Teaching Reflection
Each of the students in our class had the opportunity to teach in a classroom at West Michigan Academy of the Arts and Academics. My colleague, Kaela, and I taught in the Intro to Art class of 6th graders. We taught a lesson on tessellations and taught the students how to make their own tessellation. After teaching I wrote the following reflection:
Kaela and I taught our lesson to the sixth grade class at WMAAA on making tessellations. The lesson went fairly well, but some changes could have been made in the teaching. We quickly taught the lesson first to give the students plenty of time to work on their own tessellations. Some of the students did very well and understood the directions right away, but many of the students needed a lot of extra help as we walked around during work time. Even a few students had to start over on cutting out their shape.
To modify the lesson so that the students would understand better, I would have taught it in parts. First I would have had the students pick a shape and cut it out on the tagboard to create their tracing shape. Once every student had done that I would have them regroup and listen to the next part of directions. Now I would explain and demonstrate how to cut out the sides. I would be cutting out a oversized shape in front of the students to demonstrate how the cutting works. I would make it clear that it is very important to save each piece they cut out and only make one cut at a time. After each cut I would explain where and how to attach the cut out back to the shape. Then I would give students time to work on their cuts while walking around the class. Once every student had their shape finished and ready to trace I would ask a few students to share their shape and estimate what transformations would appear when they trace. Lastly, I would demonstrate and explain how to trace their shape on a piece of paper while I traced my shape on the board. Then I would allow students to trace their shape on their paper and walk around to help while they do so. Then students could color and design the tessellation they made and a few students would have the opportunity to share their final product with the class.
This modified lesson would walk through the lesson alongside the students for better understanding. I enjoyed teaching the students about tessellations and I thought they were a good group to work with. It was very interesting to see the range of understanding of the students.
I think it was very valuable to get the experience of teaching a geometry lesson to this class of sixth graders. It was interesting to see how some students really got the lesson right at the start and others needed a lot of help and guidance. Here are some examples of students work from our lesson:
This student made a shape that worked really well and used translation to trace. (pictured below on the left)
Kaela and I taught our lesson to the sixth grade class at WMAAA on making tessellations. The lesson went fairly well, but some changes could have been made in the teaching. We quickly taught the lesson first to give the students plenty of time to work on their own tessellations. Some of the students did very well and understood the directions right away, but many of the students needed a lot of extra help as we walked around during work time. Even a few students had to start over on cutting out their shape.
To modify the lesson so that the students would understand better, I would have taught it in parts. First I would have had the students pick a shape and cut it out on the tagboard to create their tracing shape. Once every student had done that I would have them regroup and listen to the next part of directions. Now I would explain and demonstrate how to cut out the sides. I would be cutting out a oversized shape in front of the students to demonstrate how the cutting works. I would make it clear that it is very important to save each piece they cut out and only make one cut at a time. After each cut I would explain where and how to attach the cut out back to the shape. Then I would give students time to work on their cuts while walking around the class. Once every student had their shape finished and ready to trace I would ask a few students to share their shape and estimate what transformations would appear when they trace. Lastly, I would demonstrate and explain how to trace their shape on a piece of paper while I traced my shape on the board. Then I would allow students to trace their shape on their paper and walk around to help while they do so. Then students could color and design the tessellation they made and a few students would have the opportunity to share their final product with the class.
This modified lesson would walk through the lesson alongside the students for better understanding. I enjoyed teaching the students about tessellations and I thought they were a good group to work with. It was very interesting to see the range of understanding of the students.
I think it was very valuable to get the experience of teaching a geometry lesson to this class of sixth graders. It was interesting to see how some students really got the lesson right at the start and others needed a lot of help and guidance. Here are some examples of students work from our lesson:
This student made a shape that worked really well and used translation to trace. (pictured below on the left)
This student really struggled and did not understand how the cut outs worked to make his tessellation. His shape did not tessellate.

