Thursday, April 12, 2018

Spec Grading: An Investigation on Assessment


Introduction
            I was first introduced to Specifications grading (also known as Standards Based Grading, SBG, or spec grading for short) in an undergraduate math class at Grand Valley State University. As I had never been evaluated in this way before, it was confusing at first, but in short time I came to realize that it really made a lot of sense. What I found most impactful, and what really stuck with me since, was this core principle of SBG: by the end of the class your grade will accurately reflect the amount of course material you have mastered. I had never really thought about it before, but now it seems really silly to think that this is not a main consideration in every classroom. In most classes we take a test after we have covered a certain amount of material, which count for the largest portion of our grade, but then we never review or revisit that material again. If at some point later in that class I learn an objective that I missed on the test I receive no credit. In other words, there is no incentive for me to actually learn the material, but just get as many “points” as possible, and my grade does not often reflect what I actually did or did not learn in the class.
            Obviously there are a lot more things to consider when determining how we should evaluate and grade students, but the previous example was what initially caught my attention. I am going to be a math teacher and I am passionate about finding the best ways to educate and motivate my students. I have been fortunate to have two great teachers, Dr. John Golden and Dr. Robert Talbert that taught with varying SBG designs; as a result I have become inspired by the underlying principles of SBG and very interested in using a similar system in my own classroom. The following work will describe in further detail what SBG is, the differences between SBG and traditional grading, ways to implement SBG, and some potential concerns I have moving forward. Though I have seen evidence of SBG systems working in all types of classrooms I admit that my examples may be biased toward mathematics. As I plan to construct a similar system, I have intentionally interpreted information from that perspective.
What do Grades Mean?
            One of the biggest issues with a traditional grading system is that we have no idea what a grade actually means. For instance, I could tell you that my final grade was a “B” in algebra class, but what does that actually mean? It could mean that I showed up every day, did my homework adequately, and averaged C’s on my tests. Likewise, it could mean that I Never did homework, showed up most of the time, did pretty well on tests, and took advantage of a couple extra credit assignments. In either case I am not really sure how much of the material has actually been mastered. Does my “B” in the class indicate that I am well prepared to move on to the next area of study? Generally speaking, a “B” means that your grade is in the mid 80’s as far as a percentage, but I’m not sure that either of those cases necessarily exemplifies a student that has mastered 80-90 percent of the topics in the class. I think at this point it is appropriate to ask a few questions, like is there a way to make grades more representative of actual learning, what grade represents appropriate achievement, and is our current letter system (percentages of total points) functional?
From a historical perspective, the common letter grade was designed to give more accurate appraisal of students transferring between classrooms and institutions, not for motivating students (Schinske, 2014).  Point systems encourage students to find the quickest and easiest ways to earn points but do not reward mastering the material. I am personally guilty of paying more attention to assignments worth more points and doing lower quality work on those that reward fewer points; and I consider myself a good student. The message is that, this system is not built to encourage learning; it is built to encourage doing what is necessary to get a grade. Though we don’t talk about it, everyone knows this, and that is why GPA holds little to no weight with employers. Often, it is not a symbol of how much you know, but how well you were able to work the system, how much time you put in, or how well you were able to memorize hours before a test. Schinske (2014) observes that, “Perhaps at best, grading motivates high-achieving students to continue getting high grades.”
I want to mention a few other things to consider about our current grading system. In averaging systems, a zero holds tremendous weight. The easiest way to get a good grade is to just do everything, even if it is not done well; teachers are often generous with partial credit. Students who miss assignments for whatever reason suffer greatly. Additionally, grading on a curve is principally flawed; achievement should not be based on how well a student does compared to another student, but whether or not the have displayed understanding of the learning objectives. We should also consider that often times grading is subjective; the current system and partial credit invites grading bias and almost guarantees grading arguments with students. The bottom line is, grades dampen intrinsic motivation, reduce interest, create competition, and increase anxiety. Later on we will address how SBG confronts these issues, but I want to first think about what grading should look like.
I realize that it is nearly impossible to do away with some aspects of our current system. We practically all use the common letter system for final grades and it is still a way to appraise students as they progress or transfer. However, I think we can find ways to better link those grades to real understanding. Nilson (2015) gives fifteen criteria for evaluating a grading system, a few of them are as follows: uphold high academic standards, reflect student learning outcomes, motivate students to learn, reduce student stress, make students feel responsible for their grades, give students feedback they will use, make expectations clear, and assess authentically. Perhaps a more simplified view is Talbert’s (2017, n.p.), who maintains that  “Grades should be based on concrete evidence of students success in three different areas: Mastery of basic technical skills; ability to apply basic technical skills and concepts to new problems, both applied and technical; and Engagement in the course.” Notice that both of these systems have high standards while requiring students to demonstrate mastery of specific topics, encourage learning, and positively correlate grades to skill mastery.
 I thought it was important to take a little time to think about our current grading system and what we want grades to mean before moving forward. I think it is possible for grades to more accurately reflect knowledge, while encouraging students to learn and holding them to higher standards. We can demand competence, while offering second chances, and encourage students to focus on actually learning the material instead of just getting a good grade.

