Intrinsic and Extrinsic Motivation – Students need to see themselves in Mathematics

More often than not, I find myself visualizing my future classroom and how I will implement different strategies that accommodate to all student learning styles in my classroom. With that being said, I frequently think about my frustrations with mathematics and how to avoid making those same obstructions for my students.

This last semester, I was afflicted with one of my courses because I did not feel I was understanding the material. In addition, I felt as if the professor gave no disregard to the students who were in the same boat as myself. Fortunately, I had the opportunity to sit down with this professor and discuss the problem at hand. Unknowingly at the time, this was going serve as valuable connection with teaching mathematics. The conversation highlighted the importance of accommodating all students learning styles and abilities while at the same time emphasizing the importance of challenging the students who are understanding at a high levels. This is something that had not occurred to me in the past and in recent times have found myself struggling to find the balance between students who are understanding mathematics at a high level versus those who are a little bit slower at understanding math.

The book Strength in Numbers does a good job in emphasizing that all students are capable of learning mathematics and can be pushed to learn mathematics at a profound level, despite their prior achievement or problems in the past. However, in order for students to understand mathematics at a deep level, Strength in Numbers states the importance that students need to see themselves in the subject in order to draw meaning that motivates them to desire an understanding of the topic at hand.

Strength in Numbers does a suitable job in underlining the important role teacher’s play in fostering a sense of belonging to help students’ progression in mathematics. However, I felt that the topic needed additional ideas and strategies to support teachers in facilitating students’ ability to see themselves in mathematics. This directed me to seek additional methods to help students visualize themselves in mathematics. In due course, I came across an article on edutopia, 9 Strategies for Motivating Students in Mathematics, by Alfred Posamentier. The article focuses on the importance of intrinsic and extrinsic motivation and highlighted numerous strategies to aid teachers in motivating students’ in mathematics. However, the strategies that I felt were of the most importance incorporated methods that promoted students aptitude to relate to mathematics and visualize themselves in the topic at hand. These strategies include the importance of teachers: indicating the usefulness of a topic, telling a pertinent story, and getting students involved in justifying their mathematical curiosities

By indicating the usefulness of a topic in a practical application could spark interest in students who have trouble visualizing how they will use this math in the future. Furthermore, telling a pertinent story can help students fantasize and dream about the topic at hand by putting themselves in a situation where they solved a problem pertaining to the subject. You know what they say, when you are not dreaming about something you’re not working for something. Hence, when students’ visualize themselves in a problem they are subconsciously becoming more comfortable with mathematics and the anxieties associated with the subject. Furthermore, by getting students more comfortable with mathematics they may become more interested in justifying and discovering their mathematical curiosities and ability. The teacher plays an important role by challenging the students who think that math is too easy or too hard. When people are challenged by someone, they tend to want to prove themselves or that person wrong.

In final analysis, in order to benefit and promote students’ mathematics ability at a high level, I will take these methods into account in my future classroom by forming mathematically amusing activities for students to participate with suggestively. Only when students become self-aware in mathematics then they are able to be pushed to learn mathematics more deeply and can be challenged to test their mathematical ability beyond their own expectations.


Motivating Students

While we have been concentrating on developing our lesson plans for the Lesson Studies at the high school, we have begun to discuss the challenge of motivating students in the mathematics classroom. In class, the document that we glanced at was

Where nine strategies for motivating students are discussed. The nine strategies are:

  1. Call attention to a void in students’ knowledge
  2. Show a sequential achievement
  3. Discover a pattern
  4. Present a challenge
  5. Entice the class with a “gee-whiz” mathematical result
  6. Indicate the usefulness of a topic
  7. Use recreational mathematics
  8. Tell a pertinent story
  9. Get students actively involved in justifying mathematical curiosities

In mathematics classrooms, motivating students can be a particularly difficult challenge. Students regularly struggle to see the point of what they are learning in their math classroom. Reading through this article prompted me to think about how I plan to motivate my students in my future classroom. Motivating students can have a ripple effect and subsequently help to work toward equity and access, as well as toward other pedagogical ideals we have previously discussed in this class.

