Every week I tutor in a second grade classroom at West Kelloggsville Elementary School. I have had the opportunity to assist them with their math lessons. The classroom teacher often falls into the typical instruction of algorithms without encouraging number sense and understanding. Connecting my experiences there and with what I have learned in MTH 223 has really enhanced my perception on how to teach mathematics. Understanding and number sense is SO important, yet it is rarely the focus in math. I want to change that. I want to be the kind of teacher that puts understanding first. Students should be able to see the importance of what they are learning instead of counting down the hours until they go home.

This course has made me re-evaluate teaching and see the impact of looking deeper into mathematical concepts. Even the concepts that we consider simple and elementary level contains patterns and characteristics that we never evaluated as a child. An example would be double-digit subtraction. If a student asked me exactly why do we borrow, would I be able to give them a complete answer? When I think about it, I was never taught why and I honestly never questioned it. Math is so full of “just do it” and any questions are often answered with “because you just do.” How can you teach children to do it without fully understanding it yourself? This class has made me explore concepts deeper so that I can answer those difficult questions.

The deep thinking and math activities from this course have been useful while I am tutoring. I am able to connect ideas in a way that I never would have been able to do without everything I have learned. I never realized how complex teaching mathematics is and just how much thought it required to teach it well. I want students to develop a passion for not only math but learning. I hope that I can take all of this new knowledge to teach successfully and intentionally. I am thankful for the opportunity to take such an eye-opening class. I have never taken a class that has left me so informed, hopeful, excited, and absolutely encouraged.

Thank you Professor Golden!

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I found two particularly interesting activities on Nrich.maths.org. One provides students with an opportunity to observe properties of different 3D shapes. Students are given various blocks and after the chance to play and observe, they can be asked questions like, which of these towers will collapse?

I predict that it would be great as a teacher to witness how students come to conclusions about each shape. I also think that it would be worthwhile to come together as a class to share characteristics that they noticed. I have noticed in my classroom experience so far that 3D shapes almost seem overlooked. Three-dimensional shapes can be incorporated into measurement so that students can determine the differences between 2D and 3D. Volume and surface area are vastly different calculations that area and perimeter. I believe that it is important for lower elementary teachers to set students up for success by providing with more experiences with three-dimensional shapes.

The second activity that I found on Nrich related two-dimensional and three-dimensional object by using shadows. Students are presented with a square, circle, triangle, and rectangle. They are asked what objects on a playground would form that shadow.

I find the relationship between two-dimension and three-dimension fairly interesting. I would consider myself a visual learner and making connections to tangible objects is important to my ability to learn math. I believe that more time spent working with three-dimensional objects would give students a better understanding of volume vs. area.

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One little boy I helped was named Iyasu and he was so full of joy and energy. His favorite subject is math, so you can imagine my excitement as a math major. Each day he eagerly grabbed his homework and caught me up on what he had learned that day. We worked through an subtraction problems together using unifix cubes even though he had memorized all of the basic facts. I was able to practice posing questions about his thinking and typically he could answer completely and write out his work. His love for math and learning was truly encouraging, especially because he was always up for a challenge.

On the other hand, I also assisted a little girl named Felicia. She often lied about having math homework because it was so difficult for her. I practiced using manipulatives and number strings, but what she needed most was encouragement and praise for hard work. It had me thinking about growth mindsets and how desperately this sweet girl needed to develop one. Each day I would ask her how school was and the answer was always the same, “BAD!” I knew that there was very little I could do for her in just one short week, however, it made me realize how much of a passion I have for low-income students.

Learning should be enjoyable and every single child deserves to know how great they are instead of being constantly told what they are doing wrong. Our goal as a team was to shower them with compliments and encouragement. Throughout the week, Felicia updated me about problems going on at home, and it was evident that it had a significant impact on her. My heart broke a little each day as I would watch her and other children be more worried about getting something for dinner rather than doing homework. This is what really impacted me. The reality of these kids is that they do not always get to act like kids. They worry about not getting enough for dinner as they beg other kids for seconds. Many of them struggle with family hardships on a regular basis. In their lives, how could school be the most important thing?