What is Standards Based Grading?
Though standards based grading can be represented in many different forms, there are some core ideas that differentiate it from traditional grading. And while different classrooms and different subject matter may be better conveyed with varying strategies, I believe the core principles of SBG are a strong foundation for any classroom. In order for us to develop appropriate lessons we must know exactly what skills we want our students to master and develop reliable assessments for them to convey mastery. Additionally, it important that we find a way to motivate students while only accepting satisfactory work, accommodate for life situations, and link mastery to a final grade. Scriffiny (2008) makes a good point about demanding quality and assessing proficiency when he states that, “In the adult world, everything is a performance assessment.”
Standards based grading begins by very specifically defining what skills we want students to master during a class. Nilson (2015) claims that one way of achieving this is a ‘backwards’ approach to developing learning outcomes. In this scenario you begin by thinking about the most complex tasks you want your students to be able to complete. Once this list is generated we need to break down those complex tasks into the ‘building blocks’ that students must first understand; these all become clearly defined learning objectives. When teaching on these topics it is important that we openly share objectives with students; when they know what is expected of them they can more appropriately prepare for assessments. It can often be beneficial to students to allow multiple forms of assessment, with the main objective being that they can show clear understanding. This also means that we need very detailed rubrics that explain what is actually necessary for communicating understanding or meeting criteria. For homework or short assignments this may be a short list of objectives, but for a project or research paper the requirements may be much more significant. So far all of this sounds like common sense, and there are many classrooms that operate similarly.
One less common, yet important dynamic of SBG is grading work as pass or fail. Certainly this may not be applicable to all assignments, but theoretically it should apply to most. This is one of the major positions that sets SBG apart from traditional grading, and there are good reasons for it. It gets to the point of partial credit and what we actually want students to demonstrate in the classroom. In order to implement pass/fail systems we must first define what is passing, or acceptable work. Nilson and others generally set this at what we currently consider the “B” level. This means that in order for an assignment to be considered passing, or “good enough,” it doesn’t need to be perfect; it does however need to be good, and demonstrate that the student has satisfactory understanding of the learning objective. In addition to saving time and the struggle of awarding partial points to students’ work, it allows for more appropriate and meaningful feedback as to why the submission fails to be satisfactory; there rarely seem to be issues in determining whether or not a specific piece of work is satisfactory.
While utilizing a pass/fail system of grading assignments, most teachers using a SBG system will implement some form of resubmission or retake policy. The idea here is that we want to demand a certain level of excellence, yet be understanding that it often may not come on the first attempt; some of our greatest learning comes from taking a close look at what we did wrong and reformulating our ideas. However, this does not mean that students should have an infinite number of opportunities to correct their work so that there is no accountability on the first attempt; we should be firm, yet understanding. Many instructors find it beneficial to implement a token system with regard to resubmissions. They may give out a certain number of tokens in the beginning of class and students may be required to surrender a token for a resubmission, an opportunity to retake an assessment, or even to allow acceptance of late work. This also allows us to be more accepting of unforeseen and external situations that students may not have control of. Giving students a little flexibility with deadlines can not only reduce stress, but increase the quality of work as well. Depending on the class, teachers may also want to offer ways for students to earn additional tokens or reward them for unused tokens at the end of the term.
So how can we tie this all together to a final grade that represents student learning? The really simple answer is bundles (Nilson, 2015). This can be done a couple different ways, but basically refers to the number of passing assignments needed to achieve a particular final grade. For instance, a teacher may have the following bundles containing the corresponding amount of assignments: Definitions (30), Theorems (20), Proofs (10), Group Presentations (3). They could then stipulate that in order to get a B in the class a student must clearly demonstrate knowledge of 25 definitions, 15 theorems, 8 proofs, and have 2 successful group presentations. This is very rough but only intended to communicate the basic idea. Some teachers choose to factor in attendance and participation in a similar way. Generally, in order for students to get a higher grade they must jump more hurdles, higher hurdles, or a combination of both (Nilson, 2015); by higher hurdles we mean that they show deeper understanding of the material. Nilson (2015, p.26) mentions numerous systems of assessing level of understanding but I am particularly drawn to the simple hierarchy of Anderson and Krathwohl’s revision of Bloom’s taxonomy:
1.     Remembering
2.     Understanding
3.     Applying
4.     Analyzing
5.     Evaluating
6.     Creating

