Week 6 Reflection

In this weeks class we were responsible for reading chapter 13 in the Making Math Meaningful textbook. This chapter is focused on Ratio, Rate, and Percent. Ratio, Rate, and Percent are mathematical concepts that I feel as though I have a good grasp on. I have always been solid on these concepts and there was not much in the textbook reading that really stood out to me or made me have an ‘AH-HA” moment. Ratio T-charts and the ability to construct them have stuck with me from when I was a student in elementary school. Expressing ratios as fractions is another mathematical concept that has stuck me with. One thing that did stick out from the textbook for me was on page 305 where the textbook talks about how ratios are implicitly taught from a young age. This ‘ratio-thinking’ is ingrained in our thought process. For example, in the textbook they talk about how in kindergarten children are taught that there are two eyes for every person. This in itself is a ratio. I found it interesting that ratios and rate were being taught before students even realized that they were learning the concept.  

            In class Patricia also spoke about how important proportional reasoning is. Not only is it important to fractions and ratios, but also, it is important to mathematical understanding in general. After thinking about proportional reasoning, it can be argued that it is the basis, or the most important concept, in all of math. Having knowledge of proportional reasoning allows you to be able to apply mathematical concepts to real life situations. For example, in the text on page 304 it gives the example that if a person runs 100 meters in 17 seconds, then it can be said that they would run one kilometer in 170 seconds. That is solid mathematical thinking because if the person runs ten times longer you just multiply by 10. However, this is not taking into account the fact that the speed, or pace of the runner, will not be the same during the two distances; this is the importance of proportional thinking in real life mathematical situations. Laura’s lesson planning activity demonstrated proportional thinking very well in my opinion. She chose a lesson activity in which there were groups of children and you were responsible for picking which group had more girls, or goldfish, or squares etc. This activity caught me off guard and was a good example of how important proportional reasoning is. For example the first question involved two groups, one with 5 kids and two girls, and the other with 4 kids and two girls. We were asked to pick the group with more girls, and at first I said that there was equal amounts of girls in both groups. However, thinking proportionately, the group with four kids and two girls has more girls, because 50% of the group is girls compared to 2/5 in the other group. I think that this activity was successful in showing the importance of thinking proportionately when applying math to real life situations.

kyle.goemmer. (2010). Proportion. [Online Image] http://bit.ly/1Nuvsyz

            Lastly, Patricia also spoke about what makes a good problem again this week. We revisited this idea and had a quick class discussion on what we thought the qualities of a good problem were. We summarized that a good problem is one that has a wide base, it is relatable, challenging, and also engaging. These discussions on what makes a good problem are vital to my learning as I feel as though coming up with good problems everyday will be challenging during my first few years as a teacher, and the advice that Patricia gives us now will indefinitely help me in the future. In addition, she also brought up the fact that a good problem always has to be capped off with sufficient time for consolidation, or to bring everything that we learned together at the end of class. This consolidation helps to summarize what was learned and encourages retention; if you don’t consolidate at the end of the lesson and tell the class that we will pick it up again tomorrow, chances are the children will forget. This was a good pointer as she pointed out that even if the activity runs longer than expected, you would be better off ending the activity earlier than scheduled just to consolidate and allow the children time to absorb the information that you have explained to them.

             

Week 5 Reflection

In this weeks class we were responsible for reading chapter 14 in the Making Math Meaningful textbook. This chapter was focused on integers. However, integers are a fairly straightforward topic and we did not actually spend that much time on this topic. Instead of breaking down the intricacies of the mathematical topic of integers, Pat wanted us to move away from just trying to teach integers by teaching rules. Instead of just hammering into the students heads that multiplying or dividing a negative and a positive will equal a negative, or that two negatives multiplied equal a positive, she told us that we need to be able to explain the reasoning and thought process behind these algorithms. She also said that until we are able to fully explain these concepts, we should not be basing our lessons around teaching them; the formula is not as important as the reasoning behind it.

            We actually spent a lot of time revisiting last weeks topic of fractions as Pat said that she realized many of the students in the class, including myself, were having a little bit of trouble understanding fractions from this new perspective of us being the teacher. She began the lesson with an activity called red light green light. I really enjoy the way that Pat teaches and you can see the experience that she has in the way that she runs her lessons. She chooses an activity that gets the class physically involved and then relates it to a lesson. In the green light game you are thinking about fractions in terms of who is closer to the end, without even really knowing that you are, as it is just part of the game. You are fostering a real life connection to the mathematical concepts in the students. I also really enjoyed the Mr. Tan problem involving the broken tile. Again, you can feel Pats experience in the way she leads the class. I hope to one day be that comfortable and smooth in my lessons; she dropped the tiles at just the right moment when explaining the problem and smoothly involved the mathematics into her story telling. I think using stories to teach is a great technique, but for me, the one challenge that I find in that is being able to find an appropriate story for the lesson, and making sure that what you are trying to teach is clear and understood.

