Communicating Mathematical Ideas

If someone were to ask me the which college class I have learned the most from, I would have to say my answer would be MTH 210 Communicating in Mathematics. The course has immensely helped me understand the importance of clarity and the difficulty that comes with trying to communicate your thought process and ideas. As a future teacher, I must recognize that no matter what career path my students choose to pursue, the ability to communicate their thought process and ideas will be a vital skill. Thus, if one of the purposes of education is to prepare students for their life after school, I feel as if it necessary to foster this skill in the classroom.

In my experiences as a Math Center Tutor at GVSU, I have found that the questions I most enjoy helping students with are what many professors are calling a journal entry. Essentially, the student is given a prompt where there is not enough provided information to compute a numerical value and they are asked to find an arbitrary solution and explain the how and the why of it. For example, a journal entry prompt I have seen is something to the effect of "If a > 0, for what values of b will
ax^2 + bx + c have at least one real solution?" What makes these questions great, in my opinion, is that it asks students to apply what they have learned in a conceptual sense and write about it.

Yes, math and writing can be integrated!

Often times, when I am helping a student with this type of problem in the tutoring center, what the student needs help with is not arriving at the solution, but rather communicating their computational work. My advice to them for how to proceed is to write down their thinking process in way that someone in their class who does not understand the concept quite yet could read it and comprehend it. Ron Swanson does a terrific job of depicting the facial expression I receive from students after saying this.

Ok, but...um...how do I do that exactly?

Sometimes the reaction stems from viewing the task as "stupid" or a lot of work, but I have a few students genuinely wonder exactly how they can put what in their mind on paper as they either believe the computational work constitutes a translation of the thinking process or they have difficulties with the translation from mind to paper. In response, I say that sentences in their entry can be formed by saying that "(insert known information) tells me that (insert conclusion)" or "since we want to (insert objective), we know that (insert way to show it)." For example, related to the sample prompt above, a sentence in the journal entry would be "Since we want the polynomial to have at least once real solution, we know that the determinant must be greater than or equal to zero; thus, b^2 - 4ac >= 0." This is definitely a challenge for some students, so in my future classroom, I plan to foster communication skills through a variation of the concept of journal entries: student blogging. For more on this idea, see this previous post of mine. Just like journal entries, student blogging has students explaining the how and the why, thus developing their written communication skills. Below is one of the examples in the linked post.

However, communication of thoughts and ideas is not only written, but also verbal. To help foster verbal communication, not only do I plan to infuse group work into my lesson plans, but I am considering to incorporate what the teacher I am currently observing calls "board problems." What she does is require that once a marking period (quarters at this school), each student goes up to the whiteboard and explains the how and the why of a "challenge" problem in the homework assignment.

A written computational solution is written on the board,
but a verbal explanation is required as well.

Students have the choice of deciding when in the marking period they can present their board problem. The benefit in doing it this way is that it does not harm the student if the board problem was to be preassigned and the concept did not go over well. I realize that giving student's choice is important, however, I am not quite sure that if I use this in my classroom I will let the student decide when they present. I want students to realize they can, in fact, learn from their mistakes, learn more from wrong answers than right answers, and become unafraid to share their work and thought process. Any suggestions on which way to go are definitely appreciated!

The benefits of incorporating journal entries/student blogging and board problems in the classroom goes beyond the development of communication skills. In using these, the teacher has multiple types of formative assessments that allows he/she to get a deeper and more accurate look into the student's understanding of the content, both computationally and, more importantly, conceptually. So with this win-win situation, other than the fact that it might be more work on the teacher's end, I cannot find a reason to not include these in my future classroom.

For more ways to incorporate discourse and writing in a math classroom, take a look at this presentation on the topic from a couple Michigan math teachers.

Emphasizing Concepts Over Computation Through Standards Based Grading

From my experiences as a Structured Learning Assistance (SLA) Facilitator (if you're unfamiliar with what that is, a description can be found near the top of this post) and Math Tutoring Center tutor at GVSU, I suspect the majority of students have the wrong idea of what mathematics is. If you would ask a high school students to describe what mathematics is, a typical answer would probably depict math as the art of calculation or a set of algorithms that allow us to calculate values. I would argue that this type of answer stems from math being taught with an emphasis on computation, or being able to correctly use an algorithm to find an answer. With the exception of my AP Calculus and AP Statistics courses, I can attest to math class being taught in this manner for the vast majority of the math courses I took. I cannot recall being asked conceptual questions very often on assessments or outside the lesson; the emphasis was placed on being able to compute the desired value. 

