Yummy Math That Doesn’t Involve Eating Pie on Pi Day

If you were hoping for excuses to eat food while teaching math, this probably isn’t you blog post to read. Instead, I am writing about Yummy Math, which is a website that provides interesting lessons for students of all ages. While not the most visually appealing website, it is very useful for a math teacher looking for resources. . The lessons they have are about things that are useful or realistic (unlike the problems about buying 60 pumpkins or eating 20 candy bars). They are also aligned to standards. They have tags on every problem with the standards they are aligned to. There is also the option to search for lessons based on the standards you want to address. The lessons go from 2nd grade mathematics to high school. A one- year membership only costs twenty-two dollars and gets you access to all the materials that they have on the site.

As someone who will be student teaching next year, I’m always on the hunt for valuable resources to help me teach. I think that one of the best ways to do this is to find websites like Yummy Math. It can be a challenge if one wants lessons that are geared towards standards, challenging, and interesting to students, but we have to put in the work as teachers. It’d be great if we could come up with really fun yet challenging tasks for our students on a daily basis, but the truth is that there just isn’t always time in the day, especially if you have to teach five or six different classes each day. This is why great resources are important, and I’m happy to say I’ve found another one.

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Law of Cosines (Cont.)

As I continued to dig for lesson ideas on the Law of Cosines and the Law of Sines this week, I realized that Dr. Reins was not exaggerating when he had mentioned that there are not many truly good activities already out there. Google searches, NCTM searches, Pinterest browsing, Dan Meyer’s blog, and more yielded minimal results for lessons that demanded of the students the skills that we have been discussing the entire semester in class.

However, finding a basis for the research that we are going to do in the lesson, and having a goal for the lesson will set us up for what our lesson centers around activity-wise. Looking at what had been found for potential activities already, it became clear that to move on in our planning, we had to decide what our research topic would be. This is how we came to choose our topic as using appropriate tools strategically. The tools that we would like to focus on are the Law of Sines and the Law of Sines. In this theme, I found a lesson from the same website where I found the lesson on using the law of cosines to find the distance between two points on a map. The link to the lesson: https://hilbertshotel.wordpress.com/2013/01/27/a-better-law-of-sinescosines-project/. This project is more in-depth and the teacher wrote in the post that they would have it replace a test. The students are asked to measure the length/width of a room in their house (or anywhere) and use at least two of the following: Law of Sines, Law of Cosines, or Trig Area Formulas for a Triangle to find the area of the room. The teacher has the students find the areas of different shapes at first, like pentagons. In this lesson, giving students the opportunity to use different formulas asks them to look at the properties of triangles that they have been given. It thus asks students to use the tools (trigonometric properties) appropriately and strategically. However, it does not incorporate the use of technology as my group had discussed we were interested in doing. To remediate this, our group could ask students to scale down the room/area they picked and “map” it in Geogebra. Furthermore, the project itself is just that— it’s a project. We could easily scale it down by giving the students dimensions of the room that they will be finding the area of. This project is only one of many lessons that our group could modify to fit the needs of the students in our class.

Since the research theme will be driving the rest of our lesson, it is imperative that we find resources that fit that theme. Clay had the idea of using the steel end as a starting point for our lesson study, which I believe will fit well with finding a real world application in using the laws of sines and cosines. In the theme of finding resources that ask students to differentiate between when it is appropriate to use the law of cosines and the law of sines, I found a lesson plan (link: https://sites.ualberta.ca/~urban/Projects/Math/Play/triglaws.pdf) that uses Geogebra to have students discover that difference. At this point, I believe utilizing parts of the various resources that our group has gathered for the lesson study in order to create a lesson will give students the most opportunity to learn, make connections, and truly understand the Law of Sines and the Law of Cosines.

Not Only Should We Use Real-Life Examples, but We Should Choose Real-Life Examples That Target Interests

In order to motivate students and deepen their understanding of math concepts, we as teachers/future teachers should use real-life examples. If the only problems we ever use in our classes are basic problems consisting of numbers and variables, students will get bored very quickly. Many textbooks include problems that could occur in real life, but these problems are rarely ones that students would do on any given day. They may include buying 40 shirts or 60 pumpkins. I distinctly remember doing a problem where we figured out the length of a guy wire when given the length of a telephone pole and how far away from the pole the guy wire touches the ground. I can’t remember the last time I figured that out in real life. One of my professors preaches that the real-life problems we do actually need to be things we would do in real life. I wholeheartedly agree, but I’d like to take it a step further and suggest that we do problems that students would want to do in real life, even if they haven’t thought of it.

