A Journey of Self Discovery: The Evolution of My Scientific and Mathematical Thinking
Seeing the Connection Between Math/Science and the World Entry 1: October 20, 2015
While creating our Math Trek we became aware of the math around us. We began to see math is everywhere and to notice how intricately it is interwoven into our daily experiences. In the beginning of this task it seemed difficult to think of interactive math questions but by the end of the Trek I was noticing math everywhere. Structures like bridges used triangles for support and there were squares and circles seemingly everywhere! It is quite interesting to note how easy it is to distract ourselves from the geometric shapes around us while we go about daily activities. However, upon actively thinking about math, its presence becomes so apparent.
There are geometric shapes that can be seen in architecture, signposts, fashion, and transportation etc. Although manmade geometry is evident there are also geometric shapes and patterns that appear naturally in our environment. Boaler mentions the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, ... where each number is the sum of the two preceding numbers, and the golden ratio: 1.618:1, a recurring ratio in nature. It is derived by dividing each number in the sequence by the one before it. Boaler notes "what is amazing about this ratio is that it exists throughout nature" (Boaler 16). It is apparent in flower petals, pinecones, shells, and even in proportions of the human body.
Patterns were a key point of interest during our Math Trek. People, myself included, find patterns and symmetry to be very ascetically pleasing as well as useful in architectural planning and the layout of structures, fences, and parking lots etc. We noticed patterns everywhere from gardens to the front of buildings and the layout of telephone and flag poles. There are patterns in the sidewalks and interlocking brick patios and umbrellas. As the Fibonacci sequence demonstrates there are patterns which exist in nature as well.
During the Math Trek we discussed area, perimeter, rate of flow, measurement, circles (including pi), acceleration, force and created our own inquires through wondering about the math in our environment. During my bus ride home I began thinking of gravity, collisions, and seat belts. While walking down my street I had a near collision with a rather large man on a bicycle. I thought about the force of impact that would have occurred if I did not jump out of the way and of Newton's Laws with specific emphasis on every action having an equal but opposite reaction.
During my Community Service Learning last week my grade 5 class was discussing the importance of estimation and the role it plays in math and in our lives. We discussed buying groceries without running out of money, getting the best deal on a product, and receiving the appropriate change when making purchases. These are skills I use on a daily basis but I rarely think about with an emphases on math.
As Boaler notes: "Mathematics is a human activity, a social phenomenon, a set of methods used to help illuminate the world, and is part of our culture" (Boaler 16). It is a set of tools that allows us to investigate, explain, and navigate the world around us. Even though it is easy to become inattentive to the math that surrounds us it becomes evident when we actively think about it in a way that allows us to be creative. If we allow ourselves to wonder and to search creatively for math around us we will unveil the world of math that exists right under our noses.
Works Cited Boaler, J. (2015). What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (Revised Edition). NY: Penguin Books.
Learning By Interacting With Student Thinking Entry 2: October 20, 2015
During our class on October 6, 2015 we were asked to wonder about an interesting but simple device: a valve with one deflated balloon attached to side 'A' and one inflated balloon attached to side 'B'. The valve was shut. We were asked to examine the structure and make observations. We were to ask ourselves, our elbow partners, and our group questions about the device. To inquire about our growing curiosities.
According to the article Process of Science, curiosity is deemed especially important in the scientific community. Perimeter Inspirations discuss how: "scientists are curious. A sense of wonder and awe is essential to scientists" (Perimeter Inspirations 4). Wondering about this device enhanced our curiosity and we began to make hypotheses. We were then instructed to record our hypothesis regarding our inquiry of what would happen when the valve separating the two balloons was opened. By allowing us the opportunity to observe and hypothesize about the device prior to opening the valve our sense of curiosity was intrigued and when we did eventually open the valve I was in awe.
