This week in mathematics ... (week 3)

This week I had my first day of placement in the classroom. I am in a grade 7/8 classroom at a very nice local school. I am super excited to see what the next few weeks with this class will bring. That leads me to one of the things I want to talk about this week ... math strings!!
When the students got to class, the first thing they did was pull out a book and started doing math strings. The teacher would write out any random multiplication problem, and the students would raise their hands when they had the answer. The teacher was making them up on the spot, as he had done all the ones from the textbook they are working off of. For anyone who does not know, a math string is a tool used to encourage mental math. The problem is started with one multiplication question, but really it could be addition, subtraction, division, any type, but in this case they did multiplication. They took the first set of numbers, and found the answer. Then moved to another question, this question usually featured one of the same numerators, for example, if the first problem was 2X2 =4, the second question could be 2X4= 8, then so on and so forth. It does not have to be done so neatly, as you could just to a number like 2X14= 28, which the students could use the previous "string" to determine using mental math. The other cool thing is when the teacher called on the students, they are automatically expected to explain how they got their answer. So all of the mental math strategies are being shared right there, on the spot. They are also apparently called "number strings". This blog had a lot of interesting explanations on number strings for younger students (grades 4-6). I researched through it a little and found that they use a more structured approach to their number strings, but the teacher I was working with had done so many that he did not need to pre-plan out his strings, he just worked through the numbers as he went.

     Furthermore, I am interested and excited to see how the strategies I have been learning in math class will help me when it comes to my practicum. One of the concepts I enjoyed learning about was the "representation" concept. If students are comfortable seeing numbers as more than just numeric values, they can work with them in different ways. The example of 9 is the same as 10-1, or 8+1, or 3X3.
     One of my "aha" moments this week was a lesson we did on inverting fractions. The idea that we just invert and multiply had come back to me during a required pre-assessment course. I had remembered what to do, and when to do it, but the "why?" stuck with me. Unfortunately for me, as a math student, no one had ever taken the time to explain why, it was always just well here is what you do, so now go do it. Being explained why was really awesome. So, for the inversion of fractions, I now understand that we do it so that we can compare unalike fractions in the same way. This video works to explain it better than I can, but the gist of it is that to compare fractions, they have to be "out of" the same thing, or else the comparison is meaningless.
     One of the other things I was relieved to have learned about was that assessment for learning is not just a ranking of students, but rather a summation of their strengths and weaknesses, which are build around trying to help the students improve rather than to just judge them, slap a grade on a paper, and move on. With math, where simple yes or no right answers are available, this sometimes gets lost. I am thrilled to have the "good assessment plan" guide at my disposal now, because I think it will be really useful. It was also interesting to consider that when students do bad on an assessment, that teachers should reflect on that. Why did the student do poorly on the assignment/test? What could have been explained to them differently? What could I have done as an educator to ensure that these students succeed? I think these questions are all very important for us to ask ourselves every day, especially when our students are not understanding the concepts we are teaching.
   I also really like the idea of instant feedback in seeing if students truly understand what they have just been taught. The use of Twitter or a face-paced multiple choice test could be options. Technology would be a great form of instant feedback, that could help struggling students who do not feel comfortable speaking up in class.
    Overall, I feel I have made many strides this week, I really enjoyed reading Chapter 3 of "Making Math Meaningful to Canadian Students, K-8, third edition" in particular, because it discussed the pragmatic behind the sense work behind teaching such as the grading schemes and strategies in lesson planning. Each week, more of my math questions are being answered, and I am hopeful that in two years, I will be somewhat of an expert!
    Another exciting thing we did in math class was students doing presentations. One of the students did a really cool presentation on multiplication, which was really engaging. I liked the assignments where we broke down the concepts and used them in practical ways. One of the presentations used number place values and we had to take certain numbers and use them in their correct place value using different combinations. This ended up being a really fun assignment, as we all raced to see what combinations we could come up with. These are the kinds of assignments I hope to include when I am an educator, because we all learned about place values while having fun. I definitely enjoyed the presentations! I think it will be great to continue seeing how my peers use various tools to teach math, and I am hoping to borrow some of those when I am an educator.

Citation

 Small, M. (2013). Making Math Make Sense to Canadian Students K-8 (Third ed.). University of New Brunswick: Nelson Education.

Another week, another reflection (2)

          One of the resources I found out about this week was called a “Settler”  a short quiz that features multiple “simple” math questions to see where students are in the curriculum. For example, the settler I completed had a series of subtraction questions that led to a variety of answers that added up to give one total answer. We had a short amount of time, maybe two minutes, to complete the settler and move on. It seemed like a good way to start students off in thinking about subtraction, which we could have built on in the lesson and done division or something similar. It also contributed to getting our brains working in a mathematical way. The only thing that worries me about such an activity is the added stress when students feel they have to complete a large amount of work (from their perspective) in a small amount of time. I know when we used to do activities like this when I was young, it caused me to compare how fast I could do math to others, and when I couldn't keep up, instead of pushing through, I shut down completely and didn't even want to try. This was especially true because the quick math students were always praised and rewarded for finishing their math quizzes first. That now encourages me to seek different means of finding active learners who can persevere. I would maybe try and do these with an older group and see how they reacted first. I would also take away the blatant reward for the person who finishes first. Perhaps even get students who struggle to first work in pairs, then over time, ask them to slowly start to try the settlers on their own. I definitely think this could be a useful tool, and am still hopeful that the reaction to it afterwards is what discouraged me when I was young, so now I will try and find a way to effectively encourage and engage all students with an activity like this. Below is an example of a settler... 

