I envisioned a typical math lesson looking like a well thought out plan filled with activities, where students would have the opportunity to grasp the concept from a multitude of perspectives.

Before Mathematics Methods

I had an aha moment when I learned about the student invented strategies for addition, subtraction, multiplication, and division. This concept was very profound to me because I was taught standard algorithms when I was growing up. As an educator who understands the benefits of student invented strategies my pedagogy has been altered to include student invented strategies, because of this I believe I will reach more students.

When students use different strategies they make fewer errors because they understand.Less reteaching is required, and students develop number sense. Student invented strategies are a basis for mental math and estimation and are often more efficient than traditional algorithms. It is important for students to discover methods that work and make sense to them.

AHA Moments!

Changed how I think about

FRACTIONS

a. At what moment did your thinking change?

Changed how I think about

how I understand Math

**Make Sense of Problems and Persevere in Solving Them!**

**Measure**

of

Success

of

Success

**Time & Perseverance**

b. What qualities do you think

are representative of a good math teacher?

One good quality of a math teacher is understanding the content. Another quality would be having a good attitude towards math. Also, they need to be prepared.

c.What is your own confidence level in teaching math?

On a scale from 1-10, I would say my confidence level is a 9. I think I need to work on content, but I am confident I can teach math.

d. What do you think is

important for math students to be able to do?

I think it is important for math students to do what the standards say are developmentally appropriate. I also think all students should know basic skills: + - / x

a. At what moment did your thinking change?

**Rosalynne E. Duff, 5th grade**

**" I look for struggle in the roles I choose, struggle and perseverance"- Vera Farmiga**

Before my mathematics methods course I thought fractions were the most complex aspect of math. I could never quite comprehend how to calculate them properly. Therefore, when I saw them I was frustrated. My thinking changed about fractions when I reevaluated my attitude toward fractions.My thinking changed completely when I dedicated myself to reeducating myself in the new ways of solving problems involving fractions.

b. how will this change effect how you teach

This change will effect my teaching positively. I will be an effective teacher due to my clarity and adoration toward fractions. Since I can remember my own frustrations with fractions when I was in school, I will be eager to share/show how to overcome math anxieties.

c. how will students understanding be different

Students learning will be enriched. Overall, they will have a deeper understanding of fractions due to my own revelations about fractions. After I share my math experience, they will be confident in themselves to conqueror other areas of math as well.

d. Mathematical Practices

One of the math practices being developed when teaching fractions is

modeling with math

. Students can recognize fractions in everyday life and use fractions to solve everyday problems.

Students

attend to precision

when learning fractions. They must be able to calculate them accurately and efficiently. Sometimes they must be able to use the correct units of measurements (ex. inches, centimeters..)

Students have to

reason abstractly and quantitatively

when solving word problems using fractions. Students must be able to contextualize and decontextualize problems with fractions.

e.Valli reflective process

When this change occurred according to Valli I had a

personalistic reflection

because I knew my thoughts and feelings toward fractions could inhibit my ability to effectively teach students fractions. Therefore, in order to achieve success as a teacher I had to persevere pass my own thoughts and feels about fractions. The changed occurred after encountering a trusted voices in mathematics who was able to change my outlook on math. This is another reason why I believe I experienced a personalistic reflection.

Before my math methods course I knew the importance of math, but I did not have a rich understanding of math. Now, after reading the book, being involved in the interactive activities in the classroom, and having an effective professor who in knowledgeable, passionate, and enthusiastic I see math more clearly. IT'S ALL CONNECTED!!! The moment I changed my thinking was when I saw in the reading how multiplying and adding were connected. I already knew addition and subtraction were related and multiplication and division were related. Then I realized fractions and division were related.

b.how will this effect your teaching

Understanding the interconnectedness of math will effect my teaching by making my lessons more engaging. I will be able to bridge the gap in students understanding, showing them how having a understanding of one concept leads to a better understanding of the next.

c. how will student learning and understanding be different

Students learning and understanding will be enriched. I believe my students will learn to love math once they can see how the concepts connect but also how math is connected to everyday life. Students will hopefully have a better understanding of math once they leave my class. With a strong foundation in elementary math they will be better prepared for concepts in Algebra. The next step preparing them to be college ready!

d.Mathematical Practices

When I teach/ show students the interconnectedness of math I will be developing the mathematical practice of

making sense of problems and persevering in solving them

. For multiple reasons some students see math as a difficult subject, for them they just do not "get it". When I share my math background, show them the interconnectedness, and scaffold their learning I am developing their character to never give up, especially when things seem difficult. When presented with a problem they will be able to make a plan. carry out the plan, and evaluate its success.