My intuition told me the inflated balloon would transfer some of its air into the deflated balloon until they reached equilibrium or both balloons were of equal size - this however, was not the case. Once the valve was opened, nothing happened! We were even more curious once the suspense was over and yet the reaction we believed would occur did not. We were inspired to ask more questions and discovered that when a significant amount of force was applied to the inflated balloon 'B' eventually the air was forced through the valve and rapidly began to inflate the deflated balloon 'A'. I was stumped. My intuition had deceived me. As Kline notes: "unfortunately for us, our senses are not only limited but also deceiving. Trusting solely in the senses can even lead to disaster" (Kline 21). As scientists we cannot trust our senses or our intuition we must make guesses and then test them through experiment. In this case I was incorrect with my assumption. I could see I was incorrect yet I could not understand why.
Through peer discussion I was able to figure out why balloon 'A' did not inflate when the valve was opened. Through a back and forth discussion in my group we came to an understanding based on some student's prior knowledge in conjunction with each of our varied perspectives on the observations. The force required to initially inflate the balloon on side 'A' was not equal to the force exerted by the gas released into the valve from the balloon on side 'B' and therefore balloon 'A' did not inflate nor did balloon 'B' deflate.
In the article by Perimeter Inspirations, emphasis is placed on the importance of peer discussion in the process of scientific exploration and collaboration. They note: "science requires collaboration... collaboration allows researchers to access specialized skills or knowledge needed to solve the problem... scientists must be able to communicate effectively with each other in order to collaborate" (Perimeter Inspirations 5). During this lesson, my understanding came directly from interacting with student thinking among my peers which lead me to discover the rationale behind the experiment. Although it is on a smaller scale then global scientists collaborating, in our class different peers brought different information to the forefront and allowed for a mutual learning experience to occur.
Works Cited Boaler, J. (2015). What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (Revised Edition). NY: Penguin Books. Kline, Morris. (1985). Mathematics and the Search for Knowledge. NY: Oxford University Press. P.p. 21-37. Perimeter Inspirations 02: Process of Science (Handout). P.p. 4-6. Available online: https://perimeterinstitute.ca/outreach/teachers/class-kits/process-science
An A-Ha Moment! Entry 3: November 17, 2015
For me, November 3rd's class held a pivotal experience. Although I thought I understood the concept that manipulatives can enhance mathematical thinking and learning for students, it was not until the 3rd of November, 2015 that I truly understood how they were beneficial for myself (outside of learning fractions). It began with the Desert Car Activity.
The Desert Car Activity suggests there is a series of identical cars that can travel 50km with a full tank of gas. However, a person must get to the end of the dessert 100km away. This problem asks students to find the minimum number of cars required to take the person across the dessert and explain how this could be accomplished. It requires problem solving, reasoning, and communication.
When I began to solve this problem I immediately drew the given details so I could visualize them. I had a line drawn that represented the 100km distance across the dessert and the mark of 50km where the cars would run out of gas. Through drawing the known variables I could deduce more then two cars would be required to get across the desert. Upon figuring this out I drew four X's on my paper to denote cars but I was unsure of the math in my head. I was hesitant to make further marks on my paper for fear of making a wrong calculation or assumption and needing to restart. It was then I thought to use manipulatives.
As a group, our table puled out our Hex a Link cubes and made cars. We used 4 cubes for each car, a number that could easily be divided into 0%, 25%, 50% and 100%. With the added visual representation things became clear. It was much less stressful to attempt a trial and error method with this manipulative as we could make changes easily through adding or subtracting cubes as needed, no erasing! From here solving the problem came quickly and was a pleasant experience, as apposed to a stressful one.
This experience demonstrated that through the use of manipulatives students can further their thinking of mathematical concepts and use reasoning to problem solve while visualizing the problem. It helps make the abstract problem more concrete. Boaler insists: "students want to know how different mathematical methods fit together and why they work" (Boaler 43). Manipulatives allow for students to understand how the problem works, or how they can break the problem down to solve it. Student curiosity can be enhanced through allowing them to use creative methods to solve problems. For some students simply getting the correct answer is not enough. Furthermore, students who do not get correct answers may give up. The use of manipulatives allows student to attempt to problem solve through using a visual representation.
Some students, including myself, may be hesitant to begin a trial and error process on paper for fear they might be wrong and will make a mess of their neat and well organized notebook. The use of manipulatives allows for creative problem solving while making mistakes in a trial and error manner without having to continually scribble out work to begin anew. Learning happens through the process of attempting. Even if the answer is incorrect, the practice allows neural pathways to form which will help with problem solving in the future.
