is a simple yet extremely complex and difficult question to answer. It is difficult to suss out who and what to believe when it comes to best educational practices but it is vastly important to do so (Willingham, 2012). This is why I tend to model my teaching mainly around two learning theories - constructivism and sociocultural learning.
When I think about how I learn best I tend to fit into these two learning theory categories pretty well and I have modeled my teaching style around those. During my high school and undergraduate years I would always try to work in groups with classmates I felt that were more knowledgeable than me and learning from that more knowledgeable other was extremely helpful - not only content-wise but how they learned or studied or prepared themselves for classes. And then within those groups I could bring my ideas and work them out with my peers while they provided suggestions, improvements, or new perspectives I hadn’t considered. I found that I did this with sports as well - I always pushed myself to be surrounded by people more talented than me so that I could learn from them which I feel almost perfectly blends these two learning theories.
In my classroom I try to build a community of learners by prepping my students the first week of school on how to work collaboratively and effectively in a group. We watch videos and work through activities that I have picked from Jo Boaler’s website Youcubed.org and do the 100 numbers to get students talking activity to discuss what makes efficient and effective group members. Then whenever we start a new topic I have students randomly assigned to new groups and we always start with a discovery activity that has students push their Zone of Proximal development through collaboration with their new group members.
It is also important for me to keep in mind that just because I learn through collaboration and experiences really well my students may learn other ways better. Trying to work other types of learning styles into my everyday activities is important not only to reach all learners, but to keep each lesson interesting and fresh for me and my students.
Resources:
Misura, M. July 22, 2022. Learning Theory. [Image]. Willingham, D. T. (2012). When can you trust the experts?: How to tell good science from bad in education. John Wiley & Sons. Creativity comes in many forms as I learned through my CEP 833 class this past semester. We tend to think of creativity as Big-C creativity - big breakthroughs like the invention of sliced bread. But little c-creativity, small changes to simple objects, or noticing new or old things around you and how to appreciate them in different ways can be just as valuable and important. Students (and adults) struggle to accept little-c creativity as creativity. Or they often see creativity as an artistic endeavor rather than a scientific (or mathematical!) endeavor. I sometimes struggle with this as well and don’t often think I am very successful at being creative, but these past two weeks I pushed myself to be creative and I learned a few things about how I can improve my creative muscle. For me, taking a break from the creative task and doing something else that is completely unrelated - a chore, a walk, a shower - helps me to settle my mind and usually when I come back to the creative task I feel better about coming up with new ideas. Another thing that truly helps me to be more creative is to collaborate with others. Last week while trying to come up with an activity based on the “bodily sensations, reproducible patterns'' tool in Root-Bernstein & Root-Berstein (2001) “tools for thinking” I created the Magic card below that represented the overall topic of solving linear equations. This was fairly broad and couldn’t really be used in the sense that a Magic card is (I should mention, Magic the card game is a head to head card game where players attempt to defeat each other by playing cards that make you lose health points or run out of cards to draw). Luckily I was able to collaborate with my husband (and classmate) and my professor and came up with an even better idea for my classroom that would make the Magic card idea even more dynamic. What I would do with my students is give them an equation that they would need to defeat (or solve) and then they would need to create as many cards as needed to complete that task. Typically you would shuffle up, draw a card, and then play a card and this could potentially be problematic when solving linear equations as we want to solve in a specific order, but I also think this could open up really interesting conversations about if and when solving in a specific order does matter. Overall, I think it is important that I don't doubt my creativity. I want to empower and encourage my students' creativity and I think I need to do that for myself as well. Resources:
Misura, M. (July 15, 2022). Magic Card. [Image]. Root-Bernstein, R. S., & Root-Bernstein, M. (2001). Sparks of genius: The thirteen thinking tools of the world's most creative people. Houghton Mifflin Harcourt.
I feel that most educators go into teaching because they want to inspire and help students foster a love of learning. How we go about doing this, within the framework of the subject area that we teach, can be a true challenge when considering all the variables within a classroom setting. What subject matter are you teaching? How will you teach it? Can you achieve that within the means of your classroom? And when students enter the equation there is a completely new set of obstacles to consider. Will they care about your subject matter? How can you help them to care? How do you help them to work hard and self motivate? How do you build up the 21st century skills that are so important to our ever changing workforce? Amongst all these questions, and hundreds more, it would seem reasonable that we could try to use the scientific method to theorize, test, observe, and repeat what approach would work best for all students; however, what works best for all students is likely the issue as students are all so different.