Another mathematical practice I will develop in students when I showcase the interconnectedness of math is

modeling with math

. Research states students retain information when concepts are related to real life situations. I believe in teaching with this proven research in mind. Therefore, students will recognize math in everyday life and use the math they know to solve everyday problems. Making the connection between math a everyday life makes math relevant.

When students look at math concepts in general they will demonstrate how they have

attended to precision

when solving problems and communicating their ideas. Research states when students can speak, write, and teach the subject they truly understand. To get to the point of truly understanding they will have to see how math concepts are connected.

e.Valli's reflective process

When my mind changed to see the connections in math, I believe I had a

technical reflection

. My mind was focused on instruction, and when it changed it was based on research as to how to instruct. I was sitting in my math methods course when I realized fully the connections from elementary math to basic algebra. Therefore, this is the reason why I believe I had a technical reflection.

Another aha moment I had stems from my new attitude towards fractions. After I changed my mind about fractions I was met with the concept of a whole number multiplied by a fraction. On my own, I was able to make sense out of why a whole number is equal to the whole number over one. For example 5=5/1. I did not understand this as a child because fractions were always "hard" or too complex. I now understand that the denominator tells that there is only 1 part, then the numerator tells how many parts to shade.

Another way to think about this is looking at the fraction 5/1 as an improper fraction. To convert it into a mixed number you divide the numerator by the denominator. The answer is 5.

My new found understanding has already altered my pedagogy by allowing me the ability to deepen my students understanding of this concept.

Another aha moment for me happened when I was reading our math book. In chapter 9 Developing Meaning of the Operations. Before reading and seeing the structure of addition and subtraction problems I was doing math by rote, never having a full understanding.

I now understand the structure of an addition or subtraction problem. I can identify the change, start, and the result. I understand how word problems can be set up missing one piece of the structure, therefore resulting in you having to find the missing part, and how this is related to algebra. I understand the importance of differentiating my lesson to explicitly teaching the structure to students with disabilities.

This deeper understanding will alter my pedagogy by allowing me to present addition and subtraction problems in a deeper way for all students to understand the why we do not just the how to do of math. Result being a stronger mathematical foundation to build onto for the future of my students opening the doors to algebra and beyond.

End Slide

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After Mathematics Methods

One of my favorite resources I really enjoyed learning about this semester was the Illuminations website. I read about it in the book and saw Dr. Bush use it in class, the very next day I used it with 1st graders to add using the ten frame applet. The students seemed to enjoy it.

I have used it with fourth graders to create, compare and contrast fractions. It is also a great resource for lesson ideas. I can definitely see myself continuing to use the Illuminations website in the classroom.

Resources

Another resource I enjoyed learning about was the activities from the book. Some of the most useful have been the ten frames, the rods and the blocks, the games, and following student invented strategies.

In class, I found exploring the student invented strategies the most enjoyable, because I had the opportunity to put myself in the shoes of a student who looks at the world in a different way. As a teacher this was a very valuable lesson to be open minded for the sake of my students. I am looking forward to using more of the activities and student invented strategies in the future.

One other resource I loved learning about was the math proficiencies. Since I am already in the classroom on a daily basis these are of benefit to me. I have placed the math proficiency posters inside the classrooms I go into to teach math. When Dr. Bush introduced in class I had received the posters in an email, but had not had a chance to study them. However, Dr. Bush did an excellent job of incorporating them into every lesson. Now I feel like I understand them more, this is the reason I enjoyed learning about them.

a. If I was asked by a principal what my math lesson would look like I would tell the principal my lesson would include a formative assessment for example an entrance or exit slip. The lesson would be connected to a standard appropriate for the grade level. It would include rigorous application and concept development as well as engaging hands on activities related to real world situations.

b. Qualities of a good math teacher

I think the qualities of a good math teacher are the qualities of a good teacher of any subject. They have to exhibit patience, perseverance, and have strong content knowledge of their subject.

Math teachers have to be flexible, open minded, and have a positive attitude toward math. They have to appreciate math, be able to see the relevance of math in day to day operations.

c. Confidence Level

After taking my math methods course my confidence in math is at a ten!!! I have absolutely embraced math to the point I can even say I enjoy teaching, learning, and doing math.

Since taking this course I have taught a small group math course and am now co teaching a 4th grade math class. Everyday I am thrilled about teaching math, because I "get it" I can do it, and I will try my hardest to lead my students to have a strong understanding of math.

d. Students should be able to...

After the math methods course I still think students need to have an understanding of the operations (+, -, x, and /). However, I think students should be fluent in addition and substation by 1st grade and multiplying and dividing by 2nd in order for students to become fluent in all the other areas of math.

Students need to understand the basic fundamentals of math before they can progress into abstract ideas of math, like algebra. In short, students need to be able to do all the math proficiencies and the basic operations before they can progress.