Gifted Students + Desmos

Throughout my teacher assisting semester, I have taken a closer look into not only different resources and pedogogical theories and methods for education in general and in mathematics, but I have also examined those for two step equations in middle grades mathematics. My target audience has been the eighth grade students I have had the blessed opportunity to be their teacher assistant at Grandville Middle School. It truly has been an honor to serve the 120 or so students as they learn mathematics and grow as mathematicians.

In my mission to be the best teacher for my current and future students, I want to provide opportunities for all students to develop and extend their knowledge of the learning targets and content standards. Gifted students are one set of students that I will need to attend to. In order to provide differentiated instruction for this student population, albeit small, I can use Desmos, an online math education resource known for its online graphing calculator and activity modes for both the teacher and the student. The resource can be used either through a browser or an app.

Specifically related to two step equations, the standards are seventh grade standards and within the context of an equations and expressions unit would not necessarily mean that graphing the equations are included, as was the case with the mini-unit I designed and implemented under the guidance of my cooperating teacher. Therefore, a gifted student who would benefit from an extension could receive one in the form of exploring the graphical nature of two step equations while other students continue to practice solving them algebraically. Connections between the graphical and algebraic representations of the equations will come in subsequent units and courses.

For example, algebraic problems in the two step equation mini-unit might look something like this.To extend the gifted student's learning, I would ask the student to forgo the algebraic solving that the other students are doing and to instead graph the left side of the equation and the right side of the equation separately. Next, the student would be prompted to communicate what they noticed in completing this process. Once the student arrives at the importance of the intersection, I would prompt the student to seek out the connection between the graphical representation and the algebraic representation (i.e. the intersection point contains the input value that is the same as the value solved for algebraically).

Take the first problem as a sample problem for this extension: -15 = -4m + 5

If you were to solve it algebraically, one way to solve it looks like this:

Instead, if you were to examine it on Desmos graphically, you would arrive at something like this.

The student would be moved to see that intersection between the two lines occurs at the value m = 5, which is the same value solved for algebraically. Once this connection is noticed, I would have the student verify this observation experimentally using the practice problems sheet as a guide.

Additionally, through this exploration, students are able to better understand the concept of equality and--hopefully--further dismiss the idea that an equal sign must indicate a solution of the form m equals a number, since the items entered in Desmos are seen as lines.

This is just one way a gifted student who could benefit from an extension of their learning would experience that extension. Students could also explore multi-step equations (likely to be the next presented concept), how the graphical connections hold there, and even delve into the special cases of no solution and infinitely many solutions both graphically and algebraically, among others. Lots to offer the gifted students in the classroom, and Desmos seems to cover those bases.

UDL Multiple Means of Representation Resources for Math and Other Classess

Here I have embedded a YouTube video of a screencast I created using Screencastify. This demonstrates how these resources use UDL and are easily able to integrated in any secondary classroom. Enjoy!

If the embedded video does not work, try this link. Alternatively, https://youtu.be/w1FjEQxgLe4 .

Fair Use Laws + Teaching

Common Definition for the Fair Use Laws
The federal government offers this definition for the laws.

"Fair use is a legal doctrine that promotes freedom of expression by permitting the unlicensed use of copyright-protected works in certain circumstances. Section 107 of the Copyright Act provides the statutory framework for determining whether something is a fair use and identifies certain types of uses—such as criticism, comment, news reporting, teaching, scholarship, and research—as examples of activities that may qualify as fair use.  Section 107 calls for consideration of the following four factors in evaluating a question of fair use: (1) Purpose and character of the use, including whether the use is of a commercial nature or is for nonprofit educational purposes, ... (2) Nature of the copyrighted work, ... (3) Amount and substantiality of the portion used in relation to the copyrighted work as a whole, ... [and the] (4), Effect of the use upon the potential market for or value of the copyrighted work."

