Aliens living on stars probably wouldn't be able to do geometry or draw triangles. Counting would be difficult for them as well.
Friday, July 26, 2013
Context Matters
Aliens living on stars probably wouldn't be able to do geometry or draw triangles. Counting would be difficult for them as well.
Book Review: Brain Rules by John Medina
Brain Rules, subtitled 12 Principles for Surviving and Thriving at Work, Home, and School, by John Medina, deals with the above topic through the author's use of anecdotes, examples, and simple explanations of the science that make the stories relevant. The book is a fairly quick read, although slow because of the breadth of the topic at hand. Medina breaks the topic into 12 rules: exercise, survival, wiring, attention, short-term memory, long-term memory, sleep, stress, sensory integration, vision, gender, and exploration.
John Medina is uniquely qualified to discuss this topic, given that he researches the molecular basis of psychiatric disorders. He writes that he "occasionally would run across articles and books that made startling claims based on 'recent advances' in brain science about how to change the way we teach people and do business. And [he] would panic, wondering if the authors were reading some literature totally off [his] radar screen." But there was no need for panic, as the science cannot yet tell us how the brain knew how to pick up a glass of water. Instead, Medina wishes more research would be done.
In this way, he does not ever demand that we follow these rules, merely points to the results of going with or against them. With exercise, he pointed to sharp intellectuals still active in their nineties vs "couch potatoes" who were fading or already gone by seventy.
This book is helpful to us as teachers in that it shows some of the science behind why our students behave the way they do, perform the way they do, and ultimately need our help in the areas they seem to be weak at. Using these rules can help us as educators adjust our teaching to fit our many different learners.
John Medina is uniquely qualified to discuss this topic, given that he researches the molecular basis of psychiatric disorders. He writes that he "occasionally would run across articles and books that made startling claims based on 'recent advances' in brain science about how to change the way we teach people and do business. And [he] would panic, wondering if the authors were reading some literature totally off [his] radar screen." But there was no need for panic, as the science cannot yet tell us how the brain knew how to pick up a glass of water. Instead, Medina wishes more research would be done.
In this way, he does not ever demand that we follow these rules, merely points to the results of going with or against them. With exercise, he pointed to sharp intellectuals still active in their nineties vs "couch potatoes" who were fading or already gone by seventy.
This book is helpful to us as teachers in that it shows some of the science behind why our students behave the way they do, perform the way they do, and ultimately need our help in the areas they seem to be weak at. Using these rules can help us as educators adjust our teaching to fit our many different learners.
Cubism
In Two Figures (1913-14), Liubov' Popova beautifully demonstrates the artistic possibilities of a Cubist reconstruction and, at the same time, her talent to transcend simple imitation. http://myweb.rollins.edu/aboguslawski/Ruspaint/cubism.html
Sometimes students are incapable of seeing the art in mathematics. Thus, I intend to display the mathematics in art. There are two-dimensional and three-dimensional shapes in this painting. Finding out how many and their dimensions might be a challenging group project.
Advanced Mathematics: Uses of Calculus
Another answer to the "When are we going to use math?" question. Calculus provides mathematics answers to questions sometimes we didn't even know we were asking. http://www.dummies.com/how-to/education-languages/math/Calculus/Practical-Applications.html has a whole list of problems that are answered with Calculus mathematics.
Sure, anecdotal or observational information could be substituted for the mathematics, but the mathematics will stay constant even when situations change. For instance, calculus finds maximums and minimums very quickly, and thus will define the dimensions of a box based on the total area of a sheet of cardboard, through relating volume to surface area.
In this case, the math is faster and less expensive in time and materials than simply building boxes until you come to the maximum. Thus, in order to understand this application, you must understand calculus and how surface area would confine volume, since both are related to the dimensions of the shape involved (rectangles). So this is one example of how the math we learn in middle school relates to the math learned later in life.
Sure, anecdotal or observational information could be substituted for the mathematics, but the mathematics will stay constant even when situations change. For instance, calculus finds maximums and minimums very quickly, and thus will define the dimensions of a box based on the total area of a sheet of cardboard, through relating volume to surface area.
In this case, the math is faster and less expensive in time and materials than simply building boxes until you come to the maximum. Thus, in order to understand this application, you must understand calculus and how surface area would confine volume, since both are related to the dimensions of the shape involved (rectangles). So this is one example of how the math we learn in middle school relates to the math learned later in life.
Open Ended Math Problems
Group work can be an excellent source of exploration and learning, but sometimes it can be difficult to tap into in the subject of mathematics. One of the reasons why is because many mathematics questions are set up to be closed ended. You find the one right answer and you check it and you're finished. Here's an example of changing closed ended to open ended.
| Which of the following numbers are prime? 7, 57, 67, 117 | Fred thinks that 57 and 67 are prime because they both end in 7, which is a prime number. Dick says he is wrong. Who is correct and why? |
As you can see with the open ended question, the restraint to one answer is removed. Also, more information can be given and more critical thinking can be asked by using the question on the right. I could say 77 is not prime, but it ends in 7, therefore Fred is wrong. Or I could say 57 can be divided by 3 evenly so it's not prime. These are not the only rational answers to the question on the right.