This representation means that a student who is able to apply a definition has shown greater understanding than a student who just remembers the definition, and a student that is able to create new connections between models is more advanced than the one who can only analyze and draw conclusions about a single model.
            There are many ways to generate grades based on the decided learning objectives, but in nearly all cases students need to have demonstrated satisfactory understanding of nearly all of them in order to get an “A.” Likewise, this system allows us to define the minimum amount of mastery required to pass the class and set the lowest passing grade at this point. Many structures require at least some ‘excellent’ or higher level work to get an “A,” and some structures even have a completely different set of work to get the highest grade possible. However, it is more commonly a pyramid type structure such that the requirements to get a “B” are a few things in addition to those required for a “C,” etc. By letting the students know right up front what the requirements are to get each grade we can encourage them to decide beforehand what grade they want to achieve. Thus, standard based grading is transparent and motivational while more accurately representing mastery of clearly defined skills.
Specific Implementations of SBG
As discussed, there are many ways to implement a SBG system while sticking to the core theories behind it. Having read many different interpretations on SBG, and personally having teachers use the model, I wanted to add some detail while highlighting some ideas and methods that I find particularly helpful.
One thing I really like about spec grading, and I have seen used in many different ways, is the option for students to demonstrate understanding of a topic through different mediums. Sure, there are some things that are cut and dry, like knowing a definition or being able to state a theorem. However, there is a multitude of ways to show that you understand what the theorem means, how to apply it appropriately, or how it relates to another theorem. Personally, I seem to find more understanding through visuals and find that if I create something I am much more likely to gain and retain relevant information. In math this can mean doing a picture or graph based proof, or creating a model with math based software. I have also read about many teachers giving students the opportunity to explain problems they are having a difficult time representing in another way; oral revisions seem very common is SBG. I think this may actually be one of the better ways to truly judge understanding. When a teacher sits down one on one with a student they can hear an explanation and ask appropriate follow up questions. Thus, they are able to get a really good idea of the students understanding, especially if it is something they missed the first time around. Talbert (2016) sometimes requires multiple evidences of proficiency and students can choose to fulfill this orally as well. Surely, if a student can answer a question on a test and verbally explain the phenomenon, they have demonstrated understanding. Even giving students a list of optional problems to choose from or different ways to present work them gives freedom of choice and the opportunity to personalize their work, which increases motivation and a sense of ownership.
Revising work that does not at first meet standards is another great feature of SBG. Iamarino (2014) asks a question we should all be asking, “Does the grading we do pay off in terms of improved student understanding?” When we grade work as pass/fail and offer the opportunity for corrections our comments mean a lot more to students; we are not just justifying their point deductions, but giving them useful feedback that will help them create satisfactory work. Robert Talbert (2016) uses a system that I really like; when grading written work he uses what he calls a EMRN scale, which stands for Excellent, Meets requirements, needs Revising, and Not quite there. To get an “A” in his class he requires that a certain number of assignments meet the excellent mark, but ‘meets requirements’ is good enough for most. He uses a token system and requires the use of a token for any revisions of “N” work, but “M” and “R” work can be revised without the use of a token. These revision options reduce stress while still holding students to a high standard and helping them learn form their mistakes instead of just punishing them.
Dueck (2011) offers great insight on learning with these four targets and associated questions:
1.     Knowledge Targets: What do I need to know?
2.     Reasoning Targets: What can I do with what I know?
3.     Skill Targets: What can I demonstrate?
4.     Product Targets: what can I make to show what I know?


 I find this setup great from a teaching perspective because the wording helps me generate appropriate learning objectives while focusing on what is really important in the course. From a student’s perspective, it encourages them to think about what they need to do to demonstrate knowledge and gives them options in doing so. These learning targets could be used to model learning bundles as well and seem to reflect Anderson and Krathwohl’s hierarchy stated above.
Psychology
            I want to just briefly discuss a few well-studied and proven ideas from psychology related to student learning and motivation that are present within the core ideas of SBG. Nilson (2015) does a great job of succinctly touching on five of these relevant theories, which are: self-efficacy, expectancy-value of goal achievement, goal orientation, self-determination, and goal setting. First, we know that students are more motivated to achieve when they know how to attain a goal and also feel confident that they are capable so. Second, The SBG model offers a lot of freedom for students, not only in what assignments they choose and how they choose to demonstrate understanding, but also in what grade they wish to attain. Students are more motivated and take ownership when they have choices; if they get a “A” in the class it is because they chose to take that path and do the associated assignments, not simply because it is the will of the instructor. Next, SBG promotes a learning orientation over a performance orientation. With this mind set students expect a challenge and realize that failure is a step to learning, they are free to be more creative and take risks. Thus, the learning orientation helps us create a classroom community where students feel safe to be themselves; numerous studies show a positive correlation between student success and a feeling of belonging. Alternatively, students with a performance orientation are more worried about how they look in front of their peers and shy away from failure, the grade is more important than the material. Finally, as the students are setting their own goals they are naturally more motivated to achieve them. Freedom of choice, along with a challenging yet attainable goal, and continual progress and feedback, is the perfect recipe provided by SBG.

