While thinking about this topic I was trying to figure out what items are metric in our standard units of measure. Go to the grocery store and find what we have imported or what we have as far as metric items in our country. I just found a few items that we buy by way of the metric system; tonic water, alcohol and soda pop. Also we buy everything with the metric measure in parenthesis whether it be bread which is measured by pounds, has grams in parenthesis. Most other items I have found have had grams, liters, and meters for how long something is. Essentially we have the system in play in our nation is metric, but it isn’t emphasized. Even though the United States has all the metric units emplaced on different items, it is unlikely that this great nation will change soon. Many other items are pressing the U.S. government this day and age, so it is very unlikely they will put this issue on the docket once again.
However, it is imperative that we continue to teach the students in this country the "ropes" of the metric system so they understand why the units are in place on a loaf of bread. The rest of the world works with the metric system so we need our students to know the metric system. Because of the many items that are imported, it is necessary for us as teachers to help the students understand the metric's system of measurement. Also on a side note a lot of the material that the United States exports is in the standard unit of measure. The other countries that receive our goods have to learn our unit of measure.
How do you make a big nation like the U.S. switch to the metric system? Well it is plain and simple, you don't. The U.S. is the only nation that has people fighting to get across the borders to live in the land of the free and home of the brave. We have too many people that are stuck in their ways that it is not crucial for the U.S. to switch the system that has worked for centuries.
Sunday, July 20, 2008
Wednesday, July 16, 2008
Successful Measurement Experiences in Kindergarten
Reading through our chapter on measurement got me thinking about how I can teach this important concept in my own classroom someday. As a kindergarten teacher, what is the process I should take in introducing measurement that makes sense to very young children? What kinds of activities should I focus on that serve as building blocks for a lifelong curiosity of math? Are there age-appropriate ways I can introduce the topics of time, volume, area, and temperature to my students? After some digging, I discovered that the folks at www.kindergarten-lessons.com provided all the answers I was searching for.
Measurement should become a significant part of daily life for kindergartners. They should learn through play, exploration, and modeling. Measurement activities ought to be stimulating and fun and should also include the concrete and familiar. In addition, when introducing the concept of measurement to kindergartners, the teacher must make every effort to use appropriate mathematical vocabulary. This may include terms such as: taller than, shorter than, wider than, narrower than, large, small, volume, area, length, weight, time, hour, minute, second, Fahrenheit and Celsius, degree, cold, warm, hot, centimeter, meter, inches, feet, yards, distance, decrease, increase, lighter, heavier, morning, afternoon, night, today, tomorrow, yesterday, day, week, month, year, thermometer, calendar, ruler, clock and balance scale. Instead of saying, “Paul, please pass me the blocks. “, a teacher should say, “Paul, please pass me all of the green octagon blocks.” Making every effort to use descriptive and appropriate terminology will ensure that students will learn by example and become proficient in understanding measurement.
When introducing the topic of measurement, teachers should always allow children to play with the physical objects that will later be used for instruction. This could mean setting out items for play a couple of weeks before a measurement lesson commences. Giving children the time they need for exploration will allow them to become familiar with the objects and give them opportunity to begin to formulate some of their own ideas about measurement. For example, if a teacher is going to introduce the concept of volume, he or she should bring out the water table and a variety of differently-shaped containers for students to use during free-choice time. The water table is a fantastic place to learn about volume because students will see how larger and smaller cups differ in the amount of liquid they can hold.
After the teacher has allowed for a period of play and investigation, he or she can begin more formal instruction by first demonstrating what the activity involves. For example, if the class is to venture into an activity in using non-standard measurement techniques, the teacher should introduce the materials needed and then show the children what is expected. So, if the students are to learn how to measure the width of their chair seats by lining up crayons, the teacher should model the process from start to finish. First, it’s necessary to explain the reason for only using identically sized crayons. “What would happen if we used some crayons that were long and some that were very short? Would our answer be very accurate?” Following this introduction, the teacher should show the students how to line up the crayons end to end, count them out to determine the width of the chair seat, and then articulate, “This chair is 4 crayons wide.” The modeling should not stop there, though. It is important that the teacher walk through the entire measuring process a couple of times so the students understand all of the steps and are then equipped to perform their own measurement investigations later on.
After some experience in measuring concrete objects is gained, kindergartners should be prepared to begin recording their results with a picture. This process can be modeled as well. For example, if the students need to compare the weights of 2 items—a pencil and a stapler, the teacher can bring each of these items to the table, place them on the balance scale, and then show the students how the heavier object will tip the scale downward as the lighter of the two will rise up. On a large chart with a “heavy” and “light” column, the teacher can place pictures of each of the weighed objects in the correct column and then give their justification. Including charts or recording sheets will meld together the children’s’ understanding of concrete objects and abstract symbols. This is known as the picture, representational, or connecting stage.
Although having a solid understanding of measurement through the use of non-standard measuring tools is vital, it is also important for teachers to introduce standard measurement concepts as well. After many opportunities in playing with measurement activities, students should have some exposure with rulers, yardsticks, and basic vocabulary terms such as liter, gram, ounce, inch, and hour.