This student needed extra help and his cut outs were a little off. This resulted in making tracing a little more difficult, but his finished product was interesting and was a successful rotational tessellation.
This student made a shape that fit on one side of the shape, but did not work on the other side. We tried to help him with tracing, but realized eventually that it only tessellated on the one side.

Measuring the Hallway
Today in class we did an activity where we measured the hallway using our average step length. This activity was challenging and fun and we had a really good discussion as a class about measurement.
We discussed the meaning of accuracy and precision. We also discussed how we should account for errors in measurement.
Our professor made the point that a lot of money for jobs come out of measurement error. This is how businesses make a lot of money. If they account for error correctly when it comes to measuring product or price, they will do better than a business that does not do so.
I think that it would be fun to modify this activity into a contest for students. Whichever group came the closest to measuring the hallway the most accurately could receive a prize or recognition of some sort.
I found this lesson plan on the Illuminations website http://illuminations.nctm.org/Lesson.aspx?id=761.
This lesson is a good one for teaching and learning unit conversions which is something we have also talked about in class lately. The lesson uses the length of a whale to estimate how much it weighs. I think this is a good way for students to think about measurement and estimations. It is important for students to learn based on things they are interested in and I know many students are interested in animals. Likewise, the ideas of accuracy and precision could be used toward the end of this lesson. Students could discuss how close their estimations of the whale is to others in the class and how accurate or precise their estimates are.
I found another activity on measurement from the Illuminations website: http://illuminations.nctm.org/Lesson.aspx?id=916.
In this lesson, students measure each other in centimeters and then make a box and whisker plot to compare the heights in the class. This lesson could contain a lot of error considering the different ways that students are measured. Students must remember that if the student they are measuring is not standing up straight, then their measurement could be off. This could be a good way to look at and discuss error as a class after the lesson. I also think that it is important that students learn to represent their data graphically. By using a box and whisker plot, students can see the data in a concrete way and understand a way to organize the data.
All three of these lessons are different. I think it is important to remember that measurement can be taught in many different ways. As teachers, we need to look for new and exciting ways to teach measurement that will cater to the students needs and not teach misconceptions about measurement. Also conversations and discussions about error and accuracy and precision are valuable in a classroom. Students should be learning what error in measurement means and how to calculate it, because it is important.
We discussed the meaning of accuracy and precision. We also discussed how we should account for errors in measurement.
Our professor made the point that a lot of money for jobs come out of measurement error. This is how businesses make a lot of money. If they account for error correctly when it comes to measuring product or price, they will do better than a business that does not do so.
I think that it would be fun to modify this activity into a contest for students. Whichever group came the closest to measuring the hallway the most accurately could receive a prize or recognition of some sort.
I found this lesson plan on the Illuminations website http://illuminations.nctm.org/Lesson.aspx?id=761.
This lesson is a good one for teaching and learning unit conversions which is something we have also talked about in class lately. The lesson uses the length of a whale to estimate how much it weighs. I think this is a good way for students to think about measurement and estimations. It is important for students to learn based on things they are interested in and I know many students are interested in animals. Likewise, the ideas of accuracy and precision could be used toward the end of this lesson. Students could discuss how close their estimations of the whale is to others in the class and how accurate or precise their estimates are.
I found another activity on measurement from the Illuminations website: http://illuminations.nctm.org/Lesson.aspx?id=916.
In this lesson, students measure each other in centimeters and then make a box and whisker plot to compare the heights in the class. This lesson could contain a lot of error considering the different ways that students are measured. Students must remember that if the student they are measuring is not standing up straight, then their measurement could be off. This could be a good way to look at and discuss error as a class after the lesson. I also think that it is important that students learn to represent their data graphically. By using a box and whisker plot, students can see the data in a concrete way and understand a way to organize the data.
All three of these lessons are different. I think it is important to remember that measurement can be taught in many different ways. As teachers, we need to look for new and exciting ways to teach measurement that will cater to the students needs and not teach misconceptions about measurement. Also conversations and discussions about error and accuracy and precision are valuable in a classroom. Students should be learning what error in measurement means and how to calculate it, because it is important.