Questions and Concerns
I think that the core principles of standard based grading are imperative to student success. I know that I want to use some form of SBG in my classroom, but I do have some questions and concerns moving forward. Though I know that SBG is used in high school settings, most of my sources are college professors who tackle slightly different issues and also may have a little more freedom with teaching style and experimentation. I worry that parents may not understand why I don’t want to use a conventional system or that I simply won’t be allowed to by administration. Imariano (2014, n.p.) notices that the K-12 system is different from college in that “a parental and societal expectation exists that schools monitor and encourage adolescents’ social and work habits in conjunction with their academic progress.”  Though he claims that this can be monitored separately I still see it as a concern. I’m not sure how I will handle homework, and there are a ton of options. I think some homework needs to be assigned, and I agree with those who claim that eliminating the homework grade will more accurately reflect overall performance, but I don’t know how to monitor it or address homework related questions. I basically want to make sure homework is genuinely attempted without it carrying too much weight. I know that with a system like this I need to be careful of allowing too much flexibility; multiple teachers have discussed issues with procrastination and students submitting tons of ‘make-up’ work at the end of a semester. I think this can be avoided by introducing checkpoints (where everything up to a certain point has to be turned in) or incentives for on-time work. Finally, there is the issue of students not all being at the same level, which I think is a bigger issue in K-12. Amundson (2011) suggests giving students pretests on subjects; students who already have a good grasp on the material are given “alternative enrichment activities” instead of participating in the lesson. This one I’m really not sure on, but I am sure that we can find a way to appropriately challenge every student in the classroom.
Conclusion
            I know that I still have a lot to learn, I haven’t even had my own classroom yet, but I know that I want to demand excellence of my students and provide them with a system that is fair and promotes learning over performance. From what I have seen, SBG is a great foundation for achieving my goals, and as a student I found SBG functional and enjoyable. Though I know that some teachers have tried it and not been convinced, many have reported great success, and it just makes sense to me. Something I specifically found interesting are numerous reports of grades becoming more dichotomous. That is, teachers report very few students getting C’s; students are either on board and do the work to get an “A” or “B,” or they are not on board and fail to produce enough satisfactory work to pass. It sucks to accept the fact that some students will fail, but it is an unfortunate truth. I think this dichotomy only speaks the fact that, if we hold students to higher standards, most of them will produce better work and succeed. To me, this just adds to the evidence that SBG more accurately reflects learning and encourages students to do their best, which is after all, the goal. I would also like to add a quick thank you to John Golden, Linda Nilson, and Robert Talbert who have all been great sources on standard based grading. My first hand experiences with John Golden and Robert Talbert first got me interested in spec grading, and Nilson’s book (recommended by Talbert) was a phenomenal main reference point.



Sources
Amundson, L. (2011). How I overhauled grading as usual. Educational Leadership 69(3)

Buell, Jason. Monday, July 12, 2010. The Foundation of Standards-Based Grading [Blog Post]. Retrieved from http://alwaysformative.blogspot.com/2010/07/foundation-of-standards-based-grading.html


Campbell, C. (2012). Learning-centered grading practices. Leadership41(5), 30-33.


Carey, T., & Carifio, J. (2012). The minimum grading controversy: Results of a quantitative study of seven years of grading data from an urban high school. Educational Researcher, 41(6), 201-208.

Christopher, S. (2007). Homework: A few practice arrows. Educational Leadership, 65(4), 74-75


Clymer, J.B., & Wiliam, D. (2006). Improving the way we grade science. Educational Leadership, 64(4), 36-42. 


Dueck, M. (2011). How I broke my own rule and learned to give retests. Educational Leadership, 69(3), 72-75.

Guskey, T.R. (1994). Making the grade: what benefits students?. Educational Leadership, 52(2), 14-20.