As a mom of 4, I already have some experience in introducing little ones to important measurement concepts—although I never followed a step-by-step process as is highlighted on this website. We simply explored household objects that were used every day. We organized measuring cups by size, made cookies from scratch using a favorite recipe, practiced pouring sudsy water in different containers at the kitchen sink, and even measured furniture around the house with a piece of yarn. These same kinds of stimulating and useful activities should carry over into the kindergarten classroom because they incorporate recognizable objects for young children and they are easy enough to practice right at home with loved ones. As stated on www.kindergarten-lessons.com/teaching-measurement.html “Use real objects to help children understand measurement concepts.” The teacher’s goal should be to bring in familiar objects that can remain easily accessible for exploration.
When teaching about time, the teacher should discuss concepts such as longer days, shorter days, morning and night, and weekends and weekdays. He or she can also show the class how to keep track of time with an analog clock and also take the time to discuss the parts of the clock’s face. Having the students involved in a daily calendar chart and discussing upcoming holidays or school events in terms of time are also vitally important.
Volume is best explained at the water and/or sand table because students are attracted to pouring, sifting, comparing, and dumping. Here, the kindergarten teacher should provide a variety of tall, short, wide, and narrow containers, along with spoons, shovels, and even medicine droppers for collecting the water or sand. Including a set of measuring cups and spoons would be a perfect addition to the sensory table because these are likely items the children have prior experience with. While students take the time to explore the table, the teacher can observe and ask how the cups compare. “Did the short and wide cup hold the same amount as the tall, thin cup?” “How many medicine droppers are needed to fill the ¼ cup?” “If you want to fill the sand bucket to the top in the shortest amount of time, which one of these shovels would be the best choice—the long and deep shovel or the short and shallow shovel? Can you show me how you came to this conclusion?”
The concept of area can be shown by comparing 2 similarly shaped items that are of unequal size. For example, the students can learn about area by going for a walk around the schoolyard in autumn to search for 2 similarly shaped leaves that are not the same size. The children can then practice determining the area of each of the leaves by filling the surface with Legos, pencil-tip erasers, buttons, beads, or dice. How do the leaves’ areas compare? How many ___ does the larger of the two leaves need to fill the entire space? How many of that same item is required to fill up the smaller of the two leaves? This activity can be altered to tie in with many different themes, holidays, or seasons throughout the year. Providing year-long area explorations will ensure that kindergartners become skillful in this important concept.
Using a daily weather station is an ideal way to explore the concept of temperature in the kindergarten classroom. The teacher can create his or her own chart complete with the days of the week, the date, and the month along the top, and picture cues of the weather choices (cold, warm, and hot) along the margin. Precut colored strips of paper that will fit the columns can be used to record the daily weather. With each new day, a student can measure the length of paper strip to reach the appropriate type of weather and then adhere it to the chart. For hot days, the strip will be long, while warm days will coincide with a medium-sized piece, and cold days will have a short strip of paper. Students should then evaluate any differences in weather from day to day by noting the differences in lengths of paper strips. Was yesterday’s paper strip longer than today’s? How many days were hot? Were there any cold days during the week?
Successfully teaching measurement with familiar and stimulating activities early on is vital for building up a child’s love of mathematics. Through play, exploration, and modeling, students will remain curious, connect the concrete with the abstract, and become skillful enough in using the techniques they’ve learned in kindergarten to remain valuable throughout their lifetimes.
Measurement should become a significant part of daily life for kindergartners. They should learn through play, exploration, and modeling. Measurement activities ought to be stimulating and fun and should also include the concrete and familiar. In addition, when introducing the concept of measurement to kindergartners, the teacher must make every effort to use appropriate mathematical vocabulary. This may include terms such as: taller than, shorter than, wider than, narrower than, large, small, volume, area, length, weight, time, hour, minute, second, Fahrenheit and Celsius, degree, cold, warm, hot, centimeter, meter, inches, feet, yards, distance, decrease, increase, lighter, heavier, morning, afternoon, night, today, tomorrow, yesterday, day, week, month, year, thermometer, calendar, ruler, clock and balance scale. Instead of saying, “Paul, please pass me the blocks. “, a teacher should say, “Paul, please pass me all of the green octagon blocks.” Making every effort to use descriptive and appropriate terminology will ensure that students will learn by example and become proficient in understanding measurement.
When introducing the topic of measurement, teachers should always allow children to play with the physical objects that will later be used for instruction. This could mean setting out items for play a couple of weeks before a measurement lesson commences. Giving children the time they need for exploration will allow them to become familiar with the objects and give them opportunity to begin to formulate some of their own ideas about measurement. For example, if a teacher is going to introduce the concept of volume, he or she should bring out the water table and a variety of differently-shaped containers for students to use during free-choice time. The water table is a fantastic place to learn about volume because students will see how larger and smaller cups differ in the amount of liquid they can hold.
After the teacher has allowed for a period of play and investigation, he or she can begin more formal instruction by first demonstrating what the activity involves. For example, if the class is to venture into an activity in using non-standard measurement techniques, the teacher should introduce the materials needed and then show the children what is expected. So, if the students are to learn how to measure the width of their chair seats by lining up crayons, the teacher should model the process from start to finish. First, it’s necessary to explain the reason for only using identically sized crayons. “What would happen if we used some crayons that were long and some that were very short? Would our answer be very accurate?” Following this introduction, the teacher should show the students how to line up the crayons end to end, count them out to determine the width of the chair seat, and then articulate, “This chair is 4 crayons wide.” The modeling should not stop there, though. It is important that the teacher walk through the entire measuring process a couple of times so the students understand all of the steps and are then equipped to perform their own measurement investigations later on.
