Semester Reflection

Overall, I really enjoyed this semester. The readings were really helpful to me developing my philosophies as a math teacher. It reiterated what I already felt about the math curriculum and inspired me to try hard in my practice to make math accessible and fun for everyone. I really enjoyed all the practice with physical objects, as that is something I never really got to try with math before. I think it will really help the students connect with math.

Indigenizing the math curriculum

This topic is of great interest to me. My inquiry is on indigenizing the math curriculum, and how we can do that with more than token projects.

The first thing that really got me thinking was about the grid system. I've been doing a lot of reading about how Canada was before colonizers came, and one idea really struck me is that when Europeans first came here, the first peoples had taken such good care of the land that they thought it was uninhabited. I look around now, and it makes me sad to think how much has changed, and how much we've lost that we can't get back.

I thought it was really interesting to learn about Indigenous ways of farming. I think it makes so much more sense. I think we are too concerned with timelines and dates, and we stop listening to the world. Learning about math history with various cultures in EDCP 442, I've really come to appreciate how integrated math used to be into everything. Now, it seem so disconnected and without context that nobody wants to learn it. I hope to be able to give it context and relate it to the world around us again.

This helped improve my thinking of math education. It gave me lots of ways to incorporate the values into more of my classes. Although it was very geometry heavy, it gave me an idea on how to branch into more areas.

Updated unit plan

Here is my unit plan with 3 attached lesson plans. I'm still not really happy with lesson 5, but I wasn't sure about how to change it. I made some small adjustments, but wasn't sure what to change overall. The lessons that are complete are highlighted in yellow, but they are lessons 2, 4 and 5.
I did not link curricular competencies, but made my own objectives. I learned in another class about backwards planning, and I think it helps me to make the activities fit the goals instead of the other way around.
I made the unit plan in excel, and then transferred it to google sheets, so some of the formatting may be off.

Drinking party problem

To start with, I had no idea how to solve this problem. I asked Vincent a bunch of questions and still had really no idea of how to solve it. After talking to Ainsley, she gave me the hint of looking at it from a computer science context. From there, I was able to come up with an answer! Although, this question is really interesting because it seems to have many ways to solve it, which would be really interesting in a classroom context to develop logical thinking.

Once I started thinking of looking at the combination of rats that get sick to figure out which bottle is poisoned, it all made sense!

With 2 rats, you can check 4 bottles. If no rats get sick, it’s bottle 1, if rat A gets sick, it’s bottle 2, if rat B gets sick, it’s bottle 3 and if both get sick, it’s bottle 4.

Extending this to 3 rats, I found
         Rat A.    Rat B.      Rat C
1.       0.            0.             0
2.       1.            0.             0
3.       0.            1.             0
4.       0.             0.            1
5.       1.            1.             0
6.       1.            0.             1
7.       0.             1.            1
8.       1.             1.            1

These are all the combinations with 3 rats, therefore there are 8 bottles that we can check with 3 rats. It seems to be going up by powers of 2, since 2^3=8

2^10=1024, so we would be able to tell 1000 bottles by using the combination of rats that got sick to identify the poisoned bottle.

Unit plan with lesson plans

My unit plan is for Workplace Math 11 on probability and statistics. I did it in excel, so I'm not sure how well it transferred to google sheets. I can email the original if need be. The 3 lessons that are complete have been highlighted in yellow.
https://docs.google.com/spreadsheets/d/18we1w4WGhGPGMGs0cKE9dIlVYk49-06-qmtqTRRE8g4/edit?usp=sharing

Math textbooks response

As a former student, I don't know how much I agree with the article. In my experience, for the most part, reading a textbook was enough for me to be able to answer most questions correctly. When I was younger, this wasn't true and I needed more guidance, but from about 5th grade onward, I was able to start doing my math work as soon as I knew what section we were doing. However, as a teacher, I view things differently. I know that my experience was unusual. I got my passion for math from my teachers, but my understanding came easily. As a teacher, I think giving students real context for their lives and getting them to see themselves doing math is really important, which seems to be something the textbook examples lacked.

I think that the explanation of the concepts should come from the teacher. As the teacher, you should be aware of your students interests and to some extent their goals, so that you can bring their lives into math and get them to see a reason to do it. I like the idea of textbooks for work problems though. I think having some consistency with problems that students are doing throughout the province is a really good thing. And having a little review in the textbook of how to do the problems in case the students forget their notebooks at home would also be great. But I think it should be clear that it is review only and that the actual lesson should be taught by the teacher.

