What is Math?

Simple, yet loaded question. Towards the beginning studies of mathematics, such as elementary and middle school, I was taught that there was definitive answers to given problems. However, as I gained more and more experience I realized that Mathematics is no longer definitive. As people, we attempt to give objectivity to such a broad collection of abstract concepts which is where the assumed definitive aspect comes about. My answer to the question, "What is Math?," is literally everything. There is mathematics that is attached to every physical and cognitive concept in my eyes. There was mathematics used in designing, or constructing, or analyzing physical and cognitive concepts.

 Throughout history, mathematics has had  it's fair share of the spotlight. For example, evidence exists of Hieroglyphic numerals being used in Ancient Egypt. There was also the pyramids that were created in Egypt. There was a massive amount of mathematics involved in the creation of the pyramids with using resources to carry materials from place to place, or the way that they moved the materials up the pyramid to complete the construction. There was also much mathematics used to obtain the precise shapes that they formed throughout time. One can also shed light upon Euclid formulating specific outlines of Geometry that we still study to this day. One of the most major mathematical discoveries in my eyes would be the realization that the Earth was round and not flat. That needed mathematics to be proven to individuals because how else would they be convinced that the Earth was round unless they were able to travel into space and see for their eyes that the Earth was round and not flat. However, that was not possible at the time. So, mathematics was demanded to convince others.

Even in the position that I am in, studying mathematical education for 5 years now, it is hard to answer such a seemingly simple question. But that's where the beauty lies in mathematics for me. There is no simple answer. Everything can be twisted and turned since we study such an abstract aspect of life. As we travel through history more and more discoveries are found focusing on mathematics or using mathematics somehow. The exciting part is what we can and will discover with the years to come.

WR-ATH: WRiting in mATH

The wrath of math. What does this even mean!? Through my academic life which has always included mathematics, there have been numerous times in which I become extremely angry and frustrated with the math that sits in front of me. As I reminisce through those times it is evident that the problems that I had was this: I could not communicate appropriately the math that I was performing or take what I was thinking and communicate it with words to others verbally or on paper. I have taken upon myself to observe and volunteer within multiple middle school classrooms throughout the past year. The concern that has arose on multiple occasions was that the student was not able to communicate the math to me whether it be the answer to the question or where they were experiencing the most trouble. Thus, it made me consider bringing writing into mathematics in a more concrete level. By learning how to write in mathematics from an early age, students will be able to communicate more complicated math with efficiency and ease. 

You see it all over. Communication. Math. What is one of the way's in which communication can be used with mathematics? Writing. As I have progressed through my education in mathematics, the more writing I have had to accomplish. However, when I talk with my peers that are not in the field of mathematics or mathematics education, it is strange that I still write in math. When I consider the strangeness, it makes me think back to my math education classes in middle and high school. I would say that I had a small amount of instruction regarding the writing of mathematics. This is where I think we are falling short within the teaching of mathematics. Observations have proven that teachers expect multiple steps from students to show their understanding. However, what about the justification of the steps? This goes into the instrumental versus relational debate. 

INSTRUMENTAL VERSUS RELATIONAL

What does this have to do with the instrumental versus relational argument? What I consider as relational is more than just applying rules without reason. Although students are showing steps, is that justification for why they are performing these steps? This, in my eyes, is instrumental understanding. The student is still showing work but their justification as why they are choosing the certain operations is not given. By bringing in the skill of writing, the justification is solidified. Also, if writing is applied with the steps that the student takes, a teacher can see where the student is making mistakes if and when they are. Thus, the teacher is able to spend more time in certain areas depending on the students answers and justifications for performing the operations in which they have chosen to do so. 

Considering the new Common Core Standards entering the education system within the state of Michigan, we see that students need to have a more complete understanding of the material and concepts that are covered within each grade level. By bringing writing into the curriculum, I believe that the work done by the students will improve ten-fold. This will then give teachers insight to the students mind and how they are working with the problems at hand. Also, verbal communication has been on the rise, but transcribing this material has many beneficial aspects as well. This transcription provides evidence for the progress in which each student makes throughout the course of the year. By starting at an early age such as middle school, we are able to pass on the materials to future teachers so that they are able to grasp a better understanding of the standing of each student within mathematics. This also paves a path for the individual students so that they are able to communicate at a higher level as they progress through their mathematical career into high school and higher education. 

 

 

Secretive Mathematics

Curiosity should always be welcomed. Not a single person in the world knows everything which is why we should be able to ask, "Why?," or "What if?" During my free time I enjoy watching TED talks that incorporate a multitude of ideas. Recently, I ran across one that stood out to me. Randall Munroe is a web cartoonist that answers people's questions "What if.." using mathematics, physics, general knowledge, and logic. The talk starts out by Munroe answering a question about what would happen if a pitcher threw a ball at 90% of the speed of light? Basically, the surrounding city would explode from the ball without anyone seeing anything.