As, Willingham (2012) stated, “There are other vital questions in education for which the scientific method is wholly inappropriate…Who is ultimately responsible for children’s education: parents, teachers, or children themselves, and does the answer to this question change as kids get older? Educating children raises dozens of questions, and, powerful as the method may be, science is applicable to just a fraction of them.” (p89) Educating children is so much more than just how to add, compose an essay, or learn about the civil war. And because teachers, students, administrators, community members all have a voice in the education system there are many differing opinions about how to best teach our children. When talking with my mentor teachers, they mention that how to best educate children has gone in cycles with one style of teaching being best when they first started, to something different, and then back to that original method now. And thinking about teaching now with all the divisive opinions that are only reaffirmed through everyone’s own confirmation bias bubble it is even harder to ensure that students are getting the best, well-rounded education.
I am only through half of Willingham’s book and there are so many ideas that resonate with me, some of which I mentioned above, but even more questions that I have - for example, how can I know what is best for my students? When can I trust the facts presented to me? I am hoping the second half of the book begins to answer some of these really difficult questions.
Resources:
Willingham, D. T. (2012). When can you trust the experts?: How to tell good science from bad in education. John Wiley & Sons.
If starting my masters in Educational Technology and teaching during a pandemic (both hybrid and 100% virtually) has taught me anything it is that technology in the classroom needs to be considered thoughtfully and must be adding to the lesson, not distracting from it. I think in many ways last year teachers, myself included, perhaps put too much technology in front of our students last year trying to engage and connect with them. However, usually teachers are some of the best learners and I know at least I learned that if I am going to use technology to help my students through a topic it needs to have a purpose and help, more than distract.
There are many video platforms to make a video more engaging for students to watch and learn from, Prezi, edPuzzle, VideoAnt to name a few. There are countless excellent resources and lessons created by PBS, PhET, desmos, and YouCubed that you can use and adapt for your students' needs. However, one of the most interesting resources I was introduced to this week in my CEP 805 course was The Free Learning List. This is a very simple compilation of sites, podcasts, courses, subreddits, etc. that are free to use and could enhance or engage your learners outside of just how to do the task. There are so many different avenues to explore in this list which allows you to pull in specific topics that you or your students are passionate about, but this can also be a bit overwhelming. In addition to that, while they do have a score based on their effectiveness, engagement, and popularity and a short description of the content provided it is not 100% guaranteed that everything is completely “school appropriate”. Constraints aside, I see so much potential for students to have ownership over their learning by using this site to explore topics they are truly interested in. I always love to share “fun math facts” with my students outside of just learning another algorithm that we must apply in x-situation and this site allows me to find more of those fun facts to bring in and share with my students. One specific example of this was a podcast that I was listening to about music - Key Notes - started discussing The Golden Ratio, Fibonacci Sequence, and math in music (the entire episode is embedded below). The overall summary of the podcast was that key moments in some songs happen at the “golden moment” of the song, or approximately 61.8% of the way through the song and I thought, what a fun way to hook my students into a pretty mundane topic of ratios. I had a whole idea for a mini project based on the idea, but sadly because of snow days, testing, and other demands outside of my control sadly the project was abandoned because I just couldn’t see how to fit it in. It is unfortunate that my “golden moment” project didn’t happen, and it also makes me wonder how many other fun and interesting projects or activities we could be doing with our students to add value and enrich their mathematical learning experience beyond just learning the algorithm. Integrating technology can enhance this if considered carefully as to how it elevates the lesson.
Resources:
Cuchna, C. (Host). (2021, July). The Golden Ratio, Fibonacci Sequence, and...Music? (No. 5) [Audio podcast episode]. In Key Notes, Spofity. https://open.spotify.com/episode/6PkXcoiEnD7XegE5Ei8K5s?si=NWplOJ66RcSPjwhsYxgY3Q
When I think about who I am - mathematics is such a big part of that. I truly geek out about rediscovering new things in math with my students, or reading an interesting article about mathematics, or thinking about ways to engage my students in a way so that they can try to at least half match my mathematically geeky energy. Ideally I would come up with some amazing task in my Algebra 1 class and my students would be instantly hooked and we would have a great discussion about what they wonder or what they are noticing and they would go home, try out some practice problems and we would move on to the next thing. However - my students are not me and they each have their own things that they are extremely passionate about or sometimes have things going on that completely distract them from the joys and privileges of learning.
Taking time to consider each of my students' intersectionality is a vital part of being an effective educator. It would be wonderful to hold all of our students to a high standard and not budge but our students do not always come to us with the same tools and support that we would hope or perhaps have experienced ourselves. To a degree, as a female in a largely male- dominated subject area I have faced my own struggles with feelings of whether or not I belong in a math classroom. However, because another part of my identity is having a strong family support system that has always encouraged me to pursue my passions I have the attitude that "I was just going to show those boys that girls are good (or better) at math too".
