Multiplying Fractions Lesson Plan

          This week for class we worked out of the computer lab! Here we talked mostly of data management and statistics, and began by guessing how many cookies were in a jar. The jar was roughly the shape of a cube and many people attempted to answer based on a rough calculation of volume (in units of cookies!). I had taken a different approach. From where I was, I could see two types of Oreos, regular and Golden. I therefore assumed that there were two bags in the container, and went about estimating based on this assumption. I think Oreos used to come in packages with four rows when I was a kid, but the new resealable bags contain three rows (does knowing this mean I eat too many cookies?). I usually buy Double Stuf, which are arranged in rows along the width of the package, and I assumed regular Oreos were as well. This however is wrong, as regular Oreos come in packages with rows that are arranged lengthwise. With three rows, I knew the number in the bag must be divisible by 3, meaning the number for two bags must be divisible by 6. I assumed there were roughly ten cookies in each row, or 30 per package, so my estimate for the two bags was 60. If my assumptions were correct, moving by increments of 6 meant there could be ...54, 60, 66, 72... total. The actual number however was shown to be much higher than the majority of the class predicted (and not divisible by 6). It was over 100! It was revealed that there were actually three kinds of cookies in the jar, which demonstrate how making assumptions can sometimes lead you terribly astray. The stem and leaf plot for the activity showed that most of the class thought the number would be in the range of 60-70. We found the mean, median, and mode for this data and determined that it didn't really tell us anything meaningful in light of the actual number in the jar.

          We then explored the program Tinkerplots. This wasn't something I had heard of before and I found it incredibly versatile and useful. I love analyzing data, but have found more than a few times that I might waste a lot of time graphing two data sets to find no correlation whatsoever. Tinkerplots lets you include a wide array of parameters for each element of a set, and you can easily manipulate and compare each. This would be very useful in the classroom because students are often asked to make meaningless graphs, and it generally wastes a lot of time. Tinkerplots comes loaded with some data and therefore students can immediately explore the different relationships and ways of displaying them without having to find a ruler for every student.

         For one activity we used Geometer's Sketchpad, a program I had used many, many years ago in school. I was surprised that it seemed not to have changed in all these years, and the functionality was still very limited. While you can make lines and polygons and circles fairly easily, I could not find a way to adjust side lengths or angles to a certain value, and instead had to drag points around endlessly and still ended up with slightly lopsided quadrilaterals. While exploring the capabilities of the program, I right-clicked (looking for a way to make the previously mentioned adjustments) and found the animate button. It was all down-hill from there as colourful polygons danced around my screen... I found Geometer's Sketchpad to be a pretty fruitless activity, and unless I missed some very significant features, I am not sure how I might use it in the classroom effectively.

          Much of this class was spent working on a large activity Circles, Cylinders, Rectangles, and Parallelograms. This exercise involved finding different cross-sections of cylinders in order to represent them in terms of familiar shapes. For example, a cylindrical tube could be cut lengthwise and unrolled into a rectangle. The activity was a great way to explore the different relationships between 2-D and 3-D shapes, and for students to estimate and communicate their thoughts on each step of the process.

          It was interesting how our measured circumference using a string was 14cm, while our calculated value was only 12.56cm. There were a couple of possible reasons for our low calculated value, as we had used a diameter of 4cm, which I think was slightly low. The diameter was measured from the ink footprint of the tube, however after holding the tube in the ink pad long enough to be able to make a stamp, the cardboard was soft and warped when pressed against the page. As a result, the edge of the tube folded inward slightly and our stamp measured less than we had predicted it would. Another reason the values may have differed is that it is difficult to wrap a string around the tube without compressing it slightly into an ellipse.

         We cut the tubes and laid them flat into rectangles and then used our l x w algorithm to find the area. While we are working away from going straight to standard algorithms, I would assume that by the time students are doing cylinders this relationship would be well understood. The area of the paper tube was compared to the area of the sheet metal tube to find that the area was 100 times greater for the metal tube. A student might find this confusing, as we were told that the paper tube was 1/10 of the size, and we might want to simply multiply the area of the paper tube by 10. I used a whiteboard to draw out a grid for this part of the problem. If a square was to be placed on this grid, we would be able to find the area by using l x w. Making the square ten times bigger results in the length and width being 10l and 10w in terms of the original square. We can see from such a diagram that we will end up with 100 times the area, and this concept can be extended to volume, which in this case would be 1000 times greater for the metal vs. paper tube.

