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]

Math Reflection Week 11: Formative Assessment

  This was an interesting week for math.  We got the chance to play an "eggcelent" game called, you guessed it, Egg Hunt.  The game requires students to plan out a route through Ontario cities where in each city you get a certain amount of eggs for only one visit.  The catch is that you only have 4,000 kilometers that you can travel.  The objective is to get as many eggs as possible while remaining within the allotted four thousand kilometers.  For this activity we worked in pairs and it was nice to have someone to bounce ideas off of and plan with.  Obviously this activity involved a lot of problem solving to find the optimal solution, but it also involved a lot of number sense keeping track of how many kilometers you have traveled.  For the activity we were provided the distances between cities, but students could perhaps measure it themselves and use a scale to convert it into kilometers.  I also like how this activity does a great job of incorporating Ontario geography into it.   Overall the activity was a lot of fun and I think it really got people thinking and planning which is great.  This activity or something similar to it is certainly something I would like to implement for a math or even geography class.

  Before this game though I class took the time to brainstorm about some ideas for formative assessment for math.  A lot of really good suggestions were given by my colleagues such as math problems of the day, gallery walks, KWL charts, exit cards, self-assessments, and just regular conversations and questions.  I have posted a photo of all the strategies we wrote on the white board below.  I would certainly be interested in implementing some of these forms of assessment in my own classroom or during my placements. 

 

 Michael Feagan. (Nov. 24, 2016). Strategies for Assessment [photo].

  

Math Reflection Week 10: Probability and Data Management

  For this week our class was focused on data management and probability.  For me personally I always enjoyed probability and data management, I always see it as kind of a game.  For probability this outlook seems obvious as often lessons in probability involve dice, cards and spinners.  I always got a certain enjoyment out of understanding the odds or likelihood of certain outcomes in the context of games.  To me data management ties into this as it can represent the likelihood of these events in a understandable form. For this week I had the chance to develop a math question with a photo to accompany it.  So due to the timing I decided to make a probability question.  The question is simple enough, if you were to roll 5 dice and each time you rolled you removed the dice that came up as 1 eventually all dice will be eliminated.  If we were to do this process 10 times how many rolls on average will we have to make?  I quite like this problem as it can easily become a whole lesson where the students break up into groups with chart paper to keep track of their experiment.  I also like it because it lets students use dice as manipulatives.  This activity can also be easily extended to incorporate the data management strand by having students graph their outcomes.  At the end all the students can come together and the whole class can combine the data to come up with a class average.  I have personally designed this lesson with a grade 8 class in mind and as such it best fits those curriculum expectations for data management and probability.  You can read through the curriculum expectations here, for this assignment it would be the first and second specific expectation for probability.

  In class we had the opportunity to experience other data management and probability activities.  My table group was assigned the task of guessing how many times we could toss a ball into a cup and guess the likelihood of those events occurring.  We of course got the opportunity to actually toss the ball into the cup, which was fun.  I quite liked this activity because its not your typical roll rice or toss a coin probability activity.  It nicely incorporated prediction into it by giving us the predictions, with our responsibility being to guess how likely they are to occur.  

  Both of these activities I feel like can be turned into a lesson easily and are defiantly going on my list of things to consider implementing in a class during my placement.  I believe that both activities are great for teaching probability and data management as they allow for the use of engaging manipulatives and allow for collaborative group work.  I always loved learning about probability so I would love to get the chance to try and translate that passion and interest into a lesson. 

Math Reflection Week 9: Measurement and Digital Tools

  For this week in our math course we are exploring how to teach measurements to students.  If geometry was my favorite strand in math then measurement was probably my second favorite.  Students learn measurement in math in three stages throughout the grades: definition/comparison, nonstandard units, and then standard units.  Nonstandard units mean using things like cubes, while standard units is measuring in centimeters or grams.  You can explore how this translates to a grade by grade level at the math curriculum here.  Throughout the grade levels students perform two overall expectations that are generally the same.  First they develop good measurement sense using estimation or physical measuring using a variety of strategies.  Second they determine measurement relationships, the relationships among units and measurable attributes among various shapes depending on the grade level.  I believe one of the reasons why I enjoyed both geometry and measurement is because they encourage the use of math manipulative.  Having something concrete to work with defiantly helped my learning and understanding of units of measurement.   

  If you want to know what teaching measurements looks like you can watch one of my teacher colleagues present a measurement lessons here.  I quite like this activity because it makes good use of math manipulatives, having students cut out images from magazines, having students estimate which ones are larger, compare them by size, and then calculate the area of the images using graph paper as a guide.   This lesson also makes use of different means of assessment for and as learning.  Her "students" made conjectures and hypothesis while they worked and the teacher gave feedback to these questions.  As stated earlier this lesson also makes good use of manipulatives, a great idea for a grade 4-5 lesson who often need these concrete manipulatives.  You can find strategies for providing assessment for and as learning in math here.  As I stated earlier this lesson makes good use of assessing reasoning and proving and representing.  The instructor does an excellent job of providing encouragement for the student's learning asking questions to the students, encouraging communication and having the students share their strategies. 