After some experience in measuring concrete objects is gained, kindergartners should be prepared to begin recording their results with a picture. This process can be modeled as well. For example, if the students need to compare the weights of 2 items—a pencil and a stapler, the teacher can bring each of these items to the table, place them on the balance scale, and then show the students how the heavier object will tip the scale downward as the lighter of the two will rise up. On a large chart with a “heavy” and “light” column, the teacher can place pictures of each of the weighed objects in the correct column and then give their justification. Including charts or recording sheets will meld together the children’s’ understanding of concrete objects and abstract symbols. This is known as the picture, representational, or connecting stage.
Although having a solid understanding of measurement through the use of non-standard measuring tools is vital, it is also important for teachers to introduce standard measurement concepts as well. After many opportunities in playing with measurement activities, students should have some exposure with rulers, yardsticks, and basic vocabulary terms such as liter, gram, ounce, inch, and hour.
As a mom of 4, I already have some experience in introducing little ones to important measurement concepts—although I never followed a step-by-step process as is highlighted on this website. We simply explored household objects that were used every day. We organized measuring cups by size, made cookies from scratch using a favorite recipe, practiced pouring sudsy water in different containers at the kitchen sink, and even measured furniture around the house with a piece of yarn. These same kinds of stimulating and useful activities should carry over into the kindergarten classroom because they incorporate recognizable objects for young children and they are easy enough to practice right at home with loved ones. As stated on www.kindergarten-lessons.com/teaching-measurement.html “Use real objects to help children understand measurement concepts.” The teacher’s goal should be to bring in familiar objects that can remain easily accessible for exploration.
When teaching about time, the teacher should discuss concepts such as longer days, shorter days, morning and night, and weekends and weekdays. He or she can also show the class how to keep track of time with an analog clock and also take the time to discuss the parts of the clock’s face. Having the students involved in a daily calendar chart and discussing upcoming holidays or school events in terms of time are also vitally important.
Volume is best explained at the water and/or sand table because students are attracted to pouring, sifting, comparing, and dumping. Here, the kindergarten teacher should provide a variety of tall, short, wide, and narrow containers, along with spoons, shovels, and even medicine droppers for collecting the water or sand. Including a set of measuring cups and spoons would be a perfect addition to the sensory table because these are likely items the children have prior experience with. While students take the time to explore the table, the teacher can observe and ask how the cups compare. “Did the short and wide cup hold the same amount as the tall, thin cup?” “How many medicine droppers are needed to fill the ¼ cup?” “If you want to fill the sand bucket to the top in the shortest amount of time, which one of these shovels would be the best choice—the long and deep shovel or the short and shallow shovel? Can you show me how you came to this conclusion?”
The concept of area can be shown by comparing 2 similarly shaped items that are of unequal size. For example, the students can learn about area by going for a walk around the schoolyard in autumn to search for 2 similarly shaped leaves that are not the same size. The children can then practice determining the area of each of the leaves by filling the surface with Legos, pencil-tip erasers, buttons, beads, or dice. How do the leaves’ areas compare? How many ___ does the larger of the two leaves need to fill the entire space? How many of that same item is required to fill up the smaller of the two leaves? This activity can be altered to tie in with many different themes, holidays, or seasons throughout the year. Providing year-long area explorations will ensure that kindergartners become skillful in this important concept.
Using a daily weather station is an ideal way to explore the concept of temperature in the kindergarten classroom. The teacher can create his or her own chart complete with the days of the week, the date, and the month along the top, and picture cues of the weather choices (cold, warm, and hot) along the margin. Precut colored strips of paper that will fit the columns can be used to record the daily weather. With each new day, a student can measure the length of paper strip to reach the appropriate type of weather and then adhere it to the chart. For hot days, the strip will be long, while warm days will coincide with a medium-sized piece, and cold days will have a short strip of paper. Students should then evaluate any differences in weather from day to day by noting the differences in lengths of paper strips. Was yesterday’s paper strip longer than today’s? How many days were hot? Were there any cold days during the week?
Successfully teaching measurement with familiar and stimulating activities early on is vital for building up a child’s love of mathematics. Through play, exploration, and modeling, students will remain curious, connect the concrete with the abstract, and become skillful enough in using the techniques they’ve learned in kindergarten to remain valuable throughout their lifetimes.
Tuesday, July 15, 2008
Is it too late to go Metric?
Until I did a little research on the topic of going metric, I didn't realize how long this has been an issue. However, I do remember now being in math class in high school and the buzz was on whether the U.S. should go metric at that time. It is hard to believe, but that was almost twenty years ago and here we are today and we still have not changed our measuring system. I, personally do not want to see a change because I fear it will be too much work to relearn a new system despite how easy it is. However, I do see how valuable it would be to young people today for the U.S. to adopt the metric system. It seems kind of silly for us to be one of the only three countries that do not use the metric system. The U.S., Liberia and Myanmar are the only three non-metric countries. It appears to me that it would make since for everyone to be on the same page when we are talking about something that requires such accuracy and precision in most measurement circumstances.