Shape Competition Week 6 and 7
Today in class we did an activity that I loved. We did a freehand drawing competition of 2 dimensional shapes. I think what made this activity so fun and educationally valuable was the competition, but also it emphasized our need for measurement in every day life. Measurement is used each and every day. The building that you are reading this in was constructed using incredible amounts of measurement. Measurement is used a lot in construction, but it is also used in every day activities such as cooking, and shape competitions!
For the shape competition we were split up into 5 groups of 4. We had a list of 6 shapes to choose from that our group would "host" at our table. The shapes were: circle, line, square, isosceles trapezoid, equilateral triangle, and Lagon. After each group had a different shape, we had to come up with rules for our freehand competition. My group chose Lagon which is a hexagon with 5 exterior right angles and one interior right angle. When other students came to our table they would have to draw the best Lagon they could freehand. Then we had to come up with way to measure who had the first, second, and third place most accurate Lagon drawing. At first I thought this task would be easy, but it turned out to be quite challenging. At first my group wasn't aware that we could use a protractor to measure the angles. So, we tried to come up with a way to measure the angles using the corner of a piece of paper. We also came up with a new way to measure an Lagon to see if it had parallel sides. We measured the sums of the top 2 lines and compared with the parallel bottom line. If these two numbers were equal, then we knew that the shape was perfectly parallel. The problem was that we thought that this meant that the angles would all be 90 degrees as well, but we were wrong. Our professor showed us that this method only proved whether or not the Lagon was parallel. We had to come up with another way to measure the right angles. Then our professor told us that protractors were allowed. We used a protractor to measure one angle, the "corner angle", or the reflex (opposite angle from the concave angle) angle. After each group had come up with a way to measure their shape, and rules for how to get first, second, and third place in the competition, we presented to the class. My groups rules were No folding and the shape must be drawn free hand with no help from a straight edge. The lagon must be drawn to take up at least half of the paper (this way to make measuring more precise) The lagon will be measured by how close to equal the sum of opposite parallel sides are and the deviation of the concave reflex angle from 90 degrees. The difference of the sums were measured in cm and points were given for the difference. The deviation from 90 degrees was also added to the score as a point for each degree off from 90. The lowest score is the winner. Then we were given the option to submit a freehand drawing to 3 of the shapes/groups. I chose the equilateral triangle, the circle, and the square. I had to be creative and used my fingers to make sure each edge of the triangle was the same length. I ended up getting first place on the triangle! I also got 3rd place on the circle. We were awarded candy prizes, which made everything much more fun. This activity showed us how important measurement is. It taught us why standard units are so important. There is an obvious need for standard measurement when doing things like competition. In order to make the competition fair, we had to measure each shape the same and be consistent in the way we were measuring. I learned that it is very important to learn proper technique for how to measure, or students could measure the sides wrong and a person could end up losing the competition that should have won. Also, when it comes to measuring, it is important that students account for human error, and error in the tools they are using and understand that there is an uncertainty when it comes to measurement. Overall, this activity was enriching and taught me a lot about measurement. I would definitely use it in my future classroom. 
Measurement Week 4 and 5
We have started to talk about measurement in class. As in activity in class we were challenged to make a measurement journal for a day. With this journal we were to write down every time that we thought about measurement throughout our day. This included thinking about how much, how long, or how far. I was blown away by how much I use measurement in my daily life. As soon as I wake up in the morning I have to use measurement. I must measure how much toothpaste to use on my tooth brush, how much makeup to use on my face, and how much product to put in my hair. When it comes to eating, we make decisions on measurement every day. Every time I decide how much to eat of something I am using measurement. Realizing how important measurement is in our lives made me think about how important it is to effectively teach measurement in my future classroom.
