These last two weeks my mathematical focus switched from exploring measurement to data and probability - which is very exciting and timely for my professional context! I have studied both the content and technological aspects of data and probability, diving into the Common Core State Standards as well as playing around with some pretty neat online tools.
When I look at the data and probability content standards, I notice that this is the math that I actually use in my everyday life and not just because I work with numbers and data sets. Data and probability show up in many aspects of my everyday life, from knowing the probable outcome of sporting games, coin flips, or drawing a card from a deck. The Principles and Standards for School Mathematics (PSSM) for grades 9-12 encompass what many adults should know in order to have logical conversations and make logical decisions. One standard that I think I see often in my professional role is “formulating questions that can be addressed with data and collecting, organizing, and displaying relevant data to answer them” (NCTM, 2000, p. 324). Knowing I use this content often, I explored some technologies I use in conjunction with this content, particularly focusing on expanding my Knowledge of Content and Teaching (Hill & Ball, 2009) by sourcing new technologies that could be used to teach data and probability content to either K-12 students or adult learners. There is an expansive list of resources that could and should be used to strengthen data and probability learning. Because this content is so pertinent to daily life, there are many technologies in various modalities that can support student learning. A website that I explored this week was random.org. This website does a great job of sourcing commonly used data and probability technologies and provides them in a concise list. I think this is a great tool for all learners to have handy and there is large potential for its use in the classroom. Teachers can utilize this resource to support their Knowledge of Content and Students (Hill & Ball, 2009), knowing what content students need to master, how they will best master the content, and finding a support from random.org that can support their mastery. I have also been exploring the incorporation of Teaching Mathematics for Social Justice in data and probability lessons. Mathematics is a difficult subject to incorporate social justice into and, in my experience, it is omitted entirely. I find that teachers focus their social justice efforts in subjects such as ELA or social studies, but data and probability is a wonderful place to teach mathematics for social justice. Issues of social justice can be difficult to incorporate into certain content, like geometry, but an easy way for teachers to start incorporating TMFSJ is by emphasizing that all students can achieve in mathematics regardless of color, size, shape, age, or ability. Creating a classroom culture where all students are encouraged to excel can open doors for other TMFSJ content. References: Common Core State Standards Initiative (CCSSI). (n.d.). Mathematics standards. Mathematics Standards | Common Core State Standards Initiative. http://www.corestandards.org/Math/ National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. The National Council of Teachers of Mathematics. Hill, H., & Ball, D. L. (2009). The curious - and crucial - case of Mathematical Knowledge for Teaching. Phi Delta Kappan, 91(2), 68–71.
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As I continue my exploration of Mathematical Knowledge for Teaching and dive deeper into my own content knowledge, I am taking time to sit with topics that make me uncomfortable. In accepting my own discomfort, I have found that I am better able to understand what my learners may need from me and how I can provide that for them. This week as I looked at the topic of measurement, I investigated the unease that comes with teaching and learning unit relation and conversion. I am aware that my Common Content Knowledge (Hill & Ball, 2009) of this subject is not 100% and have seen how it impacts my own teaching and learning. From my previous experience teaching measurement to second grade students, I am aware that I am not confident in measurement content, specifically unit conversions, and this impacts the way I teach the content. Through the design thinking process, I worked to unearth possible resolutions to teaching unit measurement and created a rough draft prototype of one potential tool