Looking back over the quarter's math work, I've come out with a real appreciation for the merits of cooperative groupwork in math education (and all education). In fact, next week I'm going to introduce cooperative groupwork to my 4th graders (I'm using Elizabeth Cohen's Designing Groupwork) -- we'll be engaging in an activity called "Broken Circles." Each group member will have an envelope with circle sections, and each member of the group needs to finish with a complete circle in front of them. They can't talk or take pieces, they can only give pieces. I am very curious to see how this exercise goes. These kids work in partner stations during math right now, but I think the kids would benefit from strategies that foster deeper cooperation.
Our math class also highlighted for me the value of manipulatives (both traditional and online) in "higher order" math...such as algebra. I appreciated learning different strategies for exploring concepts in ways that allow kids to construct their own knowledge, as opposed to being forcefed rules and formulas via direct instruction. The Annenberg videos also gave me good ideas for how kids could construct their own knowledge, such as the video that showed kids developing a rule for calculating the volume and area of rods. Approaching math in this way also allows us to make it more engaging and interesting for kids. I was completely absorbed by the logic puzzle we worked on during our last class. The only danger is how absorbing some of these activities can be -- regrouping students could be challenging if everyone hasn't finished!
The last observation I'd like to make is the value I see in exploring our kids' mathematical identities. Finding ways in which kids can see themselves as mathematicians will help them approach math in a more positive way. As I mentioned in last week's math blog, I found the chart of mathematical activities to be particularly promising for exploring math in our everyday lives. The ideas in Complex Identities are fantastic for both students and teachers to explore their own preconceived ideas about math.
In all, this quarter's math class provided me with good information (via the readings, videos, and class discussion) about how to implement middle level mathematics curricula. The ideas, though, will also be useful to me in my 4th grade main placement class. Thanks to Robin for making math approachable, fun, and edifying!
Showing posts with label math. Show all posts
Showing posts with label math. Show all posts
Friday, March 11, 2011
Friday, March 4, 2011
complex identities
We engaged in an activity found in the article Complex Identities in our math methods class last quarter -- we explored what adjectives we would apply to someone good at math or not good at math. However, this article was about more than just the activity we engaged in, and it really shed more light on how we and our students see ourselves in relation to math. I was particularly interested in the mathematical task chart, which listed a number of different activities. Students could be asked to identify which of the tasks, such as sending a text message or riding a skateboard, were mathematical in nature. The answer, of course, is that all of the tasks in some way involve math. I believe that this is a great activity which can help our students see that math is everywhere...not just in the memorization of math facts and rules and passing math tests.
A question I have is why this question of mathematical identity isn't given more emphasis in school. When I read about the honors math student who couldn't think of herself as good at math because great mathematicians are "brilliant" and she didn't see herself as brilliant, I had to wonder if a lesson exploring mathematical identity might have helped her bypass this conclusion.
I definitely see a place for exploring mathematical identities in the classroom. In fact, I'd like to explore ideas about what constitutes a mathematical task in my fourth grade classroom. I can envision applying the same strategy to an exploration of scientific tasks, too! How many things do we do everyday, without thinking, that involve either math or science. Yet many students see math and science as their weakest, and sometimes least enjoyable, subjects. It would be particularly nice to see those kinds of attitudes turned around!
A question I have is why this question of mathematical identity isn't given more emphasis in school. When I read about the honors math student who couldn't think of herself as good at math because great mathematicians are "brilliant" and she didn't see herself as brilliant, I had to wonder if a lesson exploring mathematical identity might have helped her bypass this conclusion.
I definitely see a place for exploring mathematical identities in the classroom. In fact, I'd like to explore ideas about what constitutes a mathematical task in my fourth grade classroom. I can envision applying the same strategy to an exploration of scientific tasks, too! How many things do we do everyday, without thinking, that involve either math or science. Yet many students see math and science as their weakest, and sometimes least enjoyable, subjects. It would be particularly nice to see those kinds of attitudes turned around!
Friday, February 18, 2011
computer based math tools
Last week I learned about two more ways to engage students in math. We played with a couple of very neat math utilities: Fathom and Geometer's Sketchpad. What great classroom resources these would be! Geometer's Sketchpad looks like a particularly versatile tool that, according to its website, can help kids learn everything from geometry and algebra to proofs. For geometry, creating geometric shapes to fit over a clown's features - using only a limited number of available shapes - was a challenging exercise even for an adult! It really forced you to remember and apply your knowledge of the properties of geometric shapes.