But there is so much more to mathematics!

One possible reason for this emphasis is standardized testing. With the increasing weight placed on student scores on standardized testing, some teachers are feeling more pressured to teach to the tests. In my experience with standardized testing (MEAP, NWEA, the ACT and its preparatory tests, and the PSAT), there are very few, if any, mathematics questions that ask for a conceptual understanding of the mathematics. Although it is important to prepare students for these tests given their implications (e.g. funding, college admittance, and, for some, job security), I believe that teaching to the test only contributes to students exclusively seeking the right answer, rather than a conceptual understanding, and perpetuates the notion that mathematics is the art of calculation. If math can be boiled down to computing a solution from a big bag of algorithmic tricks, we lose the opportunity to show the beauty of mathematics, as captured in one way by the video below.

Thus, in my future classroom, I would much rather place an emphasis on conceptual understanding since I believe the computational skills will follow. With a conceptual understanding of a given topic, a student should be able to figure out and justify the necessary steps to solve a computational problem as well as demonstrate his/her understanding on a conceptual question. As a result of this belief, I am leaning toward implementing a standards based grading system in my future classroom. With a traditional grading system, I agree with Shawn Cornally that it "teaches kids to love accumulating points instead of learning material."

This is exactly what I want to prevent

With a culture where accumulating more points or receiving a higher score on a standardized test is equivalent to being more knowledgeable about a given topic, it leaves room for students to feel like Billy Madison

or Sheldon Cooper.

Oh Sheldon...

In an effort to not promote this culture in my future classroom, I am seriously considering moving toward assessing students with Standards Based Assessment and Reporting (SBARs) more and more. I believe that this type of assessments puts the student more at ease since it allows for students to demonstrate understanding through explanation and the ability to reassess.

You mean you want to grade more?!?!

Now, this might be the novice teacher in me talking, but I am willing to put in the extra effort grading reassessments, if that means students are arriving at a conceptual understanding of the content. With traditional assessments, I believe they only give the teacher a snapshot of a student's understanding of the material on a given day, so there is value in reassessment since it will allow the teacher to obtain a more accurate look at a student's understanding and what he/she really knows through assessment and reassessment. I suspect there are instances where a student does have an understanding of the concept, but the student was not able to produce it with the initial assessment. Thus, reassessment provides an opportunity for students to demonstrate that coming into the day of the initial assessment, they really did have an understanding of a concept they missed. 

It's in there...somewhere...maybe 

This leads me to the another aspect of SBARs I really like: students explaining their way to the answer. I believe it is really hard to determine, for the most part, by a series of computational steps if a students truly has a conceptual understanding of the content in the set of concepts being assessed. By requiring students to supply their thought process on a problem and justify each step they take, a problem that was once purely computational, and thus could only assess algorithmic thinking, can be transformed into a problem that can accurately assess conceptual understanding as well. However, it is important to emphasize the process when deciding how well a students meets the standard because, as John Golden (@mathhombre) writes (in the SBAR link above), "scores do not mean an answer is right/wrong, but are meant to reflect how much understanding was demonstrated. It is possible to demonstrate good understanding of a concept without even finishing a particular problem." 

You mean you don't want the correct answer?!?!

Now don't get wrong, being able to find the correct answer is great, but I believe arriving at the correct answer will be a by-product of a conceptual understanding, so there should not be a decrease in standardized test scores using this system. In fact, I predict they would increase since students can rely on their conceptual understanding on the test instead of trying to figure out which algorithm from the big bag of tricks needs to be applied. Hence, I believe using this system can only benefit the student. 

There are definitely kinks to be worked out in implementing such a system, but if I want to promote a classroom culture where a conceptual understanding is more valued than excellence in computation, then I believe using SBARs is step toward achieving that goal. I definitely plan to look through this beginners guide to standards based grading when (likely) implementing this system in the future. 

I would love to hear your thoughts and opinions on standards based grading or suggestions on implementing such a system!

Re-energized and Refocused

About a week ago, I had the amazing opportunity to not only attend, but also help present at the Math-in-Action (MIA) conference at GVSU. This math education conference brought together math educators of all levels from across the Midwest to help teachers discover new ways to improve their classroom. By the end of the day, my decision to become a teacher was reaffirmed, my passion for teaching was seemingly at an all-time high, and my itch to get in the classroom already was scratched to the point that I couldn't focus on my homework later that day.