        

Math Settler. (2013, February 10). "Settlers"[digital image]. Retrieved From http://www.scoop.it/t/mathematics-starters/p/3996398803/2013/02/10/settlers 

  I would like to continue to learn better post-lesson reactions. The main thing I am still trying to grasp is how to react to students who are bad at math without singling them out as “slower” or not able to keep up with the brightest students in the class. I think there has to be a way to alter the lessons in order to encourage learning in all students, regardless of where they start off or their basis of knowledge. With math, if students lack the basis of knowledge, for example how to add quickly, multiplication becomes extremely difficult, then if they continue to struggle with the basic multiplication, they will never be able to move on and do algebra, or statistics, or any other stream of math that they need that basis of knowledge for. The idea that students are being left behind because they are not speaking up when they do not understand something, or need more help, is something I would like to be able to spot out, and then help them learn the necessary skills to move forward. I think a good student is someone who says they need help when they do need it, but an excellent educator can spot that student out without being told. 

After reviewing the “Ontario Grade 1-8 Mathematics Curriculum” document, I was surprised to see the accountability the curriculum places on not only students but teachers, principals and parents. It was nice to see that they hold students accountable for their learning, but also encourage secondary learning sources such as parents and the principal responsible. If a variety of students are not keeping up with the mathematics curriculum, then perhaps there is a pedagogical reason. While this document contains an overwhelming amount of expectations, it seems to be an excellent guideline for teachers to follow, and I enjoyed becoming more familiar with it. 

     I also found some pretty cool resources while researching "settlers" a bit more, one of them from Statistics Canada. This would definitely be a good resource for Junior/ Intermediate mathematics. 

Reflection One (week 1)

Chapters One of Making Math Meaningful to Canadian Students K-8, by Marian Small

starts with an introduction that is very comforting, as it breaks problems down and explains how that helps students. The example of 3X5=15 from page 3 was a great way to put it. A student who understands that multiplication is asking for 3 groups of 5 will understand all multiplication, while a student who just memorizes won't necessarily develop a deeper understanding. 

             One of the other things that stood out was: "understanding is most likely to develop in classrooms that focus on problems to be solved, rather than exercises to be completed.” (Small, 3)  This is very interesting because it aligns with the kind of educator I would like to become. 

Another one of the first things the textbook talked about was math manipulatives. This is something that was stigmatized when I was younger, like counting on your fingers, or having to write out all the steps to a problem, which increased my confusion and prohibited me from obtaining deeper understanding. I am thankful research has improved and that math manipulatives are now accepted and encouraged. While volunteering I witnessed students using them with great success, specifically on the grade 3 EQAO math test. Overall, I am excited to learn more about new and innovative math manipulatives and to continue breaking down the stigma about needing that little bit of extra help with math problems. 

Math Manipulatives. (2006-2016). Rainbow Resource Center. Accessed through http://www.rainbowresource.com/product/sku/MMMSST/

The other section I enjoyed reading was “Different Ideas to Many Strands” (Small, 18). This section explained how to take a suggestion that a theorist many have about math, in general, and to apply it to all the different types of math. If a student can understand the broader idea as a whole, they will find it much easier to apply it to any different kind of math question. The example the textbook used was how certain classes of objects behave in similar ways, when they applied that thought to geometry and fractions they found a pattern in how it appears. This connection can be invaluable to students who are struggling to apply big ideas to small concepts. 

Another thing I was interested in reading was from the “TIPS4RM: Mathematics Processes” article in which it detailed some core basics involved with learning mathematics. The subtopics included things such as: reasoning and proving, problem solving, reflecting, representing and connecting. These are all considered mathematical process skills, which the article explains, can empower students to have a positive outlook on math. The idea that small changes in the way we teach students math can change their entire outlook on the subject is very exciting.  The article uses charts provide a really excellent guide that will come in handy when I try and infuse these “big” ideas into any math lesson. One of my favourite subtopics was “connecting” because that is what I struggled with most as a student. I think making the connection between the lesson and actually doing the problem on my own was the hardest part. I did not find that the teachers knew how to support me in finding links between the concepts being taught and those I was expected to enforce. This section helped me see how I could use different strategies to prevent my students from falling into the same pattern. The connections between math, real life scenarios, and other disciplines is so strong, and I think emphasizing that would be really beneficial to educators and students. 

               These strategies all seem to be working towards students having greater understanding of why we do the math that we do. Instead of abstract work and tedious word problems that do not relate to everyday life, these theories and tools have encouraged me to a more positive outlook on mathematics and how they are being taught in a more positive, connecting, reflective way to students. I am hoping to use some of these and test them out for myself. I can’t wait to see how students react to these concepts! 

Introductory Post

Hi! My name is Casey and I am currently in my first year of teachers college! I hope to teach in an elementary school one day. This blog will be for posts about teaching mathematics. I will do weekly reflections based on the "Making Math Meaningful to Canadian Students K-8, Third Edition" by Marian Small and other interesting mathematic-based readings!
 Before starting this math course I have always been extremely apprehensive about math in general, especially after I studied only English Literature for four years. I struggled with math as a students so I fear teaching it. But with one very hard pre-assessment course under my belt I am regaining a bit of confidence! I think I improved dramatically after completing the modules (so those resources are obviously excellent forms of teaching! I should have taken more notes on that...) and after attending the first class session I am looking forward to more improvement!