Further opening it up could be a question like "What prime numbers end in 7?" By asking this question, I removed another limitation, this time I removed the upper bound on prime numbers. Daring students could find primes that are beyond our usual scope of 1-100.
Monday, July 22, 2013
Estimation or Using the Test Against Itself
Hello everyone. I'd like to discuss using estimation as a test taking strategy. This is one of my favorites, but one some students that I encountered did not willingly embrace. Some of the complaints were that it was extra work, or that they were not careful enough with the estimation. I believe they may have had bad experiences in the past with estimation and rounding, as it has a few arbitrary rules that they may have been caught up in.
Estimation is a vital skill that we use every day whether we know it or not. When we're out shopping, we estimate prices all the time. 19.95 becomes 20 very quickly. This in turn, helps us calculate and budget how much should be spent on each item. It also helps us compare prices, especially when the stores don't want to give out cost per unit. 3.99 for 10 apples for example versus 0.67 for 1 apple.
Where students seem to be falling down is that they don't understand when to estimate, or what information the estimate gives us, thus why it's useful. Estimation is especially useful on problems that involve measurement, as they nearly all include multiplication, sometimes multiple step multiplication. Thus, having an easier set of multiplication to do can eliminate answers, perhaps even enough answers that the correct one is left alone.
Estimation gives us minimums and maximums. In my example above, the maximum it will be is $20. Then estimating tax of 10% gives us tax of $2, so we should have $22 ready to spend at the register for our item that costs $21.64 exactly. Thus, we got a maximum price, exceeded it so that we would get change back instead of having to leave without the item.
Likewise with measurement. We should estimate close to the measurements given but make sure we always round down to get our minimum, then always round up to get our maximum. This strategy can help students who are not careful with decimals when they multiply. Students who know their multiplication facts will also find that estimation is much faster than multiplying the decimals out. Other things to point out would be when a number is very close to an easy multiplication fact, like 2, 5, or 10.
The problem with this strategy, when the test is really mean, is that it is possible that it will not eliminate enough answers and a guess or the long multiplication is still required. In that case, remind students that their exact answer will fall between their minimums and maximums. Looking at the entire question including the answers would be the best idea to determine if they should estimate or not. If the numbers vary greater than 3-4, estimation will eliminate 1-2 answer choices and possibly all 3 of the wrong ones.
Estimation is a vital skill that we use every day whether we know it or not. When we're out shopping, we estimate prices all the time. 19.95 becomes 20 very quickly. This in turn, helps us calculate and budget how much should be spent on each item. It also helps us compare prices, especially when the stores don't want to give out cost per unit. 3.99 for 10 apples for example versus 0.67 for 1 apple.
Where students seem to be falling down is that they don't understand when to estimate, or what information the estimate gives us, thus why it's useful. Estimation is especially useful on problems that involve measurement, as they nearly all include multiplication, sometimes multiple step multiplication. Thus, having an easier set of multiplication to do can eliminate answers, perhaps even enough answers that the correct one is left alone.
Estimation gives us minimums and maximums. In my example above, the maximum it will be is $20. Then estimating tax of 10% gives us tax of $2, so we should have $22 ready to spend at the register for our item that costs $21.64 exactly. Thus, we got a maximum price, exceeded it so that we would get change back instead of having to leave without the item.
Likewise with measurement. We should estimate close to the measurements given but make sure we always round down to get our minimum, then always round up to get our maximum. This strategy can help students who are not careful with decimals when they multiply. Students who know their multiplication facts will also find that estimation is much faster than multiplying the decimals out. Other things to point out would be when a number is very close to an easy multiplication fact, like 2, 5, or 10.
The problem with this strategy, when the test is really mean, is that it is possible that it will not eliminate enough answers and a guess or the long multiplication is still required. In that case, remind students that their exact answer will fall between their minimums and maximums. Looking at the entire question including the answers would be the best idea to determine if they should estimate or not. If the numbers vary greater than 3-4, estimation will eliminate 1-2 answer choices and possibly all 3 of the wrong ones.
Product Review: Kuta Software: Infinite Pre-Algebra
Mr. V. here. I purchased a product called Kuta Software's Infinite Pre-Algebra to help me generate content during summer school. After using it all through the past few weeks, I thought I would give it a review for their information if they ever stumble over my blog, and for any fellow teachers out there.
The options on Kuta's website are Pre-Algebra, Algebra, Geometry, Algebra 2, and Calculus. All of these options would load in the same program if ordered together. As a teacher of middle school math, I limited my purchase to Pre-Algebra. Any Elementary teachers should do the same, as the content will quickly become more and more limited to the specific subject, and Pre-Algebra had the broadest set of concepts.