Wake up, Nick! I had to snap myself back into Nick the pre-service teacher who still has a lot to learn. I had to remind myself that I have yet to write a lesson plan, create my own activity other than worksheets and a KaHoot review game built from textbook problems, and initially teach students material, among other things. In just over two short years, I will be looking for a teaching position. By that time, I would like to be ready to make an impact on my future students having acquired the necessary skills and feeling the same excitement and passion that I was feeling after MIA. But what I do know is that I have many fantastic takeaways to, well, take away from MIA. The biggest of which is the necessity to incorporate technology and mathematical literacy in the classroom. I had always realized the necessity for both, but now I have tangible ways to make these a reality. One way to infuse both came from the first session I attended. Zach Cresswell, a high school math teacher in Michigan, discussed the concept of an inquiry based flipped classroom, which I am very interested in bringing to the classroom, and his experiences with it. As a part of his model, he has students blogging about various aspects of class.

Really? How had this not hit me until he explicitly stated it?! What am I doing right now? Blogging. As Zach explained, students +  blogging = getting a better idea of a student's understanding. Here are a few examples of his students blogs that demonstrate what I believe to be the power of having students blog about class. These examples show that student blogging not only gets the student thinking critically about a given concept, but it also allows the teacher to obtain a more accurate reading of the student's understanding. This benefit is something I view as a must in my classroom. Much like the SBAR assessments in my math education class, I would much rather place an emphasis on conceptual understanding than being able to perform an algorithmic computation.

Let's be real; this is NOT math

Don't get me wrong, there is nothing wrong with being able to find the correct answer. But what is it worth if you cannot explain the concept behind it? I think that a student blogging about the connections between the unit circle and the trig function from an activity that uses a program like the one below (slowed down of course) is much more worthwhile for the student (and the teacher!) than answering a few computational trig function problems for homework since it gets at something deeper than meaningless values.

This is so cool, right?

Anyways, I really like how versatile student blogging is with what a teacher can do with it. It can be used for an explanation of a homework problem, a mini project, a check for in-class understanding, and any other form of concept check,to name a few, while allowing a student to further develop mathematical literacy. When Zach (@z_cress) reads a "good" blog post and shares it via twitter, the students are shocked at how the MTBoS responds to their blogs. The students are really excited that real math people are viewing their post, not just their classmates. I love how this initiates a sense of professionalism with students. It helps to teach them that their work matters and is of worth. With all of these benefits, I cannot imagine a classroom of mine where student blogging is absent. After all, what math teacher doesn't want his or her students excited to talk about math?

The Complexity of Confidence

After a given lesson, each student's understanding of the material is unique; thus, each student represents a point on what could be called the "Spectrum of Confidence" related to the taught material.

The "Spectrum of Connnnfidennncee" according to Spongebob

Let Spongebob's right hand represent no confidence in understanding a given concept, his left hand represent complete confidence in understanding a given concept, and each "star" represent an arbitrary student in the classroom (sound like the start of a proof anyone?). From my experience of serving a SLA Facilitator (for those unfamiliar with the role, a description can be found here), I can tell you that it is not hard to find two points along the spectrum: complete confidence and complete lack of confidence. To visualize what is going on internally with these types of students, I call upon my friends Robin Hood (complete confidence)

Oh please, I got this concept.

 and Harry the hedgehog (complete lack of confidence). 

If I can't see you, you can't see my lack of understanding.

As teachers, we need to identify if our students fall into either of these levels of confidence especially. First, we will discuss what it means to look like Harry the hedgehog. While going over a topic with my SLA class, I can often identify these students by looking for the face depicted below. 

I look like I got it, right?

This blank stare signals to me that I am speaking what sounds like a foreign language to these students. For most students, it is very difficult to be honest with themselves and admit that they need help with understanding the material. Often, they choose to give up because they believe will never understand it since, as some of them say in frustration, "math is stupid and too hard for me to understand."

Did you really just say that?

