Last week in the elementary school classroom, we were teaching students how to find the area and perimeter of parrallelograms. The previous week, the students had learned how to find the area of triangles: (Base x Height)/2. The textbook has the students figure out the area and then compare it to the perimeter.
The students seemed to get the formula for area for triangles. However, when we moved on to parallelograms, the students started to revert back to the length x width formula for area, using the side lengths and not the height (which results in an incorrect answer). This made me stop and wonder why we use different language for the two formulas. Why don't we just say "height" for length and "base" for width. It seems like it would help eliminate some of the problems or confusion our students now face.
The math curriculum is structured in such a way that students learn about area and perimeter of squares and rectangles first. Then, triangles and parallelograms. Students are taught to find the height of the triangle, and learn to use this measurement instead of the side length for defining area. But for some reason, when we move onto parallelograms, the students go back to using the side length. I think they do this because parallelograms look like slanted squares or rectangles, and we use the term "length" to figure the area of squares and rectangles. Students want to use the side lengths again to find area.
The textbook we used also tried to show how different shaped parallelograms could have the same area if they had the same height and base, the angle of the slant did not matter. While we proved this to the students using formulas, I think it would have been so much better if we had allowed the students to construct two parallelograms with the same height and base but different slant angles, and then cut these figures out to move the pieces around and SHOW the students that the areas are the same.
Likewise, when the textbook asked if the students could draw two non-similar parallelograms with the same side length measurements and asked if the area stayed the same, it would have been so much more powerful for the students to have constructed the shapes with paper and brass brads to see how the area visibly changes.
Showing posts with label Math. Show all posts
Showing posts with label Math. Show all posts
Friday, March 11, 2011
Saturday, March 5, 2011
Using Voicethread to Share Math Strategies
During my Math Methods course this quarter, I was introduced to an online tool called Voicethread. This tool allows you to upload pictures, documents, or video, and then share them with others. You can comment using audio, video, or text features, and you can allow those who you share the thread with to make these comments as well.
I played around with the tool a lot this weekend and I used it to upload a teaching video that I needed to share with some other teachers. I can see the potential to use this tool among other educators as a way to challenge ourselves on our teaching craft. I also think it could be incorporated into an upper elementary, middle or high school level for certain projects.
We used Voicethread in my Math Methods course to show how some students think about solving quick image activities. These activities have you look at a picture comprised of grouping of images and you have just 3 seconds to count the images. We did an example in my Methods course with other teacher candidates, and I was surprised at how we each looked at the problem differently. During a field day placement, we used Flip cameras to record our students' thinking and then uploaded these videos onto voicethread to share and comment on their thinking amongst ourselves.
I have not worked all the kinks out yet, but I can see that voicethread will be a powerful tool to use to reflect on my teaching. It might also provide a new technology medium to use in my classroom for homework help or instruction.
I played around with the tool a lot this weekend and I used it to upload a teaching video that I needed to share with some other teachers. I can see the potential to use this tool among other educators as a way to challenge ourselves on our teaching craft. I also think it could be incorporated into an upper elementary, middle or high school level for certain projects.
We used Voicethread in my Math Methods course to show how some students think about solving quick image activities. These activities have you look at a picture comprised of grouping of images and you have just 3 seconds to count the images. We did an example in my Methods course with other teacher candidates, and I was surprised at how we each looked at the problem differently. During a field day placement, we used Flip cameras to record our students' thinking and then uploaded these videos onto voicethread to share and comment on their thinking amongst ourselves.
I have not worked all the kinks out yet, but I can see that voicethread will be a powerful tool to use to reflect on my teaching. It might also provide a new technology medium to use in my classroom for homework help or instruction.
Monday, February 28, 2011
Pentomino Lesson
Last week, I got to teach a math lesson using pentominos. In case you are not familiar, pentominos are the various shapes that result when you connect five identical square tiles along the edges. This lesson comes from the Covering and Surrounding workbook from Lappan, Fey, Fitzgerald, Friel, and Phillips. The lesson requires students to draw out all the possible pentomino shapes. To enhance this lesson, I used multicolored square tile manipulatives. The students were very engaged and enjoyed using the tiles, however, many of them quickly became more interested in building 3-D shapes with the tiles.