to be used alongside unit measurement instruction. This process not only allowed me to find my strengths and weaknesses when it comes to this particular content area, but it strengthened my Knowledge of Content and Teaching (Hill & Ball, 2009) at the same time. Taking on the mindset of a learner and teacher simultaneously put me in a space to understand what supports potential learners may need and how I could facilitate those supports. Although there are sub-categories of measurement that cause me discomfort, there are also many categories of measurement that excite me and I look forward to teaching and learning more about. When I was looking through the Common Core State Standards for grade 4, I found that even though the first and third categories of content standards left me feeling uneasy and unsure, I was really excited about the second category. The first and third categories deal with conversion of units and measurement of angles. The second category dives into representing and interpreting data (which is a lot of what my professional role is currently!). Taking my own experience with these standards into account, I would argue that it is really important that students are able to form positive relationships with all categories of mathematics, even if they have not-so-great relationships with sub-categories within those larger categories. Because I had a positive reaction to one of the three possible topics within the 4th grade measurement CCSS, I felt capable and willing to learn and teach more. Looking at this experience from the teacher point of view, I am curious how many students rule out their capabilities based on one aspect of a given topic and if engagement and abilities could be encouraged by coupling more favored topics with less favored ones? References: Common Core State Standards Initiative (CCSSI). (n.d.). Mathematics standards. Mathematics Standards | Common Core State Standards Initiative. http://www.corestandards.org/Math/ Hill, H., & Ball, D. L. (2009). The curious - and crucial - case of Mathematical Knowledge for Teaching. Phi Delta Kappan, 91(2), 68–71. Mad House Photography. (May, 2009). measure [Photograph]. Flickr. https://www.flickr.com/photos/mad_house_photography/4311409835 This week I learned about observational and social learning as well as using technology as a tool in learning (something I discuss quite often on this blog site). I also started a new job this week and subsequently have been attending a lot of onboarding and training sessions. I wanted to challenge myself to find at least 3 connections between my onboarding process at work and what I was learning in CEP 800. This is what I found:
Reflecting on these findings and what I learned this week both at work and in my studies, I noticed that one common theme is that everyone learns differently. There are so many different factors that contribute to any given learning experience, and curating the ‘perfect’ scenario for learning is nearly impossible. This emphasizes the importance of differentiated instruction and knowing your learners beyond the surface level. Resources: Cherry, K. (2019, September 6). How observational learning affects behavior. Verywell Mind. https://www.verywellmind.com/what-is-observational-learning-2795402 Cherry, K. (2020, April 28). What is the zone of proximal development as defined by Vygotsky?. Verywell Mind. https://www.verywellmind.com/what-is-the-zone-of-proximal-development-2796034 Salomon, G., & Perkins, D. (2005). Do technologies make us smarter? Intellectual amplification with, of and through technology. In R. J. Sternberg & D. D. Preiss (Eds.), Intelligence and technology: The impact of tools on the nature and development of human abilities (pp. 71-86). Erlbaum. When you think of the word “geometry”, what is the first thing that comes to mind? Is it colorful shapes and fun pictures like the one above? Or is it a cold, stiff classroom and teacher lecturing about dreaded proofs? I have been a member of the “I hate geometry” club, because my relationship with the subject fell mostly into the second category. Ironically though, when asked what my favorite aspect of math is, my response involves the ways that math appears in our everyday lives- from the shapes that make up houses, to how the sun rises and sets each day. Although I wasn’t conscious of it, THIS is geometry. This week in my CEP 805 course, I looked at geometry from a new perspective, setting aside my dislike for the subject and focusing on what it actually is.