         The last part of the exercise (the OH NO!! moment) stated that we received metal in the wrong size. That we only had a length of 80cm to work with instead of 100cm. We then explored cutting a toilet paper tube along the spiral edge, resulting in a parallelogram. This was not something I had ever done before and was a great way to visualize the relationship between the two shapes.

         The activity of measuring the sizes of different lakes was a great way to compare how shapes may differ even if they share the same value for perimeter or area. This was related to our opening exercise when we were to find two gardens where the perimeters were the same but the areas differed by 6. To do this I created a chart of dimensions and started to pick out areas that differed by six. I then wrote small notes next to these numbers for their perimeters and started to pair them off. I found that for this exercise there was a pattern where if you moved diagonally you could predict the next two cages that would share these properties. Two cages that would share the desired properties would be 1x6 and 3x4. The next two would be 2x7 and 4x5. Each number increases by one each time, and you could continue this pattern for all fences.

         My favourite activity, and not just because it had marshmallows, was comparing volumes of two tubes. Each tube was made from the same size sheet of paper, one length-wise and one width-wise, and we filled each with marshmallows. The wider tube was then emptied and placed around the narrower but taller tube, then the narrow tube was lifted so that its contents went into the wide tube. The marshmallows that had previously filled the narrow tube did not fill the wide tube, so we were able to see that the volume of the wide tube was bigger. Reflecting on why this might be, considering each was made from the same size of paper, indicated that volume increased more with width as the radius was greater. Looking at the equation for volume of a cylinder (pi)hr^2 shows that doubling the height will double the volume, but doubling the radius will quadruple the volume. This was a great exercise for demonstrating this principle, and something I would definitely try with my class sometime.

          We began last week's class by each collecting a randomly selected shape, then finding a group of people in the class with a similar shape. At this point we discussed what it meant for a set of elements to be similar or congruent. The idea that similar shapes shared the same proportions was put forth, and congruent was decided to be exactly the same. This was extended to the question of colour---do two items have to be the same in colour to be congruent? I would argue yes, as otherwise they aren't exact replicas, though the argument of relevance was also put forth.

          There was then discussion about classifying shapes. In elementary school we glossed over the idea that a square was a rectangle, but a square is also a rhombus, parallelogram, and a quadrilateral. Any more terms we could label it with? Probably. Maybe this topic was avoided---like the time I asked whether peanut butter would be classified as a solid or a liquid---because we were bratty kids, or maybe because it could confuse some students, but these distinctions (or lack thereof) represent a deeper understanding of the material.

          Class seemed to fly by---perhaps because I was somewhat distracted by tangrams. Oh, how I love tangrams. We did them almost every day in grade 4, trying to replicate shapes from a grainy photocopy. The activities for the day dealt with patterning and spacial relationships. The first involved finding the relationship between a a string of numbers. I liked the activity as a whole, as I really enjoy trying to find patterns in every day scenarios. Unfortunately for me, the activity was aimed at a grade 4 level, so it wasn't particularly challenging, but I still appreciated it. The second activity involved transformations on a grid, and used the example of moving a car from one point to another. I felt it was pretty straightforward, however other members of my group had some trouble following the transformation steps for the first part. Students were unclear about where to rotate the shape and how far to do so (instructions said rotate a half turn at the nose of the craft), and what line to use when doing a reflection. For the grade 7/8 level, I felt the instructions were probably appropriate, and students having had instruction on these concepts might not had had the same issues that we saw at our table. The next activity was about navigating bus routes in Hamilton. I thought this a rather practical idea, and liked how a map could be used to find parallel and perpendicular lines. My only concern was that the map was difficult to make out (Garth St. was missing entirely), and if students are unfamiliar with the area they might not be able to fill in the blanks.