The point of this lesson was to highlight the differences between area and perimeter, and how the perimeter and area change when a tile is added. I think it was a great hands-on lesson to help students see how area and perimeter are connected, but not identical.
My students were able to notice patterns in how the perimeter changed when a tile was added to a particular area of the pentominos. Depending on the placement of the tile, you could increase the perimeter by 2 units or keep the perimeter the same, but increase the area by one unit.
I think this lesson was enhanced by the use of the tiles. The students could easily move these tiles around to explore the different pentomino shapes. The tiles made it easier to notice when a pentomino shape was a repeat of a previously drawn shape. I also got to see how manipulatives add to the students' understanding or help them make sense of a new concept.
The point of this lesson was to highlight the differences between area and perimeter, and how the perimeter and area change when a tile is added. I think it was a great hands-on lesson to help students see how area and perimeter are connected, but not identical.
My students were able to notice patterns in how the perimeter changed when a tile was added to a particular area of the pentominos. Depending on the placement of the tile, you could increase the perimeter by 2 units or keep the perimeter the same, but increase the area by one unit.
I think this lesson was enhanced by the use of the tiles. The students could easily move these tiles around to explore the different pentomino shapes. The tiles made it easier to notice when a pentomino shape was a repeat of a previously drawn shape. I also got to see how manipulatives add to the students' understanding or help them make sense of a new concept.
Wednesday, February 9, 2011
ReThinking Addition
I have been doing a lot of thinking about math lately, specifically addition. From discussions in my Math Methods course, I am beginning to see that we all think about addition just a little bit differently. Our teacher highlights these differences by asking us to solve a problem and then probing for us to explain how we solved the problem. She is not concerned with the answer, but how we arrived at that answer.
I'll admit it has been a challenge for me to sit patiently as we painstakingly go over the problem several times. (Okay, patience and I don't have anything to do with each other in that class.) I find that I am ready to move on to the next problem after the first person describes how he/she arrived at the answer. I just check that we got the same solution and I am ready for problem #2.
What I finally realized today as we went around the room sharing how we grouped different items to count the total in less than 3 seconds was that all of us had a slightly different strategy. If you had asked me 4 weeks ago how many ways you can count to 10, I would have said, "Just one way. 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10." Today, I witnessed multiple ways. Here are just a few:
3 + 3 + 4
3, 6 + 4
3 + 3 + 2 + 2
2, 4, 6, 8, 9, 10
Seeing that there were many ways to solve this problem made me think back to a math interview I conducted two weeks ago with a second grade student (TS). TS told me that he added two digit numbers by starting in the tens column. When I was in elementary school, this was a complete no-no. We were forced to start with adding the ones column. I was even more shocked when TS stated, "You start with the tens; that is what my teacher said." Using his tens-column-first method, TS was able to correctly solve the problem. It made me realize that there are lots of ways to solve problems and as teachers we have to try to understand how our student are thinking about the math, which may be a little more challenging for those of us who were educated in the "old school" method.
I'll admit it has been a challenge for me to sit patiently as we painstakingly go over the problem several times. (Okay, patience and I don't have anything to do with each other in that class.) I find that I am ready to move on to the next problem after the first person describes how he/she arrived at the answer. I just check that we got the same solution and I am ready for problem #2.
What I finally realized today as we went around the room sharing how we grouped different items to count the total in less than 3 seconds was that all of us had a slightly different strategy. If you had asked me 4 weeks ago how many ways you can count to 10, I would have said, "Just one way. 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10." Today, I witnessed multiple ways. Here are just a few:
3 + 3 + 4
3, 6 + 4
3 + 3 + 2 + 2
2, 4, 6, 8, 9, 10
Seeing that there were many ways to solve this problem made me think back to a math interview I conducted two weeks ago with a second grade student (TS). TS told me that he added two digit numbers by starting in the tens column. When I was in elementary school, this was a complete no-no. We were forced to start with adding the ones column. I was even more shocked when TS stated, "You start with the tens; that is what my teacher said." Using his tens-column-first method, TS was able to correctly solve the problem. It made me realize that there are lots of ways to solve problems and as teachers we have to try to understand how our student are thinking about the math, which may be a little more challenging for those of us who were educated in the "old school" method.
Wednesday, January 19, 2011
Math Facts
This past week we had to write our Math Autobiography. I don't remember much about learning math in elementary school or what I thought about math. I know in high school I came to really enjoy math.