Iamarino, D. (2014). The Benefits of Standards-Based Grading: A Critical Evaluation of Modern Grading Practices. Current Issues in Education17(2). Retrieved from https://cie.asu.edu/ojs/index.php/cieatasu/article/view/1234

Levine, Marty. (2014). Advocating a new way of grading. Retrieved from https://www.utimes.pitt.edu/archives/?p=30598

Miller, J.J. (2013). A better grading system: Standards-based, student-centered assessment. English Journal, 103(1), 111-118

Nilson, L. B., & Stanny, C. J. (2015). Specifications grading: Restoring rigor, motivating students, and saving faculty time. Sterling, VA: Stylus Publishing.

Noschese, F. (2011). A better road: Improve teaching and student morale through standards-based grading. Iowa Science Teachers Journal, 38(3), 12-17.

Scriffny, P. (2008) Seven reasons for standards-based grading. Educational Leadership, 66(2), 70-74. Retrieved from
http://www.ascd.org/publications/educational_leadership/oct08/vol66/num02/Seven_Reasons_for_Standards-Based_Grading.aspx

Schinske, J., & Tanner, K. (2014). Teaching more by grading less (or differently). CBE - Life Sciences Education (13), 159-166.

Talbert, Robert. 2017-04-08. Specifications grading: We may have a winner [Blog Post]. Retrieved from http://rtalbert.org/specs-grading-iteration-winner/

Talbert, Robert. 2016-01-31. Specifications grading with the EMRF rubric [Blog Post]. Retrieved from http://rtalbert.org/blog/2016/specs-grading-emrf 


Talbert, Robert. 2016-01-31. Specifications grading with the EMRF rubric, part 2 [Blog Post]. Retrieved from http://rtalbert.org/blog/2016/specs-grading-emrf-2

Tomlinson, C. (2000). Reconcilable differences? Standards-based teaching and differentiation. Educational Leadership, 58(1), 6-11.

Winger, T. (2005). Grading to communicate. Educational Leadership, 63(3), 61-65.

Monday, November 6, 2017

10 Things I Wish I Would Have Learned In Trigonometry Class

Okay, so I'm not going to write a list, sorry to disappoint. However, during my MTH 229 review I realized that I somehow missed a lot of really cool foundational ideas from trigonometry. It is a subject that I have always found fascinating, but also confusing. I remember asking questions, like "where does the sine graph come from," and getting really terrible answers. I remember being really overwhelmed with all of the identities and rules, but I think that is because I never really developed an understanding of why things work the way they do; I was just taught to memorize a few things and learned how to manipulate them.  So what I want to do here is spend a little time thinking about the things I wish I had a better understanding of early on, and perhaps some good ways to teach those things.


As I said, I memorized a few things, basically enough to help me solve the problems I was given. As an example of this I'm going to share something a little scary and embarrassing; until about a week ago I couldn't have properly explained why a circle is represented by the equation 2 + y 2 = r 2. I could use the equation to draw a graph and vice versa, but lacked the true understanding of why a circle was defined as such. The concept isn't hard, but I don't remember being taught, and I guess I was never aware that my knowledge was not understanding. One thing I do remember is using these 'special triangles,' the 30-60-90, and the 45-45-90 triangles pictured. For some reason I always remember these ratios and could use related trig functions to solve problems as necessary. We can see that they are related to the unit circle, which I vaguely remember being taught, but the triangles seem so much easier. The unit circle is so intimidating, especially to someone new to trigonometry, just look at that thing with all those square roots...yuck. I feel like the information is accessible if you know how to use it, but having to reconstruct or memorize it seems like a useless pain. I guess I tended toward the triangles because the made more sense to me, were easy to construct, and contained much of the same information as the unit circle.

So what I want to share now are a few ideas and resources that have helped me develop a much more thorough understanding of trigonometry over the last couple weeks. I'll begin with an article that discusses a way to introduce the topic of trigonometry without mentioning trigonometric functions. The teacher here assigns their students the task of drawing angles (every 10 degrees) within a circle of radius one, measuring the x and y values where the angle intersects the circle, and calculating the ratio of x/y. Students are encouraged to use shortcuts when they find them available, but also have the option of drawing each angle and calculating each value. Though it may seem a little tedious, I think this exercise would give students an opportunity to develop real understanding of what the trig functions are, and how they may be used. From here you would obviously introduce sine and cosine functions as well, you could compare ratios as the angle changes, and make connections amongst the functions. I recently used trig tables for the first time to do some basic calculations. I found them to be easy to use, and just looking at the tables gives you the opportunity to see a lot of patterns related to the functions.