          The last activity dealt with drawing reflections. While this seemed like a simple task, it would be good practice for the grade 4 level at which it was aimed to draw reflections and identify lines of symmetry. What I particularly liked about this activity was the differentiation! Materials were available for both left and right-handed individuals, and students were provided with a mira should they not choose to free-hand the drawings. I chose to use a mira and it was actually quite surprising to see how badly my drawings turned out with it. They are probably more accurate than if I chose to free-hand, but the lines are very shaky and uneven, giving it a very crude and unpolished look. I thought this was interesting because while I was using assistive technology, I was finding that my work was not necessarily being improved by it. It highlights how not everything works for everybody and how we cannot just give students a single tool, selected by us, and assume it will work. Also, the music was a great addition for the grade level!

          In class we began with an activity where we were given a set of cards with different relationships depicted on them. There were four cards for every relationship, with a T-table, an equation, a graph, and an illustration with counting blocks for each. It was the task of each table to group the cards into equivalent representations, but to make it more challenging each set had one blank card that had to be filled in. My role was as moderator for the activity, and I was responsible for posing questions to the group as they worked out the relationships of the cards.

          The group approached the problem by comparing the T-tables to the graphs and identifying common coordinates between the two (x,y). This led to some confusion to begin with as graphs and T-tables were grouped if they shared a single point. However, relationships y=3x and y=2x+1 both have the point (1,3), and they cards could not be assigned based on that one point alone. Realizing this error, the group used the graph to compare the change in y with each unit of x to the change in each row of the T-table. Overall this was a reasonable method of comparing relationships, and the group drew from their past knowledge in order to represent each as an algebraic expression.

          The group did not really use the counting block diagrams, and they were also an unfamiliar way of representing the equations to me. I did like how the blocks seemed to be laid out so that there was a multiplication component and separate addition component for each (they were not labelled but seemed to be physically segregated to indicate they were different), as this allowed us to clearly see which portion was growing with the change in x. I liked the exercise where we constructed the relationships with blocks ourselves, and could physically represent what was happening, and even colour-code the addition and multiplication components.

          We spent some time in this class discussing how to ask effective questions. At the time it seemed a little overwhelming, as it was so different from my experiences learning math and science. The Province of Ontario document Asking Effective Questions gives many examples of how to develop student understanding and critical thinking, and the ideas were reflected in a chart that we were given in our cohort group.

          While the chart is not as thorough, it offered a clear starting point in working towards promoting student critical thinking. The factual section of the grid is what we are generally used to asking and answering in math, but these are what we might consider low-order content-based questions. They examine a student's knowledge of details and are easy to assess, but do not consider higher-level thought processes such as analysis, prediction, and application. The chart is a nice way for me to visualize these deeper questions and the phrasing (how could, why might, why would...?) makes for a much softer introduction than provincial documents! That said, once this mindset is activated, there are many other factors that Asking Effective Questions goes into more detail about, such as wait time and making questions meaningful. 

          The first activity of the day dealt with building a bridge framework out of Q-tips. Each extension of the framework resulted in two new Q-tips being added to the original three. We wrote an algebraic expression for this pattern, using given points (1,3), (2,5), (3,7), (4,9). This corresponded to an increase of two Q-tips for every step, so the slope of the expression would be 2. For the first step there are three Q-tips despite the slope indicating two Q-tips for every step. To make up this extra Q-tip we can add it as what would be the y-intercept, giving an expression y=2x+1 for the bridge supports. This expression was then used to solve for step 45, and then we were to find the number of supports for both sides of the bridge at step 45. Something I noticed for this activity was the choice of step 45, an odd number, for the total bridge length. When adding steps to the bridge, each odd step is the shape of a trapezoid for x>1, and a parallelogram for all even numbers of steps. A parallelogram would not sit well over a span, but a trapezoid would be appropriate for building a bridge support.

          The second activity dealt with a series of operations on a number of your choice. It was a fun activity because it did have the essence of a magic trick. You did a series of seemingly random operations to get a very different number than you began with, but what was hidden in the calculations was that you were more or less adding (1+9)/2 to any number, then subtracting the number you picked, giving you again (1+9)/2=5 as an answer. The third question was very similar to the first, and the second was supposed to equate to the same value forward and backward, but I wasn't able to get it to work.

