Math Reflection Week 6: Technology and Real Life

For class this week we focused on the use of technology in the classroom and how it can facilitate blended learning and differentiated instruction. My table group focused on the SAMR Model of using technology in the classroom. The SAMR Model divides technology use into four parts: substitution, augmentation, modification, and redefinition. Substitution involves using technology as a substitution with no functional change, while augmentation is a substitution with a functional improvement. Modification allows for technology that significantly redesigns the task, while redefinition allows for the creation of new tasks previously impossible. As teachers for 21st century learners we must strive to design tasks that best use modifications and redefinition of technology for learning.

Michael Feagan. (Oct.19 2017). SAMR Model [photo].

  The online activity that I did for this week involved watching a video about math in real life. I thought this topic would be the most interesting as children learning math often say things along the lines of "There's no real life application for learning math" or any other specific math topic. I like this video because at its core it highlights that the real world skill we want students to walk away with by learning math is pattern recognition. One of the most famous mathematical patterns that this video uses as an example is the Fibonacci spiral. The video highlights all the real world uses of the spiral both in nature and man-made things. This helps show students that mathematical patterns are present in all natural and unnatural creations in their everyday lives. In the end making student curious about mathematical concepts is key but giving them the big ideas (as discussed in last week's blog post) like pattern recognition helps them be curious math investigators.

Math Reflection Week 5: Big Ideas or Memorization

Although we didn't have an in person class this week we still had plenty of online activities to do. The one idea that stuck out to me, and the one that I will be spending the majority of my post focusing on, is the concept of big math ideas versus memorization of math formulas and procedures. For me this debate was sparked by a short video you can watch below.

Dr. Boaler argues that there are few general math principles that students need to understand. Students often struggle with math because they get caught up in memorizing details and procedures. With this kind of approach math is less about how many questions you can get right, and more about the knowledge and understanding that go behind the mathematical solutions you arrive at. Principles such as student intuition, making sense, and finding ways to represent problems can all contribute to a student's understanding of the big ideas.

Although I do agree with all of the basic concepts represented in this video there are some structural problems with how I feel this is implemented in an education system. Firstly I feel like this outlook, that knowing big ideas about math, is less testable. At least in the old fashion sense of math tests filling in multiple choice responses. As that method does not allow the student to show their thought process and their knowledge and understanding of wider math ideas.

It should not come to any student teacher's surprise that Ontario students are struggling in math, according to EQAO testing. With only half of grade 6 students meeting the provincial standard in math. This is very worrying, and it does not surprise me that Ontario boards are starting to emphasize math competencies more in students and educators. However, I wonder if teaching students big math ideas are really assisting them in math, according to EQAO it might not be.This is not to say that it is a bad idea, but that EQAO might be ill equipped to assess this kind of knowledge from students.

Math Reflection Week 4: Rich Tasks and Number Flexibility

This week in our math class we focused on participating in and assessing rich learning tasks in math. A rich tasks boils down to one that is: essential, authentic, engaging, and active. One such task that we performed in class was a problem involving counting the fingers on a hand. The basic problem was that if you started counting your fingers starting with your thumb, index, middle, ring, pinky, ring, middle, index, thumb, index, etc. what finger will you land on when you count your 10,000 finger? Everyone came up with different ways of solving this question but the one that I found most interesting was the one Mina shared with the class. It was a video of how someone else solved the problem by thinking of it as an eight fingered hand rather than just one five fingered hand.

Michael Feagan. (Sept.29 2017). Hand Problem [photo].

I think that all of the different ways students had to express this problem help support that this is a rich learning activity.

The final thing I did this week was watch the math mindset video on number flexibility. The exercise the video wanted us to do was to do 18x5 in our heads. I did this by knowing that if 10x10 was 100 then 5x10 would be 50 and 5x8 is 40, so 50+40 is 90. Perhaps a longer way of doing it, but I would not be surprised if I had many students in my math classes who would mentally answer this question this way. This exercise is pretty similar to doing math strings, which I think are an excellent way of bettering number flexibility.

Math strings I think are an excellent way to better a student's mental math flexibility. Recently I presented a webinar with my classmate Paula that highlights the benefits of doing math strings, you can see it in the video below.

Math Reflection Week 3: Learning Styles and Mistakes

Class began this week with a small math game. We played a version of I Have/Who Has? which involves everyone getting a card with a math problem, the sum of that number is the number you have. The card also gives you a number that someone else has to have who goes next. It's a really simple game but I think it would work really well in a math classroom. There are just a few modifications I would do to it to help better scaffold math learning. For example I would make sure every card had to do with the same type of math problems (multiplication, division, exponents, etc.). I would then make sure that this activity functions like a math string, having each question build on the skills, patterns, and knowledges of the previous questions. As a teacher I would also try and write down the moth problems on the board for every student to see and work out. I think with these small modifications the game can be more accessible and enjoyable for more students.