The strategy that intrigues me the most is #8: Tell a pertinent story. In the description, it discusses solely using a story of historical events involving mathematics to motivate students. This strategy can be modified to creating lessons around these stories. For example, in the document is a link to a story about how Eratosthenes calculated the circumference of the earth. This could be modified to give students the information (or modified more so that the lesson would guide them less) that Eratosthenes had, ask them to make the calculations and deductions that Eratosthenes made themselves. This could incorporate the real world mathematics that can be lacking in the high school classroom, while simultaneously utilizing the eighth strategy of telling pertinent stories to motivate students. Going through the process of solving the problem gives students the opportunity to see the world as the historical figure(s) did— through a lens different than their own.

Based off of this document, I began to search for other articles that discuss different ways to motivate students in the mathematics classroom. From this search, I found a few articles from National Council of Teachers of Mathematics. One of the articles I discovered focuses on writing proofs using technology. As a high school student, I remember that writing proofs in geometry was one of the most tedious, and at the time, irrelevant tasks that I had encountered. I found this article interesting and relevant to the cause of motivating students in general because proof-writing gives rigor to mathematics. However, it is difficult for high school students to comprehend its usefulness— resulting in an unmotivated throng of students. This article works to make proofs more interesting for students by tying in technology. The link to the article:

Another article that I found examines motivating students through problem solving. The article focuses on creating opportunities for higher-order thinking to motivate students. This coincides with number four on the original list of ways to motivate students: Present a challenge. Many students have only experienced the traditional classroom where opportunities for higher-order thinking are few and far between.  Higher-order thinking is challenging for students, and if a lesson is executed well, it can lead to motivating students. The link to the article:

SDCTM Takeaways

This weekend I had the pleasure of attending the South Dakota Council of Teachers of Mathematics and the South Dakota Council of Teachers of Science, however I was only concerned about the mathematical components. During my time there, I had the privilege of hearing many insightful educators and former educators speaking on a variety of topics. There was one person above the rest that had really caught my attention. Lenny VerMaas gave me a lot to think about after attending three of his lectures. However, I am going to focus primarily on his first lecture, since I found it the most interesting.

His first lecture I attended was entitled “Homework Strategies For The Mathematics & Science Classroom That Engage Students.” This lecture focused on some tips and tricks Lenny shared to integrate into the mathematics classroom. Here are some of my favorite things he talked about:

  1. Ken Ken Puzzles: This is a sudoku-esque puzzles that used different equations on the inside of the boxes to solve the puzzle. Here is a picture of one, because it is a little hard to explain:  Image result for ken ken puzzle. The inside of the segments must add, multiply, subtract, or divide to the indicated number. The numbers ranged from 1-4 when we did it at the conference. I think that this could be changed to have fill in the blank equations like 2x+_____ = 5 for x=1 or solve for x: 7x+2=16.
  2. White Out- This is a method of allowing students to answer questions in class. Students would all do the problem on their own whiteboards, and when the first student gets it right the teacher can indicate to them that they are right. Then that student can get up and check to see if other students got the right answer. The teacher then does not have to monitor the whole time and can focus on students who need more help. Also, the student can ask other students if they would like help and help them.
  3. Highlighting Mistakes on Tests- This example, I thought, was really good. Instead of grading tests, he showed an example where the teacher high-lighted things that were wrong. The students could then inquire and investigate what they did wrong and retake the test if they wanted to. The reason why this is good is instead of students just getting a grade and putting the test away and forgetting about it, they are forced to look and think about what they did wrong, and are able to correct it. Lenny is a big proponent that a FAIL is just a First Attempt In Learning.
  4. Students evaluate their skills and concepts as Confident, Shaky, or Reteach. At the end of lessons, homework, etc. he suggested that teachers allow students to to rate how they were feeling about the lesson or concepts. If the student is confident then they are ready to move on. If they are shaky, the student is starting to get it, but something is not exactly making sense or something is missing. If the student says reteach, and enough students say reteach, then the teacher needs to spend some more time on the topic and make sure that students are confident.

The second lecture I attended was “Who Is Doing The Talking In Your Classroom? Questioning To Develop Student Understanding.” This lecture was interesting, but something we have already kind of covered in class. It was essentially about posing meaningful questions. Some of the suggestions he gave, that I liked, were to ask students how confident they were of an answer regardless if they were right or wrong, and instead of asking what is the volume of the cube to investigate everything you can tell me about the cube.