In this particular class I was also able to talk through the activity as I was explaining my thought process and demonstrating how the manipulatives made the problem much clearer. Boaler notes: "students often need to talk through methods to know whether they really understand them". Vocalizing my thoughts allowed me to verify my understanding through explaining my method to others. Upon using the manipulative I felt I could better understand the problem, visualize how to solve it, and explain my reasoning to others. I am now much more excited about using manipulatives as a tool to enhance learning in the future!
Works Cited Boaler, J. (2015). What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (Revised Edition). NY: Penguin Books.
A Change in My Perspective about Math or Science Entry 4: November 17, 2015
During class on November 10th we discussed the mathematical and scientific thinking of males and females. We discussed how it cannot be scientifically proven that males are inherently better in mathematics then females. We were shown there is no apparent way to differentiate between a male or female brain once they have been separated from bodies.
However, socially things appear to be different. Prior to this class I had not considered gendered learning styles. Seeing as we socially construct gender as soon as infants are born through dressing them in pink or blue based on gender, I am not surprised to learn thinking has also been socialized. This gendered thinking takes place in science and mathematics and according to Boaler females are "kept out" as a result (Boaler 120). In terms of learning, Boaler discusses findings that females tend to inquire about what they are learning and wish to understand how methods work whereas males are content to memorize formulas to get things accomplished quickly.
In regards to how the instruction of math and science can effect individual learning, my perspective has changed. Throughout my scholastic experience math and science were taught very traditionally. Fortunately, I had a twin brother who really enjoyed mathematics. Looking back on my school experience it is interesting to note a few things.
Firstly, when my brother and I were going through school he seemed to pick up math much easier and he always finished much faster than I. At home I often asked him questions and he explained concepts and formulas to me in a less traditional manner. This strengthened his understanding and offered me the tools I needed to be successful in mathematics. When I reflect on the classes we had I remember most girls struggling to stay interested in the math we were learning whereas most boys seemed to be doing quite well. I always enjoyed math and I think it was a result of my brother coaching me. Without his help I might not have fit the pieces together and may have been less interested in math, much like other girls.
Secondly, in high school I was one of 2 females in the grade 11 physics class. My thinking at the time was there must be fewer females interested in physics. Upon reflecting on the way females think about their learning experience it makes sense, they became disinterested as they were not granted the approach they required to engage and thrive. Since it is socially accepted that girls are less interested in math then boys there was no real concern. Male learning styles continued to be nurtured and female learning styles would fall by the wayside. Upon graduation maybe two females went on to pursue carriers related to math or science, whereas quite a few males decided to pursue these subjects at a post secondary level.
In terms of female education in mathematics Boaler notes: "the approaches they experience make many girls uncomfortable and their lack of opportunity to inquire deeply is the reason that so few women take mathematics to high levels" (Boaler 125). Women and girls wish to understand their learning and connect ideas whereas males are content to memorize and move forward. The gender differences in learning should be addressed through the school system and the stereotype that males preform better in science and mathematics needs to be dismantled. My perception of math and science has changed upon reading and discussing this new information insofar as teaching methods are concerned. I would love to incorporate these gender differences into my teaching to make mathematics and science interesting and engaging for both males and females.
Works Cited Boaler, J. (2015). What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (Revised Edition). NY: Penguin Books
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Building Thinking Classrooms and Vertical Learning Surfaces Entry 5: November 19, 2015
Throughout this course vertical learning surfaces have been discussed and promoted as a tool to enhance learning and promote discussion and communication about mathematics. During class on November 17th, Al discussed vertical learning surfaces further. We were randomly assigned groups and given instruction about the activity we would do using the vertical learning surfaces.
This particular activity was about factors. Instructions about the activity were given in a vague manner to allow the students to think about the problem, wonder, and extrapolate. The idea was: there are 12 paycheques that you have the potential to choose. However, each time you chose a paycheque the tax man must also get paid. The tax man therefore collects the factors of the number you chose.