Yes, these kids need help with their homework, but what they need more is attention and support. They need to be seen as equally capable and smart as every other child. Each of these kids need opportunities to learn that are meaningful. They need a future and the tools to get them there. What amazed me the most was the joy and laughter they all have. Each of them come from broken homes, yet they can still see the good in their lives. So, why do we as teachers struggle to see the potential in all of our students? Every child needs the support and encouragement to get them through the troubles they face at home. I realized that I want to be a teacher so that I can help kids. Far too often I see volunteering at schools as something that looks good on my resume instead of a way to positively impact a child who needs lots of support. Of course I want them to learn, that is how I can give them the opportunity to be successful in their future because they cannot get there on their own. So, the biggest take-away for me was the importance of growth mindsets and supporting every student. Kids like Iyasu need just as much attention as those like Felicia. As a teacher, I want to build interest to learn in every student and I believe that starts with genuinely caring about each child so that I can teach them accordingly.

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As a student, I recall being tested on measurement even on standardized exams such as the ACT. I wondered why something so simple would be included on an exam like that, but in reality, it is not all that simple. Yes we are taught at a young age, however, were we shown how to use a ruler in multiple ways? From the instruction that I remember, I was only ever told to measure from the end of the ruler, but never told that I can start at the two and count up how many inches the object is. In a way, this relates to what we have been talking about with addition and subtraction. Students are commonly instructed with a set method and fail to see numbers in multiple ways. A majority of the time kids only know one way to solve a problem and if they are given a task that does not appear in the standard way, they are confused and feel that they are unable to solve it. The same happens with measurement. If a ruler is presented with a starting point other than zero, students even as old as seventh grade cannot answer correctly.

As I said before, another issue is the lack of comparisons that is typical of teaching measurement. How many times can you remember measuring an object printed off on a piece of paper? I can recall countless cases of this. To make matters worse, there typically was nothing to compare that measurement to. The number we got was just a number with really no meaning. We did not use it for anything other than to write it on the line provided. So what is the point?

When a task is presented as a problem that needs to be solve, connections are made and the numbers they find from measuring are meaningful. Comparing two or more objects is exactly what measurement is used for. For example, we need to measure the length of a wall and a picture so that we can hang a picture in the exact middle, or one may need to bring a couch into a room and they have to measure the couch and doorway to make sure that the couch would fit. Measuring is not finding the length the object and not using it to make a comparison or to solve a problem.

I recently found a lesson plan for a kindergarten measurement unit, titled Measurement–Length. The ideas include spanned over several days and gave younger children an understanding of measurement before being given a ruler. Students are required to compare objects, such determining what things are bigger or smaller than others. What I enjoyed most about this lesson plan was that students are developing the foundations necessary to indirectly compare two objects. Also, the common core standards of measurement in kindergarten are met by this series of instruction. By lining up markers and miscellaneous classroom items, students are directly comparing objects and placing them in between items that are slightly smaller and larger. After working as a class, children are asked to complete the same task individually on a smaller scale. Therefore, students are exploring the concept of comparison that sets them up for success when it comes to indirect comparison in later grades. Eventually these students are required to measure with different units such as paper clips, shoes, and blocks. This provides an opportunity for the teacher to explain units and their importance.

As I mentioned earlier, giving students a reason why they are completing a task is essential and motivating. Measurement is a math topic that has a clear purpose in life and should be taught as such. This class constantly challenges me to put myself in the teacher mindset and really try to understand student thinking. I strive to challenge and question why in a way that I will be able to answer the same questions for my future students. Measurement is a necessity and the way it is taught should be directed towards the real ways that it is used.

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Last year I volunteered at Wyoming West Elementary School in a kindergarten classroom and a fourth grade class. The majority of my time spent with the fourth graders was holding up flashcards for them to tell me their best guess at the answer. It is true to say that many of the kids really were guessing. Most of them did not understand how to estimate, so when I asked 3 x 5= ? they did not understand that they could count by 5 three times in order to get the answer they need. This poses a major problem to them because they are failing to understand what multiplication is and they are failing to make connections.

We have been talking in class about the importance of connections in math. Without them, students are failing to make sense of math and solve problems in a way that is meaningful to them. From class the last few weeks, there is one phrase in particular that has stood out to me.

Be intentional.

This may sound simple and obvious, but I have recently become more aware of the impact elementary math has on people. All of the foundations students learn are typically followed with memories of frustration, thus turning people away from math as they grow older. In my opinion, math should be supported by real-life problems that prompt problem solving skills and persistence. Too often math is seen as memorization of facts and equations, while understanding how they can be applied is less frequently known. As a recent assignment, I asked two of my friends to complete a long division problem. Both remember the steps and arrived at the correct answer. However, when I asked specific questions about why a certain step was important, they responded with, “Because that is what you do.” If we only know the steps to an algorithm but cannot understand why they are done, are we really improving our knowledge of math?