Finally, I attended “No One is Born With A Math/Science Brain, Everybody’s Brain Can Grow To Learn Math & Science.” This was about maintaining a growth mindset in the classroom. So, viewing mistakes as ways to learn, and having students say they can’t do something yet. He also suggested reading books to students like Math Curse. The best thing I took from this lesson was to tie effort to achievement. This is something I think is extremely important for keeping students encouraged in math. Many times I have felt discouraged, because I would put a lot of effort into something and fall short of the grade I wanted. One way he suggested to combat this was to have students write down how long they have studied for a test and assign points or extra credit for that effort.

With all of this information the one thing I wanted to know more about was just more strategies Lenny suggested and implemented. Luckily, he has a website that we keeps a lot of his work on. (Link: ). I could write many blog posts about his website, because there is so much on it. He has presentations, ideas, and suggestions for almost every aspect of teaching.  I would love to further explore his website and find things that I want to implement in my future classroom.

Critical Thinking in the Mathroom

Recently I had the opportunity to attend the SDCTM/SDSTA joint conference in Huron, South Dakota. While there, I attended many different sessions that gave tips and trick about keeping students engaged while still learning the material. As society moves away from traditional teaching, I have been trying to think of ways to incorporate more lessons in a form other than lecture. I have also thought about the need to get my students to think critically and struggle productively in the process. While in one of the sessions, I was introduced to a method called 3-ACT math tasks. These were quick lessons that force the students to think and problem solve on their own and in groups. Each tasks contains 3 steps (where the name comes from):

  1. Introduce the central problem of the task with as few words as possible.
  2. Have the students determine the information they might need to solve the problem and have them guesstimate a logical answer and reasoning for their answer.
  3. With the information at hand, the students are able to solve for the solution–then set up a sequel or extension to make sure all students grasped the concept.

The 3-ACT we did in our session was over surface area. First, we watched a video of a guy who had a filing cabinet and was covering it in numbered post-it-notes. We did not get to see him finish, but were then asked to determine what he might be doing. While there were many answers, the obvious was trying to see how many post-it-notes it would take to cover the cabinet. We all guesstimated an answer for this question. After this, we were asked what information we would need to know to solve this without physically covering the cabinet. Being a room full of math teachers, it was not difficult to know that we needed the dimensions of the cabinet as well as the post-it-notes. However, in a classroom just learning about surface ares, it will take longer for the students to decide on the information they need. After waiting for the students to decide, the teacher gives the students only the information that was asked for–making them think a little extra if they get it wrong the first time. Once the class has decided on the information needed, they solve the question mathematically. Once we all had our guess, we watched a final video showing all of the sticky notes going on and revealing the final answer. We then discussed errors made (I will not give it away in case you want to try this yourself).

These ACT’s are a great to get kids engaged in the material. Here is a list of 3-ACT math tasks created by Dan Meyer, an officer for Desmos which is an advanced calculator application that is being implemented in current testing.

Connecting Mathematics to Real Life Through Pictures

In our 7-12 Math Methods course last week, we discussed an article about reasoning and sense making, which was very good and informative. Reasoning and sense-making is very important in our math classrooms. The thing I want to discuss in this post, however, is preparing students for using math in real life. The reading we discussed in class gave three purposes for high school mathematics. We use high school math to give students “mathematics for life… the workplace…” and “the scientific and technical community.” It may seem obvious, but we need to give students the skills to use math in their daily life. Not enough students can look at objects in the world and use mathematics skills to figure out the area or circumference or something of that nature. One way to teach students how to use mathematics in the real world is by having them take pictures of things and use mathematics to figure out problems involving these objects.

The article I read that suggested this method is Mathematical Selfies: Students’ Real-World Mathematics by Kathy Jaqua. She is an associate professor at Western Carolina University. She uses this method with her students. The idea is that students take pictures of things in real life that are similar to what’s being discussed in class. For example, if a student sees a ladder leaning up against a building, they can take a picture of it, bring it in to class, and maybe figure out the length of the ladder or high up on the wall it is.