Rationale for Laws Being in Place
The Fair Use Laws are in place to protect the original work, but also allows for the work to be transformed. In this article from Stanford, it states that the ambiguous definition of fair use--"any copying of copyrighted material done for a limited and 'transformative' purpose, such as to comment upon, criticize, or parody a copyrighted work"--  is in place so that is is open for interpretation, much like free speech in the Bill of Rights. The principle is based on the idea that the public is entitled to copyrighted materials--in portions--freely. Further, it states that most use of the laws come from two purposes: (1) commentary and criticism and (2) parody. For commentary and criticism, the rationale for the laws comes from the idea that the review provides benefits for the public and enhanced by using the original work. it also brings attention to the original work. On the other hand, for parodies, the rationale is that, by nature, parodies must use the original work and it "conjure[s] up" the original work. In all, copyright material may be used in limited form if the work transforms the original work in some capacity and fits into the four factors.  

Statistic
The Association of Research Libraries offers the infographic linked at the end of this article. Within that inforgraphic, it states that "Experts estimate that industries reliant on fair use contributed $2.4 trillion to the U.S. economy in 2008–2009, or approximately 17 percent of the US GDP" This demonstrates that the laws are very important to the US economy. The industries most impacted are manufacturing, retail, and agriculture.

Teaching Resource to Use in the Classroom

This website offers a poster of this picture for purchase. The graphic could be hung in the classroom to assist students in understanding what images they can use in the presentations, posters, and more they create for assignments. As described in the comments at the end of the page, this image is one that can be used for educational purposes, but they recommend citing the source. These have been satisfied by this blog post assignment being educational, the blog being an educational blog, being labeled as a resource for teachers, and citing the source through a link.

This fair use infographic would also be very helpful to have displayed in the classroom. It provides information beyond images. The document has been label free for reuse.

Assessments and Preparation Expectations

What a week at Grandville Middle! Throughout the week, I was able to learn quite a lot about what it looks like to live the teacher life. From reviewing content in preparation for assessments and beginning to teach new units to seeing students be heavily impacted and influenced by grades and observing a student and his teachers in a few other classes, it sure was one wild ride this week!

Although it seems as though I could write a lengthy essay on what happened this week, I am going to focus on assessments and their preparation from both the student and teacher sides. In both Math 8 and Algebra 1, assessments were given over units of review material, Math 8 had a quiz over their Integers Unit, and Algebra 1 had both a quiz and test over their Algebra Basics unit. Since these units were review and in-class work and review days seemed to be going well, I was expecting that students were going to perform well on the assessments going into both quiz days. Upon grading the quizzes in both classes, I was met with mixed results. Check out these gifs below for what I found.

Two thumbs up for Math 8

Oh, yikes for Algebra...

Let's start with discussing Math 8. These classes had multiple days in class to review the content as they played Integers Jeopardy and had a quiz review worksheet. The students seemed to be doing well over the course of those days, and their quiz scores showed that. Most students scored over 80%, leaving me mostly satisfied. I wonder how much of the errors that were made were due to a lack of understanding and how much of it was feeling overconfident in the material, since the entire unit was presented as a review unit. In the days after the quiz, its content has been needed in the new Expressions and Equations unit, and it does seem as though students crammed and dumped the concepts. On a side note, the equations side of this unit is a 10-day "mini-unit" that I am so beyond excited to be developing with my partner teacher assistant!

That's me!

  Ok, now don't get too exited, Nick. We still have to talk about those Algebra assessments.

Oh, yea...

You've probably guessed it by now; they didn't go as well as we would have hoped. Well, my cooperating teacher knew that the scores would likely be the lowest average grade for the year (since they would feel overconfident in their abilities), but none of us expected them to be as low as they turned out. I mean, this is just about the reaction the students gave us once the quizzes were returned.

What?!?!?! There's no way!

Given that this class is an advanced course in combination that it was a publicized review unit, the students seemed to be expecting an excellent quiz score. However, the high score was an 85%, with most of the scores in the 60-70%. This lead to one student asking, "Well, if we all did bad on this quiz, is it really our fault?" This really is a valid question. As teachers, we must look at quiz scores like this and wonder how much of it can be explained by our teaching. But what else is important to consider is what efforts students are contributing as well. Before this quiz, the students were given a quiz review worksheet the day before the quiz and were provided an answer key with all of the problems worked out. Thus, the expectation, which was verbalized to students, was that they would complete the worksheet and check their answers to make sure they knew what was going on. What they did not know was that the quiz was essentially the same thing as the quiz review, but with different numbers. Joey, below, shows the reaction of the students when they learned of this fact.