  I also had a chance this week to read and hear about my professor's (Rebecca Bunz's) strategies for better integration and evaluation of digital tools in the classroom.  I don't really think I have the space to go over everything about her strategy here, nor do I think I would be able to do it a good service in doing so.  Instead I would like to highlight one particular stage of her plan that stood out to me.  Stage 5 of her plan focuses on searching, finding, and evaluating digital tools, and highlights that teachers need to consider the cost, rating, age appropriateness, user policy, features, and above all play all the levels.  I believe that this level of scrutiny is so important to do if you are planning on using a digital tool for instruction.  Most important of all though you must consider if this tool will help your students achieve curriculum standards and learning goals.  This feeds back into the earlier criteria, that the teacher must familiarize themselves with the tool and determine if it achieves the goals you want, and if does not can it be accommodated.   Of course this is only step 5, and although it might sounds important there are other steps you have to take to use step 5 successfully, like establishing a PLC. 

Math Reflection Week 8: Geometry and Lesson Plans

  This week our math class got the chance to dive into the curriculum strand of geometry and spatial sense.  Geometry and patterning have always been my favorite math subjects growing up, so I excited to see what my fellow student teachers would do with their learning activities.  I was not disappointed, all of the learning activity presentations this week were great.  The first thing we got a chance to do was plotting points on a grid and connecting them to form an image.  You can find that resource here if you're interested.  As I stated earlier I really enjoyed this weeks learning activities, particularly the ones which included manipulatives.  One activity had us using various polygonal shapes to fit within a large diamond shaped asteroid. 

Michael Feagan. (Nov. 4, 2016). Asteroid Shapes [photo].

  We had to to recreate this shape using different guidelines or restrictions such as it being symmetrical or using at least two hexagons.  I love these kinds of activities for students because it gets them working hands on developing their geometry and spatial sense skills.  Obviously this activity exercises fairly basic geometry skills, but I believe that it can be modified to fit higher grade levels.  The other activity that I liked involved the creation of three dimensional shapes using two dimensional nets.  I feel like this is an activity a lot of students have done even outside of math.  What I liked about how my group chose the do it, is that we used these rubber tiles and pieced them together to form a cube, pictured below.

 Michael Feagan. (Nov. 8 2016), Rubber Cube [photo].

  I always found cutting out nets from paper, gluing them together, and trying to bend it to fit into a nice cube was always a finicky task.  So for me creating it using the nice tile manipulatives saved a lot of frustration.  That being said these tiles are somewhat limited.  For example if you wanted your students to create square or triangular based pyramids these tiles would not work so well, and in that case paper nets would be best.  Still I had a lot of fun getting to manipulate geometric objects in a mathematical context again.  Unlike many things in math, knowledge of shapes, their properties, and relations to each other has never really left me and is still something I'm interested in. 

  We wrapped up our class by discussing the creation of our math lesson plans.  It is an assignment that has us getting into pairs and constructing a lessons plan for any grade, on any strand, using the Brock lesson plan template.  Many of my classes are having me constructing lesson plans and I'm definitely looking forward to it.  Making lesson plans was a skill that I felt I really needed starting this year, so I'm glad I'm finally getting some experience in doing it.  Me and my partner will be designing a lesson plan for a grade 4 class that involves the construction of 3D shapes using 2D paper nets.  So it ties in strongly with what we practiced in class that week.  Progress on this lesson plan has already come along nicely, and I look forward to any feedback on it in the coming weeks.

Math Reflection Week 7: ESL Students

  This week we had a very informative math class with engaging activities from both our professor and fellow students.  This week some of my fellow students took on the role of teaching patterning and algebra to our class.  Now many people find algebra to be an intimidating thing. particularly young students, so I am really happy to say that all of the presenters expressed it in a accessible and understandable way.  I was particularly fond of one student's presentation of patterning where she taught the Fibonacci Sequence and spiral.  Before this lesson I was familiar with the concept of the Fibonacci Spiral in art, but was unaware of the mathematical elements behind it.  I therefore found this activity really engaging for me as I got to learn something new that helped expand my schema.  I also got to thinking about how I would teach patterning and the Fibonacci Sequence in my class, and got to thinking that maybe you could do a cross curricular activity between art and math where students utilize the Fibonacci Spiral.  Just a concept but would be something I would love to look into further. 

  The subject of teaching ESL (English Second Language) students was also brought up in class.  For me as a teacher with plans of potentially teaching abroad to ESL students I was really interested in this.  It also corresponded nicely to the virtual career fair on teaching abroad which was going on the day after.  We broke down ESL learners into four types: early stage learners, foundation learners, confident users, and fluent users.  You can find the document that details what these mean and what strategies teachers can use for them here.  Our professor broke us up into groups based on the four types of ESL learners to come up with strategies for teaching them.  My group got fluent users, which you might initially think there would be no challenges to teaching these students in English, but there are.  These fluent learners often experience frustration due to higher language expectations and can still make errors with more complex or non-standard language such as math.  These students can often get frustrated when their language is corrected.  Teachers of these ESL students should not be afraid to still work one-on-one with them to ensure their understanding and make corrections. 