Even your typical story problems don’t relate to students in the way that them taking pictures can. Stories of buying twenty watermelons, walking twenty miles, or figuring the length of a guy wire aren’t going to relate to students the way that this method can. Students aren’t going to go tell their parents about story problems they did, but if they take a picture of something they encounter and use it in a math problem, they may go home and tell their parents about what they found, which means they’re excited about it.

An important part of this method is to realize what students are interested in so we can suggest things for them to take pictures of. Students may not realize what they know until we suggest it for them. A student who rides their bike may not realize that they can figure out how far they’ve ridden a bike just knowing the diameter of the wheels and how many times they’ve gone around. We need to suggest thing. One suggestion that Ms. Jaqua gave was using video game examples. If students are having trouble focusing, but know vast amounts about video games, then using math as it relates to video games can be a great way to reach these students.

Geometry is probably the easiest subject to use this for, but there are other mathematics that can be used. As mentioned with the ladder problem, it can relate easily to the teaching of the Pythagorean Theorem. There are plenty of parallel lines and shapes in the world that can be used when teaching geometry. There are other easy real-world applications of math. Going back to video games, students could use the amount of money gained on a business venture in a game over a few days and use that to find the slope, so they can figure out how much money they’ll have gained months from now. They could bring in pictures of the money differences from day to day. The same goes for real life finances. While high school students may not be buying their own food, they can still bring pictures of nutrition facts and figure out whether one type of snack is healthier than another when they eat the same amount of each. There is plenty of math in the lives of students that they can take pictures of and take to class.

Students need to learn math to do well in the world. They will be better prepared to be adults if they can recognize what math to do and when in real life situations. Using pictures that they take of things in their life will increase their ability to recognize math problems in real life. Also, this will make sure that the students can relate to the mathematics being taught, and so they will be more willing to listen and learn.

Reasoning and Sense Making Article


Making Math Memorable

We have all struggled, at some point, to remember a math concept we learned years ago whether it was completing the square, volume formulas, or even simple definition. So, how can we make it so students are able to recall that information 10 years down the road when they find out that we weren’t kidding when we said they might need this in the future?

In high school, I learned how to complete the square using the formula, as many of you probably did as well. I never knew how they came up with the idea or why. But now, as we learn about completing the square in my 400 level college course, I was taken by surprise. We used algebra tiles (which I had never used before) and completed squares and factored equations. Previously, I had forgotten the formula from 5+ years ago and was left with no idea on how to begin problems without looking it up on google. Learning the concept with algebra tiles, however, allowed me to put a visual with completing the square and make it much more difficult to forget the concept. I was curious as to just how many other things I learned and forgot because we were just given a lecture over the material.

I found an article, Five Ways to Make Geometry Memorable, by Genia Connell with a few ideas on how to make some geometry concepts stick. The article is about getting students to actually experience the different parts of geometry in real life. She begins the lesson with general definitions and make sure the students understand them. Then she brings in the math movie Geometry Story. It is about Buzz Lineyear and his little brothers Ray and Segment. This connection to a movie many of them are familiar with will help them remember the material discussed. She also uses other things such as math yoga, scavenger hunts, and jeopardy. Students are able to become the parallel lines and search the school for different shapes. All of these activities get the students up and moving and create a lesson that is easy to remember. If you think back to middle school and high school, you don’t really remember specific lectures but instead the fun engaging activities like reenacting a war with balls of paper or the quadratic formula song you learned.

While these activities may take time to produce and a couple tries to perfect, these are the kind of activities many students need to make connections that they will remember.


Strategies to Setting and Achieving Goals in a Mathematics Classroom

In class, we discussed eight effective math teaching practices. Mine was about using goals to focus teaching practices. In this writing, I’ll discuss what I learned about setting goals and how to use them when teaching. I’ll also discuss another article/blog post I read, “10 Tips for Setting Successful Goals with Students” by Nancy Barile. Overall, I hope to be able to inform readers about how to set goals and use them to their advantage in the classroom.

