The second book I read for this course was titled "Accessible Mathematics: 10 Instructional Shifts that Raise Student Achievement" by Steven Leinwand. The book focused on different areas of mathematics that teachers need to improve on in order to increase student learning. These areas include number sense, representations, review, communications, and building understanding from data and measurement. Each chapter ended with a summary for what the reader should take from each instructional shift, and how to implement it in their classroom.

The two sections that I felt were most beneficial for me to read at this point in my career were the "Picture it, Draw It" and "Building Number Sense" sections. The "Picture it, Draw it" section discussed the importance of asking students to represent their thoughts, and ask them "why?". Leinwald also mentioned that we should use multiple representations when teaching a concept, to cater to all of our student's background knowledge. The section on number sense stressed the importance of helping kids become comfortable with numbers and estimations, mental math, place value,and building number sense with each number they encounter. I believe a huge step in the right direction for mathematics educators is to help our students feel comfortable with numbers. When our students feel comfortable with numbers, they won't be afraid to struggle a little and learn on their own.

I would recommend this book to any future math teacher because it was enlightening and I will be a better teacher because of it.

]]>The two sections that I felt were most beneficial for me to read at this point in my career were the "Picture it, Draw It" and "Building Number Sense" sections. The "Picture it, Draw it" section discussed the importance of asking students to represent their thoughts, and ask them "why?". Leinwald also mentioned that we should use multiple representations when teaching a concept, to cater to all of our student's background knowledge. The section on number sense stressed the importance of helping kids become comfortable with numbers and estimations, mental math, place value,and building number sense with each number they encounter. I believe a huge step in the right direction for mathematics educators is to help our students feel comfortable with numbers. When our students feel comfortable with numbers, they won't be afraid to struggle a little and learn on their own.

I would recommend this book to any future math teacher because it was enlightening and I will be a better teacher because of it.

After completing four years in the math education program at GVSU, you'd think I would have an answer to this question. However, throughout my four years at this institution I have found that while there are many traits that are universally agreed on as "must haves" to be a teacher, everyone has a different idea and perspective when it comes to good mathematical teaching practices.

To get a feel for how non-education majors feel about this topic I decided to ask my family members, who range all the way in age from 20 years old to 77 years old. So here you have it, what it takes to be a good math teacher according to the Farquhars:

When I reflect on these answers, and compare them to the knowledge I've gained throughout my teaching experiences, the books I've read for this course, and the testimonies from other educators, it all makes sense. I feel the key to being a good teacher of any subject is to teach in a way that makes your students want to learn. I had the privilege of taking a mathematics education course with Dave Coffey, a professor and educator at GVSU. In our class, he explained to us the importance of students holding themselves responsible for their engagement. He had us write down what it looks like when we are engaged in class, and what it looks like when we are not engaged. This was an excellent exercise for me to do because it allowed me to reflect on my own learning behaviors. As the semester continued, I was able to recognize when I was very engaged with the material, and when I was not. As I think of all the times I've sat in class, not engaged, and essentially not learning, I ask myself "what was wrong with that picture?". When these instances occur, you have to be able to realize if you (as a student) were having an off-day, or if it was the instructor who did not prepare engaging materials for his or her audience. This is a great thing to do with students of all ages (tweaked for different age groups of course) because it will encourage them to become responsible for their own learning. I also feel that a crucial part of trying to make your students care about the material presented to them is the need to cater to all learning styles. As hard as it can be, teachers need to make a solid effort to reach out to every type of student during the lesson or unit, to ensure that each student has a fair chance at learning the material.

In conclusion, it is clear that my family members did not have great experiences when it came to the mathematics education they received. The common denominator in each of their answers was that the teacher needed to find a new way of engaging his or her students in the joys of math. When you are teaching in way that makes your students want to learn, it is obvious that you also care about the subject, which creates an excellent learning environment for the students. This is something that teachers of any subject and grade level should keep in mind, because whether the students are in first grade or their first year of graduate school, they will not learn if the material is not relevant to their life and presented in a creative matter.I am glad that I had the opportunity to ask my family their opinion, because as I enter my student teaching semester it is important to realize that each student has had different learning experiences and this is the time to bring it all together and make it count.]]>

My first introduction to Desmos took place in my math capstone class, which happens to be full of education majors. I think I speak for the majority of us when I say that we were in awe at the idea of Desmos and wish we had came up with the idea ourselves!

