Models in Mathematics

Going through all my moth classes starting at a very young age, there were always some sort of models we were using to help us understand the concepts better. Looking back at this, it made me realize how much a visual has helped me understand math way better. I would not be at the level of mathematics I am without the use of models and visuals. So I wanted to understand more about how visual models help this much.

According to Room to Discover, visual models support conceptual learning and understanding. They are really a tool to our learning and are there for us to use to gain more understanding. A lot of times if there is not a use of a model or visual of some sort, students are likely to forget what they learned after some time has passed. Looking at our class now, I think that the Euclidean Activity we did is a perfect example of this. We had all learned or seen the Euclidean proof, but none of us could remember what it was or even what it involved. We then went through the activity, and now I will remember it for a very long time. Before doing this activity, I did not think the Euclidean Algorithm was important or anything that I would ever use in my life. Now, I know and understand its importance because of one simple model I just learned. This is an example of just how much models and visuals can benefit students. It is important to keep these models simple to understand. A lot of times textbooks can over complicate things. I found a few tips on Room to Discover that can make sure your students get the benefits of working with visual models. The first one is to begin with explicit instruction. It is important to explain vocabulary, the rules, and what the purpose of each model is. The next tip is to use visual models as a communication tool. Effective models use labels and maybe even colors to make it very easy to communicate about them and how they are used. It is also important to make sure creativity, collaboration, and conceptual learning are all used in the classroom and used with your models.

After reflecting on all my experience in a math class both as a young child and now a college math education major, I realized just how important models are when using math. It ranges from simple number lines all the way to the Euclidean Algorithm activity I just participated in. I would not know what I know now about math without them and their presence in my education. Looking to the future, I will be using lots of models in my classroom. There are too many undeniable benefits to them to not use them in my teachings. I also never realized just how many of them I have used before. It is definitely an aspect of teaching that I look forward to using in the future.

Online Learning

This week I had the chance to catch up with some of my high school classmates. Most of whom are in their residency year or even first year teachers. It was very interesting to her what they are having to go through when it comes to teaching students online. It is another example as to why teachers have to be ready to adapt to any situation at any time. With what is going on in our world today, I wanted to look into what makes a good teacher be a good teacher online as well.

I found a site with many tips, but I found a few of them to be more important than others. The most important one I found was to be present. Teachers need to let their students know that they are there for them, even if it is not in person. Checking in with them often is important to making sure they are staying on track. With this being said, it is important to make sure you are asking your students for feedback. Since you are not able to see them understand or not understand the content in the classroom, it should be a priority to ask for their feedback. Ask them if items are making sense, or if there is anything that you can do better. It is vital to make sure they still feel that you are there for them. Another important part of making online learning successful is to make sure you are setting expectations for the students. Make sure they know what they are supposed to be doing, and that they are doing these tasks to the expectations you set. It is also important to make sure that there are not too many expectations. Especially in a time like this, it is important that teachers are understanding of personal lives. It is also beneficial to do more discussion posts and group work. With the technology we have access to, it is vital that students are staying connected to one another and still continuing to learn from one another. Teachers still need to remember the relationship aspects of teaching. It is beneficial to check in with students individually to make sure they are okay both personally and with tasks in the classroom.

I then take a few steps back and look at this type of learning from a student’s perspective. It is not easy, nor is it how we wanted to learn, but we need to find a way to continue to learn and grow. With all of our classes going online, and being a future teacher, it makes be acknowledge what professors and teachers are doing for us to still have access to an education. It makes me proud of the profession I am going into, and very excited to be one of the teachers having to adapt in the future. After hearing the thoughts and experiences of my old classmates, and now present teachers, I am certain that I am in the right profession.

Shift Problems

In many classroom settings, teachers and students are used to traditional ways of teaching. Before the past year, I figured that was going to be the type of teacher I was going to be. However, throughout the last year I learned the importance of different methods of teaching. One of these types requires more reasoning and sense making. Shift problems are lessons that help students develop higher level thinking and more mathematical reasoning. They are used to shift students’ minds from traditional learning to more abstract thinking.

In Palha’s The Effect of Shift Problems in the Mathematics Classroom, they discuss the importance of helping move the students from traditional thinking to abstract thinking. Students, nowadays, are so used to learning from lectures and books. They have fallen into habits of taking in the material but not understanding the material. Shift problems, on the other hand, provide students with the opportunity to think critically and creatively. They provide students with different steps and processes rather than the answer. They also provide help to teachers to transition from traditional teaching to abstract teaching.

Shift problems provide a lot of help for students and teachers to change their idea of school. They give students time to work with good problems and challenges. These problems and challenges help the students learn higher level thinking. Shift problems also make students participate more in mathematical discussions and thinking. They do all of this by making the lesson oriented around group discussion settings. The teacher creates a problem that needs to be fixed or solved, and the students are able to work off on one another in order to solve it. This process leads students to be engaged but also forces the students to use the mathematics language. They become more familiar with the thinking and the vocabulary needed to be successful with math. The students are able to encourage each other and help each other through the struggles of math.