We also did an activity in class where we had to use indirect comparison of books. Each table group was given a book. As a group we were to move around the classroom and rank the books based on how tall and how wide they are. We were not given a ruler so we had to figure out how to measure the books in our own way. My group used a pencil to measure the books. We found that some books were not a full pencil length so we had to use fractions of the pencil to measure. We were pretty accurate in ranking the books in the correct order. Professor Golden told us that when doing this activity with younger students, some students would use their hands to measure, only to realize that some of their hands were bigger than others. This activity teaches us how much of a need we have for universal measurement. This is why we have standard measurement like inches, feet, etc. I think it would be cool to repeat an activity like this using indirect comparison to compare other objects like lengths of stuffed animals. I would love to observe first or second grade students doing this activity to see what kind of objects they decide to use to measure. We read an article in class about teaching measurement from Constance Kamii. She emphasized the need for teaching indirect comparisons and nonstandard measurement in the classroom before introducing standard measurement. In class we talked about how it is important for students to learn how to measure using nonstandard units as well. This way they are learning unit iteration which is something that Kamii's article states many students struggle with. Examples of nonstandard units could include building blocks, paperclips, or coins. Really students could use any small object to measure. I think that giving students a choice of using what they want to measure with could make a good lesson. I found this game online:pbskids.org/curiousgeorge/games/how_tall . It is a fun way for students to use nonstandard units to measure Curious George characters. This could be a game option that students could use in the classroom during a free time or indoor recess. I found this unit plan online: https://www.nsa.gov/academia/_files/collected_learning/elementary/geometry/measuring_up.pdf. The beginning lessons use many different objects to measure and then the unit introduces standard measurement tools toward the end. I thought this lesson would be valuable to use in curriculum because it uses so many different objects for measurement. I also found this lesson plan from Scholastic: http://www.scholastic.com/browse/article.jsp?id=3758185. This is an example of a lesson using nonstandard measurement. The students are to measure a pencil using paperclips. You could extend this activity to measure other things around the room using paper clips. I also thought this lesson was valuable because it introduces nonstandard measurement in a simple, handson way. Since measurement plays such a big part in our daily lives it is important to teach it effectively to students at a young age. Using nonstandard units and indirect comparisons can cause students to become more comfortable with measurement. I have enjoyed exploring ways to teach measurement in class. 
Week 3 and 4: Hexagon Types
In class we did a very enriching activity with hexagons. We came up with our own types for classifying hexagons. This is similar to learning the types for quadrilaterals, but since hexagons do not have types, we came up with them on our own. Originally I thought this activity would be silly and easy, but as we got into debating on the hexagons as a class it became more fun. I also learned that this activity was quite challenging as we looked at the many properties of Hexagons and how they could be classified. The activity went like this:
First, we explored with the Hexagons using Geoboards and drawing on Dot Paper. It was very important and useful that we had this original exploration before jumping into the activity. This way we were able to see the diversity and wide variety of hexagons.
Next, we discussed in our small groups what possible types for hexagons could be. My group came up with types based on whether or not the hexagon was convex or concave. Then we formed 4 more subcategories for each classification of convex and concave. For example, we had concave right and convex right. We also came up with the regular hexagon as one type where all the sides are the same length. Another hexagon we thought should be a type is the L hexagon that contains all right angles (5 interior, 1 exterior).
Then each group had the chance to present their hexagon type to the class. The type was then debated by the class if it should or should not be one of our class hexagon types. We voted as a class on the type. A few of the proposals were rejected by the class, but most were accepted. We ended up coming up with 9 types. The types we came up with are:
Regular  all sides and angles are congruent
Convex  all exterior angles are smaller than their adjacent interior angle
Concave  at least one exterior is larger than the adjacent interior angle
Right  contained at least one right angle
Bowtie  had congruent interior reflex angles
3 Peas in a Pod  contained three pairs of parallel sides
Trigruent  contained three pairs of congruent sides
Lagon  had 5 interior right angles
Mirrored  contained at least one line of symmetry
First, we explored with the Hexagons using Geoboards and drawing on Dot Paper. It was very important and useful that we had this original exploration before jumping into the activity. This way we were able to see the diversity and wide variety of hexagons.
