In case you haven't seen it, I made a song parody about some of the basic derivatives rules in Calculus to the tune of Ceelo's "Forget you." Enjoy!
updated with lyrics:
Chorus:
I see you derivin' in class during calc-u-lus
And you've got f(u)
Which you want to find the derivative of,
it's just f '(u) and a du too,
it's just the chain rule,
a great cal-cu-lus tool
and you use it when...(you use it when)
When there's an inside and an outside that you can write as,
f(u)
(ooo ooo ooo)
Another derivative tool,
is the product rule.
And I know you can get it done
The slope of the product
of functions 1 and 2
is 1d2 plus 2d1.
I pity the foo-ooo-oool
who forgets the product rule
(oh, that's how you get it done)
(it's just 1d2 plus 2d1)
OoooOooOoOh
There's just one more rule
Wait... what was the chain rule again?
-Chorus-
For quotient, quotient, quotient you don't have to, have to pull out your hair,
(your hair, your hair, your hair)
Cause it's just low di high minus hi di low all over low squared
(Low squared, low squared, low squared)
To find dy dx dy dx dy dx baaa-aaa-aaaby,
But what about, that dang chain rule?
-Chorus-
Monday, May 9, 2011
Tuesday, May 3, 2011
How I see exponent rules (and log rules)
Exponent rules can be difficult to remember, and memory has never been one of my strong suits. When I was in high school myself learning exponent rules, I would get mixed up just trying to remember them individually, and had to come up with a different way of thinking to condense it into one idea. What I came up with deals with the levels of complexity of operations:
So, you've got your simple functions on the bottom, multiplication and division are a little more complex, and then exponents and roots are more complex. The actual chart above I created after the fact when trying to explain the idea to students later in life. So, basically when it came to exponent rules all I had to remember was to "go down a level" of complexity.
So multiplication becomes addition, division becomes subtraction, an exponent to an exponent becomes multiplication and a root with an exponent becomes division. The chart also helps for remembering when to distribute. Operations distribute on to the tier below them. (exponents distribute over multiplication and division for example)
My results with trying to get students to see the same thing I do has been mixed. I usually end up doing an exploratory learning exercise with exponents , then going through the rules individually and only quickly going through this chart idea on the side. While it doesn't really connect with every student, when a particular student gets the rules mixed up it can really help because it at least gives them a plan rather than just relying on straight memorization.
Later on, when I learned about logs it turned out that log rules (surprising to me at the time, not so much anymore) followed along the same lines, with exponents becoming multiplication, multiplication become addition, etc.
Anyone else ever think of it this way? Have some other strategy for helping students get exponent rules straight? Let me know!
Tuesday, April 19, 2011
Top 5 math blogs survey
On and off I've been working on a "Welcome to the Math Teacher Online Community" website to help quickly orient people to all the great stuff we have going on between blogs and twitter. One thing I'd like to include is a not too overwhelming list of quality math teaching blogs to add to an RSS reader for someone new to blogs. You can help by filling out this form and writing in your favorite 5 math teaching blogs. Thanks!
Thursday, March 3, 2011
Working With Students With Dyscalculia
Today is International Dyscalculia Day, and it's high time I come clean about something: I teach at a school for students with learning disabilities. Now if you have ever met me in person you're thinking "Yeah, duh" but if you know me exclusively through shouting at each other between the internet tubes, then you're probably a little surprised. It's been my intention not to mention this fact before in my blogging career. In a way I've been trying to make a point, if only to myself, about teaching children with learning issues. The point is something like this: Great lessons and strategies for teaching students with LD are great lessons and strategies for all students. If you get up there and teach a differentiated lesson that uses multiple modalities and engages your class all of your kids will benefit, not just those with LD. Everyone struggles with something, and the students I teach are really just regular kids that happen to struggle with a few specific things. Education would be a lot easier if more people(especially the students themselves) realized that.