The first step is keeping up with students. Right now lots of people, especially boys, are obsessed with Fortnite. Many more enjoy sports. People obviously love social media. Teachers need to take the time to get to know their students’ interests so they can create lesson plans around these interests. Next is identifying numbers within these interests. This could be numbers of calories, numbers of followers, completion percentages in football, etc. Then problems that challenge the students need to be made. It needs to be at the correct level of difficulty while also deepening understanding and encouraging discovery. Finally, the teacher needs to find a way to smoothly fit these problems into the curriculum. They should not be random problems using concepts from weeks or months earlier.

A great website that has such problems is Mathalicious. It has 135 challenging problems for all types of math concepts that help students find out interesting facts. One problem that caught my eye was about basketball. It involves finding out whether fouling an opponent on a game winning shot is a good idea or not. It has students find the probability of each team winning and losing in either scenario in a real life situation. It really interested me, because I like sports. There are also other problems about social media, food, tv shows, games, fact about the world, and much more. It’s a great resource for finding problems that interest students.

My Internship Week

This past week (12th through the 16th), I completed my secondary education internship. In case you don’t know I have a double major in secondary mathematics education (7-12) and special education (K-12). This internship was for the secondary math education major. I was at North Sioux City Middle School from 7:25-3:05 every day for a week. I learned many lessons during my time in the classroom, but a few I’d like to focus on are: Having a backup plan, making sure everyone understands, and that there are many resources out there.

The first lesson I learned is to always have a backup plan. This is especially true when it comes to technology, since it seems to fail so often. Three out of the four days I was in the classroom, something came up that forced my mentor teacher to change her plans. One day (Wednesday) it was the walkout that took place. Another time it was just that many students had a school activity going on that she wasn’t told about. The third time, the computer program she wanted to use wasn’t working properly. Every time, she seamlessly switched to another lesson. I’m not even sure the students noticed that she had changed her plan. That’s because she always had a backup plan. On Thursday of my internship week, I taught a lesson that required the students to use Desmos on laptops. Logging in to the laptops took at least five minutes, and since I had not planned for this set-back, all I could do was stand up front and ask whether they were logged in. After my lesson, when my mentor teacher and I were talking, we discussed how important space-fillers and backup plans are when it comes to technology. That’s another takeaway, along with just having a backup plan in general.

My second major takeaway is to make sure everyone understands. One thing that I witnessed, and this doesn’t really hurt anyone except the teacher, was that my mentor teacher would plan for the fastest students. I did the same thing when I did my lesson plan. When I was trying to figure out how long it was going to take, I was thinking about how long it took the few fastest students to do a similar task. In reality, learning takes time. Checking whether everyone understands takes more time than I would have thought. A teacher needs to go around and make sure that everyone knows what’s going on, so everyone benefits from the lesson.

My third takeaway is that teachers use a lot of different resources. There’s definitely many teachers that don’t, but there are so many teachers sharing what works for them. Why wouldn’t a teacher want to find lessons that work. On a daily basis, my mentor teacher was trying to find more lessons that work, and she ended up finding more interactive, life-like lessons for things she was going to teach in a less fun way. One collaborative sort of website I found was opened.com. In this website, one can search for a standard in a grade or class and find lesson plans for that standard that other teachers have shared. There seems to be a limited number of people that have shared their lessons, but it seems like a great idea. Searching the internet for lessons that can get students more interested in mathematics in worth the time.

Other notes:

It was really interesting to finally see things we’ve learned in class implemented by real teachers. For example, if someone had headphones in or their phone out, my mentor teacher would go up and whisper to them to put it away, rather than yell or pull it out of their hand or anything like that, which is something we learned in class.

It was a new experience to see the break room at the school I was at. If there’s a reason to not become a teacher, I heard it there. They complained about pay, parents, behavior, etc. It makes sense, though. Teaching is a tough profession and at some point, we all need to vent. On the other hand, it was awesome that my mentor teacher and a few other teachers told me they wouldn’t trade their job for any other one. That was a really cool moment.