Moreover, the teachers are able to bring more encouragement into the classroom through shift problems than through traditional practices.In shift problems there is no clear step by step process for the students to undergo, so there is a need for support from the teacher. It is much easier for the students to get upset and frustrated through shift problems because they are used to having the answers in front of them. It is very important for the teachers to be regulating and encouraging them during this time to help them through the struggles of mathematics.

Overall, shift problems allow students to change their mindset from taking in information to understanding information. They are able to work closely with their peers and their teachers to understand the material. They are able to explore the world of mathematics and grow in depth in reasoning. We have a responsibility as teachers to change the game. We need to shift our mindsets from traditional settings to projects and shift problems. It’s only through this shift that we will be able to see the students achieve in a different way.

Mathematical Reasoning in the Classroom

Students today seem to only understand concepts when it is beneficial for them. They only learn the concepts for a short period of time, and seem to have a tough time retaining information. As future teachers, it is important for us to teach mathematical reasoning to help students recall information needed in future classes. We need to help create habits for our students to help relate concepts together.

We need to create situations in which the students are allowed to work through the problems. A big example I was able to experience this last week, was the complete the square lesson. The professor told us that we would be using different algebra tiles to relearn concepts we learned in high school. He began it by giving us an explanation of how to use the tiles. He then gave us a problems and let us, in partners, work how to solve them. He didn’t interject much, which allowed us students to recall past concepts he explained to us as well as our own ideas to figure it out.

As future teachers, it is important for us to bring this type of learning into our everyday classrooms. I think lectures and notes have their place in a classroom, but the biggest way students can learn is by reasoning and coming to the solution by working through the problem and critically thinking about past concepts.

According to Strategies for Mathematical Reasoning, through reasoning the student is able to evaluate a problem, understand a problem, and create a solution. This handbook provides a lot of different ideas to help create more situations for students to mathematically reason. The biggest one that stood out to me was the idea of using manipulatives in the classroom. This concept means that the teacher provides objects from the real world that the students are able to play with and manipulate to understand a concept. While learning graphing, it means seeing how to manipulate the graph by changing one variable or number. In geometry, it means bringing in objects representing each shape/theorem so the students have a visual representation of what they are learning. In the lesson explained above, the professor knew that using the manipulative of algebra tiles gave us a chance to manipulate and change how to get different equations.

They say in the handbook that “there is a proverb that states, ‘What I hear, I forget; what I see, I remember; what I do, I understand.’” I think this is a great way to prepare for lessons as future teachers. We need to provide more chances for the students to learn by using objects to understand. Instead of being the teacher that talks at the students, we need to be the teachers that allow the students to be doing the learning and teaching, while we facilitate and point them in the write direction.

Overall reasoning is necessary in the mathematics classroom. Students don’t just need to remember the different concepts. They need to learn how to understand and retain information. Through manipulatives the students will be able to mathematically reason and continue to grow in their math skills.

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.

If In Search of Practice Problems…

I volunteer at the local middle school in a resource room for an hour every Tuesday and Thursday, and the last time I went, the teacher suggested that I help students prepare for the standardized testing they’ll be doing in a week. Coming up with a way to help them broadly study eighth grade mathematics without knowing what they know stressed me out a bit, but then the teacher asked if I’d ever used IXL. She showed me it, and I’d have to say, it’s an impressive tool for review. It basically gives problems for each concept you could teach in math, language arts, science, social studies, or Spanish. For example, the teacher who runs the resource room said I should work on real life examples of area and perimeter. There’s a section called “area and perimeter: word problems” that I could use. It gives problems that are exactly as described.

The way I used it was by just going through the problems up on the board, but it’s really set up for students to use on their own. It presents a problem that the students have to answer. If they get it right, it adds to their “SmartScore.” If they get it wrong, it takes away from their score and gives them an explanation for how to solve the problem. It also keeps track of how long they’ve been working. After students are done, a teacher can look and see what each student’s areas of need are, which can help the teacher differentiate for their students. IXL can present information in graphs to show “your students’ growth, trouble spots, and even their readiness for standardized testing.” IXL is built around content standards, so it’s great for preparing for standardized testing. I think that it can potentially be a great resource, especially because of this reason, but it can be easy to go overboard with it.

While it’s great to have so many problems available, they are pretty simple problems. They should be used for repetition, not teaching. I could see some of my own high school teachers delivering a quick lesson and then just letting us loose on IXL for the remainder of class, which doesn’t help the students much. The problems don’t encourage much deep thinking; they’re better for practice. This is fine, I’d just had for teachers to become too in love with it and use it constantly. To conclude, I think that IXL can be great for practice and review, and it can help teachers learn more about their class more quickly, but shouldn’t be overused.

IXL Home Page

Silent Discussions

Some major goals of mathematics are to have a deep understanding of the content and to be able to make connections to other concepts. I decided to look around the NCTM website for articles about such things, and I found one called “Visible Thinking in High School Mathematics.” This article is about two main methods: Chalk Talk and Claim-Support-Question. I’m going to focus on Chalk Today, because it really caught my interest.