So, as a future teacher, I hope to encourage understanding with a variety of methods that we have discussed as a class. I want to provide students with methods that make sense to them. It is a clear fact that all people do not learn the same way, so teaching in different ways will support more students in a class. Creating lessons and activities that challenge all students and also give them the new knowledge they need to gain better mathematical understanding are essential to promoting long-term learning. When I think back to my own elementary experiences and how I wondered why we memorized countless math facts, I now know that we need effective strategies. However, before efficiency and speed comes understanding. I see a great importance of teaching realistic division problems, estimations, and easier to understand methods before long division. Yes, long division is effective but students need to understand why we need the algorithm. More importantly, we need to give them a why in math. There should be less “It is just what you do,” and more “I am doing this because…” As teachers, we need to be intentional in a way that we have more than just the next test in mind. We should be considering the importance of providing students with tools and understanding that will benefit them for their entire lives.

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Believe me when I say that nearly all of the kids at the center can recite the name of any shape I show them, but after reading these articles I wonder if they really understand the difference between shapes. Do they only know the shapes because we as teachers show them the same example of a triangle each day? Do they truly know what a trapezoid is, or do they only memorize the name that goes with the cute, purple shape with a smiley face on it? Of course they are given geometric manipulatives to use during their daily centers, but still I find myself questioning if they understand the characteristics of each shape. I’ll ask them how they know that a triangle is in front of them and they typically answer the same way, “It has three sides and three points.” This is where I make another mistake, I assume that they know what a side or angle really is. Instead of questioning them to deepen their understanding, I congratulate them on their answer that they have memorized. Have they ever been taught what an angle even is? The answer is most likely no.

Since these kids were two years old, we as teachers encouraged straight memorization. I mean, a two year old is not going to understand that a trapezoid has two parallel sides. So, we show them the same pictures of shapes that they will see every day all the way up until they graduate from Pre-k. By the time they are five years old they are expected to perfectly recite the name of every shape, and all of the teachers conclude that they have mastered pre-kindergarten geometry. Is it fair to say that they have mastered shapes when they fail to identify a different example of a triangle?

What is really unfair is that as they move into kindergarten, they are already expected to know the different shapes. According to Clements, “By the time they enter kindergarten, most children have many ideas about shapes. Yet teachers often do not ask children to extend their ideas.” As you can see, these articles really had me questioning how shapes are taught. Young students have yet to be taught what parallel lines and congruent sides are. Clements also explains that many of our perceptions of shapes solidify by the age of six. However, at six years old we have not learned what characteristics classify each shape. After it is determined that a child can recall the name of each shape, it is expected that they know this in later grades and the idea is not really revisited until more complex ideas are introduced. I recall learning angles and congruence around sixth grade. At that time, I remember seeing different examples of each shapes. It wasn’t until years after learning shapes that I learned the difference between isosceles, scalene, and equilateral triangles. I had never known that they existed in my lower elementary education. I thought an equilateral triangle was they only way a triangle could look like. Perhaps this is why it can be sometimes be difficult for us as college students to identify different quadrilaterals.

It all starts before our elementary education. So, this brings me back to teaching at the learning center. What we assume to be enhancing their knowledge is actually contributing to the problem. The lessons I have been teaching and help teach are actually setting up students for confusion later on.

One of the best parts about Clements’ writing is that he provides various methods and activities to further students’ thinking. He talks about ways students can picture being in a shape or identifying a shape by feeling it instead of looking at it. Many of his activities that I would consider using and seem to be very beneficial for students.

In class, we created multiple representations of the eleven different types of quadrilaterals. Originally, we created some of geoboards; eventually, we drew them on dot paper and used them for games such as matching and go fish. Some of the examples are included below.

Instead of geoboards, I found a way to use popsicle sticks to create shapes on ABeeCpreschool.blogspot.com. Students can create other representations of triangles and quadrilaterals. Teachers should encourage creativity to create shapes that fit the qualities of a shape but appear differently than the standard representation.

Clements mentions providing examples and non-examples. In another education course of mine, I learned how to create and implement concept diagrams. I have included an example below that compares and contrasts several types of shapes.

As I mentioned before, the few weeks of this class and the readings we have done so far have already changed my perception of teaching geometry and mathematics in general. When I head back to the learning center I will have a new understanding of what tools children need to learn shapes. Instead of asking a child what shape I am showing them and being satisfied with a memorized answer, I will question them and ask them to explain their thinking. I plan to introduce new visual representations of each shape so that when I hold up a triangle with the point facing down I will not receive the answer, “It’s an upside down triangle.”

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