I was initially confused about what this question was actually asking but through collaboration with our group and the use of the vertical learning space we figured out what was required of us. Through using the vertical non-permanent surfaces we were able to pool our thought processes and even look around the room to get inspiration as needed. Furthermore, each group was given only one pen. This tactic ensures communication of ideas as the person with the pen is required to write what is being dictated, which fosters thought in the classroom.
According to Boaler, "there were many reasons for the success of the Railside students. Importantly, they were given opportunities to work on interesting problems that required them to think... They were required to discuss mathematics with each other, increasing their interest and enjoyment" (Boaler 66). Vertical non-permanent surfaces offer the opportunity for students to converse and get excited about mathematics and in turn invest in the problems they are solving.
Al showed us statistics that demonstrated the effect of vertical non-permanent surfaces. They were shown to have the highest percent of students spending time on task, the shortest amount of time to begin work, and the shortest time to the first notation. It also showed increased discussion, eagerness to begin, participation, and non-literacy of work (which if scored high is better for a thinking classroom). During our time at the vertical learning surfaces if we finished Al would introduce doubt "are you sure?" which would lead to further thinking to try to disprove our work. Another tactic would be to increase the challenge. For instance, asking the students to try with 30 paycheques. The notion was to keep students thinking by ensuring they were properly challenged without becoming frustrated, thus keeping the 'flow' in the classroom.
The use of visibly random groupings compliments vertical learning surfaces nicely. Students who have been grouped randomly learn to work in any group they are placed in. This offers students the opportunity to get to know their peers as well as eliminates the need to work with only certain people and also exposes people to different learning styles and feedback regularly. It also helps build communication skills as you are constantly thinking, discussing, and working with new groups of people.
There are specific things required to foster a thinking classroom. Firstly, Al noted how organization of the classroom affects learning. For instance, a front facing room with the teacher front and centre displays a power relationship between teacher and student. If this becomes apparent it hinders a thinking classroom. How teachers answer questions also plays into the power relation. The power dynamic must remain neutral therefore the teacher cannot give students answers. If a student asks if their work is correct, deflect and inquire about something. Have students become confident in their work and stay on an even playing field of power. Oral instruction as opposed to instruction from a text is apparently better for a thinking classroom. Personally, I find students are more eager to begin after oral instruction.
Works Cited Boaler, J. (2015). What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (Revised Edition). NY: Penguin Books.
Bringing the Fun Back to Math Class Entry 6: November 19, 2015
The activity I want to discuss is the Skyscrapers Activity we did on November 17th. We used our Cube a Links to mimic a cityscape but had to do so following specific instructions. Firstly, we were placed into visibly random groups. Next we were told we need to make building representations with our Cube a Links: four 4 squares tall, four 3 squares tall, four 2 squares tall, and four 1 square high. These buildings were to be put onto a 4 by 4 grid cityscape. However, like Sudoku, there could not be more then one of each size building in any row or column.
As we completed different pages we would get increasingly more difficult papers with cityscapes on which to build skyscrapers- this increased the challenge. This was a fun and engaging activity. Through the interest and amusement we created a thinking environment. Every one was learning and thinking, trying to figure out the puzzles in order to progress to the next level. As Boaler notes: "when children work on puzzles... they are having to make sense of situations, they are using shapes and numbers to solve problems, and they are thinking logically, all of which are critical ways of working in mathematics" (Boaler 174). When problem solving we use critical thinking and learn to think in different ways. Building these skills through an amusing task or puzzle helps foster ability and creativity in an engaging manner. This specific activity could be used as an intro to patterning and algebra.
Through creating an enjoyable environment students will stay on task and even get excited about math! Boaler agrees: "mathematical puzzles, and settings are the ideal way for parents- or teachers- to encourage their children into math" (Boaler 168). Through an interesting and fun activity students will build the skills required to succeed in math. The challenge in this is the teacher must be able to keep the flow of the class going. They can do this by increasing the level of challenge as student skill progresses as well as decreasing challenge of tasks if students become anxious or frustrated. Another important aspect is that tasks have multiple entry points. That is, if a student is less skilled they can participate and if a student is more skilled they can begin at a higher and more challenging starting point.