Another resource I really like includes the diagram to the right; this particular diagram shows many of the trig identities built around the circle of radius one. The article that goes along with this visual actually explains it all really well using an analogy of a dome, a wall, and the ceiling. It does this in a way that allows us to construct, as opposed to just memorize, the diagrams and functions. The whole idea here is to build the understanding by making connections to the way these functions work and interact with each other. For example, it goes on to explain that Sine and Cosine are just percentages of the hypotenuse, which I think is a great strategy. It also includes a link to an interactive graph with a unit circle, graphs of the six common trig functions, as well as numerical representations of each of the functions. It is a really cool tool for discovery and understanding.



Finally, I want to talk about this interactive unit circle for sine and cosine. When I first played with it I though, oh yeah, this is pretty cool. Then I went back to it again and tried to really take in all the information that was being displayed; I was amazed. There is something really awesome about these interactive tools that allow you to see way more information than just plotting points on a graph; the ability to simultaneously see the circle, graph, and triangle created through all possible ranges is unbelievably helpful in building understanding. This is the graphic, that now famously, gave me understanding of the equation of a circle; I was just trying to figure out everything that was going on and it hit me. The screenshot below shows just some of the information available, but gives you an idea of the deep connections that can be made from the material. I think students could definitely benefit by doing some investigative work with this and making connections between the circle and triangles built within; I think they could either be given questions to answer, or even come up with questions of their own based on correlations they see. If nothing else, this is a great way to develop understanding of the sine and cosine graphs (another thing I historically struggled with). Again, as someone who has worked with the information in this graph quite a bit, spending a little time exploring helped deepen my understanding and build new connections between representations. Of all the learning tools and techniques I have discussed thus far, this is one that I will definitely use in teaching trigonometry.








Monday, October 23, 2017

Building an Effective Interactive Math Activity

Throughout the semester we have used technology like Desmos and Geogebra to facilitate learning of different topics. Though different methods are used, it usually involves some correlation between equations and graphs, which I have found helpful in developing understanding. Specifically, sliders are something new to me; I have found them to be very helpful as I can manipulate different terms in an equation and immediately see how it affects the graphical representation. From there I can evaluate trends and build understating based on multiple sources, equation and graph.

I decided that it would fun and practical to learn how to make an activity; so, with students in mind, I built one in Desmos. One of my favorite activities up to this point has been the Marbleslide activities that we have done using a couple different types of equations, so I decided to make one for exponentials and logarithmics. I wanted to include both types with the idea that it would help students make connections between the two related forms. 

When developing this activity I realized that a lot more planning has to go into it than I would have originally thought. The first thing I had to think about was, "what do I want students to get out of this?" I decide that I wanted to design an activity that would make students think about how the different terms in exponential and logarithmic equations relate to a graph, and how those graphs relate to each other. As such, I started with two activities that allowed the use of sliders to change the graph. The first challenge was uses an exponential equation and the second uses a logarithmic, though the ball drop and stars were in the same location. I wanted them to see that, even though the functions are inverses you can still make the graphs look very close to, or identical to each other. Moving forward I wanted to make sure that the game offered an opportunity to use a variety of different graphical shapes, and in a few cases I actually stipulated that the student must use one of each graph type (exponential and logarithmic) in collecting all of the stars. I also tried vary the difficulty of the tasks, creating some that were more simple, and one that I considered quite challenging.

This was a fun learning experience for me and something that I will definitely use in the future with my students.

Here is my finished product:

https://student.desmos.com/?prepopulateCode=cw6gs

Tuesday, October 3, 2017

Thinking About What Makes a Good Teacher

MTH 229, "Mathematical Activities for Secondary Teachers," has definitely got me thinking about activities, but perhaps more importantly how to prepare and teach the activities. The material is great, and every day I find myself saying "I have to remember that." I appreciate all the different ways we are learning to do things and the connections we are making between mathematical concepts, but what I really want to make sure I remember is the strategies and the ideas behind the lessons. We have not only been talking about math, but what strategies work well in teaching math, what keeps student's engaged, and the basic mentality behind planning a lesson. So what I want to do here is start gathering those ideas in one place so that I can think about how they all work together, reflect on what I was thinking when I underlined that note and put '!!!' next to it, and have a more permanent place to reference my thoughts. 

Something we have discussed many times, and I want to be sure never to forget, is to always keep the goal of the lesson in mind when planning. And hopefully that is not just something like, 'learn polynomial division.' I want to make sure that they understand the material, but maybe more importantly how it can be useful to them. Not only in their daily lives or careers but how it relates to other mathematical topics we have discussed as well. We need to plan those connections in lessons, not just mention it and hope that a few kids pick it up. In other words, I want to make sure that I am planning ahead an appropriate lesson, to teach specific objectives, that promotes overall understanding. In addition, I think it is important to think about how we are going to measure the success of the lesson. Have the students gained what we had hoped they would from it? 