Desmos itself is a website that focuses on learning math by doing it, rather than memorizing functions and their equations. It is an interactive graphing calculator that allows students to alter their functions to see different graphs, and learn what each function looks like. Desmos also has a version just for teachers that contains many activities for students to learn different aspects of math.

One activity I found that would be particularly helpful when teaching middle school or high school students about exponential, linear, and quadratic functions is the Penny Circle activity. In this activity, students are asked to determine which type of function best explains the growth in the number of pennies that can be held in each circle. Students are able to adjust the diameter of the circle and virtually add pennies to the circle, and move the pennies around so that more will fit. The students are able to submit their estimations, and see the data from the rest of their classmates. The program also gives information such as the mean, median, and mode estimations. The reason I was drawn to this activity was because it can be done in real-life as well. Students can compare their online results to those results in real-life simulations of this activity, and work with partners.

This activity and other activities like it are appealing to teachers because it offers many benefits to their instruction as well as their students learning. One of these benefits is group work and student collaboration. These interactive activities open the doors for students to work together and feed off of each other's knowledge. Another benefit of this activity is the ability to instantly assess their students learning. The teachers are able to see the different estimations that students make right away, and track their thought process throughout the activity. Teachers can also model and ask questions along the way to steer their students thinking in the right direction. This type of activity gives the teacher instant feedback as to what areas the students are strong in and which areas they need more assistance. An additional benefit to the Desmos activities is that they are engaging and it gives the students the opportunity to use technology. In our society students thrive with technology, and by incorporating hands-on and virtual simulations of activities students are remaining alert and engaged in the learning, thus drawing their own conclusions and discovering the mathematical concepts on their own.

As a future teacher, when I think of teaching middle school and high school math, I think it is important to keep our students active in the classroom so that they are taking responsibility for their learning. Through programs like Desmos, we can provide our students with the opportunities to discover mathematics on their own and with our guidance, they will be able to make sense of the concepts and connect them to previous topics which will make their learning more meaningful. Desmos is just one of the websites we can use to make math fun again!

The Penny Circle activity as well as many others can be found at: https://teacher.desmos.com/.

]]>Desmos itself is a website that focuses on learning math by doing it, rather than memorizing functions and their equations. It is an interactive graphing calculator that allows students to alter their functions to see different graphs, and learn what each function looks like. Desmos also has a version just for teachers that contains many activities for students to learn different aspects of math.

One activity I found that would be particularly helpful when teaching middle school or high school students about exponential, linear, and quadratic functions is the Penny Circle activity. In this activity, students are asked to determine which type of function best explains the growth in the number of pennies that can be held in each circle. Students are able to adjust the diameter of the circle and virtually add pennies to the circle, and move the pennies around so that more will fit. The students are able to submit their estimations, and see the data from the rest of their classmates. The program also gives information such as the mean, median, and mode estimations. The reason I was drawn to this activity was because it can be done in real-life as well. Students can compare their online results to those results in real-life simulations of this activity, and work with partners.

This activity and other activities like it are appealing to teachers because it offers many benefits to their instruction as well as their students learning. One of these benefits is group work and student collaboration. These interactive activities open the doors for students to work together and feed off of each other's knowledge. Another benefit of this activity is the ability to instantly assess their students learning. The teachers are able to see the different estimations that students make right away, and track their thought process throughout the activity. Teachers can also model and ask questions along the way to steer their students thinking in the right direction. This type of activity gives the teacher instant feedback as to what areas the students are strong in and which areas they need more assistance. An additional benefit to the Desmos activities is that they are engaging and it gives the students the opportunity to use technology. In our society students thrive with technology, and by incorporating hands-on and virtual simulations of activities students are remaining alert and engaged in the learning, thus drawing their own conclusions and discovering the mathematical concepts on their own.