Vinayak Hedge. (2011) Porcelain Tile. [Online Image] http://bit.ly/1PaO89z

            We also covered an interesting technique that we were never taught in school. Much like last week, Patricia showed us that the algorithms that we were taught in elementary school might not have been the best way to teach the concept. For example, when dividing fractions, we were taught to flip the fractions and multiply, however, Patricia showed us that if the denominators are easily divisible, you can simple divide across, much like you can just multiply across fractions. Furthermore, if the denominators are not easily divisible, you just have to convert the fraction into one that is easily divisible. This makes the division of fractions much easier in my eyes and is something that I will keep in mind when teaching fractions in my own class. Again, this is evidence that the concept behind why you do things in math is more important than how you do it. I cannot explain why we were taught to flip the fraction and multiply, so why would I teach that to the students. This does not foster a deep understanding of math, but rather it just leads to memorization of algorithms. This style of teaching and learning may work for some students, but you can be sure that it will not work for all; if you are able to explain the concepts fully you have a better chance of making sure that everyone leaves your math lesson feeling like they understand fractions better than they did before class. 
            We also had a good discussion in class about what to do when a child asks you a question, or provides input that you were not prepared to address, or that you don’t know how to respond. Pat told us not to be afraid to ask the class about what they know and for explanations and interpretations of mathematical concepts. She says that as new teachers we will undoubtedly be nervous trying to lead these types of discussions because we are not sure of what a child is going to say. However, she told us that we cannot be rattled, and we have to be ready for a student to say anything. Our job is to try and make sense of it on the go, and to relate it to our lesson if possible. There is no right or wrong when asking students what they know.  We also explored CLIPS [http://oame.on.ca/CLIPS/] which is a great resource for teachers that identifies common gaps in student learning and gives lesson plans/teacher notes and games that students can play to fortify their learning of troublesome concepts. This is great resource and I can see myself consulting it to get lesson ideas and to fortify my own.

Week 4 Reflection

gideonking. (2008) Lesson Planning. [Online Image] http://bit.ly/1hf1YJe 

          In this weeks class we covered chapters 11 and 12, which were Fractions and Decimals in the Making Math Meaningful textbook. In addition to the lesson that was focused more on the importance of understanding the basics of Fraction than decimals, we also had presenters for our lesson planning activities. This week I was one of the presentations and I was assigned chapter 12, which was focused on decimals. After presenting my activity I was immediately disappointed in both my activity and my presentation. This was the first time that I had done something an activity such as this and I do not think that I did a very good job. The activity that I chose was 12.13 on page 294 in the Making Math Meaningful text. My activity was grounded in teaching students division of whole numbers by decimals. I posed the problem that Ahmed needed a dollar to buy a chocolate bar, but his father gave him a handful of change. I then asked the class to try and find out as many combinations of nickels, dimes and quarters that equal a dollar. I feel like my idea on how I was to convey the division of decimals by whole numbers got lost along the way in my planning. Instead of asking for combinations, I should have asked, how many quarters would Ahmed need to make one dollar, then how many dimes, then how many nickels. I should have then asked, as a bonus question, how many different combinations of nickels, dimes and quarters could Ahmed make to buy his one dollar chocolate bar. I think that this would have been a much better activity and would have demonstrated division and multiplication of decimals by a whole number much better than the activity I thought of at first. However, hindsight is 20/20. The point of this program is to learn how to be an effective teacher, and having never done this before, I think that I have learned from my critical mistakes and next time I have this opportunity I will focus my activity more; I think that I got caught up trying to make the activity from the book more original than it had to be, and thus I lost the whole point of the activity in the first place. 
          In terms of the lesson of the day I think that Patricia made some very interesting points about fractions. She made a point that most of our teachers when we were growing up probably just skimmed over fractions and that this is a huge mistake. She explained that after whole numbers, fractions are the next important building block for all mathematical knowledge. I had never thought about that before. She also explained that many of our math teachers probably didn’t understand fractions that well, which is the reason why we were taught specific algorithms and that any other method was wrong. For example, she showed us that to divide two fractions, we don’t need to flip the numerator and denominator and multiply, but rather, if the denominators are easily divisible, you can just divide across, much like multiplication. This is much easier, and makes much more sense than having to flip the fraction and multiply. This is a strategy that I will remember and will use in my own teaching. Instead of saying that there is only one way to do something, this algorithm is an opportunity to showcase to the kids that there are many ways in math to get the same answer. In summation, I learned a lot today about being a successful teacher, not only in math, but also in general. Lesson plans and activities must be clear and concise, as well pertain exactly to what you are trying to teach. Instead of getting caught up in trying to make the curriculum fit a lesson plan and activity, your activity should seamlessly fit within the curriculum. My failure in this lesson planning activity has shown me that I need to work on being clear and concise in my planning and formulating. However, mistakes foster further learning and I am looking forward to building my knowledge of how to be an effective teacher.