          The last activity was one that I found really relevant. It dealt with two workers banking overtime hours in order to go on vacation. While this may not speak to children, I liked it because it can represent most ways we budget or save our money (I worked a lot of overtime at Canada Post to save tuition). This activity provided a background story with a T-table to illustrate the relationship between the data. We then used this to find different inputs and outputs, eventually arriving at the expression y=3x+1 for Colin's overtime worked. The second question reversed the scenario and Steve worked much less overtime than Colin. It is stated that Steve worked 1 hour of overtime in the first week. Comparing this to the T-table we find an input/output of (1,4). This was a problem however because Steve had only worked one hour, not four. I thought that perhaps the input and output columns had been reversed. In this case, it implies that (4,1) is the first value in the table, and Steve has worked only 1 hour of overtime even after four weeks. With this information, it was then possible to write the expression y=(1/2)x-1.

Weekly Reflection

          Last week's class began with an activity in proportional reasoning. A scenario was given in which there was a child and a giant and each had measured themselves with their own hands. The giant then measured the child with his hands, and we were asked to find out how many child hands tall the giant was. If that is a little confusing, I am sorry! The idea was that we could use own knowledge of each of their heights in their own hands to find an equivalent fraction.

          I approached this problem by finding the relative size of each hand. The child was six child hands tall, or four giant hands, therefore we could say that 6/4 was the size ratio between the two hands. Using my knowledge that one giant hand was 6/4 or 3/2 or 1.5 child hands, I could say the giant was then 6 x 1.5 child hands tall, or 9. The class was asked to demonstrate this proportional reasoning using the document camera, something I had never used before, so I wanted to try it out. Before I could however, I had the opportunity to see a peer's strategy for finding the giant's height, which gave me a new idea for solving the problem.

           Rather than equating the size of each hand, which some may find confusing, I decided to demonstrate the problem by equating their heights instead. Working in giant hands, we knew the child was four tall and the giant 6 tall. This indicated that the giant was 6/4 or 1.5 times as tall as the child. I thought that this was a much easier way to visualize and explain the problem than my initial approach! Using this ratio, we then knew the child was 6 child hands tall, and the giant was 6/4 that value, which works out to 36/4 = 9. The latter approach is nicer in that we set up our solution by equating the height of each individual. As our height does not change, we can quickly see that there exists a proportional relationship between them that is static. Equating hands is a perfectly reasonable approach, but as discussed in class a student may find the idea of measuring in hands somewhat unclear, leading to confusion when trying to carry forward this ratio to find a solution.

          The activities for the class also addressed proportional reasoning. The first of such dealt with Hallowe'en candy taken from one student and placed in a bag with other candy. The candy of the student was distinct (this was not clear on the handout, but explained to each group) and therefore with each handful taken from the bag we were able to estimate the proportional relationship between the student's candy and the rest in the bag. This was very similar to the exercise used for the grade 6 EQAO problem solving task, except had fewer variables (two kinds of candy vs. three colours of gumballs) which I liked. The activity was a great way to investigate proportional reasoning, and could be further extended to later topics such as data management and probability.

          The second activity consisted of a series of diagrams, each having two distinct sets of objects. The student was asked to indicate which set had more of a certain element. The instructions for this were intentionally vague, which spoke to the devil's advocate in me as the task could be approached in two ways. The first was to take each situation literally by simply counting which group had more, whereas the second approach was to represent each element as a fraction of the set and compare the fractions. Each approach resulted in different answers, which could be a valuable exercise for students to explore.

          The last activity dealt with two families eating pizza and the number of pizzas each was to buy. Each family purchased their pizza based on a proportion to the number of people, and the total amount for each case was considered. The last step of this activity had the two characters eating a total of two pizzas at different rates, and the number of slices each person ate was to be found. I liked this addition to the worksheet as it not only dealt with proportional reasoning, but with rates as well. I have found that students in my practicum have difficulty relating problems to applications, and their poor understanding of unit rates is generally responsible for this.

Weekly Reflections

     When I was seventeen I was hit by a car. I was riding my bike to work early one summer morning, and the light changed to green just as I arrived at the intersection. It was probably not even one whole second later that I was halfway into the street and I turned to see that a car had run the light very, very late. It was the sort of moment where you knew things had gone wrong, but you didn't really have time to do anything about it, and all I managed at the time was a resigned "oh--" before I was mowed down by the nose of the Dodge Neon and left lying in the median, dazed, helmet split in two.