My one memory of math from my younger years involves my Grandma. She lived in a different state and so we did not see her often. One time she came to visit and she brought my brothers and I a folder she had created of math fact flash cards. She had little pockets to put the cards in for each number from 1-12. She had taken time to laminate each card and the whole folder so it would not get worn from use. I cannot tell you how much I HATED that folder. I thought grandmas were supposed to spoil you or take you out to places like the zoo. My Grandma obviously did not understand what being a grandma was all about. She viewed her visits to see her grandkids as an opportunity to quiz them on their math facts. I remember wishing she would leave, but feeling guilty for doing so.
I know my math facts to this day, and I don't know how much of it had to do with my Grandma's visits. Even though I hated them at the time, I am so glad that she instilled in me the importance of learning math.
My one memory of math from my younger years involves my Grandma. She lived in a different state and so we did not see her often. One time she came to visit and she brought my brothers and I a folder she had created of math fact flash cards. She had little pockets to put the cards in for each number from 1-12. She had taken time to laminate each card and the whole folder so it would not get worn from use. I cannot tell you how much I HATED that folder. I thought grandmas were supposed to spoil you or take you out to places like the zoo. My Grandma obviously did not understand what being a grandma was all about. She viewed her visits to see her grandkids as an opportunity to quiz them on their math facts. I remember wishing she would leave, but feeling guilty for doing so.
I know my math facts to this day, and I don't know how much of it had to do with my Grandma's visits. Even though I hated them at the time, I am so glad that she instilled in me the importance of learning math.
Wednesday, December 8, 2010
Math Reflection
Wow, I really enjoyed my math course this quarter. I learned about a great graphing website (gapminder) and how to use some pretty fun manipulatives like bean counters and Mira tools. I learned that many people have strong feelings about math, and it is hard to teach our students without realizing their "relationship" with math. I learned how important continual reflection can be for improving as a teacher. I also learned just how much I love math.
I got a little frustrated due to the playing card groups, but not because of who I was placed with. I just like to sit where I like to sit in the classroom. However, I think I liked having random groups assigned each week because it forced the entire cohort to work with new people. I think that part of being a teacher is learning to work with different people at the school, so this was good practice. I liked how straight forward our professor was and how wanted our assignments to be concise and short.
I think I will definitely employ some of the strategies I learned in this course in my future classroom, specifically working in randomized groups and using different manipulatives that allow students hands-on learning.
I got a little frustrated due to the playing card groups, but not because of who I was placed with. I just like to sit where I like to sit in the classroom. However, I think I liked having random groups assigned each week because it forced the entire cohort to work with new people. I think that part of being a teacher is learning to work with different people at the school, so this was good practice. I liked how straight forward our professor was and how wanted our assignments to be concise and short.
I think I will definitely employ some of the strategies I learned in this course in my future classroom, specifically working in randomized groups and using different manipulatives that allow students hands-on learning.
Wednesday, December 1, 2010
Broken Calculator and other thinking games
Today in math class we explored several math game tools that a teacher could use to challenge students in their mathematical operations knowledge. We first used a broken calculator program that only had certain numbers and operation buttons that worked. Using the numbers 0,6,7,8,9 and addition or division, we had to come up with the numbers from 1-15. I really enjoyed this challenge as it forced my partner and I to re-think the way we "do math".
The second activity we did was using an excel spreadsheet that our instructor created. We had fifteen problems to solve where we input a number and from the output, we had to figure out what operations were taking place. This activity taught the importance of the order of operations and scaffolding your instruction or problems.
I learned today that there are lots of different electronic manipulatives that we can use to engage students in the class, or use as hooks, or as review. Additionally, these electronic manipulatives offer teachers opportunities to teach about other computer programs (Excel, gapminder.com, etc.). I think electronic manipulatives imply that the teacher will be engaged in current technology and actively searching for ways to bring this technology into the classroom in a way that is relevant to students' lives. I also think that while it might seem initially time consuming to create an Excel file like the one we used, once made, it would be easy to manipulate to fit the unit or mathematical concept that you want to highlight.
The second activity we did was using an excel spreadsheet that our instructor created. We had fifteen problems to solve where we input a number and from the output, we had to figure out what operations were taking place. This activity taught the importance of the order of operations and scaffolding your instruction or problems.
