As my teaching geometry in the elementary classroom class is coming to a close, we traveled to a school in Spring Lake to see how these techniques we learned would really play out in the classroom. It seemed fitting that my group would be teaching in an art class where we of course would be focusing on math integration as integration into the arts as well as other subjects seemed to be a trend all semester. However, this was what made me the most excited about going into this class. I felt that I had more tools that would help students to see math in more of the light that I do—that it is a fun challenge—and I was excited to see how this would play out in the class I was going to be visiting.
Walking up to the West Michigan Academy of Arts and Academics, I wouldn’t say that I was walking in to what I had expected. Something about this school seemed different, and that started with how it was set up, and the mural that was painted on the outside of the building. As I walked into the school I noticed that there were many, many different student created pieces of art that decorated the walls. As I made my way down to the art room, I was amazed at the music that I was hearing come from the different rooms. Students were dancing, laughing, all around having fun. All I could think was that this is how students should learn.
As this was a very different environment than I felt that I had been taught in or experienced teaching in, I was both hesitant and excited that this was where I would be spending the next hour. I felt very welcomed both by the students and the teachers, as the students were eager to participate in the opening part of the mini fraction lesson that we would be giving.
After introducing the lesson, it was great to see that the students were eager and excited to get started. It was in this time when the students started working that I thought I got the most valuable experience. I spent this time answering student questions about how to cut certain fraction pieces from what they had. I had a great time with the students one-on-one helping scaffold them to think about the fractions they had made already and think about how they made these and how this could help them make the next fraction.
One student that I struggled scaffolding for was a boy who insisted that he had created a fraction that was 1/19th. After asking how he did this, he proceeded to show me some pattern of intricate folds in which he folded in half, opened up, and refolded the shape. My first thought was to ask him to count the pieces that he had created with the folds, but unfortunately the folds were not created cleanly enough, so 19 pieces were counted as large fold creases were also counted as a piece. Instead of continuing on with the student in this way I wish that I would have grabbed a larger piece of paper that the student could then use to fold easier and see that the pieces that he was creating, were not equal, nor were there 19 of them. I think that this is a great way to apply the idea of learning through inquiry to math concepts that students are learning. I thought it was great that this student was exploring and trying to create a unique fraction, however he needed a method that he could use that would help him self-correct.
Although there were many students that took the time given to them to complete the activity, I think that this lesson would benefit from an extension activity. I think that there are a few different extension ideas that could be beneficial for the students depending on what subject you wanted the students to focus in. I think that an art extension could be challenging the students to think about the colors they are using in their picture and fractions and having them focus on how they relate to one another on the color wheel, and what tone they are bringing to their picture by using colors that are opposite or adjacent to each other. I think that an interesting math extension for this activity could be challenging students to create a picture using exactly 3 and 5/8ths. In doing this students would have to know how to add fractions with different denominators and possibly even subtract fractions in order to get the exact amount of paper to complete the activity.
Overall, I was grateful for the experience and practice to be in front of a class helping lead an activity. I was amazed by these students creative abilities and their persistence to find a way to make these fractions work in their picture. Putting some of the things that I have learned to practice in this class was exciting and I look forward to the time when I am in a classroom again and use more techniques and methods with my students. I also felt that one aspect of this experience that I will take away is that learning is messy, it is loud, and it very rarely happens within the confines of the desks that students are sitting in. I know that I will not let these “typical” boundaries placed around learning hold back my students from being the very best that they can.
Over the past few weeks we have been dealing with the areas volume of different shapes. After doing a multiple classroom activities, one thing that I noticed along with my classmates was that my spatial reasoning is very poor. One activity where I particularly noticed this was when we were comparing the volumes of pyramids to that of a rectangular prisms. When I was calculating the area of these two volumes with the same base shape and height, I realized that the area was calculated almost the same, except that the area of the pyramid was 1/3 that of the rectangular prism of the same dimensions. I was simply baffled that this was true. In my head I could not see how three of the pyramids that I just measured would ever fit into the rectangular prism right next to it.
This made me think about the education that I got and how little the math related concepts were hands on activities with three dimensional objects. So often in our classes we are taught that the best way for students to learn is through inquiry in the sciences, because it is through this process that the students are able to break apart the system at which they are doing things and figure out why the system works. Why have we not adapted this into the math education as well?