  I feel like the information I have received this week will be valuable when going in to my teaching service.  Particularly if I decide to go abroad to teach ESL students I will take the advice I have received to heart and try to make a welcoming and encouraging environment for all ESL students.

Math Reflection Week 6: Math Curriculum

  One of the activities that really stood out to me today in class involved us using paper cut outs of math curriculum expectations and matching them to example activities.  A picture of the activity and the cut outs is posted below, and if you're curious you can also read the Ontario grades 1-8 math curriculum here.  I performed this activity with a partner and even with two minds working on it it was still surprisingly challenging.  This is due to the fact that some of the curriculum descriptions and expectations sound a little vague, and you think that these examples could fit into more than one grade level.  Usually it gets tricky when deciding between two grade levels that a nearer to each other, like grades 7 and 8.  Although I like how this activity had us reading through the curriculum and getting a better understanding of the goals for each grade and how assignments meet those goals it still came across as a little strange to me.  This assignments is a test of our knowledge of the curriculum and mathematical processes.  It assumes we are unaware of what grade level we are designing activities or lessons for, so we mix and match.  In reality I do not think that this would come in handy.  For example if you are hired to teach a grade 6 class you would focus on the math requirements for grade 6.  I think it would have been more useful and applicable if we were given a single grade and many different activities and were required to find the one that just applied to that grade level.  It was a little bit of an overwhelming mixing and matching activity with all the different grade requirements.  I would still considering doing a mixing and matching activity using paper manipulatives like this for my class though.  

 

Michael Feagan. (Sept. 20 2016). Math Curriculum [photo].

   Over the past week I also got the chance to try another math game.  This weeks math game that I tried was called Dirt Bike Proportions.  It's a simple game in which you race dirt bikes, and your dirt bike goes faster is you answer a multiple choice question on simple proportional fractions.  When you answer a question correctly the bike moves forward, when incorrectly it stalls.  At the end of the game it tells you what questions you got wrong and the correct solution.  It is similar in terms of game mechanics to a game a played a few weeks ago but did not write about, Integer Orbits.  Although I think using games for math classes is a great idea, I also think that you need to keep the games fresh on a deeper gameplay level, not just changing the theme and the curriculum.  

  I had a great math week overall, getting the chance to try and improve my knowledge of curriculum expectations is always welcome as that is something I will always be doing in my professional career. 

Math Reflection Week 4: Math Games and Activities

Over the past week I have had the chance to try some new math related activities.  I have completed playing and writing about my first math game.  I chose to write about a game called Spider Match which focuses on teaching children integers at a grade 7 level.  The concept is quite simply, you are given a sum which may or may not be a negative number, and then you must find a pair of number that when combined equal that sum.  The numbers given to combine can also be negatives.  What makes it more challenging is that this game is multiplayer where you compete against three other people or bots.  As far as math games go I found this one surprisingly enjoyable, I kept getting into a "one more try" mentality of wanting to try and get more combinations than last time.  I will defiantly look into integrating online math games in my classroom as an extra tool to help with automatizing mathematical processes.

 In terms of in class activities there were two that I really enjoyed.  The first one was the dice game, which at first I thought was a probability activity, but in actually an exercise in place values.  Our version used 4 dice, and when a die was rolled you had to decide where to place the number in terms of the 4 place values.  The overall goal was to create either the highest or lowest 4 digit number possible.  So for example if the goal was to create the lowest number and a 2 was rolled, you might consider placing it in the thousandths place value.  I quite liked this activity as it is both an exercise in place values and a little probability, as you have to weigh the likelihood of getting certain numbers in only a few die rolls.  I think this game could be further enhanced with the use of a ten-sided die though, just so the options are not just 1-6.  That is one thing I like about this game however, it is highly modifiable to suit your lesson and class, so I will certainly consider using it.

The second in class activity I wanted to discuss was the tarsia puzzle.  A tarsia puzzle is basically a puzzle where all the pieces have math problems on them, and you fit them together by connecting the problem with a solution.  For our class the tarsia puzzle was focused around equivalent fractions, so you had to connect a fraction with its equivalent.  I have to admit it was harder than it sounded, and the picture below of our progress should make it evident how tricky it was.  That being said it was still a fun and challenging activity.  The only concern I have with it for a classroom is that if students are working on solving it in a group or even pairs they might have a partner that solves the problems much faster, giving the others little time to think of solutions or exercise their knowledge of the topic.  Making this activity individual would also take up a lot of paper and may also take far longer if you're waiting on all students to finish.  That being said there are many different variations of this puzzle, in terms of topic, amount of pieces, and final shape all of which can be found here.  Really enjoyed the in class activities this week and am looking forward to what we will do next. 

 

Michael Feagan. (Sept, 30 2016).  Tarsia Puzzle [Photo].