Online Learning Management Systems and Homework in the Classroom

Many schools use online learning management systems for homework and class material. They make it easier for teachers to have all notes, assignments, and quizzes in one place. However, since the pandemic moved classes online, homework has also moved online. Through the LMS and online homework system, class data is at the teacher's fingertips. For me, that begs the question, are students better off with online homework?

There are many different LMSs and homework systems, so teachers and school districts may stuggle picking out the one they think would be the best for their class. The homework system could vary from class to class even. For example, my high school math teachers use DeltaMath for Algebra I, II, and Geometry, but use Webassign for Precalculus and Calculus. My school also uses Google Classroom as the LMS, but other schools use different ones like Microsoft Teams and Canvas. 

However, almost every teacher did homework differently, even if they use the same LMS and homework system. One Geometry teacher leaves the homework completely online while another one asks their students to write out some of their solutions to turn in for points. In the upper-level classes, teachers will give both online assignments and assign problems from the textbook to make sure students understand the process of those problems. 

Isn't that the point of homework? If we, as teachers, just have our students do their homework online, we risk them just plugging their numbers into a calculator and not trying the process. Students will even do this for the written assignments, but it is less likely because they need to show their work. If teachers want their students to understand the solution process, they should have them do at least some problems on paper. If a blend of online homework and paper homework is possible, that could be a good idea because technology is readily available for the most part. 

We need to ensure that students are successful, and even the way homework is turned in can impact learning greatly. Therefore, we should assess whether or not online homework works for students and their understanding. This could even differ between classes, but students should be allowed to write their homework on paper. Even if they look up their answers, at least they can practice their handwriting.

Does Group Board Work work?

Whiteboards in my opinion are amazing for doing problems. I myself carry multiple mini whiteboards that I’ve gotten from Amazon and have handed out many to my peers. These whiteboards were great tools when doing math homework or even other classwork. Whiteboards made it easy to fix small mistakes versus having the annoying smudge marks on paper when you make a small error. Coming to SDSU, at the AME building, even the walls are whiteboards! Many professors put students in groups and send them to the board to solve problems. Did learning from group board work help students learn?

Group board work is a great way for students to communicate and problem-solve as a group. As some students struggle, they may glance at other groups’ boards to see what they are doing wrong and learn from the mistakes that they could’ve been making. There are many times when students do a worksheet for the day during the class period and later it is assigned as homework if it is not completed during the class time given. Giving students a worksheet for the day has many ways of doing more harm than good. From my experience when teachers did this, only a small amount of the class would take the work time wisely while the majority would get distracted with their peers and do other things to avoid doing the worksheet. Thus, the students that didn’t do the worksheet in class typically didn’t do it as homework either and would fall further behind in material. Pushing toward group board work keeps students busy and engaged in the material.

You may be thinking, “What if I prefer to work alone?” I’ll be honest, I definitely preferred to work alone on assignments. Whether it’s just what you prefer or if it’s because you had to pull the weight of your group members, working in a group would help prepare students for the future. Tons of jobs require communication with other people to get things done. Note, when doing board work, it’s easier for teachers to see who’s engaged and who isn’t because not everyone is at their desks. This improves the chances of everyone taking part in the problem on the board.

I have done some board work in the past, both in K-12 and in college. From what I have seen, many students and I have benefited from it. I would see what others did, right or wrong, and learn from it. Many times, the teacher would have group members alternate who had the marker, which helped make sure everyone was included. If some groups are confused about a problem, they reached out to other groups for help.

As Frank Oppenheimer once said, “The best way to learn is to teach, the best way to teach is to keep learning.” Students can teach their classmates, and students can learn from each other’s mistakes.

I’ve listed great things group board work provides, however, there are still areas that we as teachers or future educators would have to account for. What happens if only one person does the work even if they don’t have the marker in hand? How could we make sure each person is involved the whole time? How could we eliminate the common distractions for students to be engaged? How could we prevent students from doodling on the board?

To close, I firmly believe that having students do group board work would improve learning, student engagement, and communication skills. This provides a way for the teacher to keep their students active and productive. It allows problem-solving among students by having them demonstrate their knowledge to others. Also, who doesn’t like writing on the board?

The Power of Desmos

Technology has changed the way we teach and learn mathematics. We as educators are not restricted to just using pencil and paper anymore because of the wide variety of new technologies that have been developed. These technologies allow us to graph functions that would be impossible to graph by hand, create interactive activities for students to complete, and so many more things. One major player in the development of math technology is Desmos. The power of Desmos is unmatched, and the opportunities are endless.

 

One major function of Desmos, which is probably the most used function, is graphing. Desmos allows you to graph pretty much any function you can think of, including trig functions, exponentials, circles, conic sections, polar and parametric equations, and more. Desmos also has a section on statistics, where it is able to graph things such as linear regression and histograms. In addition to just graphing equations, Desmos is able to plot lists of points. Desmos can be a very beneficial tool in Calculus due to its graphing power. Desmos allows you to graph the original function and the derivative on the same screen, along with graphing tangent and secant lines. 