     Today's learning activity sort of felt like that experience all over again. I had a clear direction in mind, but somewhere along the way things went amiss, and I didn't get scared or frustrated, but I didn't really persevere either. I just got dazed, and maybe a little flustered. And like the bike incident, where my helmet was split in two, my head came out of it all okay, and I learned something. On my bike it was always wear a helmet. Here, I'm not sure I can put it so elegantly into words quite yet. I feel like my presentation fell a little flat---like a botched solo in jazz band, with everyone there to see. Everyone turned their focus on me like "yeah, this is it" and I lifted my trombone like "yeah, this is it". And then I flubbed a few notes, and never quite found my place again, and with the rhythm section still chugging through the solo section, there was an awkward moment for everyone and we all just hoped it would end so we could pretend it never happened.

     Perhaps that's a little over the top, but it's curious the sorts of feelings that whoosh back in situations such as these. I was always more of a marching band kind of kid. Structure, structure! Everything by the book! No artistic liberty here! And so I have a hard time with these moments and just want to scream "just tell me what to do!" But reflection is always key.

     I had put a lot of time into thinking about my activity---what I wanted it to mean, what value I felt it had, and how I felt it should flow. None of those daydreams came to fruition. In 8F01 we would call this an opportunity for reflection, and I have many thoughts on how it could have gone differently. First, I think I may have overshot the target. Perhaps I should have stuck to a simple activity and not tackled the more complicated concept of directional vectors. Though all vectors have a direction, so that's a redundant statement. But maybe that's the point I'm getting to---I don't look at these problems in the same way others do. And I don't mean that in a positive or negative way for any party, I just get grandiose ideas of how people will respond to math, when reality is nothing like it. I could have boiled down the activity to deal only with the directions (or signs) of each value, and maybe that would have been more productive, but either way the problem didn't seem very engaging in the end. To be honest, I hadn't felt the preceding activity was particularly engaging at first. But then we totaled the ledger and had a negative value! The kid owes money to someone! That got the wheels turning. Who do they owe? How will they pay the debt? What will happen if they don't? It ended up having a very peculiar outcome that begged for an answer. 

     I heard some great discussion during my presentation, and even heard someone talking about how we might label east as positive and west as negative, and what would happen if we did it the other way around (nothing!). But Part B where students were supposed to share feedback on the questions was met with a lukewarm reception. It was discouraging as someone that really cares about sign conventions. Like, really cares. And then I did this thing where I walked back and forth like a fool to demonstrate directionality. How fun. In retrospect, it wasn't a terribly engaging activity. I had felt that there was so much potential in it, and I really wanted to share an instance where you might use multiplication and division of integers in daily life, but I really just made a Physics 101 worksheet and demanded that everyone do it. Rather traditional if you ask me.

     I want to just shrug it off and say "oh well", but obviously there is something for me to learn from this experience. And it's not that students don't like math, aren't engaged, or any number of other things. It's that not every idea can be a winner. Not everything will always go as planned when teaching, and you have to be flexible and adapt, and maybe go home and listen to some funky music when you're down on yourself afterwards. There will always be moments like these in life, but I have a cup of coffee, some good tunes, and an online forum to vent just a little. At the end of the day you move on. It's over and done, on to new horizons, and hopefully you learned something along the way.

Weekly Reflection

     I am sure taking my time writing my reflections for last week! Last class saw some new activities introduced to tackle the dreadful problem that is fractions. What child doesn't love fractions? Personally, I was one of those people that always hated them. I think it had something to do with improper fractions, but also because I loathe the Imperial measurement system, and fractions always remind me of that. Growing up, I always had a desire to make things---build things---and the Imperial system made sure to swiftly kill that urge. Toto, I have a feeling we're not in base ten anymore.