As a future teacher, when I think of teaching middle school and high school math, I think it is important to keep our students active in the classroom so that they are taking responsibility for their learning. Through programs like Desmos, we can provide our students with the opportunities to discover mathematics on their own and with our guidance, they will be able to make sense of the concepts and connect them to previous topics which will make their learning more meaningful. Desmos is just one of the websites we can use to make math fun again!

The Penny Circle activity as well as many others can be found at: https://teacher.desmos.com/.

For this week's blog, I decided to write on a book entitled *The Mathematician's Lament* by Paul Lockhart. Why is this a Communicating Mathematics post? The post falls under this category in a unique way, because although I am writing a review and critique of *The Mathematician's Lament, *Lockhart's writing is centered around our educational system and its misconception of mathematics, which largely includes how the concept of math is being communicated to people across the board.

*The Mathematician's Lament *is broken into two parts, the first part titled "Lamentation" and the second titled "Exultation". First, let us discuss what a "lament" is. Although this may be common knowledge for some, there are still a few of us who need to widen their vocabulary. According to Dictionary.com, lament can be defined as "a passionate expression of grief or sorrow". This definition perfectly aligns with the mood of the first part of Lockhart's book.

Lockhart starts this portion of the writing by introducing his views that math is an art, and by saying that our culture does not view the subject in such a way. He states that mathematics has been transformed from adventure and imagination to facts and procedures. And while I agree with his statement, I would also like to point out that the "facts and procedures" and the rigidity of the subject are what many people enjoy about math. Personally, I find comfort in the structure of mathematical topics such as algebra. However, I also like the freedom of a proof. A particular quote from Lockhart in the book struck a nerve with me because I can relate to his claims. It says "Many graduate students have come to grief when they discover after a decade of being told they were "good at math", that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, its about making new directions". I connected to this statement in the text because as I progressed through my mathematics courses at Grand Valley, I was presented with new challenges that required a new way of thinking. It continues to be a struggle to this day because of the fact that I used to view math as "black and white". While I do not feel that my K-12 mathematics education is completely to blame, I do feel it plays a vital role in this situation.

In*The Mathematicians Lament, *Lockhart refers to teachers and educators as two different things. What does he mean by this? This brings up an interesting segway into the key issues that Lockhart believes are part of our K-12 educational system, in regards to mathematics. The issue he discusses most is the mathematical reform that is taking place in schools today. Lockhart voices his opinion that "math doesn't need relevance, it's already interesting", when referring to the way teachers try to relate math to students lives in an effort to keep them engaged. His opinion does not sit too well with me, but more on that later. He also says that the problems we are doing with our students in schools are merely exercises, not real mathematical problems that are good with our brain. However, Lockhart does make a valid point when he poses the question: Why do we let people who don't have a passion for math teach the subject? He states that "If teachers themselves are passive recipients of information and not creators of new ideas, what hope is there for their students?" Lockhart also said in the book that "You learn things by doing them, you remember what matters to you". This point is contradictory to his comment about relevance earlier in the text. If students remember what matters to them, then why does he say that math does not need relevance? By making the student's experience with math relevant, we are making it matter.

Lockhart then goes on to discuss the issues behind high school geometry, which he refers to as "the instrument of the devil". I believe his most significant point in this section was that a students first introduction to mathematical arguments is not the place for formal proofs to be taught. This idea goes hand-in-hand with his argument that most teachers do not give there students the opportunity to create mathematics, only practice the procedures with drills and tests. I think that this could be why many students, myself included, struggle with mathematical arguments, because we never had to explore the "why" of so many topics.

*The Mathematician's Lament* closes with Lockhart's second official part of the book, which is titled "Exultation". Exultation means a feeling of triumphant elation or jubilation. In this section Lockhart proceeds to tell the reader all of the fascinating discoveries that his vision for mathematics can uncover, as well as giving examples of how easy it is to explain math in a simple, elegant, and logical manner. He also states a less-than-inspiring quote that probably makes some teachers question their career choice, which was "School has never been about thinking and creating, school is about training children so they can be sorted. Its not surprising that math is ruined in school, everything is ruined in school."