I believe that every student has the capability to not only understand the material, but also should feel confident that he/she can learn and understand anything presented to them Thus, in my future classroom and currently as an SLA Facilitator, I want to instill self-efficacy into each each student, especially those who utter that dreaded phrase that sounds like nails on a chalkboard. However, I must be cautious in doing so as it is important they do not lose the ability to think critically. In other words, a student should not be so confident as to think their initial understanding will always be the correct one. An example of a student who is confident in his understanding, but not have an accurate understanding of the material is Colin from "Quadratic Functions: Students' Graphic and Analytic Representations." In the article, Colin, a tenth grade honors student, is interviewed on his knowledge of quadratics. He confidently completes each task presented to him without questioning his answers, since he believes the connections he has made are correct. However, his understanding of quadratics is very flawed. Whether going from an equation to the graph or vice versa, he made connections that he believes are mathematically sound, but are actually completely false. As teachers, we must also pay attention to the students who seem to understand the lesson because they could be just like Colin. These students will probably look like Lea Michele below, but without further investigation, there would be no way of discovering this false sense of confidence and severe misunderstanding of the concept.

See Mr. Schweitzer, I understand the material!

In other words, I want to prevent myself from finding out that a student has a severe misunderstanding of a concept before a summative assessment, and looking like this guy when seeing their test or quiz.

But it seemed like you were getting it!?!

Therefore, on my quest to instill self-efficacy into my students, I must use an abundance of formative assessment and check-in with each student. In doing so, I hope that when my students leave my classroom, they will be ready to take on not only the material in their other courses, but also everything they encounter during their lifelong learning. 

Learning from Mistakes

Throughout my elementary and high school years, I cannot recall more than a handful of times being asked how I thought through a given math problem. On the other hand, in my math education class at GVSU, one of the most frequently asked questions is, "Well (insert name), how did you get that?" In this class, the question comes after being prompted to solve a problem we would have encountered in one of our high school mathematics courses or to the desired value in a counting circle. (For more information on what a counting circle is, see Jennifer's post.) My good Game of Thrones friend, Margaery, does an excellent job of depicting how I feel when I am asked to answer the question.

Oh, let me tell you how I solved this!

But much to my surprise, I am often met with the question, "Ok, but how exactly did you do that?" Wait, you want me to explain every little detail about my thought process? My other good GoT friend, Tyrion, perfectly captures my feelings.

But, but don't you already know what I'm thinking?

As tedious as it may seem to articulate every single step in solving a problem, some of the most profound understandings or insights can come from this. One of the few times I was asked to show my thought process in high school was in my AP Calculus class where my teacher, Mrs. Cortes, asked the class how to solve a revolution of solids problem from the previous night's homework. Since the problem was an odd and the answer in the back of the book matched the answer I had found, I gladly raised my hand to volunteer to work out the problem on the board. Yes, Mrs. Cortes chose me!

Yes!!! I get to show Mrs. Cortes that I understand this tricky topic!!!

Jumping out of my chair, I walked up to the board and began to write the integral and draw the graph that I used to solve the problem. Once completed, Mrs. Cortes asked the question I hear so often nowadays, "Well Nick, how did you get the integral from your picture?" I began to explain how certain features of the graph led me to the integral I had created. Seeing there were serious gaps in my understanding, Mrs. Cortes stopped me from going any further. In my mind, I was thinking, "What? Why would she need to stop me? I'm pretty sure I got this. She must want to point something out to the rest of class so they can understand it too." She went up to the board and started to point out where I went wrong. Any excitement left from being called up was definitely gone now as I could not believe how I could be so confident in being so wrong.

Whoa, look at all of those mistakes!

After she explained how I should have analyzed the features of the graph, the process of creating the integral need to calculate the volume became much clearer, but now in the right way. In retrospect, I should not have been so process oriented and algorithm dependent. 

You mean the algorithm in the notes
does not apply to every problem?

Since the lesson over revolution of solids didn't go over so well, I can tell I was trying to memorizing a process instead of trying to conceptually understand what was going on with each problem. Clearly, I can now tell that this form of learning is not the type of learning a student should be employing.

Cheer up Nick, there's hope!

What I need to keep in mind, and wish I knew back in high school, is a mantra of Dr. Karen Novotny's: "Often, you learn more from wrong answers than you do from right answers." Now that I am in the process of becoming a secondary mathematics teacher, I plan to ask the question, "How did you get that?" frequently in the hope that it can be a type of formative assessment in the classroom. In asking this question, I hope that my future students will be able to learn from their mistakes, since I will be able to check for conceptual understanding, and they will not have to revert to a memorized process to answer a problem. In other words, I believe that if we as teachers make the effort to see what is going on in our students' minds, we can make a profound impact on their learning. Particularly, I think that if math teachers employ this practice, we can positively affect the student's experience with mathematics by increasing their self-efficacy. In the process, we might even begin to break down the stigma that math is always difficult and hard to understand. Also, we may help some students begin to think that math can be just as much fun as this shark had dancing at the Super Bowl this past weekend!

