Technology in the Classroom

     Through this assignment I have seen how beneficial technology can be in the classroom. From my own postings, and reading the blogs of other teachers and students, like Keeping Up With Trends, I have seen how much information can be relayed to a wide span of people. Through a  blog like this one, teachers can keep students and their families up on all the classroom news, homework, and other topics related to the classroom, but the best thing about blogs is the pictures, videos, and links! They are easy to put in and so helpful when trying to explain a topic, or just fun and entertaining.

     Implementing technology into the classroom through projects, visuals, and using tools like smart boards and ladibugs help teachers and students get the most out of their classroom. I have to admit that I am, for the most part, technologically unsophisticated, but I have learned a lot about the tools and resources that are available and how easy they are to use and how beneficial they can be. I definitely plan to incorporate a blog like this into my classroom, and I'm sure I will pick up plenty more tools and resources over the course of this semester.

Notation Styles of Numbers

      Numbers have many different forms in which they can be written, but no matter what the form, the number itself stays the same. Numbers can be written in standard form, like 3,491 or 27, they can be written in word form, like three thousand four hundred ninety one or twenty seven. Numbers can also be written in expanded notation, like 3000+400+90+1 or 20+7, or numbers can be written showing their place values, like 3 thousands, 4 hundreds, 9 tens and 1 one. This website, What's Your Name?, is a great online exercise for students mastering changing notation styles of numbers.

     Notation of numbers is very important for students to learn at an early age because it is crucial for students to understand that the number sequence of 427 really means four hundred twenty seven. Notation is also crucial when that student starts taking science and algebra later. If students learn and practice the concepts from an early age they will have no problems expanding on them later.

The Diversity of Algorithms

     Algorithms are the short-cuts and tricks we use to complete operations in math quickly, and often without even thinking about it because we have done them so many times. The National Council of Teachers of Mathematics said in 1996 that, "Algorithms are generalizations that embody one of the main reasons for studying mathematics- to find ways of solving classes of problems. When we know an algorithm, we complete not just one task but all tasks of a particular kind... The power of an algorithm derives from the breadth of its applicability." We have been using algorithms from the time we started learning math, and often don't even realize that there are many ways to do a problem because we are so used to taking the same steps. As educators it is very important for us to remember that there are so many different algorithms out there, and they are very difficult to learn and perfect, and that not all of our students were taught the same algorithm the same way.
     In most American classrooms children are taught subtraction using what we like to call the Standard Subtraction Algorithm. This algorithm uses a method of borrowing tens if necessary to always have a larger number as the minuend (top portion of a subtraction problem) and a smaller number as the subtrahend (bottom portion) in each place value. For example lets look at this problem...

In this case we are subtracting 32-15. The Standard Algorithm begins work with with the column that is furthest to the right, which is in this case the ones column. If we were to only look at the ones column as its own problem we would wind up with a negative number because the minuend is smaller than the subtrahend; however, since we are not only working with the ones column, we borrow a ten, literally. We take one ten out of our tens column, turning 3 into 2, and add that ten to the ones column turning 2 into 12. Now we simple subtract down each column. 12-5=7, 2-1=1 to wind up with 17.

     Now if we were to do this same problem using the Austrian Algorithm, the technique that most European students learn when first learning how to subtract,  we will do things a little differently, but still wind up with the same answer. With this technique, when we borrow from the tens column, instead of decreasing the value of the minuend in the tens column by one, we increase the subtrahend in the tens column by one.

With this algorithm, the one that is written in the center of the problem serves as the +1 for the subtrahend of the tens column, and the +10 for the minuend of the ones column. After placing the one in the middle you complete the problem by subtracting the subtrahend of the ones column from the new minuend of the ones column, 12-5=7, then subtracting the new subtrahend of the tens column from the new minuend of the tens column, 3-2=1. Then you put the numbers in their appropriate place values to make your final answer 17.

     Using unfamiliar algorithms can be very frustrating at first, but once you start to feel comfortable with one, you can see the similarities between them. This website, Alternative Subtraction Algorithms, breaks down several different algorithms, including the Standard Subtraction Algorithm, and the Austrian Algorithm and shows how both work using different visualizations of the problems, and is very helpful when comparing algorithms.

Word Problems: Friend or Foe?

     Word problems can be very daunting, you are supposed to be doing math, but instead of numbers all you see are words. You must determine relevant information out of a paragraph, figure out what function you will be using (addition, subtraction, multiplication, etc), and then answer the question using whatever tools necessary. Here is an example of a "Unite Sets" word problem that I made, where you start with 2 separate sets, determine what to do with them, and end up with one set.

When working with children, as a teacher you must get them involved in the story of the problem, so that they can picture every aspect of the story in their head, and asking leading questions is a great way to get them focused an involved. Some good questions to ask about this problem would be, Who are the characters in the problem? What are they doing? Can you draw a picture of Jen and Maddie picking apples? Asking general questions about the problem helps younger children stay focused and visualize the problem, but with older children who are more accustomed to word problems you can skip this step and move on to examining the numbers. In this stage of the problem solving process you must get the student to pick out the relevant information. In this problem, ask the students how many apples Maddie picked and how many apples Jen picked. Ask the student to draw a picture of Maddie with her apples and Jen with her apples. This step allows the student to take what they have visualized and now see groups of numbers. 