Two questions I have are: 1) in what way can Geometer's Sketchpad facilitate algebra learning and 2) how much does it cost (I think this was mentioned, but I've forgotten). The next time I have free time on campus, I think I'll try and explore ways in which Geometer's Sketchpad can help students learn or practice algebra. If the algebra functions are as engrossing as the geometry exercise was, I think they'd be very useful in a middle level math class.
One implication for classroom practice is how to best utilize Geometer's Sketchpad when you only have a couple computers available for classroom use. Perhaps it would be best utilized as enrichment or remediation tool in that case - although if the software has a high pricetag, you'd probably want every student to have access.
Two questions I have are: 1) in what way can Geometer's Sketchpad facilitate algebra learning and 2) how much does it cost (I think this was mentioned, but I've forgotten). The next time I have free time on campus, I think I'll try and explore ways in which Geometer's Sketchpad can help students learn or practice algebra. If the algebra functions are as engrossing as the geometry exercise was, I think they'd be very useful in a middle level math class.
One implication for classroom practice is how to best utilize Geometer's Sketchpad when you only have a couple computers available for classroom use. Perhaps it would be best utilized as enrichment or remediation tool in that case - although if the software has a high pricetag, you'd probably want every student to have access.
Sunday, February 13, 2011
tangrams
This past week we had fun with yet another completely engaging math lesson - firing cotton ball "frogs" from paper clip launchers, then tracking the absolute and linear distances they traveled. While this way of teaching about distance was both fun and new to me, what I really appreciated about last week's class was learning how to use tangrams. I'd never worked with them before and knew nothing about them. In class, we used a set of tangrams to create a giraffe figure, then tried to construct a giraffe twice the size with multiple sets. This was not easy!
As I mentioned above, I'd never used tangrams before this class, and one question I have is about other classroom applications. They seem to do a good job of fostering spatial thinking. Do they have any applications at the 4th grade level? I'm wondering about a use that might help my main placement kids remember geometric shapes and their properties. Since the tangrams are geometric shapes they've studied, it might be a useful way to use them in my classroom.
The implications for classroom practice that I see are more engaging ways to get kids to connect to math. The exercise of doubling the size of the giraffe was very challenging -- my group never got there, in fact -- but was also very engrossing, no one wanted to stop trying!
As I mentioned above, I'd never used tangrams before this class, and one question I have is about other classroom applications. They seem to do a good job of fostering spatial thinking. Do they have any applications at the 4th grade level? I'm wondering about a use that might help my main placement kids remember geometric shapes and their properties. Since the tangrams are geometric shapes they've studied, it might be a useful way to use them in my classroom.
The implications for classroom practice that I see are more engaging ways to get kids to connect to math. The exercise of doubling the size of the giraffe was very challenging -- my group never got there, in fact -- but was also very engrossing, no one wanted to stop trying!
Saturday, February 5, 2011
paper folding
Last week in math class I learned that paper folding is very mathematical! It was really interesting to see how the simple act of folding an origami box uncovered so many mathematical challenges. It was a truly rigorous way to explore geometry and proofs - I personally had difficulty expressing the reasons why I knew a given shape was what it was! Particularly challenging was finding a way to articulating why one folded line was parallel to another.
One question I have is how best to elicit the proofs. Is it best to have the whole class work independently and volunteer answers? Or would it be better to have students work in teams or groups? I'm personally inclined to say groups, because I know I could have used someone to bounce ideas off of when we did this in class.
I think this is a great way to explore geometry and practice articulating mathematical proofs in the classroom. While we engaged in this activity as a way to explore and provide proofs, I'm personally planning to adapt this activity into a lesson for 4th graders. My kids had some trouble with geometric shapes when they were learning about them last fall. I think (and my master teacher agrees) that examining geometric shapes while folding an origami box will help them better understand and remember what they learned. It helps that most of my class is rather origami obsessed, too!
One question I have is how best to elicit the proofs. Is it best to have the whole class work independently and volunteer answers? Or would it be better to have students work in teams or groups? I'm personally inclined to say groups, because I know I could have used someone to bounce ideas off of when we did this in class.
I think this is a great way to explore geometry and practice articulating mathematical proofs in the classroom. While we engaged in this activity as a way to explore and provide proofs, I'm personally planning to adapt this activity into a lesson for 4th graders. My kids had some trouble with geometric shapes when they were learning about them last fall. I think (and my master teacher agrees) that examining geometric shapes while folding an origami box will help them better understand and remember what they learned. It helps that most of my class is rather origami obsessed, too!