Overall, the main idea that Paul Lockhart was trying to convey to his readers in*The Mathematicians Lament* was that teachers and people in general just "need to play!" His wish is that students are able to discover mathematical concepts on their own, and play with the numbers and topics until they feel comfortable. But my question for Lockhart is: what is your solution to our current system? I agree that the joy of math needs to be reintroduced into our schools, but what suggestions does he have to ensure that students are meeting the grade level standards and developing the skills necessary to pass the standardized tests we have in place? This book was eye-opening to the issues we have throughout the math education system, and although it was difficult to read at times, I would recommend it to any future teacher so that one can have the opportunity to reflect upon their own teaching methods.

Lockhart starts this portion of the writing by introducing his views that math is an art, and by saying that our culture does not view the subject in such a way. He states that mathematics has been transformed from adventure and imagination to facts and procedures. And while I agree with his statement, I would also like to point out that the "facts and procedures" and the rigidity of the subject are what many people enjoy about math. Personally, I find comfort in the structure of mathematical topics such as algebra. However, I also like the freedom of a proof. A particular quote from Lockhart in the book struck a nerve with me because I can relate to his claims. It says "Many graduate students have come to grief when they discover after a decade of being told they were "good at math", that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, its about making new directions". I connected to this statement in the text because as I progressed through my mathematics courses at Grand Valley, I was presented with new challenges that required a new way of thinking. It continues to be a struggle to this day because of the fact that I used to view math as "black and white". While I do not feel that my K-12 mathematics education is completely to blame, I do feel it plays a vital role in this situation.

In

Lockhart then goes on to discuss the issues behind high school geometry, which he refers to as "the instrument of the devil". I believe his most significant point in this section was that a students first introduction to mathematical arguments is not the place for formal proofs to be taught. This idea goes hand-in-hand with his argument that most teachers do not give there students the opportunity to create mathematics, only practice the procedures with drills and tests. I think that this could be why many students, myself included, struggle with mathematical arguments, because we never had to explore the "why" of so many topics.

Overall, the main idea that Paul Lockhart was trying to convey to his readers in

To see my notes for *The Mathematicians Lament, *please click on the link below.

the_mathematicians_lament.pdf |

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibBio.html

http://www.ms.uky.edu/~sohum/ma330/files/eqns_2.pdf

I thought this picture was quite fitting when discussing the role of our number system, because numbers rule our world! Numbers are involved in everything we do, from purchasing something in a store, reading a book, driving a car, to doing mathematics! This society and universe we live in is centered around numbers.

Before today's written number system was developed, there were many different ways to communicate numbers and amounts. The Incans, Mayans, Greeks, Egyptians, Romans, and other civilizations all formed their own number systems that they were using to communicate across their groups. These number systems were composed of lines, dots, and symbols that represented different number amounts. However, we still use tools today to count that were used thousands of years ago, such as our hands, sticks, rocks, and other small objects that people use to count with.

By 500 B.C., we had begun to use the base-10 numerical system, which is composed of the ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The creation of this system can be credited to the influences of the Chinese, Indian, and Arabic number systems (History of Chinese Invention). This is the number system most commonly used around the world today.

Numbers are seen in all aspects of our universe, and are used in a multitude of different ways. In fact, numbers are used in far more areas of our life than just mathematics. Take a moment to pause what you are doing, and observe the two-foot radius around you. How many different uses and forms of numbers do you see that you do not consider "mathematics"? In the two-foot radius that surrounds me as I sit at my desk writing this blog, I have counted 13 different uses of numbers. One that I found is particularly interesting is the numbers on a fan. We often designate numbers to represent the different speed or intensity levels of the functions of everyday objects. While this can be viewed as mathematical, it is not something that I would normally consider to be "math in my life". This personal example demonstrates how the number system has a role in my everyday life, and how anyone reading this blog can consider how the number system impacts their own life as well.