     Next we have to solve to problem. Ask the student what happened to Maddie and Jen's apples? Where did they put the apples? This question leads the student into the addition part of the problem; because the problem never actually says to add or subtract or multiply, it is necessary for the student to determine this themselves. Thats the trickiest part of word problems for most students, which is why visualizations and manipulative are so helpful in these situations because they allow the student to see and to the function, then recognize it. In this particular problem, Jen and Maddie put all of their their apples together in a basket. Once the students know that put them all TOGETHER, they know the function necessary is to add. Once the student knows what to do they can still come up with the answer to the problem. Now all thats left to do is write out the problem and the solution as a numerical expression, in this case 4+3=7.

     Word problems seem time consuming, tedious, and tricky, but they are very beneficial to students because they teach students the concepts behind the functions and also how to determine what information is relevant. The National Council of Teachers of Mathematics said in 2000 that, “Problem solving is the cornerstone of school mathematics. Unless students can solve problems, the facts, concepts, and procedures they know are of little use.” And like so many other valuable skills, practice makes perfect. Here's a link to Thinking Blocks, an interactive tool for students to practice word problems using different functions. Check it out and good luck!

Base Blocks are Amazing!

     In our numeric system, we use base ten to count and as our written representation of numbers, however not all cultures use this system. People can count in base four, base two, base five, base twelve, ect. but a great was of representing, teaching, and visualizing these alternative systems, and our own base ten system, is with base blocks. Base blocks are a manipulate used to organize units to create a whole number and can be used in a variety of base blacks. They consist of at least three different types of blocks, a unit, a rod, and a flat. In base ten which we are most familiar with, a unit represents one, a rod represents 10, and a flat represents a hundred. For example, this picture shows the numbers 77 and 145 in base ten blocks:

Base ten seems pretty simple to us as people using the system everyday, maybe without knowing it, but to break it down, lets look at the number 145. In this case lets start out imagining 145 single units, the smallest squares, which each represent one. Now if we simply had a bunch of units sitting there together it would take a while to count them all up to figure out how many we have total, so instead we will group them into sets of 10  (base 10, get it?). Now we have 14 sets of ten (rods) and 5 remaining units. Since we are using base ten, can we make any more groups of ten? Yes! We can make one group of ten out of our rods to make one group of what would be 100 individual units (a flat). When you lay our 1 flat, 4 remaining rods, and 5 remaining units together, you still have 145 units in separate groups, and them we can write it out as 145, where the 1 represents the number of flats, the 4 represents the number of rods, and the five represents the number of units. 1-4-5.

     When each step to our base ten system is explained, it sounds pretty complicated, but once you have the system down, you can do the same thing in any base! For example, lets try base four using the same number. if we have 145 single units, how many groups of four can we put them into? We can make them into 36 rods, with one unit remaining. Ok, now how many groups of 4 can we put those 36 rods into? 9! So now we have 9 flats, 0 rods, and one unit. In this case we are going to need a block that is bigger than a flat, so we have our cubes. Now how many groups of 4 can we group our flats into? 2, with one flat remaining. So our final base blocks are going to be 2 cubes, 1 flat, 0 rods, and 1 unit. So if we were to describe the number we like to call 145 in base four instead of base ten it would be 2101!

     Switching bases can be pretty confusing at first, but I found this great online tool, Base Blocks, where you can have virtual base blocks of any number and it will even give you problems to practice on!

Patterns and Inductive Reasoning

     We see patterns every day in all aspects of our lives everyday, from the woven fabric of your clothes, to the street-lights, to the aisles at the grocery story. While patterns often hold together parts of our daily lives, they are also the essential keystone to mathematics. Dictionary.com defines the word "pattern" as, "a combination of qualities, acts, tendencies, etc., forming aconsistent or characteristic arrangement." Without patterns, we would have no idea how to solve even the most basic problems because we would not have the ability to carry the knowledge from one problem to another.
     Inductive reasoning is the tool we use to transfer patterns between different situations by making generalizations about a certain pattern, then recognizing those generalizations in another problem, then applying and modifying techniques in order to solve the problem. A good example of the inductive reasoning process can be seen in this worksheet, Number Patterns Worksheet. This worksheet uses skip counting, or arithmetic sequences,  starting with counting by doubling to establish a pattern of counting by other numbers than one. It gets progressively more difficult and incorporates subtraction. This worksheet is a great example of both patterns and showing the inductive reasoning process because students have to assess the problem, determine the pattern, then move on to a more difficult kind of pattern using the same techniques.
     Patterns are an essential part of mathematics at all levels, and teaching kids how to observe patterns and recognize their differences is a tool that will help them succeed in all levels of math, as well as other subjects.