So, all that being said there are particular strategies that I've learned that can really help when you come across students with Dyscalculia. This post is going to detail some of the strategies I've found to be successful when working with students with those students. I may also share some ideas about students with dyslexia or ADD/ADHD on my blog in the future as well.
Dyscalculia is not as well known as it really should be, which can be seen in the fact that blogger's spell check doesn't recognize it as a word despite the fact that it is estimated around 1 in 20 people have it. If you aren't familiar with it try to imagine dyslexia, but only with math. Students with Dyscalculia will likely struggle with new material, but also have a lot of gaps in their background knowledge. Many things that are helpful for most kids in learning math such as mnemonics, hands on lessons, extra practice, talking out problems, diagrams and manipulatives become necessary for students with Dyscalculia. The term itself is broad, covering a wide variety of math issues kids can have. I'll try to address some commonalities, but understand that each student is different, and determining exactly what is giving the student trouble is the key to helping the student move forward.
I would argue that for high school students(and probably even middle school) students with Dyscalculia, their actual math disability isn't their biggest problem. You may be saying to yourself "Whaaatttt?!?", but stay with me here. Try to put yourself in the shoes of a high school student with dyscalculia. You've been forced to take a subject that you have struggled with every school day for 10+ years. You've probably been told by peers, parents and/or teachers that you're just plain bad at that subject and it's possible that no one expects you to get any better. Skills that you are supposed to have mastered, but didn't (multiplication, division, solving equations, etc) are now smaller parts of more and more complicated problems. Countless times that you thought you were right, you were told that you're wrong(possibly just due to an arithmetic mistake, but to you getting an answer 2% wrong feels the same as 100% wrong because the answer is just not correct). Now imagine trying to solve a somewhat complicated math problem given all that. Many times it ends up a juggling act between cursing math, feeling incredibly anxious and doubting oneself at every step. Actually breaking down the problem and going through the steps becomes nearly impossible with all those other nagging feelings. So at some point the original issue of struggling with math facts, skills and formulas becomes overshadowed by extreme lack of confidence and anxiety. This can often lead to times when students actually get the right answer, but then change it and make it wrong because they simply have no confidence in their math ability.
The successes I've had are a result of significant one on one work, which means the student must either be in a school with small class sizes, be assisted by a good tutor or at least having time in the day to meet with their teacher for one on one support. As an educator, the most important tools in your arsenal are patience and the real belief that the student will be able to improve. The more time you can spend actively believing in a student, the more he will start to believe in him or herself. A really important strategy is to recognize each improvement a student makes, and make them aware of it. They may get 10 similar questions wrong with help in between each one, but chances are they are getting closer and closer to the right answer each time if good explanations and help are being given. Students with math anxiety aren't going to recognize this gradual improvement, and will immensely benefit from you pointing them out. "Awesome, you're getting this part down, now let's look at that next step." "Oh nice, you just made a little mistake with that negative, but you're really starting to get this!" Word choice and tone of voice can make a big difference between a student with Dyscalculia getting up and trying again versus shutting down completely. Confidence will build as the student begins to show success, even if it is little by little. While this kind of positive affirmation may seem silly or cheesy, the results I've gotten from it have been tremendous.
I've also dealt with a few students with Dyscalculia who actually do pretty well with concepts in high school, but have such serious gaps from early math that it can be hard to recognize that they understand the newer material. Identifying and working with those earlier concepts can make a big difference in certain cases. My first year I worked with a student who was diagnosed with dyscalculia and would have complete breakdowns during any math exam. After working with her a few times, her Trig teacher saw me and said "I don't know what you're doing with her, but she's doing amazing now!" The truth was we would spend about 1-2 minutes on Trig, and then work on essentially arithmetic for the rest of a tutoring session. She had a really bad experience with her times tables when she was younger, and working her through that helped build her confidence and her number sense almost overnight. This added confidence led to a huge improvement in math class, and in her ability to get through her future math exams.