After spending time with the Common Core State Standards for geometry and looking at the standards for each grade, I realized that my Common Content Knowledge, as described by Hill & Ball (2009), of this subject actually spans more content than I initially thought it did. As stated above, my initial thoughts about what geometry content is jump to proofs, postulates, and theorems. But geometry content includes correctly identifying and naming shapes at the kindergarten level, describing the partitioning of shapes using the correct language at the first grade level, finding rows and columns at the second grade level, and so on (CCSSI, n.d.). These standards are the foundations of geometry that make mathematics so exciting for me and many other students, but somehow the excitement of the subject is lost between these young grades and later grades. The first part of my initial thoughts about geometry involved the setting in which it is taught. I thought of cold, stiff classrooms and teachers lecturing. When I imagine the standards of the younger grades as mentioned above though, I picture them being taught with bright colors, comforting settings, and involving lots of play. Hill & Ball (2009) describe this as Knowledge of Content and Teaching in relation to a teacher’s Mathematical Knowledge for Teaching (or MKT). In other words, how teachers teach what their students need to learn. Lee et al. (2020) argues that there is a need to “make room for play and enjoyment in math education” (Lee et al., 2020) and not just in younger grades. A large part of why geometry in older grades is so dreaded is because once content becomes more complicated and students grow older, play disappears in the classroom and teachers turn to less interactive/ creative pedagogies in order to deliver content. There are many reasons for why teachers use particular pedagogies with older students, but the pedagogies that are implemented tend to result in rote memorization. Oldridge (2019) states that “play moves math instruction beyond rote memorization to a more expansive understanding of mathematics” (Oldridge, 2019), which could be a key factor in bringing the excitement back into geometry education for older students and giving purpose to the content. I challenge you to bring the excitement and play back into geometry in your own life with this 2-minute game: How many shapes do you think you can find? Set a timer on your phone for 2 minutes. Start the timer and count how many shapes you see in your direct line of vision. Play with a friend and compare! References: Common Core State Standards Initiative (CCSSI). (n.d.). Mathematics standards. Mathematics Standards | Common Core State Standards Initiative. http://www.corestandards.org/Math/ Hill, H., & Ball, D. L. (2009). The curious - and crucial - case of Mathematical Knowledge for Teaching. Phi Delta Kappan, 91(2), 68–71. Lee, C., Wongkamalasai, M., Thompson, N., & Jasin, L. (2020, January). Designing for playful math engagement across learning environments [Conference Paper]. International Society of Learning Sciences Conference, Nashville, TN, United States. Oldridge, M. (2019, July 24). The playful approach to math. Edutopia. Fdecomite. (April, 2011). Great dodecahemicosahedron [Photograph]. Flickr. https://www.flickr.com/photos/fdecomite/5580577168 This semester I am going to be spending some time looking at how I can better serve the MAET / GC community through my position as the Student Advisory Council representative for the MAET program. Currently I send out 1-2 surveys per semester to the students enrolled in this program. These surveys ask for feedback on the student experience and provide space for open responses regarding any questions or concerns learners may have in relation to the program. I have found that these surveys lack high quality responses, and oftentimes are only completed by a handful of students. My goal is to work through the design thinking process to empathize with these students, define the true problem and its’ root cause, ideate, prototype, and test a possible solution to this issue. This week I worked through the first stage- empathizing with this population of students. Due to my current role as both a full time student and MAET student representative, I identify with the group that I am looking at here and have been involved in multiple conversations surrounding this topic at a higher department level. This puts me in a unique position where I am able to empathize with both learners and faculty/staff. I put my department-level-brain aside, and stepped into the shoes of my peers. The first way I did this was by completing a Journey Map (Stanford University d.school). This journey map took me through the process of receiving, completing, and submitting one of the surveys. In order to fully immerse myself in the experience of my peers, I sent myself the exact email that was distributed to them, read through the email, and recorded all steps that I took in order to complete the survey. This inadvertently led me to complete a different type of empathy research - putting myself in the position of the user. As I went through the survey I recorded all of my thoughts and feelings. When I reached the end of the survey and the end of my journey mapping I sat with my experience before reflecting on what this empathy research taught me about my potential users. My reflection is as follows: I noticed that this survey is really only focused on asking about my identity and if my identities are represented in the program. There is a space to provide other comments or feedback as well as a feedback form in the initial email, but I felt that I wasn't quite sure what I was supposed to comment or leave feedback on. As the creator of this survey, I understand why specific questions were included, but a user may think these questions are random and don’t flow sensically. Also, as someone who feels that they are represented in the program pretty well, it seemed that my response to the survey was kind of pointless, since it only centered around representation. There were no really intriguing questions or questions that align with feedback I would like to give to the program. I would have appreciated a broader range of questions that were not just focused on representation in the program. Progressing into the definition stage of the design thinking process and having placed myself in the position of my peers completing these surveys, these insights will help me define the root cause of my problem. References: Stanford University d.school (n.d.) Design Thinking Bootleg. https://dschool.stanford.edu/resources/design-thinking-bootleg This week in my CEP 800 course, I dove deeper into various theories of learning and the research and history behind habits and behaviors. Throughout this semester I am going to be working on implementing the habit of reading into my daily life/ routines. Before starting this journey, I needed to understand what a habit is, how it works, and the different aspects of behaviorism that will help me implement this habit.