The article that listed the eight effective math teaching practices discussed some important details teachers should keep in mind when forming goals. One important detail is the concept the students are learning. This may seem obvious, but really it’s important to think about what exactly you want them to achieve in the end so you, as a teacher, can tell them what you hope to achieve by the end of the lesson. It’s important for students to be involved in these goals, so being able to explain what concept they’re supposed to know by the end is important. Another detail to consider is why the students need the knowledge. Students learn better when they know that what they’re learning is important. This ties into the next thing to consider, which is where the lesson is leading. When teaching, it’s important to keep your next lesson in mind so that you give them the information they need to succeed with the next part of their learning. The final detail that was listed was what prior knowledge they will use for the concept. Goals should build on what they know while also building towards future concepts. Those are the four most important takeaways for me from the article, with additional details mixed in. Next, I will discuss the article I read on my own, “10 Tips for Setting Successful Goals with Students.”

The first tip given is “Use verb-noun structure.” This means that goals should be written in a way that includes an action that need to be taken, not just an accomplishment. Examples would be to study three hours a night or complete five math problems per day.

A second tip given was to “Plan strategically and tactically.” To plan strategically is to make long term goals, such as a grade in a semester or the ability to complete an activity a few months in the future. To plan tactically is to make shorter-term goals that relate to the long-term goals. If a long-term goal is to finish the Harry Potter series, thinking tactically might involve setting a goal such as reading two chapters a day.

The third tip given was “Recognize when help is needed.” If a teacher realizes that a student is not on tract to accomplish a goal, help may be needed. It’s okay to not achieve a goal, but we should do all we can to get students to their goals.

The fourth tip was “Stop and reassess.” The time to look at progress towards goals isn’t just at the time that someone’s supposed to have accomplished them. There are multiple times along the way when a teacher and a student should sit down, see where the student is at, and determine whether they’re on pace to accomplish their goal. This largely goes with the previous tip on recognizing when help is needed. We as teachers can’t really provide help towards reaching a goal if we don’t assess where the student is at. There isn’t really a definite rule for this, but progress towards goals should be checked several times, especially with long term goals.

The fifth tip of “10 Tip for Setting Successful Goals with Students” was “Review action plans regularly.” The article advised that students keep their plans for achieving their goals somewhere that they see regularly, even on a daily basis. They might not need to see them that often, but students should be reminded of what they decided they were going to do to achieve their goal so they don’t forget or fall behind.

Tip number six was to create a timeline for accomplishing the goal. There may not always be specific things that can be checked off on the road to completing a goal, especially with short-term goals, but if there is, students can create a timeline so they don’t get behind on finishing steps along the way. A timeline will remind students of what they need to accomplish and help them do these things in a timely fashion.

The seventh tip that was given was to “Identify obstacles to success.” I think this tip has quite a bit to do with us teachers/future teachers getting to know our students. If a student isn’t doing well, there’s probably an underlying reason for it. It could be home life, a bad group of friends, relationship issues, etc.

Tip number eight is to “Include parents and family.” Sometimes the information from school doesn’t always get home to the parents. The quickest way for something to change with my study habits in high school was for my teachers to tell my parents that I was messing around too much or struggling in a class, and after that, my parents would be badgering me every day about whether I had homework. Unfortunately, not all parents take that much interest, but still, even if parents just ask their child every once in a while how they’re progressing toward a goal, it may motivate a student more.

The second to last tip is to “Aim for progress, not perfection.” Progression toward a goal is sometimes slow. As teachers, we need to recognize progress that is made so students feel that they’re doing the right thing by putting effort in. It takes a lot of perseverance to reach a long term goal, so when a student gets recognized for their work that they put in, it makes it easier for them to persevere.

The final tip from the article “10 Tips for Setting Successful Goals with Students” is “Have fun.” Schools don’t have to be as boring as most people view them to be. We can make it so students get to do something fun when they achieve their goal. We could also make goals for fun things just to help students get the idea of setting goals and accomplishing them.

I’ve now discussed what should go into creating solid goals from two different articles. I thought that both articles had great ideas, and I will hopefully use them in my classroom. As Dr. Reins said, everyone seems to think that using goals is obvious, and yet many teachers do not use them. If we can set goals and use them properly, we can really help our students succeed.

10 Tips for Setting Successful Goals with Students