Though one on one work seems to be essential at first, my experience has shown me that as a student's confidence improves he or she will become more and more independent as well. Students who I've had in class have needed to come in for extra help less and less and eventually not anymore. Students who I've tutored quickly begin to pick up more and more concepts and skills up from their classroom teacher and apply them without my help.
To sum it all up, any student with a severe lack of confidence in their math abilities is going to have trouble learning at every step of the way. Older students with Dyscalculia are almost certain to have anxiety issues with math and little to no confidence in their abilities, and helping to make improvements in these areas can have a profound effect on their performance when doing math at all levels. The go-to strategies such as hands on lessons, talking out problems, diagrams and manipulatives all can significantly help. However, the best chance for students with Dyscalculia, especially those with low confidence, is to have them work one on one with a patient teacher or tutor who can help them to see their progress and build their confidence bit by bit, showing them that while it may not come easily they can do math.
So, all that being said there are particular strategies that I've learned that can really help when you come across students with Dyscalculia. This post is going to detail some of the strategies I've found to be successful when working with students with those students. I may also share some ideas about students with dyslexia or ADD/ADHD on my blog in the future as well.
-Keep in mind, this is all from my admittedly limited interaction with children who have Dyscalculia and not the necessarily the result of years of studies and research.-
Dyscalculia is not as well known as it really should be, which can be seen in the fact that blogger's spell check doesn't recognize it as a word despite the fact that it is estimated around 1 in 20 people have it. If you aren't familiar with it try to imagine dyslexia, but only with math. Students with Dyscalculia will likely struggle with new material, but also have a lot of gaps in their background knowledge. Many things that are helpful for most kids in learning math such as mnemonics, hands on lessons, extra practice, talking out problems, diagrams and manipulatives become necessary for students with Dyscalculia. The term itself is broad, covering a wide variety of math issues kids can have. I'll try to address some commonalities, but understand that each student is different, and determining exactly what is giving the student trouble is the key to helping the student move forward.
I would argue that for high school students(and probably even middle school) students with Dyscalculia, their actual math disability isn't their biggest problem. You may be saying to yourself "Whaaatttt?!?", but stay with me here. Try to put yourself in the shoes of a high school student with dyscalculia. You've been forced to take a subject that you have struggled with every school day for 10+ years. You've probably been told by peers, parents and/or teachers that you're just plain bad at that subject and it's possible that no one expects you to get any better. Skills that you are supposed to have mastered, but didn't (multiplication, division, solving equations, etc) are now smaller parts of more and more complicated problems. Countless times that you thought you were right, you were told that you're wrong(possibly just due to an arithmetic mistake, but to you getting an answer 2% wrong feels the same as 100% wrong because the answer is just not correct). Now imagine trying to solve a somewhat complicated math problem given all that. Many times it ends up a juggling act between cursing math, feeling incredibly anxious and doubting oneself at every step. Actually breaking down the problem and going through the steps becomes nearly impossible with all those other nagging feelings. So at some point the original issue of struggling with math facts, skills and formulas becomes overshadowed by extreme lack of confidence and anxiety. This can often lead to times when students actually get the right answer, but then change it and make it wrong because they simply have no confidence in their math ability.
The successes I've had are a result of significant one on one work, which means the student must either be in a school with small class sizes, be assisted by a good tutor or at least having time in the day to meet with their teacher for one on one support. As an educator, the most important tools in your arsenal are patience and the real belief that the student will be able to improve. The more time you can spend actively believing in a student, the more he will start to believe in him or herself. A really important strategy is to recognize each improvement a student makes, and make them aware of it. They may get 10 similar questions wrong with help in between each one, but chances are they are getting closer and closer to the right answer each time if good explanations and help are being given. Students with math anxiety aren't going to recognize this gradual improvement, and will immensely benefit from you pointing them out. "Awesome, you're getting this part down, now let's look at that next step." "Oh nice, you just made a little mistake with that negative, but you're really starting to get this!" Word choice and tone of voice can make a big difference between a student with Dyscalculia getting up and trying again versus shutting down completely. Confidence will build as the student begins to show success, even if it is little by little. While this kind of positive affirmation may seem silly or cheesy, the results I've gotten from it have been tremendous.