If the U.S. were to adopt the metric system it would essentially eliminate the need for conversion factors. We would no longer need to remember that 12 inches are in a foot or 4 quarts to a gallon, all we would need to know is how many spaces to move the decimal point. However, I don't believe that it would be something we could just changed overnight, obviously. If you stop and think how many things are actually measured and rely on measurement, it would take a great deal of time to finalize the adoption. Even things such as industrial machines would need to either be changed or fixed to measure the conversions.
I came across a little article that had an interesting reader's opinion posted at the end of it. It put a twist on the whole idea of the U.S. adopting the metric system. The reader thought that the time has come and gone to adopt the metric system. Since just about everybody has access to calculators and computers there is no need for a base 10 system anymore. He believes that there is just not enough interest or desire now when every problem can be solved using a device other than our own minds. If you are interested, you can read this article at http://news.softpedia.com/news/Why-Didn-039-t-Americans-Adopt-the-Metric-System-57707.shtml. After reading his theory I would have to say I agree. What would be the point now?
If the U.S. were to adopt the metric system it would essentially eliminate the need for conversion factors. We would no longer need to remember that 12 inches are in a foot or 4 quarts to a gallon, all we would need to know is how many spaces to move the decimal point. However, I don't believe that it would be something we could just changed overnight, obviously. If you stop and think how many things are actually measured and rely on measurement, it would take a great deal of time to finalize the adoption. Even things such as industrial machines would need to either be changed or fixed to measure the conversions.
I came across a little article that had an interesting reader's opinion posted at the end of it. It put a twist on the whole idea of the U.S. adopting the metric system. The reader thought that the time has come and gone to adopt the metric system. Since just about everybody has access to calculators and computers there is no need for a base 10 system anymore. He believes that there is just not enough interest or desire now when every problem can be solved using a device other than our own minds. If you are interested, you can read this article at http://news.softpedia.com/news/Why-Didn-039-t-Americans-Adopt-the-Metric-System-57707.shtml. After reading his theory I would have to say I agree. What would be the point now?
Monday, July 7, 2008
Geometer's Sketchpad
While searching for lesson plans this past week I kept coming across the term Geometer’s Sketchpad. I’ve never seen or heard of it before this week. I looked it up online and found that it’s a software program that allows students to investigate mathematical concepts and build objects, such as three-dimensional shapes, figures, graphs, and diagrams. It can be used for a number of grades (3-college) as well as for a number of concepts, such as numbers and operations, geometry for all grades, measurement, calculus, etc. It can also be hooked up to a LCD projector, classroom computer, and SMART Boards.
Basically it’s a tool used to visualize concepts that are difficult. I like that it allows teachers to incorporate technology in the classroom and it seems to be extremely beneficial for students. It’s hard to explain so I downloaded a Sketchpad Workshop Guide and took a chunk off there to try and illustrate some of its concepts for you. Here is one of many examples of what a Geometer’s Sketchpad is about:
Sometimes students will have trouble remembering or understanding a particular theorem. Discovering the theorem themselves or actively exploring its consequences can make a huge difference in their level of recall and understanding. This tour challenges you to discover a theorem. As you solve the challenge, you’ll learn many new Sketchpad features.
When the midpoints of the sides of a quadrilateral are connected, the resulting
shape is always a __________________.
What You Will Learn
• How to construct a polygon using the Segment tool
• How to show an object’s label
• How to measure lengths and angles
• How to create a caption
• How to apply formatting options to text
http://www.dynamicgeometry.com/Instructor_Resources/Evaluation_Edition/Download.html
It then goes into a step-by-step procedure teaching the above concepts. It’s really quite simple and not very time consuming, and I think it’s pretty cool. Check out the above website to learn more about it. It’s really sweet.
Basically it’s a tool used to visualize concepts that are difficult. I like that it allows teachers to incorporate technology in the classroom and it seems to be extremely beneficial for students. It’s hard to explain so I downloaded a Sketchpad Workshop Guide and took a chunk off there to try and illustrate some of its concepts for you. Here is one of many examples of what a Geometer’s Sketchpad is about:
Sometimes students will have trouble remembering or understanding a particular theorem. Discovering the theorem themselves or actively exploring its consequences can make a huge difference in their level of recall and understanding. This tour challenges you to discover a theorem. As you solve the challenge, you’ll learn many new Sketchpad features.
When the midpoints of the sides of a quadrilateral are connected, the resulting
shape is always a __________________.
What You Will Learn
• How to construct a polygon using the Segment tool
• How to show an object’s label
• How to measure lengths and angles
• How to create a caption
• How to apply formatting options to text
http://www.dynamicgeometry.com/Instructor_Resources/Evaluation_Edition/Download.html
It then goes into a step-by-step procedure teaching the above concepts. It’s really quite simple and not very time consuming, and I think it’s pretty cool. Check out the above website to learn more about it. It’s really sweet.
Saturday, July 5, 2008
Victory through the Van Hiele Levels of Geometric Reasoning
Improving the ways we teach young children geometry is something I feel very passionately about. Although this area of mathematics is introduced at a very young age in many of our nation’s schools, countless students never claim victory with a firm grasp of spatial reasoning. Instead, these unfortunate students find solace in constructing a shallow understanding of geometry that’s riddled with holes and dead-ends. Regrettably, I was one of these deficient students. Instead of achieving mastery in geometric reasoning, I claimed only failure and incomprehension. When my classmates were climbing toward a rock-solid understanding of central geometric concepts with past experiences serving as a dependable compass, I was left clinging to the outer boundaries led only by a frail and desperate grasp of the subject matter. When others were filling themselves with the fruitful mathematical skills of spatial concepts, I continually found myself starved, drained, and dreadfully confused.