This was the reason that I particularly liked the activity where we calculated the volumes of several three dimensional shapes with water. After being given the hollow shapes, a measuring cup of water and a graduated cylinder, we were told to find out how the volume of each of the shapes in milliliters. After noticing that there were a few different ways that these volumes could be measure we decided that the best way for us to measure the volume would be to fill the shapes, then pour out the water that fit in the shape and measure this amount. Unfortunately, we were disappointed to realize that although this was likely the easiest way to measure the volume, it was not the most accurate as there was a lot of space that was missed in the plastic that was used to create the shape.
What I like most about this activity for elementary age students is the fact that this gets at the heart of what volume is and what it is measuring. Because of this students would definitely walk away with a concrete understanding of what volume is. Another aspect that I really think is great is that this is such a good integration activity between math and science. Measuring these volumes accurately with the tools given can be highlighted as a key skill. Also this leads for discussion of an alternative way to measure the volume by displacement.
I saw this activity, however, to be more versatile than just being able to be used in the elementary to understand volume. I think that this could be a great upper elementary or even middle school activity if you could help guide the students to make connections between the measurements that the make two dimensionally on the shapes to the three dimensional volumes. This would be good practice for students to do unit conversions. I also think that this practice would be great for students to begin to develop and refine their spatial thinking.
Over the last few weeks in my math class that focuses on Geometry in the classroom has been exploring measurement. We have had some challenging conversations to say the least, as we were given activities that asked us to measure and we quickly realized much more assumptions are made in measurement than we remembered. These activities were meant to strengthen our core measurement ideals as well as finding ways we can create a lesson that challenges and strengthens the student’s fundamental measurement skills as well.
A key question that needs to be addressed before any measuring was done is why do we need these measurements? This is a foundational thought that teachers need to address before we even start measuring so that students realize that what they are doing has purpose and is used for accurately communicating to other people. Whether that amount is time, or length is up to the specific problem, but we are always asking “How much?”
One activity that we did that I thought was beneficial was we were told to measure the length of the hallway. We were given a tape measure and told that we were to use our stride length in order to determine our length. To start off, when we were told to measure our stride, my group decided that it might be best to find the length of 10 strides, then divide by 10 in order to get the average length, since each stride length could be slightly different and we could use the average. Next we started at one end of the hallway and walked to the corner where the other hallway started. After doing this we multiplied our stride length in centimeters by the number of steps that we got in the hall to find out the length of the hallway.
After finding our hallway lengths and as we began to talk about the vast differences in we were seeing in each-others data, one thing that we realized was that we did not come up with an agreed upon starting and stopping point for how we would measure the hallway. Some of us started at the corner and measured to the next corner while others were measuring wall to wall. Another aspect of measurement that we began to discuss was what an appropriate range for error would be for each of our measurements. This was an interesting concept to begin to talk about as many in the classroom, especially those with little science background were unsure at what an appropriate error would even look like for a problem like this.
By doing this problem in this way, my classmates and I were able to be in the shoes of the students that we will someday be teaching and were able to come across some important questions that students need to ask themselves and understand in order to measure proficiently. Where is the starting and stopping point? How do I use the tool that I have to measure properly? Am I measuring the same way that it was intended to be used to measure? These are all questions that students need to ask so that they are measuring accurately. I think that this lesson did a good job at doing that and I think that it could be adapted so that students could go through the same process. By giving them the guidance of a worksheet that could help them walk though these steps and discover that these are important things to be doing the same so that we all get the same measurement. The worksheet below was created with this purpose. First off, I wanted to give the students a reason to measure. However, the goal of this worksheet is really to focus in on step five where they find their differences. In seeing these differences students will be able to see the importance of starting and stopping in the same place, walking in a straight line and using the tools they have correctly. Ultimately, by learning through inquiry in this way students could have a lasting understanding of measurement and how it can and should be done correctly.