I suppose this seems like a good point to transition to the idea of homework. I can see it as a way to assess where the students are and if they have understood a lesson. It can also serve as extra practice with the material. However, we must consider individual student needs and the fact that certain students have life situations that don't permit time outside of school for extra work. Something that stuck with me from our class discussion on homework is the question, 'is it supporting the students that need it most?' Given all that, if there is homework, the question of whether or not to grade it, and how to grade it comes to mind as well. My first inclination is to say that I would not want to attach a grade to homework. 


Grades in general offer even further debate. It seems to be a growing trend, that I agree with, that grades should reflect an overall understanding of the material, not necessarily how well you do on a single test. It wasn't until late in my college career that I had a math professor who allowed us to re-assess on learning objectives that we had not yet shown understanding for. I will never forget how brilliant I thought that was. Not only does it reduce stress for students in so many ways, but it hits much closer to the point of assessments. If by the end of the class, you can show thorough understanding of the material, your grade should reflect that. A grade should not suffer because you don't test well, were sick, got buried in other classes for a couple weeks, or just life.


Another thing I have been thinking about as a result of this class is group work and whole class discussion. I think that group work can be greatly beneficial in different ways. Working on your own and then sharing with a group gives you time to think on your own before collaborating and seeing other ideas. Working with a group on a white board (VNPS) gives everyone the opportunity to participate in real time and see other thought processes immediately. I think that group work can create a safe space for students to share ideas and build relationships. Whole class work is a slightly different dynamic. This provides opportunities for small groups to present to each other, methods of standing with thumbs up (down, side, etc..), or clickers to poll the group for answers. I think that small groups presenting to the larger group helps create safe space and community, while thumbs and clickers provide anonymity and offer the instructor a good baseline of class understanding. I think we have to consider the right method of group work for each situation, and remember that perceived randomization of groups is important for students.


Finally, I want to talk a little bit about methods of discovery and ways to make things more fun. I think a discovery approach to mathematics can be really fun and engaging. By that I mean lessons where students are guided to find the answers for themselves but never specifically told how to do the problems. I experienced this a little bit in class just the other day with the Rational Function Puzzle. I slowly pieced this back together as we discussed it in class and other students brought things up. It is always really rewarding when you feel like you are figuring math out on your own without someone telling you how to do it, and I think this generates better understanding as well. Technological tools are great for this as well. I had worked with geogebra some in the past and it is cool, but Desmos is my new favorite. Not only is it easy to use and intuitive, but the sliders make understanding what happens when you change the value of a variable so much easier to see for me. Additionally, the games and programs people have made for both geogebra and Desmos are fun and functional learning tools ( Quadratics game, marbleslide, Rational Function Puzzle).


I realize that I basically brought up a bunch of ideas but didn't really settle on anything; at this point I am unable to say, 'this is how I am going to run my classroom.' I recognize that I have a lot more questions than answers, but I think that's okay. I don't know what the lesson plans will look like, or how I am going to assess the success or failure of a lesson plan. It is a little intimidating and sometimes overwhelming to think about having my own classroom and being somewhat responsible for at least the mathematical ideas of these kids. But on the other hand, I think it's good that I'm a little scared, and I know it is only because I care. I just want to make sure that I put as much effort into learning how to be a good teacher as I do into learning the math. Spending time thinking about the things listed here is a good start to being where I want to be. I am going to make mistakes, but I am going to keep trying to figure out the best way to solve the problem, it's what mathematicians do. 





Saturday, September 16, 2017

Mathematical Autobiography

The idea of a writing a blog is not only completely foreign to me, but also invokes a little anxiety. As such, I suppose the best way to get started is to share a little bit about myself and how I ended up where I am. 

My earliest memory of a math class is memorizing multiplication tables in third grade. I remember not wanting to just memorize the tables, but to try make sense of them. Specifically, I remember noticing patterns with multiplying by nine. Something like, 'oh, just multiply by ten and then subtract, that's way easier.' I thought that was the coolest thing. A few years on down the road I remember doing algebra and geometry and always enjoyed solving a tough problem and finally getting the right answer. Geometry was always one of my favorites, and still is; I like being able to see what I'm working with, it seems more tangible perhaps. I was never the really smart kid who could just look at things and get it. I had to study, but when I did I always enjoyed math and it came easier to me that it did for others. I think this actually help me explain things to others. 

Unfortunately, I severely underachieved in high school. Though with little effort I could have taken all the math classes my school offered and surely done well, I only made it through trigonometry... barely. There are a number of reasons for my lack of effort there, but that's a different story all together. I guess the point is, math is something I always liked but high school was almost the end of my mathematics career. 