The main idea is to have a variety of posters around the room with questions on them, generally sounding something like “What do you know about (concept).” Forever however many posters there are, say five, that many different colors of markers are distributed among the students. Students with all the same color markers are sent to a poster, and are told to write what they know about the concept. This is a totally silent activity, which is why the author called it “Silent Discussion.”Students then rotate around the room and either respond to what other students wrote or write their own new idea.

Chalk Talk gives students the opportunity to look at other students’ ideas and get their questions answered at least partially by other students. For a question such as “What is a quadrilateral,” a student may have thought of a square, but with Chalk Talk, they can get the opportunity to see a non-square rectangle, rhombus, parallelogram, or any other quadrilateral, possibly with a picture and description. It gets them thinking outside the box. If they aren’t sure about something, they can ask, and the next group at the poster won’t even know who wrote it, and they can get an answer for their question. That’s ideal.

The posters really end up looking like a mess, but the teachers can somewhat gather what the class knows and doesn’t know, as well as where the class should go next. Even if questions get answered, it still shows that students might not quite feel comfortable with a concept. On the other hand, a question asking about a possible future direction from their new knowledge can make for a great transition into the next topic. Also, students enjoy getting out of their seats, and this is a productive way to do that. It is a great idea overall, in my opinion.

Visible Thinking in High School Mathematics

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.

10 Strategies to Highlight Strategies

Angela Watson’s Article

So often, students just want to find the answer to problems so they can get done with the assignment and do something else. I’m guilty of this myself. When I took Calculus 1-3, for online assignment, I’d look at an example problem, find where the numbers in the problem came from, switch in my answer, and move on. However, a few weeks after the test, I could not remember how to do the problem. How can we expect students to do well on standardized tests when they get homework done by taking shortcuts while gaining no real understanding of the mathematics? In our 7-12 Math Methods class, we’ve been talking about having students explain their strategies and recognize that there are multiple available strategies for many problems. We need to employ strategies to help students understand…well…strategies.

To learn how to help see students understand all strategies more effectively, I read an article by Angela Watson titled “10 Classroom Strategies to Get Students Talking (and Writing) About Math Strategies.” As the title suggests, there are ten different strategies that Ms. Watson suggests to get students to talk and write about their strategies and also just to ensure that they know.

1. Start a lesson by talking about possible strategies, rather than with an easier problem. This strategy involves having a more difficult problem that you want them to solve, and instead of requesting an answer, the teacher has them think about how they might solve it, then get together with a partner and discuss, and finally share potential solution strategies and use them to solve the problem.
2. Along with number 1, another strategy to use is to have students do problems without finding the answer. This has students recognize the importance of the process. The teacher can also have students discuss their strategies, and if there are multiple strategies to solve the problem, this method can help students realize it. I agree with this idea, because it will help students remember what they did to solve a problem rather than the answer to a problem they’ll never do again.
3. Ask students about what they did throughout the process and about their work, not just how they arrive at an answer. Ms. Watson came up with 100 questions to ask to students to start conversations about their mathematics. These include questions asking about real-life examples, where certain values came from, how one could estimate answers to similar problems, and, obviously, many more. I might have to use some of her questions in my classroom.
4. Discuss different strategies for solving problems and why people chose the strategies they did. According to Ms. Watson and probably many other teachers, students get annoyed when doing math because they can’t figure out the one single solution strategy. Showing that there is more than one strategy for solving the problem can make students more eager to do math, because they can do it in their own way. I believe in this strategy for teaching mathematics, because we do it in our methods class, and I’ve learned about connections between different types of mathematics that I never knew existed.
5. Use math journals. In math journals, students can do their work for problems, as well as state what they were thinking at every step. They could write about what went right, where they got confused, and reasoning for it all. With math journals, we as teachers can figure out where they’re at mathematically while they have to explain their reasoning and understand it more. I might be open to using math journals, although it seems like it might take a lot of time if I were to actually read all of them.
6. Have a smaller set of questions you can use all the time. These may be similar to the 100 questions mentioned earlier, but there are much fewer of them, and they should be used often enough that students start to think of them on their own. Some of the questions I might use are: How did you get your answer? How do you know it’s correct? What are some other strategies for solving this problem?
7. Use effective techniques for getting students thinking. There are many ways to do this. Asking the right questions is one way. Another is to walk around and encourage or help students. Selecting the right order to address solutions in for discussions is another one. Getting to know all the ways could make me a very effective teacher.
8. Play math games. I think that Ms. Watson might have had longer classes than the 50 minutes ones I had when I was in high school, because she suggested breaking up a class period with a 10-15 minute math game, but I think that would probably take some valuable time if done on a daily or nearly daily basis. I think I would use this strategy once a week or so to check understanding and provide an incentive for the students to learn the information.
9. Use a program that allows you to see their work as they’re doing it. I think the main takeaway from this is that there are apps and such for which teachers can have students record their screen as well as anything they say while they’re working on homework, so they can see the whole process. This gives a good idea of where students are at with the concept.
10. Have students create posters that contain all the strategies they’ve seen. They can start making these posters at the beginning of their learning about a concept. As time goes along, they can add the teacher’s strategies, other students’ strategies, and their own strategies.