On several occasions, I have heard about how educational initiatives are like a pendulum. Schools go from one initiative to another, just to have it come back again several years later. Why does this happen?
The first thought I have is about how schools claim to use research based practices. This is one of the first fallacies because research is not on a pendulum. Research is supposed to build on previous research. For example, often researchers in the social sciences will use quantitative research is meant to check the validity of claims done with qualitative research. Many journals are refereed, so people read over them and make sure that the methods are sound and information is accurate.
If research is not like a pendulum, why would people in education feel like initiatives are like it? Many schools and districts do not stay with initiatives long enough to see them through. If they are not given enough time, schools cannot tell if the initiatives are being effective or not. It also makes teachers feel like they are bouncing from one initiative to another. If the teachers do not get to see the initiative come into fruition, then they will believe it did not work, which leads to teachers believe that an initiative that seems like the opposite should be followed.
Some school leaders agree and say that initiatives are like a pendulum. Leaders are the ones that have the opportunity to shape initiatives. Granted many principals must follow directions from assistant superintendents and superintendents, but school districts need to be able to find a way to truly practice research based initiatives. To be able to have research based initiatives shows there needs to be people that are well versed in the research.
Another problem with initiatives is that many people that implement professional development, do not help make connections between previous initiatives and current initiatives. This gives the illusion that initiatives are disconnected and that schools are bouncing from one initiative to another. It also makes staff members feel like before one initiative is mastered, the school is focusing on another.
Initiatives in school should not seem like they are pendulous. The ways many schools approach research and new initiatives helps lead people in education to view initiatives as disconnected and going back and forth from one to another. When leaders view initiatives in the same manner, it is discouraging since leaders are the ones that can help the faculty make connections between initiatives and can make sure initiatives continue to grow throughout initiatives.
This year Guy 2 decided to start the year with Data and Statistics. Students will have the opportunity to gather data throughout the year and all of the other units throughout the year will relate back to this initial unit. This will make the math during the year more connected and less disjointed that it may seem if each unit was taught in isolation. We challenge you this year to help students make more meaning out of math and have students see that math is more than just computation. Have a great year!
Geometry is often treated as shape recognition in elementary school classrooms. It is a separate unit often put off until the end of the year, squeezed in at the last minute right before state testing. Further, I feel that state standards (at least in Virginia) are partially to blame. For example, in fifth grade, students are expected to know the definition of circumference, chord, diameter, and radius. That's it for circles, four definitions. When they are assessed, they simply have to recognize which of these is labeled in the picture. Geometry is so much more than labeling shapes.
Geometry is the measurement of the world around us, and how things fit together. It also provides visual constructs for making sense of complex mathematical concepts. In Euclid's Elements, he wrote entire sections on algebra, and many have the explanations rooted in geometry. Subtraction is not only having 7 objects, removing 3 of them and counting how many are left. Subtraction is also the distance on a number line between 7 and 3.
When I teach, I try and use geometry to help students explore and understand the mathematics. Number lines are used to explore computation. Area models are used for exploring fraction concepts as well as algebraic properties such as the commutative, associative, and distributive properties. Geometry can help students understand why "completing the square" is a way to solve quadratic equations. Geometry can help a 5th grader use calculus to understand how we are able to find the area of a circle when we do so by filling it with squares.
In my ideal classroom, mathematics would start with the explorations of geometry and from the study of shapes and figures comes the discover of computation.
As the temperature starts to increase, many people will be doing activities outside such as jogging, hiking, and playing other various sports. Many people that participate in these activities buy shirts that are supposed to keep them cooler. Do these shirts soak up more or less sweat than normal cotton shirts? Take a shirt of each type and soak them in water. Pull the shirts out of the water and let the water drip off the shirt until it stops dripping without wringing it out. Then wring the shirts out and measure how much water comes out of them. Which shirt holds more water?
By the time it becomes mid March, many people are ready for spring. The United States has a tradition where a groundhog tries to predict how much longer winter will last, but exactly how long is winter? Is it the same as the other seasons? How does Daylight Savings Time factor into how long winter is? How does February having less days factor into it?
There have been some studies that indicate that the average house collects about 40 pounds of dirt in a year. How much dust would a house average in a day? How much dust would a town of 75 houses collect each day? In a year?
What better way to celebrate World's Maths Day than with a math problem? March used to be the first month of the year. In 1752, many calendars changed to January being the first month. If you could rearrange the calendar, anyway you want, how would you do it? Things to consider. The number of hours in a day and the number of days in a year should not change since they are based on Earth's movements. Should each season get the same amount of days? Is there a reason why the seasons start when they do? Should all months have similar amounts of days? Why is the way you created the best?