Saturday, January 29, 2011
engaging math
Last week in math class, I learned strategies for creating group math lessons that engage students and are inquiry focused. Since so much of the math in my main placement classroom seems to be whole class direct instruction, it's refreshing to see ways that math concepts can be taught -- and taught intuitively -- without being stultifyingly boring. The group work we did in class the last two Mondays using the online statistics generator was fun, interesting, and engaging. It didn't really feel like we were learning, but we were using math skills such as statistical analysis and data graphing.
My ongoing question about group work seems to be how to best implement group lessons. Exactly how much groundwork will be required to familiarize my students with group roles? With a pacing guide, will I have that time? Will students in my groups really be responsible for the learning of everyone on his team?
I believe the implications for classroom practice are that group inquiry lessons, if implemented thoughtfully, can greatly enhance our math curriculum. Using absorbing, thought-provoking lessons like the one we experienced in our class the last two weeks can be an effective alternative to more "traditional" methods.
My ongoing question about group work seems to be how to best implement group lessons. Exactly how much groundwork will be required to familiarize my students with group roles? With a pacing guide, will I have that time? Will students in my groups really be responsible for the learning of everyone on his team?
I believe the implications for classroom practice are that group inquiry lessons, if implemented thoughtfully, can greatly enhance our math curriculum. Using absorbing, thought-provoking lessons like the one we experienced in our class the last two weeks can be an effective alternative to more "traditional" methods.
Sunday, January 16, 2011
math inquiry
In our last math class, my "aha moment" came when we learned how to use algebra tiles to uncover algebraic concepts. It took a couple of example problems, but once I saw how the algebra tiles worked I realized their potential as a learning tool - especially as a visual representation of rules like "FOIL" - rules that we were simply given and expected to memorize back in the day.
A question I have about using inquiry based techniques like algebra tiles for teaching math concepts is how to introduce them to a classroom with a scripted, direct instruction based, curriculum. My gut feeling is that these techniques would increase comprehension in my main placement classroom. However, I can see an inquiry based approach like algebra taking up more teaching time, therefore, the idea would meet with resistance.
In my main placement classroom, as stated, my feeling is that using manipulatives like algebra tiles would help some of our struggling kids "get it." Simply re-iterating the rule isn't working for them. We're currently using a FOIL technique called multiplication wrestling to teach double digit multiplication. The kids take each double digit number in the problem, separate it into 10s and 1s, and then multiply all the numbers together using FOIL. For instance, the kids would write out 32*49 as (30+2)*(40+9). Then the 30 and the 2 need to wrestle the 40 and the 9 (30*40, 30*9, 2*40, 2*9). The results are added together to get an answer. Many kids get this, but many do not, and going over the rules again isn't helping things sink in. I'd be interested to see whether these kids are helped by using a manipulative technique.
A question I have about using inquiry based techniques like algebra tiles for teaching math concepts is how to introduce them to a classroom with a scripted, direct instruction based, curriculum. My gut feeling is that these techniques would increase comprehension in my main placement classroom. However, I can see an inquiry based approach like algebra taking up more teaching time, therefore, the idea would meet with resistance.
In my main placement classroom, as stated, my feeling is that using manipulatives like algebra tiles would help some of our struggling kids "get it." Simply re-iterating the rule isn't working for them. We're currently using a FOIL technique called multiplication wrestling to teach double digit multiplication. The kids take each double digit number in the problem, separate it into 10s and 1s, and then multiply all the numbers together using FOIL. For instance, the kids would write out 32*49 as (30+2)*(40+9). Then the 30 and the 2 need to wrestle the 40 and the 9 (30*40, 30*9, 2*40, 2*9). The results are added together to get an answer. Many kids get this, but many do not, and going over the rules again isn't helping things sink in. I'd be interested to see whether these kids are helped by using a manipulative technique.
Saturday, January 8, 2011
patterns and rules
This past week, I learned how pattern recognition can help students find a rule and develop algebraic thinking. In honesty, while I've certainly detected patterns to solve problems before, using patterns to define rules is not something I've spent much time doing. Growing up, rules were given to you. You memorized them. It did not matter why or how they worked. Using inquiry methods to derive a mathematical rule from a perceived pattern - and expressing that rule algebraically - isn't something I ever remember doing in school. While I developed a real appreciation for inquiry methods in last quarter's math methods class, for some reason last week's activity "Crossing the River" (in which we determined how many trips, subject to certain rules, it would take to ferry a given number of adults and children across a river) really reinforced their value.