This revelation was made the day between the unit quiz and the unit test. Also, they were given a test review that day to complete in preparation for the test the next day. This time, too, they were given a worked out answer key. I had the opportunity to score both the quiz review and the test review, and it was definitely interesting to see how students responded to these quiz scores. For the quiz review, only two students out of the combined 64 had a perfect score. For the test review, that increased to about 12 of them, with another 20 only a couple questions behind. What this told me was that students still were not looking at the provided answer key, even after being explicitly told to do so.

To explain this occurrence, I can offer my best guess. Students worked more carefully in completing the rest review but still did not check their answers, since it was made known to them that the score in the gradebook would be based on completion. After all, the test scores were beyond better; as a matter of fact, they looked a lot like the Math 8 distribution as there were a good chunk in the upper 90% range and one student even received a score above 100% after one question was thrown out.

So what really happened? I'm not sure. I can only really know if I polled the class and they gave honest responses. However, what I do know is that this tells me that there is a disconnect between what my cooperating teacher, my partner teacher assistant, and myself were expecting of students. In other words, it is apparent that the students are not preparing in ways that are beneficial to them, even when they are explicitly given a method to help them out. To me, this tells me in another capacity that teaching is more than delivering content. We must also be helping out students as much as possible with test preparation strategies, among many other things. After all, we cannot assume that students enter our classroom with every tool and strategy that will allow them to be successful in our class. Thus, I will take this lesson and keep in mind throughout the rest of this year in both of my placements as well as in my future classroom.

With another assessment coming in just over a week, I will look to be more intentional helping students learn what they can do to assist them in brushing up and learning the material in order to be successful. Here's to another week full of lessons to be learned about teaching!

Is Student Engagement Easy to Evaluate?

One of the many common educational beliefs teachers can have is that student engagement is easy to figure out. That is, scanning the room quickly can determine whether or not a particular student, a group of students, or the entire class is engaged in the task at hand. However, being able to make this determination is much harder than we think.

Huh? 

I know, I have definitely fallen into thinking that I can just look at the class to see if their eyes are at the front or see if they are completing the task they have been given to complete in a group or alone and determine from there whether or not they are engaged. What I have been looking for is not engagement, but rather compliance; engagement and compliance are not the same thing.

When a student is engaged, they are actively thinking about the concepts being presented to them and thinking about them at a level that challenges them to grow in their understanding of the concepts in a captivating way. I emphasize the "and" because students are not engaged if the content is too easy or too challenging for them. With the former, the thinking that is taking place is simple recall that is relatively mindless; thus, their engagement to the task in very minimal. On the flip side, if the task is too challenging, the student is likely to become frustrated and becomes disengaged as a result, I understand that these arguments are seemingly only pertaining to activities given to students. However, the same logic can be translated to any classroom discussion or lecture. For students who already understand the concepts, they can appear to be engaged by looking at the teacher, but mentally they could be checked-out due to boredom. On the other hand, students who do not have the prior background knowledge cannot be appropriately engaged in the dialogue happening as they are unable to process what they are hearing. Despite this phenomenon, students are learning how to adapt their behavior and appearance in class so that they can look to be engaged (Price, 2014). We must then work to captivate students, hopefully just as much as Jon Stewart is here.

Now how do we accomplish this? Ellen Skinner and Michael Belmont offer a motivation model for the psychological needs of a students in order for the student to become engaged.

Skinner & Belmont, 1993 

They also offer an argument that there is a reciprocal relationship between teacher behavior and student engagement that justifies this model. Take a look at the abstract of the article below to see a summary of the article's premise and findings related to the model. The entire article does a nice of explaining this phenomenon.

In a different study, Helen Marks looked to examine student engagement at the elementary, middle, and high school levels through three driving questions. The first concerned how the student's orientation to school, or their perception of school, impacted engagement. Marks found that "at all grade levels, positive orientation toward school, as reflected in school success, solidly predicts engagement; negative orientation, as reflected in alienation, just as solidly predicts disengagement." For the second question, Marks examined the efficacy of authentic instructional work and social support. It was found that, for all students, authentic instructional work contributes strongly to student engagement as well as social support of learning (i.e. a positive classroom environment and parental involvement). Finally, with her third research question, it was found that the subject matter influenced engagement at the elementary and high school levels, but not at the middle school level. It is important to keep these findings in mind when evaluating student engagement.  