Middle school is an odd age to me. In some ways, they don’t even seem close to adults. They don’t take care of their hygiene, they behave like middle-schoolers, and they love games and throwing things. On the other hand, some have the same interests as me. There’s also always a group that thinks they’re too cool for me. I’m not saying I don’t like it, it’s just an interesting age.

Those are my takeaways from my internship. I definitely learned a lot, so it was a very valuable experience. It’s great to get out of the college classroom and get some time doing what I want to do the rest of my life.

Cognitively Demanding Tasks

This past week, we read about the different levels of cognitive demand that can be implemented in a mathematics classroom. Typically, in a traditional mathematics classroom it appears that there is a low level of cognitive demand. For example, the ticket and annulus problem that we did in class was a higher level cognitive demand activity but when we were given an opportunity to look at the original questions presented in the textbook about the problem, the questions led the students to the answer rather than asking them to problem solve their way through the question– wondering what information they would need to know in order to answer the question. Although the strategy of leading students to the answer is psychologically sound as it can be seen as scaffolding, especially if students are still in the zone of proximal development where they would need help to solve the problem, it does not contribute to developing essential problem solving skills in students.

As discussed in Principles to Actions, there are four different levels of cognitive demand for students:

  1. Lower-level demands that include memorization
  2. Lower-level demands that contain procedures without connections
  3. Higher-level demands that contain procedures with connections
  4. Higher-level demands that include doing mathematics

Many of our lecture-based mathematics classrooms only, at the most, employ higher-level cognitive demands where the students are executing procedures while making connections between them. However, even more common are students who simply memorize the formulas but later forget their purpose and do not truly understand them, such as making the connection between proportions and the area/arclength of sectors of circles. This reading, and discussions in class prompted me to begin to think about how frequently teachers employ tasks that are cognitively demanding at a higher-level where students have a full understanding of the mathematics that they are doing.

From this, I was able to find a dissertation entitled “Teacher challenges in implementing cognitively demanding tasks in the mathematics and science classrooms” (Monarrez). This dissertation studied the professional development opportunities available to teachers as they attempt to implement cognitively demanding tasks in the mathematics and science classrooms. As a basis for the study, Principles and Standards for School Mathematics (NCTM, 2000) was cited, as well as the TIMSS Video Study that was discussed in class, where a majority of United States mathematics teachers only required students to regurgitate formulas rather than engaging them in cognitively demanding tasks. In discussing the challenges that teachers face, the study found that many teachers cited the students, content knowledge, and external factors as their main challenges when creating cognitively demanding tasks rather than the mathematical task itself (Monarrez, p. 128). Previously, cognitive demand had been discussed as part of the mathematical task, which is why it is significant that teachers discussed other outside factors as roleplayers in the challenges in construction of the tasks themselves (Monarrez, p. 128). This information gave me an opportunity to reflect on what I will need to consider as a future teacher when creating/implementing cognitively demanding tasks for my students. In any given classroom, students of various levels of content knowledge will be given the same task. This is where differentiation of mathematical tasks can be employed, and a vital tactic in ensuring that students are able to get the most out of the tasks assigned to them. Using the information given in the dissertation, I believe that pre-testing students would be useful, and tracking their performance and levels of understanding in previous tasks when creating lessons. This also displays how imperative it is that a teacher gets to know his/her students so that they are able to be aware of how to best help certain students. In a kindergarten classroom where I volunteer, the teacher frequently takes notes over what each student is struggling on (i.e. sounds of letters) so that she can better help them in the future. Practices such as this would be helpful in combating the challenges that arise when planning and implementing truly cognitively demanding tasks for students. In the future, I can use the information found both in Principles to Actions, and the dissertation to better plan cognitively demanding tasks, and to remember the importance of them in the mathematics classroom.
Monarrez, A. M. (2017). Teacher challenges in implementing cognitively demanding tasks in the mathematics and science classrooms (Order No. 10278789). Available from ProQuest Dissertations & Theses Global. (1924679389). Retrieved from http://excelsior.sdstate.edu/login?url=https://search-proquest-com.excelsior.sdstate.edu/docview/1924679389?accountid=28594