Geometry and I have never experienced anything but a critically strained relationship. As a result of my past frustrations and failures, I want to do all I can to shield my future students from the evils of a hollow understanding of geometry. I want them to leap over the same hurdles in their mathematical journey that hindered me for so long and then experience geometry in ways that I could have never dreamed possible in my own youth.
With a mastery of geometric understanding, students are equipped with a lifetime of possibilities. Understanding the hierarchical levels of geometric reasoning that every student progresses through is the key to unlocking these possibilities and setting every young boy and girl on a course toward victory. IMAGES, found at http://images.rbs.org , is a website devoted to the connection between understanding geometry and improving the strategies teachers use to instruct, and it is a valuable resource for understanding the process every child must go through to attain geometric mastery.
According to the instrumental work of Dutch educators Pierre van Hiele and Dina van Hiele-Geldof, children progress through 5 levels of special understanding that advance them toward geometric thinking. These 5 levels, labeled the Van Hiele levels of geometric reasoning, are not age-dependent, but hinge upon rich geometric experiences that are developmentally appropriate.
The first level in the Van Hiele model is visualization. Children at this stage are able to recognize and name shapes solely based upon their appearance but cannot pinpoint the properties of the shapes. Likewise, students may see some of the characteristics of the shapes, but do not necessarily use them for sorting or recognition. This level is based upon perception rather than reasoning of geometric shapes.
Level two is analysis. It is at this level that students grow to understand the properties of shapes and some of the related vocabulary that describes the properties. However, students are not yet capable of understanding the relationships between these properties. For example, when a child describes a geometric object, he or she may be able to list properties of the shape, but will not necessarily know which properties are important enough to use.
The third level is based upon informal deduction. Students at this level of maturity are able to recognize relationships between properties and are capable of creating meaningful definitions while forming logical arguments to justify their reasoning.
Next is the deduction level of the Van Hiele model of geometric reasoning. Students passing through this phase do not just identify the characteristics of geometric shapes, but are also equipped to construct proofs that use both postulates and definitions together.
The highest level in the Van Hiele hierarchy is labeled the rigor stage and students who have reached this final stop along their way to geometric reasoning understand the formal aspects of deduction, such as creating and comparing mathematical systems. A college-level geometry course functions at this level of thought and students who have reached the rigor level are capable of using proof by contrapositive as well as indirect proof. Furthermore, students at this level can also understand non-Euclidean systems.
These five levels (visualization, analysis, informal deduction, deduction, and rigor) must be experienced in sequential order if a student is to gain mastery of geometric concepts. Failure to progress through each of the levels will assuredly lead to gaps in spatial comprehension. When teachers understand each of these necessary levels and how to effectively mold their classroom experiences in geometry around each progressive level, students will claim the prize that I relentlessly sought in my younger years: a fruitful and rock-solid relationship with geometry. Victory will be theirs.
Geometry and I have never experienced anything but a critically strained relationship. As a result of my past frustrations and failures, I want to do all I can to shield my future students from the evils of a hollow understanding of geometry. I want them to leap over the same hurdles in their mathematical journey that hindered me for so long and then experience geometry in ways that I could have never dreamed possible in my own youth.
With a mastery of geometric understanding, students are equipped with a lifetime of possibilities. Understanding the hierarchical levels of geometric reasoning that every student progresses through is the key to unlocking these possibilities and setting every young boy and girl on a course toward victory. IMAGES, found at http://images.rbs.org , is a website devoted to the connection between understanding geometry and improving the strategies teachers use to instruct, and it is a valuable resource for understanding the process every child must go through to attain geometric mastery.
According to the instrumental work of Dutch educators Pierre van Hiele and Dina van Hiele-Geldof, children progress through 5 levels of special understanding that advance them toward geometric thinking. These 5 levels, labeled the Van Hiele levels of geometric reasoning, are not age-dependent, but hinge upon rich geometric experiences that are developmentally appropriate.
The first level in the Van Hiele model is visualization. Children at this stage are able to recognize and name shapes solely based upon their appearance but cannot pinpoint the properties of the shapes. Likewise, students may see some of the characteristics of the shapes, but do not necessarily use them for sorting or recognition. This level is based upon perception rather than reasoning of geometric shapes.
Level two is analysis. It is at this level that students grow to understand the properties of shapes and some of the related vocabulary that describes the properties. However, students are not yet capable of understanding the relationships between these properties. For example, when a child describes a geometric object, he or she may be able to list properties of the shape, but will not necessarily know which properties are important enough to use.
The third level is based upon informal deduction. Students at this level of maturity are able to recognize relationships between properties and are capable of creating meaningful definitions while forming logical arguments to justify their reasoning.
Next is the deduction level of the Van Hiele model of geometric reasoning. Students passing through this phase do not just identify the characteristics of geometric shapes, but are also equipped to construct proofs that use both postulates and definitions together.