I am a mathematics education major, so it is apparent that I feel our number system is important and matters in our everyday lives. However, I asked my roommates what they think of our number system, and if it matters to them. Here are the responses I received:

*"Numbers are numbers. I don't see how we could count or keep track of things without numbers. There are numbers on everything and to take those away or use a different system would kind of screw everyone up I think. It would probably result in a lot of mass murders and devolve quite a bit."*

*"I think our number system matters because its a tracking system for literally everything. But I don't think that means its the only right way to do it. If we switched I think it would be fine eventually but cause a lot of problems at first."*

Their responses were similar to what I expected they would be. The number system we have currently matters because of that exact point, it's what we have right now. Many things in our world are built off of the number system, including all of our current technology, so if we were to change the system, I believe it would cause a bit of an uproar. However, the world is constantly changing and evolving, so there is no way of knowing if our future generations will take our discoveries to the next level and invent a number system that is more efficient.

In conclusion, our number system is basically what makes the world go 'round. In my opinion, I believe the concept of a "number system" is what matters more than the actual numbers and symbols used in that system. The system itself is what keeps things organized and functioning, but it could be represented with any symbols.

It matters because it plays a role in everything we do, and has been around for many centuries,

*Citations:*

Number Systems: Where Did Numbers Originate?

http://www.math.wichita.edu/history/topics/num-sys.html#hindu-arabic

History of Chinese Invention - The Decimal System of Number Representations

http://www.computersmiths.com/chineseinvention/decimal.htm

]]>Before today's written number system was developed, there were many different ways to communicate numbers and amounts. The Incans, Mayans, Greeks, Egyptians, Romans, and other civilizations all formed their own number systems that they were using to communicate across their groups. These number systems were composed of lines, dots, and symbols that represented different number amounts. However, we still use tools today to count that were used thousands of years ago, such as our hands, sticks, rocks, and other small objects that people use to count with.

By 500 B.C., we had begun to use the base-10 numerical system, which is composed of the ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The creation of this system can be credited to the influences of the Chinese, Indian, and Arabic number systems (History of Chinese Invention). This is the number system most commonly used around the world today.

Numbers are seen in all aspects of our universe, and are used in a multitude of different ways. In fact, numbers are used in far more areas of our life than just mathematics. Take a moment to pause what you are doing, and observe the two-foot radius around you. How many different uses and forms of numbers do you see that you do not consider "mathematics"? In the two-foot radius that surrounds me as I sit at my desk writing this blog, I have counted 13 different uses of numbers. One that I found is particularly interesting is the numbers on a fan. We often designate numbers to represent the different speed or intensity levels of the functions of everyday objects. While this can be viewed as mathematical, it is not something that I would normally consider to be "math in my life". This personal example demonstrates how the number system has a role in my everyday life, and how anyone reading this blog can consider how the number system impacts their own life as well.

I am a mathematics education major, so it is apparent that I feel our number system is important and matters in our everyday lives. However, I asked my roommates what they think of our number system, and if it matters to them. Here are the responses I received:

Their responses were similar to what I expected they would be. The number system we have currently matters because of that exact point, it's what we have right now. Many things in our world are built off of the number system, including all of our current technology, so if we were to change the system, I believe it would cause a bit of an uproar. However, the world is constantly changing and evolving, so there is no way of knowing if our future generations will take our discoveries to the next level and invent a number system that is more efficient.

In conclusion, our number system is basically what makes the world go 'round. In my opinion, I believe the concept of a "number system" is what matters more than the actual numbers and symbols used in that system. The system itself is what keeps things organized and functioning, but it could be represented with any symbols.