One question I have about using patterns to develop algebraic concepts is how to differentiate instruction for varying ability levels. For example, some students may not be able to detect a pattern when solving a problem like "Crossing the River" and members of their learning group may not be able to successfully explain it. Perhaps there is another entry point for this exercise, but I am not sure what it would be.
In the classroom, while we may choose to do an exercise such as "Crossing the River" and be resolved to allow as much time as is needed for every student to achieve comprehension, the reality is that we may not be able to. Running out of time seems to be a common occurrence, in my classroom experience thus far. On the other hand, if we're able to successfully differentiate an exercise like "Crossing the River," small group or math station work lends itself to differentiation quite nicely. This strategy might better enable every student to succeed.
One question I have about using patterns to develop algebraic concepts is how to differentiate instruction for varying ability levels. For example, some students may not be able to detect a pattern when solving a problem like "Crossing the River" and members of their learning group may not be able to successfully explain it. Perhaps there is another entry point for this exercise, but I am not sure what it would be.
In the classroom, while we may choose to do an exercise such as "Crossing the River" and be resolved to allow as much time as is needed for every student to achieve comprehension, the reality is that we may not be able to. Running out of time seems to be a common occurrence, in my classroom experience thus far. On the other hand, if we're able to successfully differentiate an exercise like "Crossing the River," small group or math station work lends itself to differentiation quite nicely. This strategy might better enable every student to succeed.
Tuesday, November 30, 2010
the khan academy
The Khan Academy bills itself as "The free classroom for the world." This useful website has video lessons and interactive exercises galore - math is highlighted, but there are lots of lessons in other disciplines (science, economics, history...) and they range from introductory to advanced. Take a look at this PBS NewsHour feature, then then go check it out yourself!
I tried out a few of the interactive math exercises - addition and multiplication. There's a helpful map on the first screen that organizes the math exercises, so students can see the progression of math concepts and select the appropriate exercise. The math exercises I investigated were all drills, so they would mainly be useful for children who just needed to bone up on their math facts. It could be that other exercises are more sophisticated, but I haven't delved deeply into the exercises section. Of course, those kids with comprehension problems can take a look at the lessons, too.
The main drawback of this site is that it requires a computer with internet access to utilize. While we assume that everyone is wired in this day and age, I have found that a lot of kids in my main placement school face significant challenges gaining access to computers. For your kids who do have computers, I'd recommend taking a look at this helpful site.
I tried out a few of the interactive math exercises - addition and multiplication. There's a helpful map on the first screen that organizes the math exercises, so students can see the progression of math concepts and select the appropriate exercise. The math exercises I investigated were all drills, so they would mainly be useful for children who just needed to bone up on their math facts. It could be that other exercises are more sophisticated, but I haven't delved deeply into the exercises section. Of course, those kids with comprehension problems can take a look at the lessons, too.
The main drawback of this site is that it requires a computer with internet access to utilize. While we assume that everyone is wired in this day and age, I have found that a lot of kids in my main placement school face significant challenges gaining access to computers. For your kids who do have computers, I'd recommend taking a look at this helpful site.
Friday, October 22, 2010
rappin' math
I've been enjoying our math articles these past few weeks. They've been giving me great food for thought about mathematizing and approaching math in new ways. I recently received a e-newsletter from Edutopia about math in the classroom, which included information about writing your own math raps. I enjoyed the rap they posted, PEMDAS Boss by Rappin' Mathematician Alex Kajitani, so much that I thought I'd share another one with those of you who follow my blog. This one's called The Number Line Dance. Enjoy!
I don't know if I have what it takes to be a rap master of math, but I'm willing to consider it (or at least use the Rappin' Mathematicians CD ;)). If you want to take a look at the Edutopia resource, visit How to Write Your Own Math Rap. While you're there, look around. Edutopia has tons of fantastic resources!
I don't know if I have what it takes to be a rap master of math, but I'm willing to consider it (or at least use the Rappin' Mathematicians CD ;)). If you want to take a look at the Edutopia resource, visit How to Write Your Own Math Rap. While you're there, look around. Edutopia has tons of fantastic resources!
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