The highest level in the Van Hiele hierarchy is labeled the rigor stage and students who have reached this final stop along their way to geometric reasoning understand the formal aspects of deduction, such as creating and comparing mathematical systems. A college-level geometry course functions at this level of thought and students who have reached the rigor level are capable of using proof by contrapositive as well as indirect proof. Furthermore, students at this level can also understand non-Euclidean systems.
These five levels (visualization, analysis, informal deduction, deduction, and rigor) must be experienced in sequential order if a student is to gain mastery of geometric concepts. Failure to progress through each of the levels will assuredly lead to gaps in spatial comprehension. When teachers understand each of these necessary levels and how to effectively mold their classroom experiences in geometry around each progressive level, students will claim the prize that I relentlessly sought in my younger years: a fruitful and rock-solid relationship with geometry. Victory will be theirs.
Thursday, July 3, 2008
Golden Ratio
I find the idea of the golden ratio very facinating. Two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approx. 1.6180339887.
The golden ratio was first studied by the ancient greeks because of its frequent appearance in goemetry. The golden ratio can be seen in architecture such as the Acropolis, Parthenon and Great Mosque of Kariouan. It can be seen in works of art by da Vinci (Mona Lisa) and Dali (The Sacrament of the Last Supper). It is also found in music pieces such as Debussy's Image, Reflections in Water and in nature in such things as starfish, animal skeletons and tree branch distribution.
I like this description of it. Adolf Zeising, mathmetician and philosopher, wrote in 1854 " the golden ratio is a universal law in which is contained the ground-principle of all formative striving for beauty and completeness in realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form". (http://en.wikipedia.org/wiki/Golden_ratio)
What a fun concept to teach to students--nature's grand designs and special ratios.
The golden ratio was first studied by the ancient greeks because of its frequent appearance in goemetry. The golden ratio can be seen in architecture such as the Acropolis, Parthenon and Great Mosque of Kariouan. It can be seen in works of art by da Vinci (Mona Lisa) and Dali (The Sacrament of the Last Supper). It is also found in music pieces such as Debussy's Image, Reflections in Water and in nature in such things as starfish, animal skeletons and tree branch distribution.
I like this description of it. Adolf Zeising, mathmetician and philosopher, wrote in 1854 " the golden ratio is a universal law in which is contained the ground-principle of all formative striving for beauty and completeness in realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form". (http://en.wikipedia.org/wiki/Golden_ratio)
What a fun concept to teach to students--nature's grand designs and special ratios.
Wednesday, July 2, 2008
Manipulatives and Geometry
I came upon a study involving the use of manipulatives, especially geometry. After using manipulatives in recent course work, I believed in the learning benefits so I was intrigued by this study.
Studies have varied on the value of using manipulatives but general theories about state that using manipulatives promotes learning as it allows students to physically adapt and interpret their environments. This is called "physically distributed learning". Many believe manipulatives can be even more beneficial in geometry because the manipulatives resemble the geometric shapes that are being studied. Children don't need to translate "the spatial representation to the numeric representation to solve the task".
The study included to experiments; one arithmetic and another geometry. The arithmetic problem involved solving addition problem using a picture of objects and then physical objects. The geometry problem involved making matching triangles using lines on a paper or pieces of pipe cleaner.
The results of the study showed that in both experiments, students benefited from using manipulatives. Of course, the study went in a lot more detail. If you wish to check it out yourselves, go to http://www.scientificjournals.org/journals2007/articles/1073.htm.
Studies have varied on the value of using manipulatives but general theories about state that using manipulatives promotes learning as it allows students to physically adapt and interpret their environments. This is called "physically distributed learning". Many believe manipulatives can be even more beneficial in geometry because the manipulatives resemble the geometric shapes that are being studied. Children don't need to translate "the spatial representation to the numeric representation to solve the task".
The study included to experiments; one arithmetic and another geometry. The arithmetic problem involved solving addition problem using a picture of objects and then physical objects. The geometry problem involved making matching triangles using lines on a paper or pieces of pipe cleaner.
The results of the study showed that in both experiments, students benefited from using manipulatives. Of course, the study went in a lot more detail. If you wish to check it out yourselves, go to http://www.scientificjournals.org/journals2007/articles/1073.htm.
Tuesday, July 1, 2008
When Should We Introduce Geometry?
Our first week assigned reading, "A Coherent Curriculum" by William Schmidt, Richard Houang and Leland Cogan really left me disturbed and confused. I am having a difficult time understanding why the U.S. curriculum is the way it is, basically a big unstructured mess. There were many issues that puzzled me, one of them being the fact that in the U.S. we start introducing Geometry right away in first grade. I suppose the reasoning behind this approach is based on the idea that the earlier you learn a skill the easier it will be and therefore the student will be able to build upon that skill with more complex concepts. This method may be effective, however, it is difficult to measure being that the U.S. curriculum is the way it is. The article refers to it as being a mile wide and an inch deep, referring to the vast array of concepts introduced and the little time or depth spent on each one. This is clearly visible when comparing the two figures provided in the article. The "A+" countries figure is so streamlined in contrast to the U.S. figure.
The article states that the "A+" countries wait until seventh grade to introduce Geometry. That is a significant difference when compared to the U.S. I would imagine the idea behind this approach is to give the students the time and opportunity to truly master the basic skills so that they are using these skills much more efficiently and effectively when they begin to do and learn more abstract mathematics. This approach is obviously working being that the "A+" countries are producing such "A+" results.