As the talk progresses, Munroe receives an interesting question. The question regards the estimate of punch cards that Google goes though. Google is a company that releases limited information except what is expected from them. With research, Munroe comes up with a staggering number of punch cards. The estimated amount would cover the region of New England that would rise 3 times higher than the glaciers height during the last Ice Age. I'm not sure how others reacted to these findings, but I know my mind was blown. Interesting enough, Google ended up hearing about these findings and sent Munroe punch cards in which a code was needed to define what the punch cards held as a message. Instead of ruining the talk for my audience, check out the video for yourself included here.

After watching the talk, I started to think about this concept of punch cards. To those that are not within the Computer or Mathematics world, it would come as a surprise to some that mathematics plays an important role in encryption. I, for one, have never done much work with coding or encryption but the idea is rather enticing. My mind started to wander on how great this would be for students to get intrigued by math. There have been many articles and research done for the idea of starting class with an interesting concept to bring students into the math world. For starters, it might be interesting to see if codes could be used at the beginning of class to introduce concepts or topics that will be covered that day or that week. Or there could be an elongated code that goes across multiple days in which the students gain knowledge about specific hints that they would find useful for the topic that has been chosen for that week or however long the lesson is planning to take. 

As I progress through my education, I either learn or find ways in which math becomes "easier" in a way. Or I find a way to connect my earlier teachings of mathematics to my teachings now or the way that I plan to teach a concept or topic to my students. A goal that I have always had was getting the people around me to enjoy mathematics instead of being scared of it. This might be one of those ways. By presenting mathematics as a secretive concept, there is a chance that it will involve some of those who have distanced themselves away from it. The best part about this is that the equations that can be involved in developing and solving these secretive codings are unlimited. They could stretch across all different age groups or concepts and involve all types of students no matter their learning styles.

But...don't you want the right answer?

Coming from a family of teachers has many advantages. Recently, I had the opportunity to visit my mother's 5-8 grade classroom. Her students know that I am obtaining my degree in mathematics with an emphasis in education. The students insisted on impressing me with their math skills so I put some challenging problems in front of them and told them that I wasn't looking for a correct answer but wanted to focus on their process. I loved their skeptical looks. Was I testing them? Was I really looking for the right answer but wanted to make it seem as if it was okay if they didn't get the right answer the first time? The amount of effort that they put into the work brightened my day.

Some of the students were working with decimals, some working with word problems, some wanted to be given some questions regarding my research Set, and some just wanted to be challenged with some of the work that I have been doing in college (of course I had to improvise and make it so that they could understand the gist of what I was talking about). As they attempted the problems, I went about the room peeking over the shoulders of some of the students. When the students got frustrated and wanted hints or the path to the answer, all I would say is try your best. I cared more about their process than the answer anyway. Even when they had the right answer and asked if it was correct, I would shrug my shoulders and say, "Can you find a different way to get an answer?" And if they thought that they could, then I would say, "See if you get the same answer in that case." The students may not be satisfied at first, but once they find another way, they answer their own questions. When compared to the previous answer, if they are the same, then the student feels satisfied that they were able to find it in two distinct ways. They can then go on to see the similarities and differences between the two ways that they approached the problem. Now, if they didn't get the same answer, they checked their own work and found their mistakes. What better way for students to learn from their mistakes?! They are defining where they are, how to change them, and how to avoid them next time.

From time to time the students would ask me, "If you don't want the right answer from me, then why am I doing this?" I had to sit and think about this one for a second. We, as people, do things on a daily basis in which we do not know the correct answer to. If we knew the correct answer all of the time how would we better ourselves? We wouldn't be able to because we would be too scared to try anything that we didn't know the answer to. Mathematics is the same way to me. How would we ever better our understanding and comprehension without taking the plunge into the unknown world? I know that we live in an education world where the right answer counts on tests. We obtain money from these tests to further our education and there is no alternative for that quite yet. However, our students can't be afraid of the math that lies before them. My phrase for the day with the students was, "take the leap," and that's just what they did for me.

My decision of becoming a teacher is defined more and more with each passing moment that I share with students. The students have been working with my mom for more than a year now in which they know that she does not look for right answers from them, but cares about their processes and how they believe to go about a problem. It shocks them that I didn't care about the right answer though. Maybe they thought that my mom was a rare teacher that doesn't expect the one correct answer, but it needs to be voiced more to students. Instead of being scared about what lies before them, they take that leap and better their understanding and comprehension without something holding them back. And that right there is what we want from our students.

Are you PREpared?