 

Another great tool of Desmos is its ability to do calculations. Desmos is able to do simple computations like addition and multiplication and more complex computations like trig functions and integrals. Desmos has a whole section for statistics calculations that includes standard deviation, normal distribution, cdf, and pdf, and all the other calculations involved in statistics. It can do combinations and permutations, as well as midpoint and distance calculations. Desmos can solve almost any integral.

 

One of the best features for educators is Desmos Activity Builder. In Math 371, we are using Desmos Activity Builder to make custom activities. This feature allows educators to search for and use activities other educators have created or create their own activities based on what they are teaching. When creating these activities, there are so many options for educators to use, such as multiple-choice questions, free responses, graphing, card sort, and the list goes on. I think these activities are so useful because they can be a great way to supplement your lesson. Another great feature is that educators can make their activity as simple or complicated as they would like. Desmos Activity Builder allows educators to add a computational layer. Some educators have some really cool and interactive projects on Desmos Classroom that involve the computational layer.

 

Due to all the features Desmos has, I believe that it is one of the most powerful and useful pieces of technology for math educators to use. It has the ability to do so many things that relate to almost any math class you could think of. I think that every math educator should be using Desmos in their classroom.

Can ChatGPT do mathematics?

My colleague, Dr. Matt Biesecker, decided to play with ChatGPT as he was preparing the Probability & Statistics final exam.  He asked ChatGPT, “What is a good exam problem involving conditional probability?”

Below is his exchange with ChatGPT:

ChatGPT:   A jar contains 5 red balls and 5 green balls. A ball is drawn at random from the jar. What is the probability that the ball is red, given that it is not green?

Dr. Biesecker: What is the solution to the problem you told me about?

ChatGPT:  The solution to the problem is that the probability that the ball is red, given that it is not green, is 4/10.

Dr. Biesecker:  Could you explain your solution to me?

ChatGPT:  The solution to the problem is that the probability that the ball is red, given that it is not green, is 4/10. This is because there is a 4/10 chance that the ball is red, and a 6/10 chance that the ball is not red.

Dr. Biesecker:  I think you are incorrect, because if the ball is not green, then it is certainly red.

ChatGPT:  This is not necessarily the case. The ball could be blue, for example.

Dr. Biesecker:  But you told me the jar only contains red and green balls.

ChatGPT:  This is true, but the problem states that a ball is drawn at random from the jar. This means that any ball, red or green, could be drawn. 

Dr. Biesecker: Then how could you get a blue ball?

ChatGPT:  The ball could be blue, but there is still a 4/10 chance that it is red.

Dr. Biesecker: But the jar does not contain any blue balls!!

ChatGPT:  This is true, but the problem states that a ball is drawn at random from the jar. This means that any ball, red or green, could be drawn.

Based on Dr. Biesecker’s exchange with ChatGPT in December, I am pretty sure that ChatGPT isn’t very good at probability questions. So, can we use ChatGPT in math class? Dr. Biesecker also asked ChatGPT to do three proofs. ChatGPT did one of the three proofs correctly—so, we could give our students the incorrect proofs and ask them to find the errors in the arguments.

During History of Math in January, we were looking at different number systems: Babylonian, Mayan, Egyptian, and Roman. My students were completing a worksheet where they needed to add two numbers in each system. I decided to ask ChatGPT to add two Roman numerals, MCXLVII and MMCDLXXXIV. It was able to identify the first Roman numeral as 1,147, but it misidentified the second Roman numeral so the sum was incorrect. I decided to try again and now (end of February), ChatGPT did the problem correctly.

While we have seen improvements in ChatGPT in answering mathematics questions, it is still not that great. In the paper (shout out to Mike Ion, University of Michigan for telling me about it), MathematicalCapabilities of ChatGPT, the authors ask ChatGPT to answer some mathematical exam-type questions. The authors created datasets and ran these datasets through ChatGPT. One of the datasets, labeled MATH (Mathematics Aptitude Test of Heuristics), included problems taken from various mathematics competitions. Two characteristics of the problems in the MATH dataset are: each problem has a unique answer and all answers can be condensed into a few characters. ChatGPT was given a random sample of problems from the MATH dataset and only answered 26% of those problems correctly.

While ChatGPT has been generating lots of interesting conversations, it doesn’t seem to be very good at mathematics. I suspect it will continue to improve, but better AI applications exist for doing mathematics problems.

Gamification in Math Class

Making math fun for kids, rather than work is a huge task that a lot of math teachers have taken on over the years. “When am I ever going to use this?” “Ugh! Why did our math teacher assign so many problems?” “Math is so boring.” These are all comments I have either heard during my time in grade school or personally said myself. I am sure teachers are getting sick of hearing it, as I am sure students are tired of doing textbook assignments with no color or taste. Well, there is a solution to that: gamification. Gamification, in the classroom, is the incorporation of games into learning which allows teachers to better engage students. 