I learned today that there are lots of different electronic manipulatives that we can use to engage students in the class, or use as hooks, or as review. Additionally, these electronic manipulatives offer teachers opportunities to teach about other computer programs (Excel, gapminder.com, etc.). I think electronic manipulatives imply that the teacher will be engaged in current technology and actively searching for ways to bring this technology into the classroom in a way that is relevant to students' lives. I also think that while it might seem initially time consuming to create an Excel file like the one we used, once made, it would be easy to manipulate to fit the unit or mathematical concept that you want to highlight.
Gapminder and Telling Data Stories
We explored the statistical data sets in a program called gapminder.com. At first, we looked at sugar consumption by country, by year, and compared to the country's gross domestic product. The gapminder website allows you manipulate the graph to show information you are interested in viewing. Whenever you see graph information, one has to be careful not to confuse correlation with causation. I think it is really easy to make this error, and so the gapminder graphs provide a great way to teach students to really question the data.
To prove this point, my group specifically chose two data sets that one would generally assume influence each other. The y-axis was percentage of the population living on less than $2/day. The x-axis was women's literacy levels. We searched for data where women's literacy increased, so did the percentage of the population that lives on less than two dollars a day. One could look at this data and argue that women's increasing literacy has a negative effect on those living in the country.
I learned how powerful data can be, but also how we need to be cautious in using data and not make it say more than it does. I think the correlation/causation lesson is important for students to learn, so I see the relevance for our classroom instruction. I would like to know where the data sets come from and if they are all a high caliber.
To prove this point, my group specifically chose two data sets that one would generally assume influence each other. The y-axis was percentage of the population living on less than $2/day. The x-axis was women's literacy levels. We searched for data where women's literacy increased, so did the percentage of the population that lives on less than two dollars a day. One could look at this data and argue that women's increasing literacy has a negative effect on those living in the country.
I learned how powerful data can be, but also how we need to be cautious in using data and not make it say more than it does. I think the correlation/causation lesson is important for students to learn, so I see the relevance for our classroom instruction. I would like to know where the data sets come from and if they are all a high caliber.
Friday, November 19, 2010
...a teacher in training...: Permutations & Combinations
...a teacher in training...: Permutations & Combinations: "1. What did you learn? I have always had a tough time remembering the difference between 'permutations' & 'combinations'...for which does or..."
In response to Mr. Linger's blog post about the difference between Permutations and Combinations - I can totally relate! I don't recall learning this during my years of high school or college math. I was introduced to P&C as I began to study for the WEST E Middle Level Math test. I liked how Mr. Linger found a way to keep the two straight using mneumonic devices. I have a similar one that I use for Multiples and Fractions. This might seem silly because if you just look at the names, it seems fairly apparent that multiples implies many and fractions implies parts of wholes. However, I have found that on tests and in classrooms, students are mostly exposed to GCF and LCM. The F and M don't have the same meaning as Fraction and Multiple to students. I always say to myself, Fraction is Fewer and Multiple is More.
Thanks for the helpful tip, Mr. Linger!
In response to Mr. Linger's blog post about the difference between Permutations and Combinations - I can totally relate! I don't recall learning this during my years of high school or college math. I was introduced to P&C as I began to study for the WEST E Middle Level Math test. I liked how Mr. Linger found a way to keep the two straight using mneumonic devices. I have a similar one that I use for Multiples and Fractions. This might seem silly because if you just look at the names, it seems fairly apparent that multiples implies many and fractions implies parts of wholes. However, I have found that on tests and in classrooms, students are mostly exposed to GCF and LCM. The F and M don't have the same meaning as Fraction and Multiple to students. I always say to myself, Fraction is Fewer and Multiple is More.
Thanks for the helpful tip, Mr. Linger!
Wednesday, November 17, 2010
Blogging Reflections Update
I had a very clarifying experience, and it seems only right to share about it here as it has to do with blogging. A few weeks back, I wrote a post about why people blog. I understood the need to keep a journal or record of your thought process, and the need to be reflective, but I did not understand why people would want to post this kind of information online for anyone to read. The truth is that blogging is sort of a professional discipline. It is allowing yourself to be vulnerable to the thoughts of others. It is also creating a portal between you and another person so that you can learn from each other, and sort of hold each other accountable.