Shedding more light on student engagement, Adene Klem and James Connell worked to show the relationship between student engagement and teacher support. They are summarize their findings well below.

"These results indicate teacher support is important to student engagement in school as reported by students and teachers. Students who perceive teachers as creating a caring, well-structured learning environment in which expectations are high, clear, and fair are more likely to report engagement in school. In turn, high levels of engagement are associated with higher attendance and test scores - variables that strongly predict whether youth will successfully complete school and ultimately pursue post-secondary education and achieve economic self-sufficiency. Links between teacher support, student engagement, and academic performance and commitment hold for both elementary and middle school students, providing further support for an indirect link between student experience of support and academic performance through student engagement" (Klem & Connell, 2004).

These results indicate teacher support is important to

student engagement

in

school as reported by students and

teachers. Students who perceive teachers as creating a

caring, well-structured learning environment in which

expectations are high, clear, and fair are more likely to

report engagement

in

school. In turn, high levels

of

engage-

ment are associated

with

higher attendance and test scores

-

variables that strongly predict whether youth

will

successfully complete school and ultimately pursue post-

secondary education and achieve economic self-

sufficiency.” Links between teacher support, student

engagement, and academic performance and commitment

hold for both elementary and middle school students,

providing further support for an indirect link between

student experience of support and academic performance

through student engagement.

With all of this in mind, I would have to agree that evaluating student engagement is definitely complex. We can see from the research that it depends on variables outside the classroom; thus, these studies provide evidence that glancing around the room is not sufficient in determining classroom engagement. Even though it may be challenging to reach an accurate conclusion of the engagement of learners in the classroom, it does not change the fact that we, as teacher, should still strive to create lessons that are accessible and challenging to all learners (i.e. differentiated instruction) and captivate their attention in an effort to engage them in the learning process. If a teacher would say that student engagement is too complex to strive for, I would say that we are doing a disservice to our students and not giving them the learning opportunities we deserve. We should be constantly searching for the many ways we can engage learners, even though it might mean leading lessons in an array of formats so that we reach all learners. C'mon, who wouldn't want to see smiles like this everyday from their students? 

References

Klem, A. M., & Connell, J. P. (2004). Relationships matter: Linking teacher support to student engagement and achievement. Journal of school health,74(7), 262-273.

Marks, H. M. (2000). Student engagement in instructional activity: Patterns in the elementary, middle, and high school years. American educational research journal, 37(1), 153-184.

Price, D. (2014).  Are your students engaged?  Don’t be so sure.  Retrieved from http://ww2.kqed.org/mindshift/2014/01/21/are-your-students-engaged-dont-be-so-sure/

Skinner, E. A., & Belmont, M. J. (1993). Motivation in the classroom: Reciprocal effects of teacher behavior and student engagement across the school year. Journal of educational psychology, 85(4), 571.

Transcending Engagement Beyond Review Games

This past week, my partnering teacher assistant and I were observed leading a lesson for Math 8 students where we reviewed their review unit on integers by playing Integers Jeopardy, a Jeopardy game the two of us created. The lesson itself went really well, only having to work out a few kinks. What I noticed most about this lesson was that student engagement in this lesson was the highest I have seen thus far. Every single student seemed hooked into the game! Granted, it is not surprising that it would come during the first game we played. However, it did get me thinking as to how I can bring 100% engagement into the classroom daily. After all, wouldn't we all like to see each student raising their hands as eagerly as Hermione is?

Sara Van Der Werf's article on achieving 100% was definitely helpful in learning about ways to increase daily engagement. I'm going to summarize a few of her methods here that I found the most helpful and will look to implement here, but please feel free to check out the article or her video presentation for more methods.

Unfair to Assume
The first method she hits on is to model the engagement we are looking for. She makes an excellent point in stating that we cannot assume students know how to function in our classroom. We must teach and model how to engage in group work, partner work, and other social norms in our classrooms so that students can be successfull in our classrooms and we can see the engagment we want to see. For me, when I have my own classroom, I will take time at the beginning of the year to teach these norms in the first few days of the school year with a rich, accessible task (task TBD).