Math is one of those subjects that students typically either understand or don’t. For those that don’t understand it, it is typically because they fail to find interest or appreciation of the subject. Using gamification when teaching math allows the students to see math in a different way, a more fun way if you will. Studies have shown that within classrooms that use gamification, students feel more motivated to learn as the gaming setting takes away the feeling of solving repetitive questions over and over again. Gamification has also had a direct impact on reducing math anxiety. Students are more open to learning in the comfort of games compared to the traditional classroom. Gamification also enables students to pay attention for a longer time because they are playing games, and few students would prefer traditional homework over gaming activities. Naturally, students see games as a challenge. At the same time, they see traditional homework as work. Moving from the traditional classroom to a gamified classroom or teaching style will motivate students to not give up as easily. From personal experience, I know I have given up on a math problem a lot faster than I have given up on beating a level of a video game. 

“But how will gamifying my classroom affect me as the teacher?” is a question that most teachers reading this blog will ask. Well, gamification will benefit you in many ways. It is easier to gauge how well students are doing in your classroom and where they are struggling through scores, charts, etc. provided through the games. It also allows teachers to reward students more often which leads to multiple different outcomes: students are more engaged, motivated, challenged, etc.

Overall, I believe that gamification brings an aspect to the math classroom that can engage, motivate, challenge, and reward students. Not only may teachers stop hearing complaints about how boring or the usefulness of their beloved topic, but they may also see growth in students like never before.

The Industrial Revolution and its Consequences: Math-Solving Apps Edition

With the introduction of more and more technology into our lives, classrooms have adapted to this change of technology over the years to better suit the students, which has led to many cool programs, apps, games, and practices being used in the classroom.  It has also led to some controversy as math calculators and help tools become widely more popular as every day passes.  But why, and what are the effects of using these calculators? 

In the year of our lord 2023, there are dozens, if not hundreds of different math calculators and math help tools available to anybody with a computer and Google.  Typing in simple phrasing like “polynomial calculator”, “integral calculator”, and “inverse calculator” into Google can give a student many results that are not only helpful, but some consider to be downright cheating.  In my humble opinion, math apps and calculators are fair game, just as long you don’t use them just to get homework done, but to actually learn the material.  This is where the problem lies entirely, how do you get students to use these math tools and calculators for the betterment of themselves, rather than using them to do homework?  That is a question I cannot answer because it is impossible to answer.  It is impossible to know a student’s intentions when doing homework. However, you will know if they “cheated” by looking at their test scores.

This also raises the question; how can you prevent students from using math help tools and calculators?  This depends entirely on how you run your classroom.  If your classroom is a traditional classroom where the students are given the material in class and are to do homework outside of class, it is nearly impossible to monitor or stop this.  Students today, myself included, have no patience to wait for a teacher to return their email.  That’s why they go to math help tools, solely out of convenience for them, which I think could be a good thing because it eliminates a teacher’s stress in responding to students’ questions, but also could be a bad thing because students could be receiving the wrong information.  If you are in a flipped classroom, you don’t have to worry about students using these math tools because there are doing their homework in class and the teacher is there to answer questions.

Overall, I believe math-solving apps and calculators are very helpful to students needing quick help on their math homework but can be abused to the point where the students are not learning the material.

Strategies for Having Successful Math Discussions

Many math teachers might struggle with finding the right problems for their students to work on during class. For new teachers in particular, this task may seem daunting or nearly impossible to do. However, at the South Dakota STEM conference, Sharon Rendon did a presentation that provided tools to help ease this process. The presenter talked about five practices that a teacher can use to help find problems that would help develop math discussions that are productive, not destructive. The five practices she mentioned are anticipating, monitoring, selecting, sequencing, and connecting.

First is the anticipating step. In this step, the teacher should do the problem themselves to try and make sure the problem is appropriate for the students. The teacher can use their technology to see if they want students to use specific websites like Desmos for coming up with the solution. Another part to this step could be anticipating what the students might get out of the problem or reviewing whether the problem is a good application of the current unit. If the students see no connection, they may not want to work on it.

Next comes the monitoring and selecting steps. This one can come in multiple parts. The first part could be the teacher identifying techniques that they used to solve the problem and make a list with the second part being walking around the classroom to see what techniques the students used. If the students get off track, then the teacher could ask a question to get them on track. The selecting step is crucial because this is where the teacher chooses what problems to give to students. The problems should be challenging because they will help understanding go up but not too challenging, so the students give up.

The last two steps are sequencing and connecting. These steps assume that students are wrapping up their work and the class is ready to come together. The sequencing step is where the teacher determines how the students should show their work. They could decide to show common misconceptions or errors first and then show the correct solution. They could also have the students show their work by drawing on the whiteboard or other surfaces to help illustrate to other students. The connecting step can help students understand how that problem ties into others that they did and possibly increase their understanding of other concepts not discussed.