It matters because it plays a role in everything we do, and has been around for many centuries,

Number Systems: Where Did Numbers Originate?

http://www.math.wichita.edu/history/topics/num-sys.html#hindu-arabic

History of Chinese Invention - The Decimal System of Number Representations

http://www.computersmiths.com/chineseinvention/decimal.htm

I was experimenting with tesselations on GeoGebra, and I thought to myself "how cool would be it to illustrate a children's book using only tesselations?". The picture you see below is an example of an illustration that I created. This illustration could represent two animals, a fox and a wolf, that do not get along. This particular shape allows the characters to take on a defensive stance, demonstrating a conflict in the story. If you were to illustrate a story using only tesselations, you could create the shape to represent any mood the characters are feeling at that point in the story. For this picture, I also used the shapes to resemble other parts of the setting, such as a the sky, grass, and sun. I also used segments of the shape, as well as other geometric figures to create a flower in the picture.

I have been brainstorming ideas for my final project for this class, and so far, this is my favorite idea. I think it would be so awesome to write a children's book and illustrate it with my own tesselations. This could also be something I could use in the future in my classroom, depending on what grade I teach. If I were to teach lower elementary, this could be a book that I share with my students when learning about patterns and shape properties. If I were to teach upper elementary or middle school, I could have my students create a similar project in which they create their own tesselation and write a story about it. This is a great assessment idea for a unit, and it could assess the students learning of more than one subject. For the illustrations, I may create them on the computer or by hand, I have not decided yet. Overall, I think this project would be a creative use of my time that I would be able to use throughout my future career! ]]>

The dictionary definition of axiom is "a rule or statement that is accepted as true without proof". In simple terms, an axiom is something we know to be true without questioning it.

There are two different types of axioms,

An example of an addition and multiplication axiom we know to be true:

Let

Then

There are axioms that we assume to be true for a variety of mathematical content areas, and these axioms help us develop important theories. Axioms are the building blocks of mathematical proofs, as they are helpful when processing information and assess what we already know to prove mathematical conjectures. Axioms also define unknown terms, such as line, point, etc. Axioms play a crucial role in mathematics, as they give us a basis for proofs, and give us information to use when solving mathematical problems. When people discuss using "logic and reasoning" to determine the results of a problem, it is my opinion that they are using axioms to determine those results. Axioms are directly associated with the logic and reasoning we use in many different subject areas today, not just mathematics.

**To create this post, I used my own knowledge of axioms as well as information published on Princeton University's webpages.**

]]>

One of my favorite things about math is that it teaches us to ask WHY something is the way it is, not just accept that it is true. I think one of our biggest problems as a society is that the majority of people accept things to be true without further exploration. As a future teacher it is my goal to encourage my students to dig deeper into what they are learning, and ask questions! Not only in math, but in all subjects they encounter. This will develop their reasoning and logic skills and help them become independent thinkers.

There have been many great mathematics discoveries, but these are the five that I feel are most significant to my future as an educator:

1. The discovery of numbers and counting. Numbers themselves are the backbone of everything we do. Without numbers, we would not have math. This discovery has led to our ability to sort, calculate, and draw conclusions among other aspects of life.

2. Shapes and their attributes. I feel this particular aspect of geometry is very important. The discovery of the different shapes and all of their characteristics is crucial to our understanding of their relationships to other shapes and how they fit into the world. We use shapes for many things, like road and traffic signs, building, and cooking.

3. The four basic operations. I feel that this discovery is pretty self-explanatory. The four basic operations: addition, subtraction, division and multiplication, are just one of the ways we integrate numbers and use them in our daily lives. Without these operations, math would be quite different from what most people know it to be.

4. Proof Writing. This is the aspect of mathematics that pushes us to find truth in things we already accept to be true, and things that we want to decide if they are true or not. It encourages us to ask the why and the how, rather than just knowing the answer.

5. Calculus. I feel that calculus is a beautiful aspect of mathematics. Newton and Leibniz developed a subject that can be applied to many different situations and is not arbitrary.

I chose these five mathematics discoveries because each one relates to how I want my students to view mathematics, no matter which grade I teach. If I teach first grade, I want my students to recognize the math that is in our classroom, such as the shape of the clock, the progression of days on the calendar, the way the table is built. If I teach high school mathematics, I want my students to be able to connect all of the different aspects of math together, and realize that it has a purpose. ]]>