Now I know it isn't as easy as flipping a switch, however, if these "A+" countries have such outstanding results then why doesn't the U.S. at least try to adopt this idea of a solid structure based curriculum? I believe that in order to understand a complex subject such as Geometry, a student definitely needs to have this solid base of arithmetic skills mastered to be successful.
The article states that the "A+" countries wait until seventh grade to introduce Geometry. That is a significant difference when compared to the U.S. I would imagine the idea behind this approach is to give the students the time and opportunity to truly master the basic skills so that they are using these skills much more efficiently and effectively when they begin to do and learn more abstract mathematics. This approach is obviously working being that the "A+" countries are producing such "A+" results.
Now I know it isn't as easy as flipping a switch, however, if these "A+" countries have such outstanding results then why doesn't the U.S. at least try to adopt this idea of a solid structure based curriculum? I believe that in order to understand a complex subject such as Geometry, a student definitely needs to have this solid base of arithmetic skills mastered to be successful.
Thursday, June 26, 2008
Extending Math Beyond the Classroom
Do kids like math? I don’t know, but I think many kids are turned off by math because they find it confusing and not related to real life situations. Math needs to be seen as a way of making sense of things; it needs to extent to other areas of life and not just left in the classroom. Another reason I think kids are turned off by math is that they are afraid of getting the answer wrong. Math isn’t art where anything goes, it’s not language arts with open-ended questions, or science- where it’s okay to predict the wrong outcome. Math has a formula; it isn’t always clear.
There are a number of ways to engage students in math that is motivating, sensible, and extends beyond the classroom. Below are some ideas for parents, and as future teachers, it’s a good way to get parents involved in the classroom. These ideas can also be modified for the classroom.
· When measuring for cooking or baking, have your kids make the measures for you. Once they get used to the whole and fractional measures, limit what they can use so that they have to reason how to get the measure they want. (For example, measure a cup of flour using only the ¼ or 1/3 cup measures.)
Good way to integrate home economics and math
· Use colored candies to teach fractions. Ask how many are a certain color and what fraction of the whole batch that color represents.
· Estimation is a powerful tool in mathematics. The next time you are traveling, have your kids estimate when you have traveled a mile, then verify it with the car's odometer. At the restaurant, challenge them to estimate the total bill.
This can also be done on field trips.
· Talk about the shapes of 'stop' signs, 'yield' signs, 'mileage' markers, designs on buildings or sidewalks, or even the lug nut on car wheels or fire hydrants.
Maybe you can get the bus driver involved.
· Take apart cereal boxes to see how the 3-D shape is made from 2-D shapes.
Ideas taken from: http://www.pbs.org/parents/experts/archive/2005/04/sharing-everyday-math-experien.html
This website has a number of other excellent ideas for extending math to other areas of life. Take a look. :)
There are a number of ways to engage students in math that is motivating, sensible, and extends beyond the classroom. Below are some ideas for parents, and as future teachers, it’s a good way to get parents involved in the classroom. These ideas can also be modified for the classroom.
· When measuring for cooking or baking, have your kids make the measures for you. Once they get used to the whole and fractional measures, limit what they can use so that they have to reason how to get the measure they want. (For example, measure a cup of flour using only the ¼ or 1/3 cup measures.)
Good way to integrate home economics and math
· Use colored candies to teach fractions. Ask how many are a certain color and what fraction of the whole batch that color represents.
· Estimation is a powerful tool in mathematics. The next time you are traveling, have your kids estimate when you have traveled a mile, then verify it with the car's odometer. At the restaurant, challenge them to estimate the total bill.
This can also be done on field trips.
· Talk about the shapes of 'stop' signs, 'yield' signs, 'mileage' markers, designs on buildings or sidewalks, or even the lug nut on car wheels or fire hydrants.
Maybe you can get the bus driver involved.
· Take apart cereal boxes to see how the 3-D shape is made from 2-D shapes.
Ideas taken from: http://www.pbs.org/parents/experts/archive/2005/04/sharing-everyday-math-experien.html
This website has a number of other excellent ideas for extending math to other areas of life. Take a look. :)
Saturday, June 21, 2008
Discovering the Hidden Treasure
It's interesting how my entire outlook on mathematics has transformed over the course of my lifetime. What was once a debilitating fear has slowly become somewhat of a passion. I now have a desire to seek out a greater depth of knowledge so that my inabilities and inhibitions will someday become strong skills and confidence. This new found passion of mine has been born and bred from a desire to help others so that they never experience the mathematical voids that I have faced during my lifetime, and it is my hope that the lessons I have learned thus far will help my future students in their own quest for mathematical understanding.
I have noticed this passion of mine reaching into all aspects of my life. For instance, I now spend my precious free time skimming countless pages in search of exciting mathematical concepts and activities that I can collect in my own treasure chest of mathematical ideas. This cache has quickly become heaped full of exciting resources that will surely be useful for me in the near future. During one of my early morning quests for mathematical treasures, I came across a profound website that is worthy of a peek or two. It is www.livingmath.net and is brimming with wealth.
Julie Brennan, the creator of Living Math and homeschooling mom of 4, bases her instruction on the foundation of natural math. Natural math is about seeing math in every way and every situation. It provides the bridge that spans the gap between "math" and all other areas of life and school. With natural math, concepts are no longer isolated islands in the sapphire sea. Rather, each concept is connected with one's real life and the tangible objects that are all around. Students come to understand mathematics more fully and deeply because the math is naturally a part of their individual lives. Natural math is a wonder to behold: a true treasure buried just under the surface.