In my class, we tackled a stack of fraction cards, and put them in order from least to greatest. Many teachers show their students number tricks to quickly determine which of two fractions are greater:
(A shortcut would be to multiply the 6 and 7 to get 42. Then multiply the 8 and 5 to get 40. Write these numbers above each fraction, the greater number is above the greater fraction.)
While the example above is how I compare fractions as an adult, I also understand conceptually what I am doing and why it works. Most students don’t have this conceptual understanding so after they learn a “trick,” it’s quickly forgotten. I believe it’s important for students to understand the concept behind a trick, and then they can often apply the trick to new situations, such as algebraic concepts that involve fractions
Therefore, to build student conceptual understanding, here is the process we followed as a class to compare fractions such as 5/6 and 7/8.
First, we create a rectangle using the denominators of the two fractions as the length and width. For this problem, we create a 6 by 8 rectangle:
Next, after drawing two of the same size rectangles, we shade the first rectangle to illustrate the first fraction (5/6) and of then we shade the other rectangle to illustrate the second fraction (7/8):
Finally, we count the number of squares inside the bigger rectangle that are shaded for each fraction. The fraction with more squares shaded is the greater fraction.
In essence, we use geometry to find the common denominator. At first, I don’t worry about getting the least common denominator, but instead focus on providing a method that will work for any two fractions. It is important that students make solid connections between geometry and numerical process whenever possible. This visual method helps provide meaning when students perform more abstract numerical techniques for finding the least common denominators or LCM. Further, the practice of making rectangles (an area model for fractions) will help students in later lessons when learning to add and subtract fractions.
What methods do you use to help build a conceptual understanding for comparing fractions?
This December I received a Kindle Fire and purchased a few electronic books. On one hand, the ebooks are really cool. I have purchased an electronic version of a book that will last forever. The pages will not tear, I can’t spill coffee on my book (the Kindle is a different story), and my family can all read the same book at the same time. However, it is hard to give up holding a good old, hardback book. I am still adjusting to the electronic books and still prefer vintage bound pages.
My biggest complaint of electronic books on ereaders is the lack of page numbers. It is important for me to know how many pages are in the book, when I have reached the 100th page, when I am half way through the book, and how many pages I have left. When reading an ebook, the page numbers are gone. They have been replaced with percentages. So I no longer can mark my milestones in a book with page numbers, I now have to mark them with percentages.
Later, I am teaching my class about fractions, percentages and decimals and I make a connection to the percentage at the bottom of my Kindle. What a great, real world application of mathematics. Here are some thoughts I have on how to apply my Kindle percentages to math class:
Concept: Understand that percentage is a portion of a whole.
Question: How is 50% of War and Peace similar and different to 50% of Green Eggs and Ham?
Concept: Percentages are multiplied to determine portions.
Question: How many pages have I read in a 765-page book when my Kindle says I have read 34%?
Concept: Divide the part by the whole to determine a percentage.
Question: What percentage of a book have I read, if I have 100 pages left in a 456-page book?
“Fractions!” This one word strikes fear in many a young mathematician’s mind. Such a simple concept, yet few things cause so much confusion to a young mathematician. Their world is turned upside down as a 6 is no longer greater than a 3. We use whole numbers to represent parts of a whole:
1/6 < 1/3
When the whole numbers are in the denominators, the "rules" for greater and less than are reversed.
Further, when we add fractions, the old rules don’t apply:
1/6 + 1/3 isn't 2/9
Fractions are different. Fractions are hard. Fractions are real. We do not live in a world where numbers stay nice and whole; they get messy. So what are we to do? We need to make sure that students have a strong conceptual understanding of fractions and how they work in mathematical operations. The concepts of addition, subtraction, multiplication and division do not change when working with rational numbers. When teachers rely on numerical “tricks” and manipulations to teach fraction concepts, students are left with a confusing stew of math symbols that have no meaning. It is important for young mathematicians to explore fractions concepts to develop deep understanding of how rational numbers fit into their understanding of whole numbers.
When learning fractions concepts, teachers need to slow down and give students time to develop their own understandings. They should play around with many different models and figure out how operations work with these models. There are many math kits that help kids explore fraction concepts, but teachers do not have to buy commercially prepared materials. Egg cartons, clocks, number lines, rulers, candy bars, blocks, marbles, money and beans are a few examples of readily available tools that can help students explore fractions. These representations can help students explore fractions as part of a set, part of a region, and part of a linear length. Giving students the opportunity to explore each of these different representations is important for helping developing minds work through fraction concepts showing them different ways fractions are used in the real world.
In closing, when struggling to comprehend fractions concepts, students need to be given the time and the tools to explore. They need to build on their understandings of math operations when working with fractions. Reliance on numerical fraction tricks will only confuse and stunt the growth of young mathematical minds.