Scales problem

My first thought when I read this problem is that 1 should be one of the numbers. My second thought was that in order to get all the numbers, I would have to use the difference between 2 weights to get all the numbers between 1 and 40.

I started with
a+b+c+1=40      which means that
a+b+c=39

I also thought about the lower numbers. I realized that with 1, I would either need 2 or 3. I realized that 3 gave me more options, and so I figured that one of my numbers would have to be 3, which gave me
a+b+3+1=40     which means that
a+b=36

I also can say that:
a+b+3=39
a+b+3-1=38
a+b+1=37
3+1=4
3=3
3-1=2
1=1

From here, I thought about how to make 5. Since I could only use the weights once each, I would need a larger number and subtract 3 or 1 or both from it. My first thought was that since 1 and 3 are small, it would be better to use the largest number possible, which is 9. If one of the last 2 numbers is 9, the other is 27 from a+b=36. Now I have:
27+9+3+1=40
27+9+3=39
27+9+3-1=38
27+9+1=37
27+9=36
27+9-1=35
27+9+1-3=34
27+9-3=33
27+9-3-1=32
27+3+1=31
27+3=30
27+3-1=29
27+1=28
27=27
27-1=26
27+1-3=25
27-3=24
27-3-1=23
27+3+1-9=22
27+3-9=21
27+3-9-1=20
27-9+1=19
27-9=18
27-9-1=17
27-9-3+1=16
27-9-3=15
27-9-3-1=14
9+3+1=13
9+3=12
9+3-1=11
9+1=10
9=9
9-1=8
9-3+1=7
9-3=6
9-3-1=5
3+1=4
3=3
3-1=2
1=1

I'm sure that there are more ways to do this problem, although I'm not sure of what they are. Hopefully someone in class found a different solution! I think that I don't understand this problem deeply enough to come up with an extension for it. This problem was the most challenging to me of the ones we've done so far, it took me by far the most amount of time. I would have to spend more time understanding why powers of 3 work in this case to understand how to extend it.

3 curricula all schools teachs

The first thing that made me stop and think is the idea that the honours classes can be a negative thing to students. At my school, we didn’t have extra credits for being in the honours program, but it was a more elite class. I was always really proud that I was in all the honours classes my school provided as well as French immersion. I know the other students saw me and my friends who were in all the same classes as smart, even without knowing our grades or really talking to us. I never thought of it from other students point of view if they wanted to get in, but didn’t. I’m also realizing how much of school is teaching children something specific unintentionally, and I need to keep those things in mind to make sure students have the freedom and creativity to learn the way that works for them. The article in general just made me stop and think about what we teach kids and why we teach them that way.

I think the new curriculum does a better job of being open and teaching students in a better way. I think it gives teachers the opportunity to be less strict and incorporate more into their classroom. I would love to be able to do units on art and music and with the new curriculum, I could work that in. I think the new curriculum gives us more opportunity to incorporate all the wonderful aspects of life into any subject that we teach. I also think that because it is so open, we have to be more careful about our own implicit biases and how we are teaching, and allow students to develop on their own with our biases having as little effect as possible.

Group Microteaching Reflection

I had a few takeaways from this teaching.
First of all, it was easier to look at the positives of the presentation and judge myself nicely when I presented in a group. Even though it didn’t go the way we had planned, I still felt like we pulled it together and I was happy with our group. Sue and Vincent were really great to work with, I felt we had similar values in teaching math, we cooperated well, and it was a really positive experience. I’m excited to have them both as colleagues in the future.

Secondly, we really just had too much information for 15 minutes. Looking back, it would have been wonderful to get the class more involved and have them discover the connections better, but we just ran out of time and didn’t get through everything we had planned. I know from this that I really need to work on my timing, especially for areas of math that I’m not as comfortable in, like trigonometry. Even after we cut down a lot of stuff we had intended to do, it still felt rushed and we didn’t get to all the activities we had planned for the class.

Thirdly, I’ve noticed a big difference in my confidence levels when I’m teaching vs presenting. In presentations, I’m nervous and unsure of myself, but in teaching all of that fades away. I’m excited for the practicum and to get started teaching. I want to practice with students and work on my communication skills. Even though I’m more confident teaching, I still always worry that I’m not explaining things in a way students understand. I’m looking forward to working on that and on timing my lessons and unit plans with an experienced teacher.

Overall, I really enjoyed this. I enjoyed learning from others and I really enjoyed working with Vincent and Sue. Actually, I’ve really enjoyed working with everyone I have been with so far in the math cohort. I’m excited for the future of math education and for working with teachers who share my values.