Living Math incorporates a vast assortment of ideas into everyday objects and situations. With Living Math, students can see how to uncover the mysteries of geometry, algebra, calculus, arithmetic, and countless other necessary concepts important in daily life. Through the guidance of Living Math, with natural math as its foundation, math shows itself to be something more than abstract formulas and confusing thoughts found within the dusty pages of a hefty textbook. It is a living, breathing entity that lies hidden in nearly every aspect of our lives. Math can easily be found in our home, the store, the toys we play with, the books we read, and even in our other subjects such as art, social studies, music, language arts, and physical education. With the ideas behind Living Math, students will come to understand and appreciate the richness and beauty of mathematics because it can be found in every daily experience.
Natural math requires an appreciation for the hidden concepts all around us along with a nugget or two of creativity. As teachers and parents, we must first understand and acknowledge the presence of mathematics that envelopes us. We can uncover it just about anywhere if we know where to dig. It's in our paintings, our classical music that streams from the radio, the storybooks on the shelves of our library, and in the activities we choose to do in our free time. Stop and look all around. Investigate the surroundings. Where can we find the hidden treasures of mathematical concepts that await us? They are there if we only step back and use some imagination in our quest.
On www.livingmath.net, Julie explains the idea of "Strewing", which is to inundate an environment with objects, materials, and resources that are rich in mathematical concepts. Some valuable suggestions include puzzles, art supplies, activity books, video and audio tapes, such as the Schoolhouse Rock series, measuring cups and spoons, recipe books, tape measures, board games, Legos and dominoes, and mathematically rich textbooks and storybooks. Providing these resources within easy reach for children will ensure that they are wholly involved with mathematics and outfitted for personal discovery and investigation.
The ideas behind natural math and the work of Julie Brennan in Living Math are exciting for me as I seek jewels to add to my own mathematics treasure chest. When children are shown the math in their real life in very tangible ways, they will uncover the gems of mathematical enlightenment that are buried just under the surface. They will become confident and appreciative of the beauty that surrounds them everyday, and that, I believe, is truly a priceless treasure to seize.
I have noticed this passion of mine reaching into all aspects of my life. For instance, I now spend my precious free time skimming countless pages in search of exciting mathematical concepts and activities that I can collect in my own treasure chest of mathematical ideas. This cache has quickly become heaped full of exciting resources that will surely be useful for me in the near future. During one of my early morning quests for mathematical treasures, I came across a profound website that is worthy of a peek or two. It is www.livingmath.net and is brimming with wealth.
Julie Brennan, the creator of Living Math and homeschooling mom of 4, bases her instruction on the foundation of natural math. Natural math is about seeing math in every way and every situation. It provides the bridge that spans the gap between "math" and all other areas of life and school. With natural math, concepts are no longer isolated islands in the sapphire sea. Rather, each concept is connected with one's real life and the tangible objects that are all around. Students come to understand mathematics more fully and deeply because the math is naturally a part of their individual lives. Natural math is a wonder to behold: a true treasure buried just under the surface.
Living Math incorporates a vast assortment of ideas into everyday objects and situations. With Living Math, students can see how to uncover the mysteries of geometry, algebra, calculus, arithmetic, and countless other necessary concepts important in daily life. Through the guidance of Living Math, with natural math as its foundation, math shows itself to be something more than abstract formulas and confusing thoughts found within the dusty pages of a hefty textbook. It is a living, breathing entity that lies hidden in nearly every aspect of our lives. Math can easily be found in our home, the store, the toys we play with, the books we read, and even in our other subjects such as art, social studies, music, language arts, and physical education. With the ideas behind Living Math, students will come to understand and appreciate the richness and beauty of mathematics because it can be found in every daily experience.
Natural math requires an appreciation for the hidden concepts all around us along with a nugget or two of creativity. As teachers and parents, we must first understand and acknowledge the presence of mathematics that envelopes us. We can uncover it just about anywhere if we know where to dig. It's in our paintings, our classical music that streams from the radio, the storybooks on the shelves of our library, and in the activities we choose to do in our free time. Stop and look all around. Investigate the surroundings. Where can we find the hidden treasures of mathematical concepts that await us? They are there if we only step back and use some imagination in our quest.
On www.livingmath.net, Julie explains the idea of "Strewing", which is to inundate an environment with objects, materials, and resources that are rich in mathematical concepts. Some valuable suggestions include puzzles, art supplies, activity books, video and audio tapes, such as the Schoolhouse Rock series, measuring cups and spoons, recipe books, tape measures, board games, Legos and dominoes, and mathematically rich textbooks and storybooks. Providing these resources within easy reach for children will ensure that they are wholly involved with mathematics and outfitted for personal discovery and investigation.
The ideas behind natural math and the work of Julie Brennan in Living Math are exciting for me as I seek jewels to add to my own mathematics treasure chest. When children are shown the math in their real life in very tangible ways, they will uncover the gems of mathematical enlightenment that are buried just under the surface. They will become confident and appreciative of the beauty that surrounds them everyday, and that, I believe, is truly a priceless treasure to seize.
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