In our math class last week, our professor talked about her teaching experience and how she improved as a teacher when she reflected on her teaching methods. She saw a gap between the curriculum and what her students were asked to do on tests, and so she created an activity to fill the gap. Her improvement as a teacher and her decision to add to the curriculum would not have come about if she had not been reflecting. When she told us that, a huge light bulb went on in my head. I have noticed that each quarter we are asked to write papers reflecting on our teaching process, reflecting on the experience, reflecting on ..... whatever. The goal of the teacher cert program I am enrolled in is to create a cohort of reflective educators. That is their "gift" or what sets this program apart. In previous quarters, when I saw the reflection assignments, I would sort of roll my eyes and grit my teeth. I am not a super reflective person by nature, you see, and so writing reflective papers is not on my list of favorite things. However, after class last week, I am starting to see the benefits and having a little change of heart. I am going to look forward to being reflective, and blogging, and see these as opportunities to improve myself and my teaching craft.
In our math class last week, our professor talked about her teaching experience and how she improved as a teacher when she reflected on her teaching methods. She saw a gap between the curriculum and what her students were asked to do on tests, and so she created an activity to fill the gap. Her improvement as a teacher and her decision to add to the curriculum would not have come about if she had not been reflecting. When she told us that, a huge light bulb went on in my head. I have noticed that each quarter we are asked to write papers reflecting on our teaching process, reflecting on the experience, reflecting on ..... whatever. The goal of the teacher cert program I am enrolled in is to create a cohort of reflective educators. That is their "gift" or what sets this program apart. In previous quarters, when I saw the reflection assignments, I would sort of roll my eyes and grit my teeth. I am not a super reflective person by nature, you see, and so writing reflective papers is not on my list of favorite things. However, after class last week, I am starting to see the benefits and having a little change of heart. I am going to look forward to being reflective, and blogging, and see these as opportunities to improve myself and my teaching craft.
Thursday, November 11, 2010
Do you see the pattern?
For math class this week our instructor revisited ways to teach students how to add and subtract positive and negative integers. One method she showed us that I found really clever was to introduce a pattern. For example:
3 - 3 = 0
3 - 2 = 1
3 - 1 = 2
3 - 0 = 3
3- -1 = 4
3- -2 = 5
3- -3 = 6
If you start with what kids know (subtracting positive numbers from positive numbers) and then move into a more challenging concept (subtracting negative numbers from positive numbers) it helps students to see how the sum is calculated and follows the pattern. If a teacher then asks students to see if they can identify what is happening when a negative number is subtracted, it is more likely that they'll see the result is addition of that number.
I can absolutely see using this in my classroom and I would be excited to see if the students were then able to come to the conclusion that subtracting a negative number results in adding that number. I really like this technique.
3 - 3 = 0
3 - 2 = 1
3 - 1 = 2
3 - 0 = 3
3- -1 = 4
3- -2 = 5
3- -3 = 6
If you start with what kids know (subtracting positive numbers from positive numbers) and then move into a more challenging concept (subtracting negative numbers from positive numbers) it helps students to see how the sum is calculated and follows the pattern. If a teacher then asks students to see if they can identify what is happening when a negative number is subtracted, it is more likely that they'll see the result is addition of that number.
I can absolutely see using this in my classroom and I would be excited to see if the students were then able to come to the conclusion that subtracting a negative number results in adding that number. I really like this technique.
Bean Counters
I neglected to write what I had learned after class last week. We had a hands-on tutorial of using manipulatives (in this case, black and white beans) to add and subtract integers. I had never experienced doing this kind of work in an actual class, and I found the use of two colored manipulatives for counting to be very helpful.
We were given a number of problems to solve, and we had to use our beans. One color represented positive numbers, and one color represented negative numbers. Using the two colored beans helped for solving problems such as -5 - 4. In order to solve this problem, you first had to add 4 positive and 4 negative beans to your stash of negative 5 beans. The addition of the +4/-4 beans did not change the value of beans because the negatives canceled the positives. Then, when you took the 4 positive beans away, you were left with -9 beans. If the problem were -5 - -4, then you would take away -4 beans from your stash of -5 beans, resulting in -1 bean.