Group microteaching

https://docs.google.com/document/d/14_ELNbqpjlDr3S3QB8YEs5Ev5DmnBZ2Q/edit

A geometric puzzle

To solve this problem, I looked at a clock with markings for minutes. Since there are 60 minutes in an hour, I said 7 would be where 14 is on the clock, and what's across from that? 44 was across, so 44/2 is 22, and so my answer is 22.

An extension could be to have 40 or 50 markings instead of 30, those numbers don't go as well into clocks, which is a circle with markings that we are the most familiar with, so the students might have a harder time visualizing the answer.

An extension that has no right answer would be for example 31. Because the number of markings is odd, there is nothing across from any number. A harder extension would be with a bigger odd number, where it might not be as easy to see that there are no markings across from each other.

I think that there is some value in giving students impossible questions. I did a lot of disproving in the first few years of my degree. I learned to think critically and trust my gut, and to apply logic to show why something wasn't possible. My favorite questions were the ones that gave a statement and said prove or disprove. I think it's important to learn not to take things at face value and challenge what you read and what people tell you. I think it's important to be curious and question everything, and that that will serve students well in the future.

I'm not too sure about what makes puzzles geometric rather than simply logical. If I had to guess, I would say that I think geometric puzzles have a visual aspect to them that logic puzzles don't necessarily have. For example, I think this is a geometric puzzle because to solve it, I had to either visualize the circle or draw it out. With logic puzzles, I often don't visualize anything, usually I jot down some ideas, let them stew in my brain, and come back to it a few days later and solve it.

Battleground schools reflection

The first part of this article that really struck a chord with me was when it is talking about how many math teachers are really specialists in other subjects and don't understand math as well, and also about how people think that math is about memorizing and using formulas. I couldn't disagree more with this! The majority of people that I know that don't like math don't like it for this reason exactly. Their teacher didn't really understand what to do or how to do it, so if they had an original way of doing something, the teacher would mark it as wrong, and then the student gets fed up and starts disliking math. The dislike for math is, in my opinion, not without good reason. It has been separated from logical thinking and defining the world and now is only about doing well on tests. I am so grateful that I had wonderful math teachers in high school who really understood the topic and inspired me and other students to delve deeper. My grade 12 math teacher actually won a Prime Ministers award for teaching in STEM, and he is the one who inspired me and other students from the same high school, to get math degrees and to pursue teaching as a career. I am grateful to him all the time, especially now that I am in this program and can really see how amazing he is.

The part in the article about the Bourbaki French mathematicians wanting to stop teaching geometry and other visual representations of math blew my mind a little bit. To me, math is such a beautiful thing! Being able to see things represented geometrically really helps me understand them better. In the math art project, the one that struck me the most was the visual representation of irrational vs rational numbers. It is such a clear thing; oh this one has a pattern and all rationals will have some sort of pattern vs oh, this one is messy and hard to decipher, all irrational numbers will be this way. To me, teaching math in only abstract concepts will alienate a lot of students, and seems counterproductive to the goal of having more scientists and astronauts that was around at that time.

Microteaching reflections: Dutch Blitz

My biggest reflection on this teaching is that I think I am too hard on myself. I tend to think people aren't understanding me when they are following really well, and while I am fine with that during the teaching process, afterwards I am always thinking of different ways to teach it that would have been better. I also think in hindsight I could have picked a game with less rules. Although this game is my favourite, I know a lot of games that have less complicated rules.

I think my strengths here were timing and engagement of learners. I was able to explain the rules, we played a game, and then I was able to go over some extensions to the rules and right as I finished, time was up. I also felt that since it's a really interactive game and you always have something in your hands, that people were really engaged the whole time.

One thing I would change is that I wouldn't play the game next time. I debated a lot about playing and in the end, I decided to, but I think it would have been simpler if I hadn't, and that way I could have been more focused on the students and made really certain that everyone understood. I didn't notice one of the students had gotten confused until it was really too late to help them fix the error without starting the game again.

The dishes problem

"How many guests are there?" said the official.
"I don't know.", said the cook, "but every 2 used a dish of rice, every 3 used a dish of broth, and every 4 used a dish of meat between them". There were 65 dishes in all. How many guests were there?

Taken from A puzzle from 4th century CE, China from the Sunzi Suan Jing 孙子算经

My first thought when reading this is that the number of guests must be a multiple of 12, since that is the least common multiple of 2, 3 and 4. 

My next thought was to write out that every guest had half a bowl of rice, one third of a bowl of broth and one quarter of a bowl of meat. Changing the wording helped me understand the problem better and find a formula to solve it.

I ended up with the equation:

guest + guest + guest = 65

  2           3            4

I simplified that and found that there were 60 guests total.