To start, I looked at what a habit actually is and how one is formed. According to Duhigg (2012), habits “emerge because the brain is constantly looking for ways to save effort” (Duhigg, 2012). When I look at the root of my daily routines and habits with this statement in mind, I can see why so many of my ‘bad’ or unwanted habits have formed. Focusing on my goal habit of reading each day, I situated this habit to (hopefully) replace an unwanted habit within my life. Currently I am taking the hour before I go to sleep to scroll through social media. Because my days are so busy, this is usually one of the only times that I have completely to myself and naturally I am drawn to use this time on social platforms. Duhigg describes the habit loop as a cue, routine, and reward, and as this loop is repeated it “becomes more and more automatic… until a powerful sense of anticipation and craving emerges” (Duhigg, 2012). Currently my cue is getting into bed, routine is picking up my phone and scrolling social platforms, and reward is forgetting the chaos of the day and being calm and ready for bed. So how do I change this? Duhigg states that in order to change your habit into something new, “you must keep the old cue, and deliver the old reward, but insert a new routine” (Duhigg, 2012). In this instance the cue is getting into bed and I need to maintain the same reward of being calm and ready to sleep. What needs to change is the habit of picking up my phone and instead picking up a book to read for an hour. A challenge that I am expecting to encounter is becoming bored with reading, or continuing to crave the connection on social media that I will now be lacking. If this becomes an issue, I plan to enforce the Premack Principle. This principle takes a less valued action and pairs it with a highly valued action. The Premack Principle can be used to reinforce Operant Conditioning, where a stimulus leads to a behavior, which results in a consequence (good or bad) (Miller, 2020). This is similar to the habit loop and can be used in the same ways. Using the Premack Principle, if I am struggling to read for an hour I will break it into smaller chunks and reward myself with time on my phone. More specifically, if I finish reading two chapters I can go on my phone for 15 minutes. Reflecting on behaviorism and habits this week, I took some time to acknowledge how prevalent the theories of behaviorism are in my life currently, beyond just trying to implement this new habit change. I recently adopted a puppy and while he has been a joy to train, there has been a lot of effort that has gone into building and maintaining his habits and behaviors. As I'm sure you can imagine, teaching a dog new commands involves repetitive cue-routine-reward cycles all day, but there is a lot of behavior training that goes into a puppy as well! Using the same theories of Skinner’s (1937) Operant and Pavlov’s (1897) Classical Conditioning, my dog is learning acceptable behaviors, such as waiting at the door to go outside, and unacceptable behaviors, such as stealing socks out of the laundry. My dog has shown me how much learning truly relies on the foundations of behaviorism! References: Cherry, K. (2019, September 5). What is classical conditioning?. Verywell Mind. https://www.verywellmind.com/classical-conditioning-2794859 Duhigg, C. (2012). The power of habit: Why we do what we do in life and business. Random House. Miller, K. D. (2020, March 25). Operant conditioning theory: Examples for effective habit formation. Positive Psychology. https://positivepsychology.com/operant-conditioning-theory/ This week I learned about the differences in empathy and sympathy and attempted to empathize with any individuals who may experience hearing loss. Using a hearing loss simulator, I listened to a short conversation between a man and his grandchild at different levels of hearing loss and attempted to transcribe the whole conversation. I consider myself to have pretty good listening skills and do not normally have any issues with communication, so listening to this conversation on the ‘moderate’ hearing loss setting was a bit shocking at first. As you can see in my transcriptions below, I was able to make out most of the conversation, but a lot of this had to do with inferences I was making to make the words make sense. When I listened through a second time on the ‘mild’ hearing loss setting, I realized how much I had actually missed the first time through. I noticed that I had heard some words wrong and missed entire sections completely. I finally listened a third time through on the ‘normal’ setting (no hearing loss) and was shocked at how different some of my inferences were from what the man and child were actually saying. I noticed that there were certain tones that were harder to understand than others. For example, I was able to understand almost all of what the man said due to his low and powerful tone, but the child had a higher pitch voice and was much softer when communicating which made understanding them more difficult. This experience made me empathize with some of my older relatives