Starting at the beginning of October in 2013, the Professional Readiness Exam was instated within Michigan. The bottom line of this test is to measure students' knowledge and understanding that the state demands within the education system. Within the state of Michigan, there have been laws that mandate a test to be given to students' as part of the teacher certification process. Throughout the student body, mix feelings start to spread questioning the reliability and validity of the selected material on the test. During my time at Grand Valley studying education, discussions upon discussions have arose within  the classrooms relating to standardized tests and how many feel as if they have to teach to these standardized tests because their jobs rely upon them. We, as a community of experienced and newly structured teachers, need to stray away from this idea as a whole. Standardized tests are not the answer within the educational world. The methods and measures that are derived from these methods are not an accurate description of the students knowledge as well as falling back on the teaching strategies that are structured within the classroom. As we attempt to move away from the idea of teaching to standardized tests, the state mandates the Professional Readiness Exam, (*standardized test*), to assess students' understanding that are trying to go into a field in which they should not be teaching to these standardized tests. A tad bit hypocritical if I do say so myself.

Those who are studying Elementary as well as Secondary Education take the same test prior to going into the College of Education here at Grand Valley State University. This test includes sections Reading, Mathematics, and Writing. These sections assess the students' understanding and knowledge that they will need in order to teach within a Michigan classroom. Is this test showing understanding and knowledge of the content, or is this showing test-taking skills? Public information about the PRE states that this test focuses on the students' ability to show their knowledge and understanding of the concepts that are covered as well as test-taking skills. Why? What is important about test-taking skills that is needed for the teacher to bring into a Michigan classroom? Do we need these because the community of teachers needs to teach test-taking skills to the students? The test-taking skills does not provide sufficient evidence of the students' understanding and knowledge of the concepts that we, as teachers, are teaching to them.

Briefly, lets focus on the mathematics portion of the PRE test. The subarea test areas included are: quantitative literacy and logic, algebra and functions, geometry and trigonometry, and statistics/probability. Within colleges across the state, there are different requirements for each student to take math classes. These requirements could range from only needing one class to multiple classes. If you were one of the students who were only required to take one math class through their whole college career, are they going to be well PREpared for this test? Not only do they need to focus on all of their other classes in order to obtain good grades so that they graduate in a good manner than just skimping by, but they need to teach themselves these concepts all over again that they haven't had experience within 3 or more years. Sound fair?

 So, why make these points? If Education is a path that one decides to travel, shouldn't they just have to do what they are told and take the test? I feel as if there is more to it than just that. We have been taught for years that assessment gives justification for understanding. We are able to take the time in class or outside of class and assess one's skills. The PRE does just this so it places a teacher in the student's shoes. Since we have to take the test to further our education in the field in which we have wished to go into, we must make the best of it. It may seem like a lot of work, but everyone should be able to gain something out of it. Test taking skills are essential in education as it sits today. As one studies for the PRE, they typically remind themselves of concepts that they had learned years ago. One benefits from regaining knowledge, thus making them a better overall teacher. Also, concepts that had been learned years previously might make more sense during the present time. Students would be able to make more connections to the concepts and would be better teachers in the end.

The point of this was not to bash the PRE test for the new teachers that are coming through the educational system within the state of Michigan because I do not have a simple solution to bring to the table. However, my point that I hope was elaborated was the fact that the test seems to lack reliability and validity in assessing students' understanding and knowledge. It is a test that points more in the direction of assessing test-taking skills in which the state believes that teachers will then bring into the classroom. That is what the community of teachers is trying to move away from though. We are coming into a new mindset where teaching to the standardized tests is hindering our own students' understanding and knowledge. The same is happening to us when we are mandated to take the PRE test in order to further our studies within the educational world. We need to move away from test-taking skills and standardized tests in order to enhance our students within the classroom above and beyond what they have in past history. 

Spacing out on Proportions and Fractions

As we progress through our education and enhance our knowledge, we then must take a step back and place ourselves within the shoes of students. What are some strategies and tools that they would use to understand a problem? How would they go about solving it? Little did I know that this would be one of the hardest things to do within my college classes. During a recent class, we focused on problems that were related to proportions and fractions. The strategies that we used were on a level that middle grade students would have a hard time wrapping their mind around. The real issue is that we need to realize that the ways that we solve a particular problem may not be on the same level as our students. We, as teachers, need to have enough tools to formulate multiple strategies for our students in order for them to master the concepts presented to them. This post has been created to put in writing certain strategies and tools that I plan to pass along to my students referring to proportions and fractions so that they are not simply getting frustrated and spacing out.

Within the recent class period, the problem referred to the diluting of blue chemicals for blue jeans. The problem was asking which solution was "bluer" based on how many blue bottles and clear bottles were in each solution. The strategy that I found most useful was cancelling to find out what was left in the larger solution to see which one was "bluer." However, I soon realized that this is not a well thought out strategy to use. By using this manipulation, the proportions are changed which would be hard for a student to understand and realize. Although it worked for me and I was able to determine which solution was bluer, if I were teaching this to students, it would not be the strategy that I would focus on to teach. So what are some other ways that students would be able to have "click" in their head? After this class, some thoughts raced through my mind that I would try to implement into my lesson plans.