As we have continued our conversation about shapes and shape properties one activity that I found particularly engaging was the fold and cut activity. The premise of this activity is that any shape can be made by folding a piece of paper and making one cut. The challenge we were given was to make as many different quadrilaterals as we could by doing this our self. I started doing this activity by making a square, and after realizing that it was difficult to visualize where the lines of the shape would be after I had folded it a few times, I began by drawing the shape with a sharpie so that I could see the lines on the front and the back sides of the paper. After doing this step I was quickly able to come up with the square by seeing the lines that could be overlapped by folding. From this I knew I could get to a rectangle by folding the long sides of the rectangle on top of themselves in an accordion fold until it resembled a square, then following the same steps as a square to end up with the rectangle. Using this same type of process I was able to first find the folding pattern to create the rhombus and then made the accordion folds in a parallelogram to make it resemble the rhombus, and cut to create the parallelogram. After these shapes I found that other quadrilaterals were more difficult. One that I found after a few tries was the isosceles trapezoid. I think that this shape was easier to find because it had a line of symmetry. By being able to fold it down the center and have then lines that immediately overlapped, this helped make the other folds more natural. I also think that this is one of the reasons I was not able to figure out how to make the right trapezoid. In this shape there was not a line of symmetry that could be folded down and other lines just line up on top of each other. Because of this I found myself trying to make awkward fold that would not go through the entire length of the paper, however I do not think that this would end up being the correct way to create this shape.
Personally I really enjoyed this activity because it was very easy to fold a piece of paper, make a cut, and then unfold it to see what I had come up with. It was quick to see if you had done it correctly and if you didn’t, I could try again right away. I also thought that this activity challenged my understanding of shape properties. As I was folding these shapes, in order to create the shapes correctly, it was necessary for me to understand different shape properties such as symmetry, angles, right angles, and parallel sides. For these reasons I think that this activity would be beneficial for elementary students. First off, it also allows for quick self-assessment. Students are able to try one way that they think that they could correctly create the shape, unfold, and see if they were correct. This is a great way for students to begin to see that this is a critical aspect of math. Very often when we are trying to find an answer, we will try one way and see if that leads us to the correct answer, but if it doesn’t, that is ok, we can try again. Another aspect of this activity that I thought was beneficial for students was the fact that students would be challenged in their shape properties as they would complete this activity. It is imperative that students understand lines that are parallel to each other, lines that are perpendicular, or if they are on a different angle, how they could fold the paper so that these lines would overlap. Because this activity was a bit challenging even for me when I tried this with quadrilaterals, I wonder if this could be used in an elementary classroom with different triangles or even if they were just challenged to make a square and rectangle. I think that only having three different lines to worry about or making familiar shapes like squares and rectangles could make the activity a bit simpler. However, overall I think that some form of this activity could help extend student’s thinking and help further their geometric thinking.
For the past two weeks we have been doing an interesting activity in my math class that focuses on teaching geometry for elementary students. We have been creating hexagons and classifying them. At first I was uncertain about what I would learn about teaching geometry to elementary students from creating and coming up with names for these hexagons that we had created however I found the process to help my understanding and move me forward in my Van Hiele thinking. I am certain that if adapted to their level, this could be a very beneficial activity for elementary students to help them better understand the concepts associated with classifying quadrilaterals rather than memorizing the properties that are commonly used in classifying them.
The first thing that we did as a part of this activity was we created as may different hexagons as we could. Our only limitation was the dot paper we were using, however even in this there were many, many hexagons that we could create that all looked very different. Of these shapes there were some that were concave, convex, had reflex angles or more than one reflex angle, there were some with right angles, and of course the regular hexagon was one of the first for almost everyone to draw. These varied shapes allowed for use to be as creative as we could and discover as many “odd” hexagons that we could until we had exhausted all possibilities.
If elementary students were able to do this activity using quadrilaterals instead of hexagons I believe that this could be an incredible activity that could help move them up in their Van Hiele level of thinking as well and allow them to discover and learn through inquiry. This of course is one of the best ways that students can learn is though self-exploration. In coming up with all the possible quadrilaterals they can I think that they could become more comfortable with some of the shapes that they do not see as frequently such as trapezoids. By giving them this time to play and explore these non-standard shapes they will become more comfortable simply by familiarity.