     There were some good activities presented. Roll to Win dealt with rolling two dice twice, to get two fractions whose product was a whole number. I liked this idea in theory, but in practice I found myself rolling over and over and over and over and over. 31 times. The optimist would say that this was excellent practice in multiplying fractions, and I would hesitantly agree. The next activity, Who Walked the Furthest? was sort of self-explanatory. I found this to be a good way to practice finding common denominators and adding fractions. I put everything over 12 and away I went. I made a mistake adding (shame on me!) but my elbow partner was there to save me. And the last activity, Dollars and Decimals entailed finding all of the combinations of quarters, nickels, and dimes that you could sum to a dollar. I laid this out into a chart with each coin as a column, and then worked my way down starting with the largest number of quarters, then to the smallest. I found 29 different ways to make a dollar (this is verified by algebra.com, though I am not certain what credibility can be attached to that). I really liked this activity! I think there are many ways you can use the idea, and there is a lot of potential for people like me to fuss over whether they got them all or not. I did not however feel it was terribly relevant to the topic of fractions, and that it fell more comfortably in the category of decimals. Oh yes, I know they are the same thing, but I doubt anyone added four quarters as 1/4 + 1/4 + 1/4 + 1/4, and instead used $0.25 + $0.25 + $0.25 + $0.25. There's nothing wrong with this---I think we'd call it a strategy---but it somewhat defeated the purpose. But perhaps I liked this activity the best because it had the least to do with fractions?

     In preparing for my own presentation this coming class, I read the material weeks ago. Having read it all, I had some ideas, but I had no first-hand experience directing an activity in our classroom, so I sat on it for a while to see how others approached it. I was really set on doing the integer football activity in the van de Walle text, however I think it's a little confusing. I think it's a great application of integers, but having two teams moving in two different directions, with a sign given to each, and only one team has the ball, and...and.... And it concepts get buried in trying to figure out who has possession of the ball! At least that was how I felt about it all.

     With integer football out, I decided to do something based on the idea, but hopefully more straightforward. A person is standing in the middle of a number line and has to choose between two trips---one east and one west---and we'll discuss distance and time in the process. My thought was that directions are more intuitive than simple positive and negative values, and could be a physical way of showing something move up or down a number line. I think the most common statement in math next to "I hate math" is probably "when am I ever going to use this?" So I think that in showing multiplication and division of positive and negative numbers in a context will be useful. I won't say they're positive and negative, just east and west to start, and build from there. Integers are just opposite numbers after all! More on all of this once I see how it goes...

Weekly Reflection

     Today kicked off the string of class presentations for the term. It was fun to work through some activities and see how other approached the problems. I always considered math my thing in school. I had an exceptionality and fast-tracked all of my high school math courses, but I never considered math easy. I always thought that people that were good at math had to be good at mental math, and I always felt I was terrible at it.

     We played a game in grade 4 called Math Castle, in which alien ships would shoot lasers at your castle if you weren't fast enough at solving increasingly difficult math problems that would float up in the clouds. I wasn't very good at the game. I was slow and didn't know my times tables past ten. I felt embarrassed and discouraged, and my confidence took a blow. It took another in grade seven when variables were introduced. What was x? What can we do with it? Simplification didn't seem all that... simple.

     I worked hard at math though. Math was supposed to be my thing and I sort of felt that if I couldn't do math that I shouldn't be in a gifted program. I spent hours studying math at home, worked on algebra on weekends, and those courses I fast-tracked in high school? I spent hours each night slowly working my way ahead. I didn't have a gift, I just worked at it. And when I started university and had a little more freedom, I stopped working so hard. And I sank.

     I excelled at math within the traditional confines. I did my worksheets, learned my tools, memorized my terms, and practiced using identities. But my exceptionality wasn't in math per se, it was in problem solving. I think sometimes that this, combined with my dedication, was what made me a strong math student. Math was a puzzle for me to solve, and for the most part, I loved it.

[Note: My IEP also said I had no social skills, so that's where I found all the time to work on math]

[Content removed for reasons of pessimism]

     I saw my abstract problem solving come into play today in class. Perhaps I spent too much time watching Dan Meyer talks on youtube this morning when I couldn't sleep, but I found myself looking at our math activity and thinking "there has to be a better way!" This was in regard to the horse racing activity in which we had to add all of the numbers on the cards to see how far our horse had gone. Ten three digit numbers, no calculator, and little patience. My problem solving nature kicked in and assessed the situation. The numbers differed by regular intervals, with no gaps, so one could "add" them by finding the average value and multiplying it out. But I didn't bother calculating the average longhand, I listed the numbers from highest to lowest and paired them off, highest-lowest, next highest-next lowest, etc. This works in from the outside towards the middle of all of the values, and you are left with 147 and 157. The average of these two will then be the average for all of the numbers: 152. No long calculations. And if there had been an odd number of values, the average would have simply been the middle value, or the last one remaining when pairing them off.