Fast forward a few years down the road and I was in a job that I didn't hate, but most definitely didn't love. I could have made a decent living, but the one thing I knew for sure was that I didn't want to get stuck there for the rest of my life. To me, that meant I needed an education. I quit my job, moved to Grand Rapids, and started college fresh at the age of 29. Wow, was that a change of pace. 

Back to math. I started out at Grand Rapids Community College and had to get all of my basics out of the way. That means starting with college algebra, I had forgotten everything in the ten years I was out of school. I was pretty sure that I wanted to be in the sciences somewhere, probably physics, maybe engineering. Teaching had kind of always been in the back of my mind, by I wasn't set on it. Through all my work experience I had always ended up in training roles, and I just have a knack for helping people learn new things. 

As I made my way through the classes at GRCC I rediscovered what had drawn me to mathematics as a child. People look at me like I'm crazy when I say it, but math is fun. I love solving puzzles and figuring out how and why things work. That's math. The 'refresher' courses were a breeze for me, but I really did need to see all that again. I finally got to calculus, new material for me, and it blew my mind. I thought calculus was pretty much the coolest thing ever. It was interesting because I was using this tool that I had such a limited understanding of, to gain a deeper understanding of things that I had been using for ever, but had taken for granted. I'm thinking of how we derived a bunch of formulas for area and volume. Later on I got a deeper understanding of how calculus worked and my mind was blown again.

It was somewhere in that first year of calculus that two things happened which would both have a big impact on the direction I chose to take in education. First, I decided that math was too cool all by itself and I didn't feel the need to focus on a different place to use it, like physics. Though applied mathematics is great, I just wanted to focus on the math itself. Second, I started tutoring math at GRCC. I remember someone came to our class to put the word out about tutoring and kind of recruit people to do it. I thought it sounded like something I would be good at, and fun. I was hired in the tutoring lab part-time and loved it. 

I could probably write a whole paper just on my tutoring experiences. It never felt like work and the time always flew by. Helping empower people to solve problem and seeing them have those 'ah-ha' moments when they realize they figured something out are priceless. I'm sure there were times when it was frustrating, but honestly all I remember is feeling really satisfied every day I left work there. I felt like I was doing something important, and that made me feel good. And not only was I able to help others, but my knowledge in the material was growing every day as I tried to find new ways to explain things that someone different would understand. It was through that first tutoring experience that I knew I wanted to be in the education field in some capacity. 

After I finished up at GRCC and transferred to Grand Valley I didn't have the time to tutor anymore, but somewhere in there I decided that I wanted to get my masters degree in mathematics and teach at a two year college like GRCC. I think there was a lot that went into that decision, but mostly I just had a great experience there and saw that I could make a difference in a role like that. I finished up my B.S. in Mathematics at GVSU and overall really enjoyed the process. Advanced Calculus made me feel a little dumb, but I hear that's common. 

At this point something else unexpected happened. I started doing some more tutoring work with GRCC and Grand Rapids Public Schools, which was all great, and looking into masters programs. Unfortunately, the masters programs just didn't feel right. It wasn't that I didn't think that I could do the work, it was that the work just didn't inspire the same passion as the undergraduate math did. I applied for a few jobs with student services at GRCC but nothing came of that, and once again I felt lost. I knew I wanted to be in education, and I knew I liked math, I just didn't know what to do with it. Teaching high school seems like an obvious choice but for some reason I had always had an aversion to that. I think I had this idea in my head that I wanted to teach 'the good stuff,' to people who cared about it. Well, they would either like math in the same way I did, or would be diligent enough to at least try because they needed it for their degree. 

I figured I should give the high school thing a fair shake and started doing some substitute teaching. Though it can be a little hard to pick your preferred assignments I was able to get into a few math classes, and even met a teacher that started calling me when she needed someone to fill in. Through this process I changed my mind about high school and had some realizations about what I wanted to accomplish. I realized that the education piece is just as important to me as the math. It isn't enough to be good at math and like it, I want to know how to best help others learn it. In high school I have to opportunity to help students learn to appreciate math, or at least not fear it, before the get to college. Or if they don't continue education, at least teach them enough to have a basic understanding of the math that surrounds us every day. I know this a career path that will challenge and stimulate me, and at the end of the day leave me feeling like I am doing something worth while. Plus, in high school I can coach a sport and have summers off, so there's that. 

So here I am, working on a 'teachable minor,' headed toward a teaching certificate and a Masters in Education. I am excited to teach math, and to gain deeper understanding in order to teach it well.