I've also dealt with a few students with Dyscalculia who actually do pretty well with concepts in high school, but have such serious gaps from early math that it can be hard to recognize that they understand the newer material. Identifying and working with those earlier concepts can make a big difference in certain cases. My first year I worked with a student who was diagnosed with dyscalculia and would have complete breakdowns during any math exam. After working with her a few times, her Trig teacher saw me and said "I don't know what you're doing with her, but she's doing amazing now!" The truth was we would spend about 1-2 minutes on Trig, and then work on essentially arithmetic for the rest of a tutoring session. She had a really bad experience with her times tables when she was younger, and working her through that helped build her confidence and her number sense almost overnight. This added confidence led to a huge improvement in math class, and in her ability to get through her future math exams.
Though one on one work seems to be essential at first, my experience has shown me that as a student's confidence improves he or she will become more and more independent as well. Students who I've had in class have needed to come in for extra help less and less and eventually not anymore. Students who I've tutored quickly begin to pick up more and more concepts and skills up from their classroom teacher and apply them without my help.
To sum it all up, any student with a severe lack of confidence in their math abilities is going to have trouble learning at every step of the way. Older students with Dyscalculia are almost certain to have anxiety issues with math and little to no confidence in their abilities, and helping to make improvements in these areas can have a profound effect on their performance when doing math at all levels. The go-to strategies such as hands on lessons, talking out problems, diagrams and manipulatives all can significantly help. However, the best chance for students with Dyscalculia, especially those with low confidence, is to have them work one on one with a patient teacher or tutor who can help them to see their progress and build their confidence bit by bit, showing them that while it may not come easily they can do math.
Friday, February 25, 2011
Angry Birds Geogebra
Angry Birds is a pretty popular game with the kids nowadays. My students brought up the game when we started talking about parabolas and I've been working on a way to bring that connection into a class. So, I created a lesson using GeoGebra and some screenshots from Angry Birds mixed in with some inspiration from Dan's Will The Ball Hit The Can?
I created 4 GeoGebra files each with a piece of a different Angry birds shot like so:
Using GeoGebra, students worked in groups of 2 on their laptops to place points onto the bird's trail as accurately as possible to create a quadratic models in order to decide if the bird would score a direct hit on any of the pigs. If you had 4 points labeled A-D, for instance, the GeoGebra command
Some commands that helped them place their points accurately:
CTRL= Zooms in
CTRL- Zooms out
CTRL CLICK DRAG Pans your view
Once students finished their files would look something similar to the file shown here:
We then discussed if they thought they scored a hit, what would happen when it hit, and then showed them the big reveal:
Overall, the students were engaged, worked hard to get their answers, and learned how to use GeoGebra to create quadratic models. If you'd like the files I used you can find them here: Angry Birds GeoGebra files
PS- I'm muddling through learning GeoGebra, so if you know of a better way to do things than I'm doing, let me know.
I created 4 GeoGebra files each with a piece of a different Angry birds shot like so:
Using GeoGebra, students worked in groups of 2 on their laptops to place points onto the bird's trail as accurately as possible to create a quadratic models in order to decide if the bird would score a direct hit on any of the pigs. If you had 4 points labeled A-D, for instance, the GeoGebra command
- would be FitPoly[A, B, C, D, 2]
Some commands that helped them place their points accurately:
CTRL= Zooms in
CTRL- Zooms out
CTRL CLICK DRAG Pans your view
Once students finished their files would look something similar to the file shown here:
We then discussed if they thought they scored a hit, what would happen when it hit, and then showed them the big reveal:
PS- I'm muddling through learning GeoGebra, so if you know of a better way to do things than I'm doing, let me know.