     I was pretty proud of myself for this lazy approach. Not because it was easier or better, but because I had set out to find another way of approaching it, and it worked. These are the sorts of discoveries we want our students to make. I love challenging myself with little things like this. Borrowing a term from Dan Meyer, you could say that I find these patterns perplexing and want to know more, want to find ways of breaking them down, and want to find models that can be used to represent them.

     I want to inspire students to find these connections and learn math from the ground up. Tools aren't always helpful---someone else made them up! I borrowed my neighbour's wheelbarrow and it was rusty and hard to push, and when I wanted to give it back he gave me a hard time because he didn't want to go back out and open his garage. This is our traditional education---or just a really bad metaphor, I'm not sure. But it's someone else's tool that they found useful and are trying to pass off on you, even though sometimes it's more trouble than it was worth.

     I used my knowledge of math and love of science to go on and study chemistry. It was in a graduate level course on electrochemical systems that I encountered the Butler-Volmer equation. This thing is gross, but it didn't just come out of nowhere. No one handed this formula down to Butler and Volmer, they studied what perplexed them, and came up with it. There are many other formulas that branch off of this equation, and I would consider them tools because most have certain stipulations in order to use them properly. You have to know how to apply them. And I think I've come full circle now because the tools we spout to students are the same. You have to understand them, and you won't unless you start from the ground up. Let students be perplexed and discover strategies that work for them, even if they're longer. There's nothing wrong with using a little more paper than the person next to you in class.

Weekly Reflection

     Today's class began with an activity counting manipulatives. Yellow and orange blocks were used to represent a fictional currency that a certain "Matt" had accumulated while at summer camp. The exercise was for us to work with our elbow partner to count how many blocks we had been given in a plastic baggie. We quickly divided the allotment into groups based on size, or denominations, and then went about counting the number in each group. It was fairly straightforward counting our sets of ones, tens, and hundreds. We had 272 blocks total.

     After tallying up our sets, I thought we might want to further divide them into colours. We had been given no information about the meaning, or lack thereof, of colour to the exercise, but we thought it might become important. Did different colours have different values? Did they come from different activities? Were some blocks simply left out in the sun too long and they all should have been orange? The role colour played was never discussed, but I like to image that one paranoid camper kept them closely guarded in their pockets through rock climbing, kayaking, and even nervous moments around the campfire. But I digress.

     The consensus in our class was that everyone had taken a similar approach to the counting problem, counting largest to smallest. When asked to add two groups together, students took one of two approaches: adding the new blocks to the initial group from largest to smallest, or by regrouping all of the blocks and counting all over again. Here we saw a clear demonstration of how two students might approach a problem differently. We were then asked to add two numbers the way we have always been taught, have always practiced, and have always thought of doing so. Ones, tens, hundreds.... Smallest to largest? A clear contradiction from the method we had all used in our camp economics class.

     This new method made sense from an approach using manipulatives--I would never count up all of the money in my piggy bank by starting with the pennies. Pennies don't even exist anymore! So why is this different from what we all know and practice? Because we have been taught an arithmetic tool for evaluating such expressions. But it is no way for students to learn how to add. I know, use, and love this tool, but I certainly don't add smallest to largest place values when doing mental math!

     So what do we do with these tools? We ignore them. They're great when you don't have reams and reams of paper for writing longhand, but the purpose here is to understand the principles of what we are doing. Any other time we can use a calculator to save paper.

     Reading parts of the curriculum, one thing in particular piqued my interest: that all children are capable of doing math. Just as children read and write, math is its own language and follows a set of rules that are, unlike many constructs of the English language, quite logical. We then talked about brain plasticity, a concept that I am quite familiar with from courses in psychology. It was an idea that I had never considered before. That one's ability to do math, much like speaking a foreign language or playing a musical instrument, requires practice to maintain and improve upon. Unfortunately, most people don't exercise those synaptic pathways enough, and worse yet, are often discouraged to. Math is "geeky" and people "hate" math. But math plays a role in everything we do. What is time? What is money? What is a kilogram? These are questions that math is important for, and the kinds of questions that I absolutely love to think about.