who have a hard time communicating with me or my mom, but seem to hear my brother and father with no issue. Even though I try my hardest to speak clearly around them, I now understand that this higher and softer tone of voice is just more difficult to understand with hearing loss. My (rough) transcriptions are included below. These transcriptions were completed while listening to the conversation all three times. I have also included the real transcription from the conversation for comparison. As you can see, the real transcription and my ‘normal’ transcription are pretty similar, an indicator that I do not experience any hearing loss. What is not indicated on any of the transcriptions is the ability to hear inflection and personality within the conversation. When listening to the conversation with mild and moderate hearing loss, I was only able to hear and focus on the actual words spoken, but when this factor was taken away, I heard the snuffles, laughs, brightness, and inflection in the conversation. References: Clare-Rothe, J. (2011). Hear [Photograph]. Flickr. https://www.flickr.com/photos/imagengine/6245030138 This week I explored pedagogy a little deeper. Two pedagogical strategies that I explored in depth were modeling and think-pair-share. In a future creation I am going to be using these two pedagogical strategies to facilitate a professional development session for teachers, teaching them a few different pedagogical approaches for teaching fractions (that’s a lot of ‘teaching’ and ‘pedagogy’!). If you need an intro (or refresher) to pedagogy, this video by Team Satchel is a wonderful place to start! The first pedagogical approach I explored was modeling. The University of Louisville’s College of Education describes modeling as “the teacher [engaging] students by showing them how to perform a skill while describing each step with a rationale”. Modeling is a very common pedagogical strategy that can be used with almost any content and in any classroom. In my future PD session I am going to be teaching my learners how to use a different pedagogical strategy for teaching fractions by modeling the strategy for them. A very common example of modeling can be found in math classrooms- the teacher explains an operation, models the operation, and students follow suit. While this is a very basic (and somewhat boring) example of modeling, it can be spiced up and reformatted for more learner engagement! One constraint of the modeling approach, specifically for a PD session, is that it has the possibility of fading into more of a ‘lecture’ style pedagogical approach. With a proper plan and strategic execution, modeling can be very engaging for learners and facilitate active learning. My goal in incorporating this pedagogical approach in my future PD session is that I facilitate it in a way that places the focus on the learners instead of the teacher.
The next pedagogical approach which I will use in my future PD session is think-pair-share. I have seen this specific approach grow increasingly popular throughout my education, and can count on it to be used in every classroom I am in. Because of the ease of this approach, it can be incorporated into almost any lesson or learning experience. The think-pair-share method gives students time to think about a prompt, pair up with a partner, and share their ideas with their partner. It can be taken a step further and students can share their pair’s ideas to the whole group. While this pedagogical approach works well for older students, it may not have the same benefits when working with a younger group of students. Because this approach involved some student-led discussion, teachers of early elementary aged students may need to structure this a bit more. With these approaches in mind, it is important to note that “many curriculum decisions are made at the school or district level and lie outside the province of the classroom teacher” (Kilpatrick et al. 2001). There is a limitless list of different pedagogical strategies, each with their own benefits and constraints. It is the job of teachers to understand their learners, know the content well, and incorporate the pedagogies that serve their students best. References: “Modeling- Elementary School.” Modeling - Elementary School - College of Education and Human Development. The University of Louisville. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) (2001). Chapter 9: Teaching for mathematical proficiency. Adding it up: Helping children learn mathematics. National Academy Press. Team Satchel (2020, Feb 13). What is Pedagogy? 4 Essential Learning Theories [Video]. YouTube. https://youtu.be/QcpwEoW1uY8 I read an article this week about the difference in instrumental understanding and relational understanding and what each of these means (Skemp 1978). I have learned about these concepts in multiple courses throughout my undergraduate and graduate career, but did not know they had a name until now. In short, instrumental learning is more ‘surface level’ or the "what," whereas relational learning is deeper, understanding the "what" as well as the "why."