This year has been a struggle to embrace and try to connect with some of my freshman students in Algebra 1 who come to me with extremely difficult home situations, despite this being a suburban school district. No one at home to wake them up to get to the bus, or bring them to school, no one at home when they return to encourage them to do their homework or help them if they get stuck - or even make them dinner. This year I have kept a stash of snacks in my classroom because many of my students do not have an opportunity to eat breakfast in the morning or bring lunch to school. Because my students are coming to me with all of these much greater issues, how can I then motivate them to factor the quadratic so that we can see the zeros of the graph in the equation? While my love and passion for mathematics will always be a huge part of my identity - it is true what one of my undergraduate professors said - if you aren’t going into teaching for the love of your students you are doing it for the wrong reasons. This year more than ever I have really had to care for my students and make sure they have other needs met before we even attempt to factor that fabulous quadratic equation. After a few weeks of considering how to integrate social awareness into my Data and Statistics unit for my end of the school year unit, I decided to have the unit culminate in a project. In this project, students will need to research some issue using the website Statista and then create their own survey to distribute to some of our student population to see how the data that they collect compares to that data that they have researched. The will need to display that data using multiple modes, and I also want to incorporate some big questions about what that data is telling them about the world around us. I had had an idea to do a project like this for a long time, but always struggled with the research aspect of this for myself and for my students. I think that researching a specific topic for specific data points can be a real challenge and giving students the freedom and wide-openness to do this I think would be overwhelming, so the project always sat on the back burner. However, thanks to my amazing classmates in my CEP 805 class I was introduced to the site, Statista. On their website you can search for almost any topic and find a brief overview of the topic, and then tons of different kinds of data points that you can view as line graphs, bar graphs, or get related information to that topic. I love that all of my students' research can be done in one place. That being said, one of the major downsides to Statista is that you need to sign up for an account to access the information and even after that some of the data is still behind a paywall. I haven’t decided yet if I want to ask my students to use their school email to sign up for an account because there are always privacy issues to take into account and asking students to sign up for things they aren’t comfortable with is unfair. I could simply use my login, but then I would feel like students aren’t really doing the research so there is a lot to consider. With that too, because there is so much information under one category it could be difficult for students to sift through what is important to them for their project and not. Overall I feel like perhaps the benefits outweigh the negatives here because incorporating some real world data into a mathematics project about collecting, analyzing, and displaying data in a meaningful way would be so powerful for my students to experience. Bringing math to life for them so that they see the value in what they are learning is a key factor in exciting students to learn more about that topic. In my current district I feel like I have a unique and different approach to teaching mathematics. Whether it is because I am one of the ‘younger’ teachers or because I learned math through an integrated approach or because my love of math goes far beyond just the algorithms we teach to the beauty of it that appears in nature and the world around us I always try to bring interesting or different topics to my students about math as well so that hopefully their love of mathematics can grow. However, an area that I have always struggled to integrate an interesting or different approach is Data and Statistics. This week in my CEP 805 class I was able to give myself some time to improve my Specialized Content Knowledge (Hill & Ball 2009) in this area through the lens of social justice. This was truly eye opening for me as I’ve always wanted to integrate other subjects into my math classroom so that students can see that subjects have interconnections and are not just separate entities. It seems obvious that you could integrate science and math, but integrating something like social change or social awareness always seemed so difficult to me and I felt I would just have to leave that to the History and English teachers in my building. After investigating lesson plans on Dr. France’s Harpers site, solving world problems, that integrate data, probability, statistics and other mathematical concepts to promote social awareness and justice, I was so excited to borrow or modify these lessons and bring them to my classroom to share with my students. That being said, while these topics absolutely need to be taught to students, the current climate of education has me a bit worried about push back from administration, parents, and students. Being married to an English teacher who teaches kids social awareness through the lens of empathy for others on a daily basis I see the stress and worry that it brings him. But if the focus is instead that “culturally relevant pedagogues understand that their job is not to make students think like they think, but rather to have them think” (Clark 2021, p27) then I don't think there could be much room for people to complain. After seeing ways that I can integrate social awareness into my mathematics lessons and reading about why it is so important that underrepresented groups are seen and heard in all subject areas I feel more confident to tackle these important issues. While looking for lessons that integrate social justice and mathematics I found ways to improve my Knowledge of Content and Teaching (Hill & Ball 2009) through sites like PBS Learning Media and the always amazing YouCubed. Resources:
Hill, H., & Ball, D. L. (2009). The curious - and crucial - case of Mathematical Knowledge for Teaching. Phi Delta Kappan, 91(2), 68–71. Clark, C. P. (2021). An opportunity or change: Groundbreaking scholar Gloria Ladson-Billings on culturally relevant pedagogy and why education as we know it needs to be transformed. Literacy Today, 38(5), 24–27. |
AuthorMarissa McGregor, high school math teacher extraordinaire. I love my husband, daughter, and family dearly. Archives
August 2022
CategoriesThis work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. |