I think it's time for a math dance party!

Wouldn't Have It Any Other Way

As I was reading "Algebra for All: Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts" by Annette Ricks Leitze and Nancy A. Kitt for my MTH 229 course at Grand Valley State University, the sentence, "Teaching is a challenge, and most teachers would agree that we would have it no other way," really resonated with me. Even though I am an undergraduate student, I have been given the opportunity to be placed in a quasi-instructor role at GVSU as a Structured Learning Assistance (SLA) Student Facilitator for MTH 097 Elementary Algebra (very similar to concepts taught in a high school Algebra I course) and MTH 110 Intermediate Algebra (very similar to concepts taught in a high school Algebra II course). Now one might ask what exactly is an SLA Student Facilitator. Well, it is my duty to attend all of the class lectures, create one hour course content support workshops twice a week, and meet with the professor to discuss how both the workshops are going and what we can do as a team to close the achievement gaps of each of the students. Until a few weeks ago, I also defined my role to be almost synonymous with a glorified tutor, per say. But when I had said that in passing to my partnering professor, she stopped me and said something like, "No, Nick, you are not just tutoring these students. You are teaching the students."

Mind = blown

Boom! I just had an epiphany about what it means to be a teacher. Since beginning to tutor my peers in mathematics in high school, I had thought that tutoring was simply helping my peers complete an understanding of the mathematical concepts they were learning. Also, I  did not consider myself to be teaching them because I thought to teach is to introduce the topics and initiate the learning process with the goal to also complete the learning process. In some regards, I could not be more wrong. According to the Google dictionary, to teach is to "show or explain to someone how to do something" and to be a tutor is to be "a private teacher."  So all of this time I really was teaching the SLA class since I was showing and explaining those in my SLA class how to do and understand the mathematical concepts of MTH 097 or MTH 110, even though it came in the middle of the learning process. 

You mean I was this guy all along? 

Now back to our regularly scheduled topic: teaching as a challenge. To say that being an SLA facilitator is easy would be an immense lie. For example, no matter how much planning goes into one of the workshops I plan, it rarely goes according to plan. Often times, I am either amazed (and excited!) at how what I have prepared goes really smoothly and nearly all the students finish the activity early or I discover that they are struggling immensely with the activity. When the latter occurs, I scramble to determine the root of the problem while fielding the seemingly endless questions the class has on the material. Most of the time, the students have some variation of the same question, and I then will explain to the class what they seem to be missing. But this is not always the case. Sometimes a student is really hung up on a given concept, to which I give my best go at explaining the concept in another way. Usually, my partnering professor has given one explanation, so that leaves the way I would explain the concept. However, this explanation sometimes fails to close the student's achievement gap. When this occurs, I am sometimes reeling for another way to explain the concept.

Nope, another explanation isn't coming to me.

Challenge accepted. Although on the outside I may look put together while thinking of another way to explain the concept to the student, inside I am a little more all over the place.

Just like these guys.

Despite all of this craziness going on mentally, I somehow find a way to alter what I have just said in an attempt to clear up the confusion. I wish I could say that I have a 100% success rate in adapting my explanation so that the concept now make sense, but I do not. This is where I have found teaching to be the most challenging thus far. Everyone is different. Although that comes with many benefits, it is a double-edged sword in the sense that it also means that everyone may require a unique explanation to understand the topic. As Oliver Wilde once said, “I may have said the same thing before…but my explanation, I am sure, will always be different.” This quote embodies how I try to approach my explanations. Each time I discuss a certain topic, I will try to change at least one aspect of my explanation. That way, I hope that it will spark something in someone's mind that will allow them to understand, or at least better understand, the material. Speaking of the spark, that is precisely why I choose to accept the challenge that is teaching. There is nothing like seeing the "light bulb" go off in a student's head. 

Or the meter flipping too, I suppose.

To see this spark go off is such a gratifying experience, and it brings me joy to know that I can make this type of impact on a student and further their learning process. This is exactly why I accept the challenge and wouldn't have it any other way.