This was employed in the Skyscraper activity as there were multiple sheets of paper to increase the challenge level. If someone needed to increase the challenge it was easy, if it became too challenging it would also be easy to revert back to an easier level to reduce the challenge. Another method would be to swap group members. If one group is excelling at each challenge and one is getting frustrated maybe mixing them up will be beneficial for everybody.
Once everyone was getting the hang of the activity we had a group discussion where we identified some patterns we noticed. Once we discussed patterns we could apply them and develop a strategy. That is, where would be a good first move for paper number 10? what would be a good second move? Boaler emphasizes the idea of communication as key in math. She notes: "...puzzles are incredibly useful in mathematical development, especially if children are encouraged to talk through their thinking and someone is there to encourage their logical reasoning" (Boaler 175). Through discussion the ideas learned are reinforced and reasoning clarified.
The skyscraper activity we completed followed this structure as we first had a chance to wonder, then used our reasoning and logic to participate. During the activity we communicated with our group members and upon completion of the activity we had a class discussion to solidify the information each group gathered and explored. Through using puzzles and games in math students will be engaged and thinking about mathematics and learning while being interested and enjoying themselves.
Works Cited Boaler, J. (2015). What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (Revised Edition). NY: Penguin Books.
Ah-ha Moment: Patterns in Numbers! Entry 7: November 29, 2015
What an epiphany I had on November 24 in class. We were in random groups using vertical learning surfaces and were asked about patterns in numbers. We were to investigate the patterns involved in multiplying by 11. We began writing the equations from 11*10=110,11*11=121, 11*12=132... in order to determine a pattern. We had equations up to 11*20= 220.
We finally figured out a pattern. When you are multiplying a number by (11 eg. 11*54=?) simply write the first digit of the number which will be the first digit in the answer (5). Next, add the digits of the number (5+4=9) this will give you the correct middle digit in the answer. Lastly, write the final digit (4). Therefore, 11*54= 594 .
This was an a-ha! moment for me as I have never really considered finding patterns in numbers, unless it was a puzzle or a question from a math book. This was a really fun and interesting activity that lead to self directed discovery. It would be a great activity to get students involved with the mathematical process instead of simply memorizing numbers. It offers the chance to investigate and find patterns. Boaler notes: children "need opportunities to play with numbers... and develop a 'number sense" (Boaler 145). Through playing with patterns we can solidify our understanding of what numbers are and how we can be flexible with using them. The minimal direction offered in this instance allowed us to be self directed and advance at our own pace. We gained a special insight into the relationship between these numbers.
When we began writing multiplication tables on the vertical learning surfaces we initially had an error. We had written 11*12=121, 11*13=132. We had incorrect answers for three different equations. We noticed this was incorrect once we had found the pattern and tested it out only to get a different answer. We quickly realized our error and moved hastily to fix it. This activity offered the opportunity to fix our mistakes through self regulation. We did not feel embarrassed by our blunder and learned how double checking our work was an important step in problem solving.
This problem was great as it offered multiple entry points for students which easily allows for accommodating students will difficulties as well advanced learners. This problem could easily have been continued if groups finished their task through advancing from two digit multiplication to three to four digit multiplication. You could also find different patterns such as:
11*11= 121 111*111= 12321 1111*1111= 1234321 11111*11111= 123454321
This pattern is really visually appealing and simple. I had no idea this pattern existed but am thrilled to learn of its beautiful symmetry. Students will be particularly enthralled by their discoveries. This could also be extended through examining other multiplication tables for patterns, for instance multiplying by 9. Boaler notes: "research on learning tells us that when children are working on their own ideas, their work is enriched with cognitive complexity and enhanced by greater motivation" (Boaler 172). Through allowing students to recognize patterns in numbers through self directed learning they become invested in the problem and as such get far more out of the process.
When working in a self directed fashion and discovering a pattern such as this the a-ah moment is more significant! In addition, I feel I will be more apt to recall this pattern years from now. I love that I discovered a tool that will allow me to do complex multiplication (by 11) without a calculator. It is a thrill knowing 11*562876= 6191636 without the use of a calculator.
Works Cited Boaler, J. (2015). What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (Revised Edition). NY: Penguin Books.