The use of examples that do not have an emphasis on numerical values. One of the main problems that have rose to the surface through my volunteering and observations of classrooms and other mathematical education activities, is the emphasis on correct numerical answers. What if we used questions that required more reasoning? Consider an example that talks about the concentration of orange juice. When mixing with 3 cans of water, would a container of orange juice with 3 cans of orange concentrate have more or less compared to a container with 4 cans of orange concentrate? These types of questions require much more in-depth understanding and comprehension compared to a question asking strictly about a comparison of numbers.

When I was in mathematics classes, nothing excited me more than money. Since this was the case I sincerely believe that by including more examples that the students would be able to relate to and be excited about, they're more apt to stay within the learning part. An example including this would be an exchange rate. The students would be able to use hands on money that have different exchange rates. They could then write out proportions and fractions that model what they are trading with one another based on a system that tells which exchanges they are allowed to make. For example, we could say that 3 US dollars are exchanged for 2 British pounds. As described in the previous strategy and example, the students would have a more in-depth understanding and comprehension by using these hands-on activities regarding proportions and fractions. Below are a few examples that could be used throughout the activity. It is important to note that planning questions are not concrete. Students may struggle or fly through specific parts of an activity, which is why the teacher must be adaptive.

1.  Represent the exchange rates as fractions for US currency and British Pounds. 
2. How many US dollars would you have if you had 14 British Pounds? 
3. How many British Pounds would you have if you had 26 US dollars? 
4. If you had 1 British Pound, how much US dollars would you have? 
5. Create your own proportion. 

Overall it is known that proportions and fractions are a hard concept for a lot of students to wrap their head around. It is our responsibility as teachers to provide different strategies and tools to our students so that they are able to comprehend at a deeper level. Using the above strategies and many more that are not elaborated on within this post, the students would be able to manipulate more on their own with the problems and be able to apply the correct operations when working with non-numerical and numerical problems.

Should I take your word for it?

One of the shortest yet most complicated question that arises within a mathematics classroom is...wait for it.."Why..?" We can all relate to this situation whether it be that we are the one asking the question or if we are the one being asked the question. It is noted that everyone is different in their own unique and positive way. Not all teachers are going to be the same. Not all students are going to be the same. We must be able to adapt while interacting with others especially in close quarters such as classrooms that we will be spending a rather large amount of time together. With each and every person being unique in their own way that must mean that we all learn and teach in our own unique way. There may be similar patterns between people but overall, it is not all that concrete. So I ask the question, are we answering these, "Why..?," questions appropriately?  Pertaining to all loaded questions such as these, there are a multitude of answers. Included in this blogpost are my own experiences and personal thoughts in regard to these loaded questions. 

There are many different ways of learning where teaching is only one of them. Growing up my mother was a teacher even at our home. However, not much teaching went on. I was pushed to find my own answers when I had questions. This method was soon engraved within me. When I had a question at school that my teacher had a hard time answering in front of the whole classroom leading to finally being told to just believe them about it, I did further investigation. I made it my goal to find a better answer than that. Another benefit to the act of finding answers on my own without it being spelled out to me was the fact that this kind of learning sticks with you for longer and is more in depth when finding it on your own typically. The desire to actually want to learn about certain concepts to make my understanding more clear and concise pushed me in the direction to want the same thing for my students. 

Having the ability to experiment with their questions allows students to be more involved in their learning. This makes them want to learn and they will retain information without them even realizing! Teachers should all be on the same page in regard to the fact that the objective of education is learning, not teaching. When we are questioned about the concepts and ideas that we are teaching to our students, we should have a firm grasp on said concepts/ideas. Did we just gain this understanding by listening to lectures, or do we find time to investigate these on our own to deeper our understanding? The latter should be the answer 10 out of 10 times. The other part of this is the goal of being able to have the student follow our explanations and leave the classroom knowing more than they did before entering. Their understanding should be enriched so that they are able to go out and investigate more on their own or even teach another peer about the concept/idea. This does entail effort for the teacher to be able to manipulate their own understanding so that the student is able to grasp what the teacher is trying to relay to them. That should be our ultimate goal: to ensure that all of our students enhance and enrich their understanding and comprehension of the concepts that we teach.

So, the next time that a student or peer ask you that loaded question, "Why..?," take a moment to fully digest the situation. If you don't have enough background knowledge of the concept to teach to it in a different way, come back to it at a different time after you learn even more. Because that is the point of education right? Education is for learning, not teaching. For everyone.