I learned a new method for teaching students how to add and subtract positive and negative integers. I think this is very applicable to the classroom and it can help students who struggle to understand the number line. When I was in middle and high school math, the number line was the only tool we were given to help solve these problems. I like the idea of using familiar items, like beans, instead of fancy colored chips because it shows kids that they can use regular items to help solve math problems.
We were given a number of problems to solve, and we had to use our beans. One color represented positive numbers, and one color represented negative numbers. Using the two colored beans helped for solving problems such as -5 - 4. In order to solve this problem, you first had to add 4 positive and 4 negative beans to your stash of negative 5 beans. The addition of the +4/-4 beans did not change the value of beans because the negatives canceled the positives. Then, when you took the 4 positive beans away, you were left with -9 beans. If the problem were -5 - -4, then you would take away -4 beans from your stash of -5 beans, resulting in -1 bean.
I learned a new method for teaching students how to add and subtract positive and negative integers. I think this is very applicable to the classroom and it can help students who struggle to understand the number line. When I was in middle and high school math, the number line was the only tool we were given to help solve these problems. I like the idea of using familiar items, like beans, instead of fancy colored chips because it shows kids that they can use regular items to help solve math problems.
Wednesday, October 27, 2010
A fraction of a fraction is....a fraction
So, math class today was very interesting. I learned two things:
1. We all solve problems differently.
2. If I have an expectation about what the answer should be, then I don't trust my math.
We were presented with a problem about two parks, and had to decide which park would have more blacktop area. The first park devoted 3/4 of the area to playground space, and 2/5 of that to blacktop. The second park (having equal dimentions) devoted 2/5 of the area to playground space, and 3/4 of that to blacktop. I started out with math (multiplying to find the square yardage) and then multiplying by the fractions. I also started out with the preconceived idea that one would be bigger than the other. When the math calculations were finished and both had the same blacktop area, I quickly resorted to drawings, figuring that somewhere along the way my math was off.
After some drawings and taking a step back from the problem, I began to see that what was coming into play was the commutative property. A (the area of the park) x B (3/4) x C (2/5) was the same as A x C x B.
I liked getting the opportunity to hear how other classmates solved the problem and it was fun to see different lights go on in different people's thinking as some of us shared. I saw the value in having groups present their ideas for problem solving to the class. I also saw how this type of problem is a perfect segway into multiplying fractions and I think it would be great for use in the classroom.
I also like how this in-class activity highlighted the teaching model discussed in our reading for the week. I thought the reading was cumbersome because we had to continually flip back and forth to different figures in the text. This real-live example was much easier to understand. The only thing I have questions about or see as a potential problem to using this method is step 1: anticipate possible solutions or methods that the students will use. I find it difficult to anticipate how others will solve a problem. I tend to see it from my viewpoint. Also, students will at times use the wrong operations or math to solve a problem. I don't know how to anticipate for those "wrong" solutions.
1. We all solve problems differently.
2. If I have an expectation about what the answer should be, then I don't trust my math.
We were presented with a problem about two parks, and had to decide which park would have more blacktop area. The first park devoted 3/4 of the area to playground space, and 2/5 of that to blacktop. The second park (having equal dimentions) devoted 2/5 of the area to playground space, and 3/4 of that to blacktop. I started out with math (multiplying to find the square yardage) and then multiplying by the fractions. I also started out with the preconceived idea that one would be bigger than the other. When the math calculations were finished and both had the same blacktop area, I quickly resorted to drawings, figuring that somewhere along the way my math was off.
After some drawings and taking a step back from the problem, I began to see that what was coming into play was the commutative property. A (the area of the park) x B (3/4) x C (2/5) was the same as A x C x B.
I liked getting the opportunity to hear how other classmates solved the problem and it was fun to see different lights go on in different people's thinking as some of us shared. I saw the value in having groups present their ideas for problem solving to the class. I also saw how this type of problem is a perfect segway into multiplying fractions and I think it would be great for use in the classroom.
I also like how this in-class activity highlighted the teaching model discussed in our reading for the week. I thought the reading was cumbersome because we had to continually flip back and forth to different figures in the text. This real-live example was much easier to understand. The only thing I have questions about or see as a potential problem to using this method is step 1: anticipate possible solutions or methods that the students will use. I find it difficult to anticipate how others will solve a problem. I tend to see it from my viewpoint. Also, students will at times use the wrong operations or math to solve a problem. I don't know how to anticipate for those "wrong" solutions.
