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.


Ways to Motivate Students

When I tell people that I’m going to be a math teacher, the general reaction is something along the lines of: “Gross. I hate math.” Math is arguably the most disliked subject in high school. People may dislike writing papers, but they can write about things they like. People might dislike science, but they at least might enjoy doing experiments and such. Students don’t see the reason to like math if we don’t properly motivate them. I found an article recently called 21 Simple Ideas to Improve Student Motivation that has some good ideas, in my opinion.

One general idea that I like is giving students some control. This could be through letting them in on decisions like what type of assignments they do or it could be giving them responsibilities or positions in the classroom. Another idea I like is to create variation in most things. One point the author made was to change up the scenery. This could be how the classroom set up or leaving the classroom to go outside or just to another room. Another variation that could be done is in what type of lessons are done. Switching between lecture, group work, projects, etc. can be very beneficial, because students will show up eager to find out what kind of activity is being done for the day. A third idea I like is to just try to make things fun. Rewarding students both verbally and with physical objects is fun for the students. Being exciting as a teacher and making jokes is something I really believe should be done more. Students also love a little competition in class too. I think that these are some things that I will include in my own classroom so my students will be as motivated as possible.

Setting the Stage for Productive Struggle

A growing interest in teaching with productive struggle is a hopeful sign in the math classroom. Productive struggle can lead to deeper understandings, connections, and motivation in students. So, why are some teachers still hesitant to use it?

Recently, I was able to present a lesson over the Law of Sines and Cosines in a local high school. Normally, these students learn in the traditional way of lecture. However, we challenged ourselves to create a lesson that caused students to struggle a little before they came to the solution. When we presented it, many of the students seemed disinterested and unengaged. I was worried about this when creating the lesson, but I did not think it would occur to the extent that it did. So, I have looked into articles that provide ways for a teacher to slowly introduce and transition students into lessons that incorporate productive struggle.

The article, Beyond Growth Mindset: Creating Classroom Opportunities for Meaningful Struggle, gives tips on what to do and what to avoid when teaching with struggle. Years of research has proven that student learning is enhanced when they have to be persistent to reach success–this concept goes all the way back to the educational reformer, John Dewey who “described learning as beginning with a dilemma.” One major key to a successful productive struggle lesson is to have the goal of the lesson be focus on getting students to have a deeper understanding of the material instead of just focusing on creating struggle. One common way to do this is through real world applications. For example, in the lesson we created for the geometry class, there was an activity that involved drones and using them to deliver pizza. Although this is not necessarily happening today, it is something that could take place in the near future. When this activity was introduced towards the end, there seemed to be a switch that flipped. Almost all of the students were working together to figure out the problem. And, they were able to recall previous math knowledge to help them solve the Law of Sines problem. This is another helpful tip when causing productive struggle–having students use math that they are familiar with can help produce productive struggle instead of frustration. Having the knowledge needed for part of the problem gives students a boost of confidence. However, students also need the time to realize these connections. As a teacher, you should be sure to give your students enough time to think about the activity before giving them any hints or guidance. This time is crucial for allowing students to make connections and deepen their understanding of the math at hand.

The article provides the following list of key elements for providing productive struggle:

  • Determine timing and placement for productive struggle within the unit or curriculum—lessons that are “preparing students to hear something really important.”
  • Align struggle activities with clear, specific learning goals.
  • Design struggle tasks based on assessment of students’ prior knowledge and skills.
  • Foster a safe environment that encourages student inquiry and exploration of important ideas.
  • Use probing questions to solicit student thinking and provide strategic assistance to nudge students through their zone of proximal development–the zone of students’ thinking just beyond the level they can do completely on their own.
  • Follow-up each struggle episode with carefully structured lessons that build on students’ ideas, address misconceptions, and help students forge new understandings.
  • Assist students to reflect and articulate what they learned as a result of productive persistence.

In the future, these tips will definitely help to transition into a classroom with productive struggle much better than the geometry class I just experienced. For instance, I think that the students would have gotten much more out of our discovery activity for the Law of Sines and Cosines if we gave them more thinking time and created an environment that allowed students to talk about their ideas without fear of being wrong.

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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.

Motivating Students in Math

Recently, we have been discussing ways to make students more motivated in math because for many interest is lost between elementary school and high school. This interest is generally not regained until college or beyond. Janelle Cox wrote an article, How to Motivate Students to Love Math, that teaches teachers how to get students to be more interested.

One of the first things she discusses is building on skills that students have already mastered. Whether this be something as simple as addition or as complex as the Pythagorean Theorem, letting the student use knowledge they feel comfortable with will make them less frustrated and more likely to remember the new connections they make. Then, work towards an achievable and personalized goal. When students meet one goal, they are more likely to strive for more and also more difficult ones. So, be sure the first goal they set is somewhat short-term and achievable–you may have to help them with this. Incorporate things that pertain to their life into the lesson such as real world examples and applications as well as the use of technology. Some of this may mean you have to do a little research and learn the latest technology and websites for math, but it will benefit your students tremendously.

And finally, make math interesting! It does not always have to be lecture and learn. Use music, games, and be enthusiastic when teaching and discussing math. Students pick up on your tone, so if you are bored teaching and do not seem to have a passion for the content, they will not be interested either. One more way to intrigue students would be with a magical math problem. This means finding math in the real world that might give an unexpected or interesting result. As we have talked about in class and the article talks about, the birthday problem. This problem talks about the probability of two students in the class having the same birthday. Proving this in your own classroom may be the one thing that sparks an interest in math for your students.

While it may be difficult to be creative with teaching every lesson, incorporating as many motivational lessons as you can will both show your students you care and keep them engaged. These ideas and tips will help create a more effective teacher and get students more interested in math and it’s applications. InspirationalMathQuote3.1-Instagram.png

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

5 Practices to Support Active Learning

All through my life I now realize that I have always been an “active learner”. Meaning I would much rather work with materials and be able to talk and discuss things while discovering the learning rather than sitting and listening to lecture, reading, or writing. It was always much more important to me to see myself as being able to do something with the knowledge I was obtaining. Many other students are like this, and it is something that I want to be able to incorporate well into my classroom.

I found this article on NCTM called Using the 5 Practices in Mathematics Teaching by Keith Nabb, Erick B. Hofacker, Kathryn T. Ernie, and Susan Ahrendt. It gives five practices to use during the lesson to support active learning, and one for beforehand preparation.

0. Identify the Goal or Objective:  Make sure before class you know what the goal of the lesson is going to be.

1. Anticipating: Predict how well they think students will do on a particular problem or lesson

2. Monitoring: Identify different strategies students are using in order to solve a problem. Ask and answer questions in order to further understanding, and document who is doing what on a particular task.

3. Selecting: Pick specific groups to share their work or aspects of their work.

4. Sequencing: Make sure that the sequence of lessons or material makes pedagogical sense. There should be coherence between lessons.

5. Connecting: Make direct connections between strategies and approaches and different content. Do this through questioning or focusing, either directly or indirectly.

The article then does on to show this method in action in two different calculus classrooms. Overall, I really liked the article. I definitely want to incorporate the five practices into my classroom. Specifically, it would be nice when taking a reasoning and sense making approach so that there is a more laid out way on how to guide students through learning and self discovery of the math.