The next thing that we did in our class exploration with hexagons was that we took all of the hexagons that we created and defined a “type.” We then proposed these types for the class which as a group we either decided was a good type or we vetoed it and decided not to make it a shape. As we were doing this I found myself asking the question “What makes a good type?” After some deliberation in my group we thought that the answer to this would be that a good group has very specific characteristics that would include a population of hexagons but would also eliminate a significant portion as well. As we thought about our definition of what a type should be, I believe this influenced how we thought about the propositions of what types we thought that we should keep and which one we thought were not necessary.
I love this aspect of the activity for elementary age students. By creating their own categories and deciding what they think would be a good type could help them in as they learn what the mathematically accepted types of quadrilaterals are and why we have chosen these to be types. I also like the critical thinking that is involved as students propose types they must think about the specific examples of what rules will fit the shapes they have created as well as the additional rules they must come up with to include or exclude other specific shapes that can be made.
Now that we had our categories the next thing that we did was we were asked to create shapes that would fit in exactly 3 categories, 2 or more categories, and exactly one category. I found this to be one of the most challenging activities that we did with this activity. I think that this is because we had created all of rules, it was difficult for me to find one shape that fit exactly in one of my categories because I was unsure as to what I was trying to not create as well. After recognizing that we had both concave and convex as a category, I realized that in order to be just one category, the hexagon would have to fall in either of these categories yet none of the others. After discovering this it became easier for me to come up with an example that would fit just one category. I also tried to create a venn diagram for the shapes that we had created and I found this incredibly challenging as there was a difficult aspect in knowing what to do with categories that overlapped in some categories but not in others. After modeling my venn diagram after that of quadrilaterals it became somewhat easier to create yet still proved to be a mathematically challenging activity. I think that this could be an extension for some students who show that they are proficient in their math abilities in the previous activities with quadrilaterals. As students categorize and organize these categories they will begin to question and further examine their definitions and ask themselves if their definitions are inclusive or exclusive. This would lead to a good discussion to the mathematically accepted definitions of quadrilaterals that are inclusive and why they think this could be. Students could also use this time to see if they would like to eliminate certain categories that they had created and come up with reasoning why they should eliminate it. Some categories could be redundant or only cater to a small population of hexagons that may seem irrelevant or grouped better by another definition.
In participating through these activities, I would think that these critically thinking questions can help students to move up in their thinking. Hopefully by carefully analyzing and looking at all properties of quadrilaterals, students could possibly begin to think formally and continue to realize that very frequently it is the shape properties are what we analyze in mathematics.
I recently started another math class for elementary education major and am continuing to love what I will get to go into when I graduate. The topics that we have been exploring for the past few days have ranged from Van Hiele levels to different ways students learn shapes to noticing and recognizing attributes. One thing that I have particularly enjoyed learning about was attributes. What I particularly enjoyed through this process was that I was beginning to notice how I was finding patterns. I think this was important as a future teacher as this was not something that I have really thought about much but needs to be as noticing patterns is a critical step in school that elementary age students make. Often students do not know how to even begin. From this thought what stuck out to me the most was how important it was to have both examples and non-examples. I found both of these to be important so that I could come up with a possible rule then use the examples and non-examples to either confirm or deny that that was the actual rule.
One activity that I found online at http://www.crazyforfirstgrade.com/2011/12/attributes-and-sorting-fun.html was a game that could help students develop this ability called “guess my rule.” This would be a game where students are given a worksheet with a circle on it and develop a rule for shapes that should go in the circle. Students would then use shape blocks and place these in the circle according to whether or not they belong inside the circle. What I really like about this activity is that students get to practice creating the rule and using their judgement on if the shape they have fits in the circle or not. I also like how this activity could be use multiple ways as students can spend time creating the rule as well as trying to figure out what fellow classmates came up with as their rule. If done in partners and the student guessing finds another rule that would also fit this group of shapes or if there was a shape in their circle that contradicted their rule this could lead for a good discussion. Students could talk about why the shape they have breaks their rule or if there was another shape they could add that would help define between the two rules the students came up with. One thing that I would like to modify about this game would be to create a space where students could put shapes that do not fit the rule. This would help emphasize the idea I discussed earlier about the importance of examples and non-examples. This could also lead to good discussion as to why particular shapes are not “in.” Students could also discuss what the shapes outside the circle have in common and notice they all lack the attributes of those inside the circle. All activities help student develop critical thinking and problem solving skills that are critical in elementary age students.