Microteaching lesson plan

Micro teaching topic

I will teach my group members how to play the game dutch blitz

Wordy Problem

Brothers and sisters

Have I none - 

But that man’s father

Is my father’s son!

Who is speaking: The only child of his father

Who is ‘that man’: The son of the speaker

What makes this problem difficult/interesting: I think what makes this problem interesting is that it’s really purely a logic problem. There’s no background information other than understanding what a brother, sister, father, son relationship is. It’s different from other math problems in this way, in most math problems you need to be familiar with definitions and theorems to be able to solve them, but with this you only need to be able to think through a question. What I found really great about this problem is that it’s a great introduction to logical thinking. Since you don’t need any background information, it would be a great problem to start the class with to get kids thinking mathematically!

Reflections on the art project

I really enjoyed doing this art project! We chose an art piece by Margaret Kepner, which is based on Ulam's spiral of primes. Pictured to the right is our recreation of her original work. I thought it was really interesting to see numbers represented visually as triangles drawn out in a spiral. The coding was quite a bit of work, and if Kyle hadn't figured out how to do it, we probably would have needed to figure out a different medium to represent this. My job in this was to comment on every line of code what number that triangle represented. It took a long time and was kind of frustrating, but after it was done, it was really cool to notice patterns of where numbers were and I felt like I had a better sense of the artwork.

Choosing how we were going to extend the project was, to me, the most interesting part. We were researching different kinds of numbers and came across Lucky Numbers, which at first I just thought was an interesting type of number, but then we found the connection to the primes and the sieve of Eratosthenes and it was even more interesting and relevant to the piece. We also chose perfect squares as something we could apply to the classroom during a unit on exponents perhaps, this could be a good activity to see numbers represented in a different way. Then we found the similar spirals between the perfect squares and triangular numbers and it was all the more interesting!

This was a really interesting way to see numbers represented in a way you wouldn't often see them. I really enjoyed the process of figuring which triangle represented which number and finding patterns in the artwork.

Locker Problem

When I first looked at this problem, I had a lot of ideas rolling around in my head. I started off thinking about multiples, but that was only a quick thought as I thought there might be a better approach. I realized early that person 501 and onwards would only touch one locker, so I would only need to figure out the first half of the problem and then flip the last half of the lockers.

I decided to work on the first 10 lockers and see if I could spot a pattern.

1:cccccccccc

2:cococococo

3:cooocccooo

4:coocccccoo

5:coococccoc

10:coocooooco

From this, I started to see a pattern that 1, 4 and 9 were all closed. I wanted to try 20 to see if my suspicions were correct and that the perfect squares were all closed.

I got 20: coocoooocoooooocoooo

From there I saw that 16 was also closed, and this was enough for me to say that all the perfect squares are closed, and all others are open.
I thought about this and wondered what makes perfect squares have this property that all of them would be open. I thought back to when I was teaching math last year and how we went over perfect squares and one of the interesting things is that they have an odd number of factors. Every other number will have an even number of factors, so they will all be touched an even number of times, so they will go back to their starting positions. Perfect squares will be touched an odd number of times, so they will change from their starting position.

2 letters from former students 10 years from now

Short note from a student who thought you were the greatest, and a student who struggled in your class. And a summary afterwards of hopes and worries

Ms Ludeman,
I just wanted to let you know how much you impacted my life. Your enthusiasm when teaching math really helped me get engaged. Because I was engaged and wanted to understand, you helped me see that I am good at math! I went on to University in (a math-related field), something I wasn’t sure I would be able to do even though it is something I was always interested in.
Thank you so much for your help!
Future student 1

Ms Ludeman,
Looking back on high school, I realize now where my aversion to math came from. I didn’t understand why you always insisted on going into such detail when I only wanted to learn the formulas, I never cared about the why! I thought you made class much more difficult than it needed to be. I’m good at memorizing formulas and knowing how to apply them. All this other work you had us do was so unnecessary and made me confused. I wanted to let you know, because I think you should simplify your lessons and go back to the proper way of teaching math.
Sincerely,
Future Student 2

Writing these 2 letters, I noticed it was a lot easier for me to write the second one, which leads me to think I have more worries than hopes at this moment. I’m worried that trying to introduce students to logical thinking and getting away from memorizing formulas where possible will be detrimental instead of helpful. What I am most hopeful of is that I can learn from my mistakes and that if I try something and it doesn’t work, I don’t keep trying it. I’m hopeful that introducing students to logical thinking will engage them and get them excited to learn.