Changing the Language
Instead of asking "What questions do you have?," what if we started asking "What do you notice?" or "What are you wondering?" What kind of responses would we be getting? My bet is that we would see more than the same few students responding and it would lead to some rich mathemtical converstations that would lead to the breakdown of misconceptions and thus deeper conceptual understanding. Further, what if we changed "explain" to "convince me"? Again, I would bet that the responses would allow us to peer deeper into the minds of students and engagment in the mathematics would soar, since they are being required to display conceptual understanding.

Stand and Talks
I'm guessing that most of us are familiar with Think, Pair, Share and that a good chunk also use it in their classroom. Now, does it happen exclusively with students adjacent to each other? If so, what do you think about adding movement to it? Instead of the activity being essentially sit and talks, they would now be transformed into stand and talks. Adding this movement piece gets student up and moving, which can only benefit students. The movement engages the students not only with the mathematics, but also more students within the classroom.

Partner Work, Not Group Work
One of the chief complaints of group work is that one student does all the work and/or other students not doing any work at all. Van Der Wark argues that changing it up so that students work in pairs will work to resolve this issue. In supplying the partnership with one set of materials, the students must rely on each other and thus must both be engaged in the task and discussing the mathematics at hand. This method makes it difficult for students to hide.

With these in mind, I can see how the tasks already going on in my teacher assisting placement can be easily transformed to tasks that have a stronger impact on students. The unit I am planning will definitely includes some of these elements in the math workshop style lessons. I cannot wait to test these out and work toward bringing engagement levels to 100% on a daily basis!

Questions: Unquestionably Important

With one on my goals for the semester being to promote productive struggle, the types of questions I am asking students becomes a critical component in reaching this goal. So far, I am finding myself being successful in some situations but stuck in others, both in leading parts or all of a lesson and assisting students when they ask question. Some of these questions are leaving students like this.

Uh oh, that's not what I want at all. Rather, the questions I am asking should be leaving students thinking deeply and productively so that they can make sense of the mathematics at hand and have a reaction like Andy Dwyer after taking the time to think about the question.

"Oh, I understand it! It makes sense!"

Given that this reaction has not always been commonplace as of late, Steven Reinhart's article "Never Say Anything a Kid Can Say!" was definitely a helpful read! There were many takeaways from this article, and I would like to share a couple of them here: one involving leading a lesson and one about helping students.

"Be Patient. Wait Time is Very Important."(Reinhardt, p. 480)
As Reinhardt mentions earlier in the article, it is important for students to do the talking and explaining and the teacher to do this listening if students are to really learn mathematics. Thus, if students are taking more than a couple seconds to answer a question, I must become comfortable with the awkward silence, which may contain students looking this this.

Students may simply just need more time to process the question. After all, shouldn't we be asking some though-provoking questions, anyway? If I am not waiting and the same students are answering the questions because they process them quicker, I am doing a disservice to the students who take a few seconds more to process by taking away their opportunity to learn through making the connection the question is directing them to make. Better student responses and responses from more students will come from waiting. Thus, I will work to raise the wait time.

"Never carry a pencil." (Reinhardt, p. 483)
I have found that, when assisting students, I like to draw the pictures, write the expressions or equations, etc. I believe this practice stems from my 2+ years in the Math Center at GVSU, since we have lots of whiteboards and whiteboard tables at our disposal. Although this method is not inherently bad, since it allows me to make my responses not exclusively verbal, it is important that students be the one's creating the representations of their thinking as I assist them in making sense of what their struggling with. Carrying a pencil or using a whiteboard marker tempts us to do the thinking for the student or can unintentionally supply too much that we giveaway the opportunity for the student to make sense of it themselves. Thus, I will strive to ask thought-provoking questions that will allow students to develop a deep conceptual understanding of mathematical topics and to have the same reaction as Tony Stark does here.

These two takeaways, as well as the many others from the article, will definitely be taken into account as I answer and provide many questions for students as they learn, I hope to improve in asking thought-provoking questions that allow the students to do the thinking and sense-making themselves. Any feedback or methods that work for you are most definitely welcome! After all, asking better questions allows us to teach mathematics better.

"It [is] not enough to teach better mathematics; [we have] to teach mathematics better" (Reinhart, p 478)