Wednesday, October 20, 2010
Show me your parallelogram
Today in Math class (college), I learned (or rather, reviewed) the difference between all the members of the quadrilateral family. It was fun and challenging. I learned that I have forgotten some of those terms we use to describe the quads. Things like bisect, perpendicular, congruent... Oh, would that be more vocabulary? Perhaps that is why I cannot access it in my memory.
I also learned that I can get REALLY excited when I understand a math concept. I learned I need to check myself so I don't overwhelm my cohort mates (or future students). I need a little bit of Math filter. :)
From the readings this week, I learned the importance of using manipulatives (especially in the middle level) and some strategies for incorporating literacy in Math class. I actually think integrating these two is a good strategy because students can use writing to explain the depth of their thinking about math concepts.
My major question at this point is: How do we inspire the kids? Or how do we motivate them? My question stems from my experience in the classroom. The middle school class where I observe has a daily morning work assignment that they work on for 7 minutes and then go over the answers for 7 minutes (so about 15 minutes, or a third of the 45 minute period). It is my understanding that this work is not correct, graded, collected, reviewed, etc. I go around the classroom, trying to help students, but some are not working on their morning work. When we go over the answers, some are following along, and some are not. And my question is, why should they? They will not turn in the work or receive a grade for it. The problems might show up on a future test or even the MSP, but that is not enough of an incentive. I don't know that grading these, or attaching some kind of reward or punishment to the morning work is the direction I want to go. So, my question is: how can I motivate the students to do the work?
I also learned that I can get REALLY excited when I understand a math concept. I learned I need to check myself so I don't overwhelm my cohort mates (or future students). I need a little bit of Math filter. :)
From the readings this week, I learned the importance of using manipulatives (especially in the middle level) and some strategies for incorporating literacy in Math class. I actually think integrating these two is a good strategy because students can use writing to explain the depth of their thinking about math concepts.
My major question at this point is: How do we inspire the kids? Or how do we motivate them? My question stems from my experience in the classroom. The middle school class where I observe has a daily morning work assignment that they work on for 7 minutes and then go over the answers for 7 minutes (so about 15 minutes, or a third of the 45 minute period). It is my understanding that this work is not correct, graded, collected, reviewed, etc. I go around the classroom, trying to help students, but some are not working on their morning work. When we go over the answers, some are following along, and some are not. And my question is, why should they? They will not turn in the work or receive a grade for it. The problems might show up on a future test or even the MSP, but that is not enough of an incentive. I don't know that grading these, or attaching some kind of reward or punishment to the morning work is the direction I want to go. So, my question is: how can I motivate the students to do the work?
Wednesday, October 13, 2010
A lightbulb went on today
So there I sat, in math class (at college), using flat squares, rectangles and small squares as a way to model multiplying algebraic expressions. I had never done anything like that before, but it made perfect sense and I could see using that in my math class (at middle school). Then, our professor began answering some of the questions we had posted, and the bomb dropped. She stood there and told us quite frankly that we would NOT do everything right as teacher and that we were bound to make MISTAKES as we taught. At first I was horrified, or maybe just surprised. You know that moment in Oz when Toto pulled back the green curtain to reveal who the Wizard really was? It was sort of like that. I had thought that there was some teacher curriculum or education program that would teach you how to do it right EVERY time. I remember feeling at the end of some of my past classes a sense of disappointment. I wanted to know when we (my cohort and I) were going to learn all the TRADE SECRETS. Today, I learned that there is no magic teaching formula that will allow you to always get it right. There are certain things that will make your teaching better, but every year you have a new batch of kids and every year you will need to tailor your teaching to fit THEM. You are bound to screw up some of the time. So, today, after the bomb dropped, I felt this HUGE sense of relief. I'm a teacher and I am still learning.
The implications of this realization will of course filter down to my teaching. I think it will take some of the pressure off and allow me to be excited about learning from my mistakes instead of dreading that I will make any. I also think it allows me to see my master teachers in a new light. I can watch what they do and learn from them. I can record what I like, the things that they model well. And I can give them a break because of course they won't be perfect all the time, they are still learning too!
My only question for the day is where can one purchase cool manipulatives like the multi-colored shapes we used for solving the algebra expressions today?