Math and me

The first thing I remember about math is being in Grade 1 and being the worst in my class at math. I was the only one who ever got homework and it was really discouraging. I remember the very few lessons where I actually finished the work in class and could play with my friends and being so excited and happy that I understood something. I guess my math progressed throughout elementary school, because the next distinct thing I remember is getting my first set of straight A’s in grade 7. I had a teacher in Grade 7 who had a math degree, and she taught us so well and I think that was the first time I felt good at math and like I understood it.

Moving on to high school, I kept getting better. I think the logic and reasoning needed in high school math really struck a chord with me and allowed me to excel. I got into math honours all through high school and took calculus and got top marks. I had several really amazing teachers who made math approachable and made me feel like I was really good at it. That confidence really helped me when I was picking courses for University.

In university, I originally took general studies which included a math class. I did so well in the math class, my teacher talked to me afterwards and suggested I think about doing a math degree. And then I did. I really enjoyed all the theoretical learning (except for topology) and especially classes I took in group theory. My first math teacher in university offered me a job running tutorial sessions for precalculus and I absolutely loved it. It helped me stay grounded and remember practical applications for what I was doing in my theoretical courses. It was a voluntary class, and I had students who wouldn’t miss a single one. I felt really confident running those tutorials, and I would go after class and do a bunch of homework because I just felt so energized and excited about math.

I then worked for an after school math program in Calgary after I graduated and taught Kindergarten and Grade 9. The Kindergarten class didn’t really want to learn math, and they were understandably tired after a long day at school. My grade 9 class was amazing. The students were smart and excited to learn, and didn’t get discouraged when they didn’t understand something. It was another amazing experience, and I would go home so happy that I could share my excitement about these topics. I know that teaching in schools won’t be the same as teaching at voluntary after school programs and tutorial sessions, but I think that if I can make students as excited as my teachers made me, I’ll be happy.

Mathematical understanding and multiple representations

The thing that convinced me the most in the authors argument was the study comparing students who learned analytically, visually and representationally. The students who learned representationally, or as a mix of both analytic and visual, did by far the best. It makes me think back to my high school days and how I always appreciated it when teachers showed an alternative way to solve problems if the class wasn't working well with the one we had been taught. One of the things that I love about math is that although there is only one answer, there are a million different paths to get there and they are all valid. I think the more methods students are exposed to, the more flexible they will be in their thinking. Math can often involve a lot of shifting ideas. If you've chosen one way to answer a question and get partway through and find that it won't work, you have to be able to switch to another method. This method of teaching will help students develop that mental flexibility.

The main types of mathematical representations in the article compare analytical learning to visual learning or learning with objects such as blocks. One thing my high school math teacher did really well was relate things to humans. When we learned about exponential growth, he showed us in human population and explained why it was so important to understand it. He used diagrams, but his explanation and the way he related it to people made it feel so important to learn, and also so clear. I think this is a strategy that is underutilized in math. Most people don't use exponential growth every day, but I think most people in my class still understand what it is and why it is so important to understand just from a 5 minute discussion in class. Relating things to human beings can really help with understanding, and explaining why it is so crucial to understand really helps with motivation.

Relational Understanding and Instrumental Understanding by Richard Skemp

The first part of this article that made me stop and think is the explanation of instrumental understanding. It made me stop and think back to my own high school days and I felt very grateful for my teachers, because I wanted to learn relationally and that is how they taught. I know a lot of people who get frustrated because their teachers taught them instrumentally, and they didn't understand a lot of things. The next part that made me stop came right after, in the football analogy. I too held the belief that relational is a better method of teaching, and it is interesting to look at instrumental and try to understand it. The last part that really struck a chord with me is in the devil's advocate part when Skemp talks about how one reason it could be argued that instrumental understanding is better is that it is a faster way of teaching. This reminded me of a reading I recently did for another class, which was about learning objectives. According to the article, if students spent half an hour on each learning objective that they are given in school, it would take them 9 extra years of schooling to complete all of them. Skemp talks about an over-burdened syllabi, and this other article talks about making clear and concise learning objectives. I think the 2 could work together.

I think that for the most part, I agree with Skemp. There were so many parts in the article where I stopped because I felt such a connection to what he was saying. So many of my friends dislike math because they had teachers who taught them instrumentally, because they learned instrumentally and didn't understand how to teach relationally. These friends are all people who regularly question things and want to know why they're learning these things and how to apply them and can they be used in these situations. I think math should be taught because it helps develop logical thinking, not because people need to know how to find the area of a triangle. I think teaching relationally can lead to logical thinking much easier than teaching instumentally.

Hello!