     The "Handshake Problem" was then posed. If you have n number of people in a room, how many handshakes must occur for each person to shake hands with every other individual only once? I quickly wrote on a whiteboard that it was equal to (n-1)!, and started drawing tree diagrams. But when we began talking as a class I realized I had been impulsive and mistaken. My grade 12 data management teacher would have cringed! It is actually 5C2. Woops. But we all learn from our mistakes, right?

Hollywood Hates Math

     As one of the seemingly few who has always loved math, Carolyn Johnson's article about Hollywood's "math problem" spoke to me; Indeed it seems all too common that people take pride in their inability to do even basic arithmetic; But math plays an important role in many aspects of our society; When elections are won on economic promises, how can one make an informed decision if they have no concept of number sense;
     Ms. Johnson points out how the so-called math problem diverges from what we might see as a language problem; While being bad at math is a boastful opportunity, a narrow vocabulary is not seen in a similar light; Nor, I imagine, is illiteracy;
     I prefer to compare the math problem to a grammar problem; I don't have to know how to properly use a semicolon in my everyday writing because no one expects me to; But that semicolon is calculus, and no one is asking you to master it;
     Having no number sense would be similar to never constructing a proper sentence; But we do pretty well when it comes to using commas, and periods, and quotation marks; So let's make sure we give math its commas, and periods, and quotation marks and give it the chance it deserves; And you can leave the calculus to me; And I'll leave the semicolons to you;

Weekly Reflection

     Simple activities can be thought-provoking. Allowing students an opportunity to consider all of the aspects and approaches to a what may seem like a straightforward problem can be used as a demonstration to show that each student possesses a unique capacity for problem solving. As discussed in the Dan Meyer TED Talk, being less helpful to students may benefit them by allowing them the opportunity for discovery. Real life, practical questions are often the most engaging. Their basis in reality and not formulas or arithmetic may engage even the least inclined students. In order to accomplish this however, it is recommended that "noise" be removed from problems, as this extraneous information may detract from the focus of the problem at hand. Assignments should never be rooted in formulas as this does not teach any of the underlying critical thinking and understanding of problem solving. This "black box" approach to arithmetic generally provides all but one piece of information for a given formula, and asks students to simply plug the values in for the answer. Students have become focused on finding "the answer" as a result, despite the fact there is almost never a single correct method for any exercise.

     My experiences as a univeristy-level teaching assistant in chemistry highlighted how misconceptions about mathematics are ingrained in students. The focus of student questions and discussion dealt with a single value, such as a mass, or volume, or pH, and how exactly their solution matched an answer key. Students often neglected consideration for the process by which they got their answer (beyond using a formula that they were familiar with) and rarely considered whether their reasoning or solution were logical. It was obvious that students were always taught in a very singular, traditional approach where methodology was downplayed and students were all expected to think and learn in a very monochromatic way.

Introduction

     One of my earliest memories of math is an older boy from my neighbourhood that I used to catch the bus with. He was known as "The Human Calculator" because even from a very young age he was exceptionally good at mental math. Questions that I may struggle with even now, he was able to compute in his head immediately. So it came as a shock to me when in grade three I was identified as gifted, and found myself sitting next to The Human Calculator. I had always loved math, but despite the claims in my IEP that I was exceptional, I knew I was nowhere near as good at math as The Human Calculator. I didn't understand it at the time, but I eventually came to realize that we all think and learn differently, and that being exceptional at mental math may not necessarily translate to being exceptional at problem solving also.
     My high school's self-paced program allowed me to fast-track every math class offered by the department before applying to McMaster to study math. I however had a last minute change of heart and enrolled in the sciences instead, graduating with a degree in chemistry. My math skills have since become a little rusty, but here I am, hoping to learn how to teach math effectively. I am particularly excited to explore different methods of problem solving and seeing what others come up with and which strategies and tools they find most helpful. And while math was always a strength of mine, I am excited to have my eyes opened to subtle intricacies and ways of working course materials that I had never considered before!