The implications of this realization will of course filter down to my teaching. I think it will take some of the pressure off and allow me to be excited about learning from my mistakes instead of dreading that I will make any. I also think it allows me to see my master teachers in a new light. I can watch what they do and learn from them. I can record what I like, the things that they model well. And I can give them a break because of course they won't be perfect all the time, they are still learning too!
My only question for the day is where can one purchase cool manipulatives like the multi-colored shapes we used for solving the algebra expressions today?
Monday, October 11, 2010
explaining equivalent fractions failure
As I observed and participated in the middle school math class today, I worked with students to add, subtract, and multiply fractions. I learned a new method for adding fractions that have different denominators from a required course reading. I had never heard of this method before. I will try to explain it here:
Say you want to add 5/6 and 2/7. Instead of finding the common denominator between 6 and 7, cross multiply and add the products: (5*7) + (6*2) = 35+12 = 47 (this is the numerator). Multiply the denominators and their product is the denominator (6*7) = 42. So, the sum of 5/6 + 2/7 is 47/42.
Check this by doing it the other way:
5/6 = 35/42
2/7 = 12/42
35/42 + 12/42 = 47/42
You learn something new every day!
My question is how do you handle a situation where you explain to a student how to do a problem, you do multiple problems with them, explaining each step, and they still do not get it. I worked with a student for 20 minutes one-on-one explaining again and again how to find equivalent fractions, and the student did not understand. I kept waiting for the lightbulb to come on, but it never did. How do you proceed?
What do you do when you are the sole teacher and you cannot spend 20 minutes in a one-on-one reteaching?
Say you want to add 5/6 and 2/7. Instead of finding the common denominator between 6 and 7, cross multiply and add the products: (5*7) + (6*2) = 35+12 = 47 (this is the numerator). Multiply the denominators and their product is the denominator (6*7) = 42. So, the sum of 5/6 + 2/7 is 47/42.
Check this by doing it the other way:
5/6 = 35/42
2/7 = 12/42
35/42 + 12/42 = 47/42
You learn something new every day!
My question is how do you handle a situation where you explain to a student how to do a problem, you do multiple problems with them, explaining each step, and they still do not get it. I worked with a student for 20 minutes one-on-one explaining again and again how to find equivalent fractions, and the student did not understand. I kept waiting for the lightbulb to come on, but it never did. How do you proceed?
What do you do when you are the sole teacher and you cannot spend 20 minutes in a one-on-one reteaching?
Thursday, October 7, 2010
first math class
On September 29th, I attended the first session of my math class. I learned about my instructor and about myself. I could tell from the way the instructor called our class to order that she was a real teacher - that she had been in a classroom of students and knew what she was doing. I learned that I have a similar feeling towards writing papers as my instructor has to reading them. I appreciated how she encourages us to get to the point. Our whole class worked together in small groups to come up with an algorithm that would be true for any number in a series. I learned to be patient and take time to work out the problem in my head or on my paper, allowing all the group members ample time to try to find their own solution. I also learned that even though we all had the same problem, there were MULTIPLE ways to solve it. If I had jumped in and told my group how I figured it out without waiting for them to come up with their own solutions, then I would have missed learning about the multiple ways to solve the problem.
Even though this was a great learning opportunity, I am still wondering how you would proceed in a classroom where a majority of the students understood or saw the pattern quickly, and only a few did not. Or, is it important for the students to know and understand each of the different methods for finding a pattern? Is it okay if they only know one?
The implications from this lesson for my classroom - well, I would love to use this same activity as an introduction to see where my kids are at. I would be sure to encourage each group to think through the problem on their own before sharing with the group. I might even tip them off that there is more than one solution and see if they can find out multiple ways for solving the problem.
I am so excited for this course. I love math!
Even though this was a great learning opportunity, I am still wondering how you would proceed in a classroom where a majority of the students understood or saw the pattern quickly, and only a few did not. Or, is it important for the students to know and understand each of the different methods for finding a pattern? Is it okay if they only know one?
The implications from this lesson for my classroom - well, I would love to use this same activity as an introduction to see where my kids are at. I would be sure to encourage each group to think through the problem on their own before sharing with the group. I might even tip them off that there is more than one solution and see if they can find out multiple ways for solving the problem.
I am so excited for this course. I love math!
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