This article made me think about the ways that I have experienced instrumental and relational learning in different courses and I made a few connections back to some “ah-ha!” moments I experienced in my own education as well as teaching experiences (did you know "completing the square", is not just a formula, but comes from physically filling in the missing part of a square!?). In another article I read this week, a student named Benny had trouble with understanding fractions (as many students do) (Erlwanger, 1971). He had a bit of relational combined with some instrumental understanding of fractions. His understanding of fractions was a result of how he was taught. Benny had the proper foundations and he showed signs of relational understanding (he knew ½ was equivalent to 50 cents), but was unable to apply this understanding to all fractions. In one of my undergraduate courses we learned different ways to teach fractions for relational understanding (using lots of images and diagrams as representation of fractions). Although I was unaware of the title, this was building my Mathematical Knowledge for Teaching (MKT), as described by Hill & Ball (2009). The instructor was actively trying to portray that mathematics is "more than being able to solve the problems" (Hill & Ball, 2009). While this was all wonderful and I understood the representation of fractions in this way and how to teach fractions to students using these learning tools, I still found myself reverting back to the instructional techniques for computing answers that I have relied on for so many years. At the moment, I didn’t think much of the ways that I was learning and processing information. Much like Benny, I just wanted to get the ‘"right" answer and pass the class with a good grade. I learned what I needed to know, and let my brain go on autopilot to help me with what I already knew. I think this is an issue that many older and adult learners face. If something is initially learned in an instrumental understanding, practiced this way for many years, and suddenly you are expected to understand the content in a relational way, it can be very hard to switch the method of learning. This is why MKT is so important. As stated in Hill & Ball (2009), "most adults remember a 'rule'" and proceed with mathematics teaching in this way (Hill & Ball, 2009). Skimming through the Common Core State Standards, I came across a handful of standards that I wasn't even sure I understood, let alone knew how I would teach them to a student- for example, standard 4.NF.A.1, which involves explaining fraction equivalence using visual models instead of mathematical processes. For students to develop a relational understanding, the teacher must teach with this goal in mind. This requires that the teacher has a thorough understanding of the content and combines this knowledge with effective pedagogy. References: Common Core State Standards Initiative (CCSSI). (n.d.). Mathematics standards. Mathematics Standards | Common Core State Standards Initiative. http://www.corestandards.org/Math/ Erlwanger, S. H. (1971). Benny's conception of rules and answers in IPI mathematics. Journal of Children's Mathematical Behavior, 1(2), 7-26. Hill, H., & Ball, D. L. (2009). The curious - and crucial - case of Mathematical Knowledge for Teaching. Phi Delta Kappan, 91(2), 68–71. Skemp, R. R. (1978). Relational understanding and instrumental understanding. The Arithmetic Teacher, 26(3), 9-15. Whytok, K. (2014). Educational Quotation: "A change in understanding is neened to change instructional practice" [Photograph]. Flickr. https://www.flickr.com/photos/kenwhytock/15106104349 In November I took a look at depression in the workplace and how the app, Headspace, could help with the impacts of and managing this depression. While this research was not taken anywhere, there is a lot of potential for incorporating this technology in the workforce. In order to present this technology to any corporation for implementation, it has to be proven to work. While the owners of Headspace have done studies proving its benefits, I wanted to take a look at how it could impact a specific population. In this infographic, we look at using an experimental model for implementing Headspace into a population of workers facing signs of depression. This infographic shows the stages of this experiment, starting with the background of who the participants are and how they will be broken into two random groups, what the intervention will entail, how the collected data will be analyzed, and how it will be shared.
It is important to note that the data collected from this experiment will need to be analyzed with a critical eye and taking into account the makeup of the participants. Because everyone will have different outside factors contributing to their mental state during the extent of the experiment, these each need to be accounted for throughout and after the experiment. |
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