Final Blog Post

Hello all! Thank you so much for sticking with me through this and I hope you enjoy some last little musings about linear algebra in the form of my final reflection…….

Throughout this process, I feel like I learned so much, but the most valuable lesson that I took away was how to self-teach. That seems incredibly straightforward given the point of an independent study, but, more specifically, I explored how to self-teach out of a textbook. In fact, the only resource that I used for this entire study was the appointed textbook and Dr. Prudhom for questions! Having to muddle through each chapter, sometimes spending an hour on just a couple of pages of a lesson, really taught me how to start understanding mathematical notation better, and I began making analytical connections in my brain much more quickly.

 But even after all the textbook work that I did, I come away with so many questions. My biggest curiosity is seeing how this subject comes into play in actual careers, and has real world applications. For my final presentation I researched this a bit, but it would be so amazing to have a chance to see this in action. I know that some other independent studies met with experts in their fields outside of school, and it would be incredible to talk to someone working with linear algebra, whether they’re an economist, a cryptographer, or anything in between. 

Although I didn’t meet with experts, one really interesting result of my presentation was that during the questioning time, Mr. Jenzano, and Dr. Prudhom asked me what it would look like to incorporate linear algebra into the DA math curriculum; something that I talked about in my presentation. It was really cool to have teachers at DA ask me to expand upon my ideas! 

This is just one of the ways that this opportunity handed me moments to step outside of the usual curriculum and really do and try something different, which leads me into advice for future independent study learners: don’t psych yourself out of trying something new or uncomfortable just because it seems daunting: Don’t cheat yourself of any experience that makes you passionate: that is what school is for. Finally, trust the process! The way to push past those intimidating feelings is to place trust in your work ethic, your faculty advisor, how you designed the course, and the way that DA runs the independent study program! For me specifically, I had to trust the process so many times. At so many points I felt like there was so much information that I was adding to my brain each day, and that my videos were not communicating effectively: I had to slow down, and remind myself that the point was to keep reaching for something better, and that is what really taught me to believe in my own abilities. I hope that many more people choose to take math independent studies: struggling with math problems on your own helps cultivate academic resilience, and at the end of the day, it is just a lot of fun! I would like to specifically thank Dr. Prudhom for being so unendingly helpful every step of the way, and always taking the time to patiently explain things a second time, when it went right over my head the first! I appreciate you sharing even a fraction of the knowledge that you have on this topic. Also, I am so grateful to Ms. Bessias for checking in on all of my blog posts, leaving comments, checking up on me, and always making sure that things were running smoothly. I am glad that I got to get to know you a little better through this process, and I felt really supported the whole time! 


(Almost!) final reflection!

Hello all! So I know that I have not really updated everyone in about two weeks, so I am rectifying that now! I am sadly finishing my final project up in the next couple of days, and then I will be done. This has been an amazing experience, and I will be so sad to have to stop, however, I have learned so much! I will put out a final reflection in a a couple days after I have officially finished my projects. For a bit of an update on what these projects are…

1) “crashcourse,” video: explaining the basics of linear algebra in 20 minutes 

2) presentation 

To explain how these projects are progressing, I will start with number one. I have done a draft of a video, and chose to focus very briefly on these topics and in this order: 

1) vectors and matrices

2) basic facts about matrices, and the idea of linear transformations/what makes something a linear transformation 

3) inverse matrices

4) very brief intro to determinants through the concept of inverses 

5) “Types” of transformation matrices (reflections, orthogonal projections, etc..) 

6) Image+ kernel+ subspace+ dimension 

7) and finally..fundamental theorem of linear algebra 

The only issue is that I filmed the video on time lapse, thinking that it would be more aesthetically pleasing, and that it would still be easy to slow the video down to see certain parts. However, after filming I realized that the video was actually very fast, and I am not sure that it is the best way to present the content. Right now I am trying to figure out if there is a way that I can find a happy medium: the normal speed video feels too slow, but the time lapse video feels much too fast! 

#2: presentation: I am sure that you noticed my myriad of announcements, considering you most likely got an email from your math teacher about it, an announcement from Mr. Wilson, and heard a standup announcement today after the moving up ceremony. However, to explain the process a bit: here is a peek look at the slide show deck!

While the presentation is not finished, it is a good peek at how little math is involved, and hopefully it peeks your interest! 

After finishing these two things (presumably by the end of the day on Thursday), I will finish my final reflection, and then this experience will officially be over! If you have any questions for me about anything relating to this project, come to the presentation, or email me! Particularly if you are interested in doing a math independent study, please don’t hesitate to reach out!


Taylor Winstead 

Slide Deck Continuation: problem set and 6.2

Hey guys! So, continuing on with the form from last week:

here is the slide deck again: for new updates see from slide 25 to the end

A New Strategy!