Next, we discussed in our small groups what possible types for hexagons could be. My group came up with types based on whether or not the hexagon was convex or concave. Then we formed 4 more subcategories for each classification of convex and concave. For example, we had concave right and convex right. We also came up with the regular hexagon as one type where all the sides are the same length. Another hexagon we thought should be a type is the L hexagon that contains all right angles (5 interior, 1 exterior).
Then each group had the chance to present their hexagon type to the class. The type was then debated by the class if it should or should not be one of our class hexagon types. We voted as a class on the type. A few of the proposals were rejected by the class, but most were accepted. We ended up coming up with 9 types. The types we came up with are:
Regular  all sides and angles are congruent
Convex  all exterior angles are smaller than their adjacent interior angle
Concave  at least one exterior is larger than the adjacent interior angle
Right  contained at least one right angle
Bowtie  had congruent interior reflex angles
3 Peas in a Pod  contained three pairs of parallel sides
Trigruent  contained three pairs of congruent sides
Lagon  had 5 interior right angles
Mirrored  contained at least one line of symmetry
Lastly, we were able to name the hexagon types as a class.
I really enjoyed doing this activity in class. It was fun to debate and "fight for" your type to be a part of the class types. We were challenged to decide which hexagon types were necessary, and which were not. This activity encouraged collaboration between students and creativity.
I also liked that we each made a set of hexagons out of paper to classify using our types. We were able to further our understanding of hexagon types by deciding where each hexagon would fit in the groupings. Professor Golden also taught us a few games that we could play using these sets. Math games are very helpful to facilitate learning in a fun way for students.
I think that this activity could be simplified or modified to teach to younger students. I think it would be a great activity to teach triangle types to lower elementary students.
I really enjoyed doing this activity in class. It was fun to debate and "fight for" your type to be a part of the class types. We were challenged to decide which hexagon types were necessary, and which were not. This activity encouraged collaboration between students and creativity.
I also liked that we each made a set of hexagons out of paper to classify using our types. We were able to further our understanding of hexagon types by deciding where each hexagon would fit in the groupings. Professor Golden also taught us a few games that we could play using these sets. Math games are very helpful to facilitate learning in a fun way for students.
I think that this activity could be simplified or modified to teach to younger students. I think it would be a great activity to teach triangle types to lower elementary students.
This would help students to understand that there are a variety of different triangles, not just an equilateral triangle with its point upwards. It would also teach them how to classify shapes by their properties at a young age.
After students came up with their own types, the Teacher could explain what the actual types that mathematicians have come up with are. 
Week 1 and 2: Van Hiele Levels
This week in class we talked about Van Hiele's levels of understanding. The article we read outlined these levels of geometric thinking. In the first level, students must judge figures based on their appearance. This level is called visual. In this level a student would be able to identify a triangle because it looks like a triangle. In the next level, figures are described based on their attributes and properties. In this level, descriptive, a student could identify a shape by its angles or the length of its sides. Thirdly, in the informal deduction level, students can use the properties they already know to create definitions of the shapes. In this level students should know that all squares are also rectangles. Van Hiele used the same puzzle that we used in class to demonstrate this pattern of levels of geometric thinking in students. When students approach accomplishing activities using the puzzle they will go through the levels of geometric thinking as they solve. I noticed in class that when I tried to solve the puzzle, I used a love of the descriptive level of thinking as I compared the sizes and angles of the shapes. In order to make the rectangle out of the shapes, we needed to find shapes that made 90 degree angles to find the corners of the rectangles. These levels were very interesting to learn about and helped me to understand how students learn and how they accomplish geometric tasks. I enjoyed reading the article on Van Hiele and accomplishing the puzzle tasks in class.
This is a figure from the Van Hiele article of the puzzle we used in class. This is just one activity that we did with the puzzle. There were many possible activities that could be done with the pieces.