After that we had a long talk about learning styles in math. Learning styles meaning people being auditory, visual, or kinesthetic learners. I know I was definitely not the only one in class who was kind of rolling their eyes at the idea of taking another learning styles quiz or just hearing about learning styles again. As many of you may or may not know learning styles has (at least in academic pedagogical circles) fallen out of favour. Why? Well for starters the study that argued for learning styles had weak evidence to begin with and made very large claims about how catering to them could affect learning. There are a lot of studies critiquing learning styles so I won't belabor the point. A quick look online will give you some places to start for further reading. 

The last thing that I looked into this week was the idea of mistakes helping us grow in math. Scientifically you experience more neural growth in the brain when you get a math question wrong than when you get it right. This seems to make sense, as there really isn't any new information received in getting a question right, but there certainly is in getting one wrong. I think a lot of people, particularly young students, are afraid of failure. I think as educators we have a responsibility to create an environment where mistakes are good and seen as stepping stones to success. There's a phrase that I really like called "failing forward". I've heard this phrase a lot in the context of table-top games, where a failed dice roll usually means nothing happens, in a fail forward game it means that something interesting still happens. The idea is that the game still goes forward even if a dice roll is failed. I like the idea of applying this to learning. Just because you failed to give a correct answer should not mean you failed to progress, to learn, or to enjoy yourself.

Michael Feagan. (Sept.21 2017). Math Quote [photo].

Math Reflection Week 2: Math Mindsets

Our class this week largely focused on how we as students and teachers conceptualize math. The way we started this conversation was with another card trick. The card trick involved us picking a number from 10 to 20 and then adding the two place values together (for example 15 would be 1+5=6). Our instructor would draw 15 cards and then subtract 6 from the pile and draw a queen. The trick was basically one involving particular placement of the cards based on a mathematical rule that if you choose a double digit number and subtract the total of the two place value you will get the same number within certain ranges as shown in the photo below (i.e. 10-19, 20-29. 30-39 etc.).

Michael Feagan. (Sept. 15 2017). Card Trick Math [photo].

For the online portion of our class I watched a video about growth mindsets in math. The idea of growth mindsets originated from Carol Dweck and argues for students and educators to have a mindset that anything can be learned by anyone with enough effort put into it. Dweck's ideas have become very popular in the world of education, with the term growth mindset becoming entrenched in educators' vernacular. Overall I agree with the fundamental concept of a growth mindset, as opposed to a fixed mindset which Dweck asserts is the belief that an individual can only learn so much and is incapable of learning more advanced concepts. I still have some issues with Dweck's concept of the growth mindset. A cursory glance at criticisms involving Dweck's theories should leave any educator questioning how they integrate this approach in their classroom. One of the things I always thought of when hearing about growth mindsets is "what if the student is already trying their hardest?" As someone who has struggled with math I would not feel more motivated if I was told to try harder if I had already tried my hardest. Likewise if a student is already trying their hardest and can't get it, and is told that anyone can do it if they try hard, I as a student would be left wondering what's wrong with me. I feel like a growth mindset might be encouraging students to unfairly push students who are trying and still struggling with any academic concept. I therefore think that teachers need to be reflective about how they're integrating the ideas of a growth mindset into their classes and make sure that they are applying it fairly to all students.

Math Reflection Week1: Assumptions About Math

Beginning this week of our math class we were asked a lot of questions to help us think about our own experiences and thoughts about learning math. For this week I watched two videos about public perceptions on mathematics. The first one was a Discovery Education video where children and adults were asked their honest feeling about math, you can watch it here. I felt like this video did a good job of highlighting the very mixed feelings both children and adults have about doing math. From my own personal experiences I never really like math a lot. As I got older I got better at doing math, but I wouldn't say I enjoyed doing it. This might have something to do with how math in schools was presented, I never felt like it was something that felt relevant to me. I still think that not everyone has to love math, but that people need to be comfortable with it and know that there are certain problem solving skills that can be developed by doing math.

The other video that I watched this week for class was titled Hollywood Hates Math. I found this video to be really interesting though I came out of it with more questions than answers. One of those questions was this; are movie's and television's attitudes towards math reflective of larger societal opinions, or are people's opinions about math influenced by how the media presents math? I posted this question in our class forum and the responses I got were very well thought out. One response I got was wondering what the historical trends were like about mathematics to try and find when the dislike for math as a subject of study began. As someone whose field of study was history this interested me greatly. I did a little bit of research into this and found a few interesting ideas. One of which is that the feelings of nervousness or inadequacy about math is such a common thing that it is called Mathematical Anxiety. The historical roots of this anxiety might date all the way back to 19th century schools. According to teachers such as John Taylor Gatto who argues that Western industrialized schooling beginning in the 19th century was designed to create an environment of confusion and intellectual dependence on an expert. I think it's possible that a lot of people's mathematical anxieties can come from the culture of memorization in math classes.

I really enjoyed looking into this topic and hope to challenge my ideas of what teaching mathematics is about .

wecometolearn. (Sept. 11 2017) Overcoming Math Anxiety [image]