*This week, I decided to try something new: instead of a LONG video, I made a helpful slide deck: for lesson 6.1, please go to the page, entitled, lesson 6.1, which is on slide 12 and beyond: the stuff before that, is a work in progress 🙂

3.4 and More problems

lesson 3.3

Weekly (written) update

Hello all! So this week, you will have to settle with less content from me (I know you’re devastated), because I was at a two day in person debate tournament, and simply did not have the time to fully prepare good tutorials, and edit them. However, that gives me time to talk about several important things that happened this week: 

1) Presentation: this week I gave a presentation on my independent study as part of a larger display of independent studies and Jack Linger Grant projects. It was more difficult for me to present on my topic than many other studies because I couldn’t exactly just talk about math: so instead I talked about the experience overall…and was pleasantly surprised to realize that I had learned more general lessons from this than I had thought. One of the biggest, that I didn’t really even realize until being asked about it, is how rewarding it can be to explore something completely because of a passion, and (mostly), unconnected to college, and the future. Yes, I won’t lie, colleges often responding to independent studies did go into my decision, however, several people asked why I chose linear algebra, and I said the truth: that I am much more of a humanities focused person; I have my career in history or law planned out, this summer I will spend a week finishing my novel at a summer program, and three week taking several humanities college courses through another program. I do congressional debate, my sixth class is a second language…by any standard possible, I am a more humanities focused person. And yet…I chose this. So the answer that I gave, is that I chose this because it is sort of my last opportunity to really spend a lot of time exploring something that I just love, rather than an AP that makes sense for a certain college that I want to go to, or soon after that, a class that I have to take for my major. I have always loved math, and this was a way to tap into that passion, even though its not my end goal. Before this week’s presentation I had not realized that so much of my motivation for this specific study was to capitalize on the last remaining time I have to learn just for the sake of learning.

2)  Repititon and pattern: The other thing that I wanted to briefly discuss is how this week’s notes have made me feel like I have “figured out,” some patterns in how linear algebra is ordered. The textbook has become fairly predictable, and while I still struggle with some concepts, often don’t know how to approach a question, etc…I think that it is a positive sign that I am starting to make connections across chapters for how the information is presented. This realization just happened to coincide with a really cool experience: I attended my honors precalculus class, and the topic that we learned in that class was basically the honors precalculus version of a topic that we had just covered in linear algebra. Overall, I am just drawing more and more connections, and that is one of the reasons that I love math, so right now the work feels quite rewarding!

I hope that you enjoyed this post, and I hope that this, and the other general posts have made you think about what you are passionate about, and what you want to explore independently. 


Weekly Video Posts

3.1 end:
3.2 end:

Midterm Reflection:

It’s been another month and I have made a lot of progress. Above, I attached some samples of work that really shows the progress that I have made. In terms of content, I have progressed into much more theoretical territory. The current chapter that I am on is about subspaces and dimensions of linear transformation: in other words, thinking about representing the solution sets of matrices. I feel like this theory does make sense, although as I have gone along I will say that one thing I learned is that it’s not possible to get every single one of your questions answered, and still keep up with content. That is difficult to accept, and a little uncomfortable, but I think it is part of the independent study experience. One thing that I found very helpful more towards the middle of this last month or so was looking at the last few lessons and compiling a written list of terms and proofs that summarized everything. I think that this will be something that I continue to do periodically because it allows me to to synthesize more than the videos do. 

One thing that I am noticing more and more is that for the very theoretical concepts, I understand them, and the examples, and can often get onto the right track with the problem sets, but it is difficult to go that extra step of thinking outside of the box and using creative proofs. This is something that I obviously want to improve at, but I don’t know if there is any way other than continuing on and getting more and more familiar with thinking that way. 

Some things that have been really enjoyable are really diving into theory, and how to apply the basics of what I have learned in the first chapter to the second and third chapters. It’s really picked up to the point that now I am learning about the subspaces of a dimension: a topic that is more conceptual and requires imagining the spaces that vectors and matrices inhabit. This has been very enjoyable so far. 

My posts have also improved because I found a better way to film and also started adding a general update post that is less focused on math and more on the general learning experience. While this is more work, it is also really helpful in terms of appealing to my audience more. I know that not everyone wants to learn theoretical math! 

In terms of my goals for the last bit of the year, I want to get more strategic about the questions that I ask Dr. Prudhom so that I can maximize how much I get out of our meetings, and I want to clarify what the final “assessment” is. That is one thing that I am a little bit confused about, because in the beginning I thought that I heard something about a presentation to some sort of board about the project. If that is true I would love to have a little bit more information so that I can start preparing, and figure out how I would like to communicate my experience. 

Lesson 3.1 beginning

Hello! So this week I started lesson 3.1: stayed tuned for next week’s blog post finishing up 3.1 and also going over some problems from 2.4

Part one (part two)