Kuta is strong at reinforcing the mechanics of calculation, as you have near complete control over difficulty of problems through slider bars and can "regenerate" them as needed (the software will swap out one problem for another). You could also use it to try to build teaching worksheets, that move from easier problems and concepts to more difficult ones or ones that use the previous information. I found this style to be a bit more successful, as it reminded students of what they needed to keep track of when they were doing more complicated questions.
Where I found Kuta falling down a bit, and this may be just the package that I bought, was in measurement of shapes and solids. It seemed like the program's idea of difficulty was to include decimals in measurement. Basically this challenges students who's foundation of multiplying decimals is weak, and that's it. There were no Integers only check boxes for measurement, and the number slider did not seem to adjust the shapes in the same way it did other problems like fractions and order of operations.
Another area that I wasn't quite pleased with was Kuta's attempt at creating word problems. While some seemed interesting to the students, they seemed to lack that STAAR craziness quality that the students hate. It was usually very clear what the student was supposed to do. This is a strange complaint, I know, but one of my goals is STAAR readiness, and these word problems do not come as close as I'd like. While there is a way to write custom questions, this somewhat defeats the purpose of buying this software as an aid.
Overall, I would give Kuta's Pre-Algebra package an 8 of 10 and would recommend it to late elementary school teachers onward.
The options on Kuta's website are Pre-Algebra, Algebra, Geometry, Algebra 2, and Calculus. All of these options would load in the same program if ordered together. As a teacher of middle school math, I limited my purchase to Pre-Algebra. Any Elementary teachers should do the same, as the content will quickly become more and more limited to the specific subject, and Pre-Algebra had the broadest set of concepts.
Kuta is strong at reinforcing the mechanics of calculation, as you have near complete control over difficulty of problems through slider bars and can "regenerate" them as needed (the software will swap out one problem for another). You could also use it to try to build teaching worksheets, that move from easier problems and concepts to more difficult ones or ones that use the previous information. I found this style to be a bit more successful, as it reminded students of what they needed to keep track of when they were doing more complicated questions.
Where I found Kuta falling down a bit, and this may be just the package that I bought, was in measurement of shapes and solids. It seemed like the program's idea of difficulty was to include decimals in measurement. Basically this challenges students who's foundation of multiplying decimals is weak, and that's it. There were no Integers only check boxes for measurement, and the number slider did not seem to adjust the shapes in the same way it did other problems like fractions and order of operations.
Another area that I wasn't quite pleased with was Kuta's attempt at creating word problems. While some seemed interesting to the students, they seemed to lack that STAAR craziness quality that the students hate. It was usually very clear what the student was supposed to do. This is a strange complaint, I know, but one of my goals is STAAR readiness, and these word problems do not come as close as I'd like. While there is a way to write custom questions, this somewhat defeats the purpose of buying this software as an aid.
Overall, I would give Kuta's Pre-Algebra package an 8 of 10 and would recommend it to late elementary school teachers onward.
Sunday, July 21, 2013
Future Mathematics: Fractals
One thing my students tend to ask me, is "Mr. V? Where is math in our lives?" Sometimes that answer has clear ties to what we're learning in class, but other times the mathematics are more advanced. Fractals are a great example of advanced mathematics with current application. Some examples follow:
Urban Planning:
http://www.nature.com/srep/2012/120724/srep00527/full/srep00527.html
Natural Patterns:
http://webecoist.momtastic.com/2008/09/07/17-amazing-examples-of-fractals-in-nature/
Chemistry/Physics:
http://www.wired.com/wiredscience/2010/08/superconductor-fractals/
Art
http://www.oneness4all.com/?cat=44
Crop Circles
http://www.cropcircleconnector.com/Bert/3dfractals.html
3D Graphics
http://www.profantasy.com/products/ft.asp
http://specialeffectsmoviesebook.blogspot.com/2013/02/1-fractals-in-special-effects.html
Technology
http://classes.yale.edu/fractals/panorama/ManuFractals/FractalAntennas/FractalAntennas.html
Sound and Music
http://www.tursiops.cc/fm/
Urban Planning:
http://www.nature.com/srep/2012/120724/srep00527/full/srep00527.html
Natural Patterns:
http://webecoist.momtastic.com/2008/09/07/17-amazing-examples-of-fractals-in-nature/
Chemistry/Physics:
http://www.wired.com/wiredscience/2010/08/superconductor-fractals/
Art
http://www.oneness4all.com/?cat=44
Crop Circles
http://www.cropcircleconnector.com/Bert/3dfractals.html
3D Graphics
http://www.profantasy.com/products/ft.asp
http://specialeffectsmoviesebook.blogspot.com/2013/02/1-fractals-in-special-effects.html
Technology
http://classes.yale.edu/fractals/panorama/ManuFractals/FractalAntennas/FractalAntennas.html
Sound and Music
http://www.tursiops.cc/fm/
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