Measuring in a Different Way

In class last night, we measured without a ruler or yardstick. We took our hands, feet, forearm, fingers, and arms to measure objects in the room. While several measurements were close (those that chose the same form of measuremen), they were not exactly the same and the measurements were not exact. This is an interesting approach when teaching the importance of proper measurements. The thought processes behind the differences of proper and improper measurement. Below is a link where a teacher has his students completing this activity.



Measuring with Non--Standard Units

A New Way to See Shapes

Creating Stories

This is a great little video (click the header above) about a different way to use Tanagrams. Usually, these objects are used with seeing shapes differently, how shapes can be made, what fits in an area, etc.

In this video, the students are creating their own mythological story. The students need to keep along the lines of Greek mythology, but other than that, they can make up their creature (it can be a cross between two creatures, or a new god) and create it with Tanagrams. This is seeing shapes differently in a more fun and engaging way. I love this idea.

Practical Tips for Fractions

Learning to Teach

Doug M. Clarke, Anne Roche, and Annie Mitchell wrote the article, "10 Practical Tips for Making Fractions Come Alive and Make Sense." This is an article that can come in use for both new and seasoned teachers. As new teachers, we do not know everything that is out there. We have not experienced much, and our first couple of years are spent learning what and how we need to teach. Seasoned teachers have experienced a lot. These teachers have their groove and feel that they know best. In a lot of ways they do, however, even seasoned teachers can update their teaching methods. The information that we teach is always changing, why shouldn't the way we teach?

10 Tips

1. Give a greater emphasis to the meaning of fractions than on procedures for manipulating them--We need to not worry about "meeting the standards" as much as we need to worry about whether our students are truly understanding what fractions are, what their purpose is, and how to use them.
2. Develop a generalizable rule for explaining the numerator and denominator of a fraction--We introduce telling students that the denominator represents the "whole" and the numerator is the equal parts that fit in. While this may work for smaller fractions that are less that 1, it does not work for fractions that are larger than 1. Instead, the new definition that is suggested is "In the fraction a/b, b is the name or size of the part (e.g., fifths have this name because 5 equal parts can fill a whole), and a is the number of parts of that name or size. If we have 7/3, the 3 tells the name or size of the parts (thirds) and the 7 tells us that we have 7 of those thirds (or 2 1/3)."
3. Emphasize that fractions are numbers, making extensive use of number lines in representing fractions and decimals--number lines allow students to see all the variations that fractions can take, and see how fractions can compare to decimals and whole numbers.
4. Take opportunities early to focus on improper fractions and equivalences--If students are understanding fractions, they can move on to improper fractions. The students who are struggling can sometimes pick up faster by playing games and watching their classmates.
5. Provide a variety of models to represent fractions--Manipulatives, manipulatives, manipulatives.
6. Link fractions to key benchmarks, and encourage estimation--This helps students to learn residual thinking and understand the meaning behind fractions.
7. Give emphasis to fractions as division--I look at figuring out fractions by division sometimes.
8. Link fractions, decimals, and percents wherever possible--These are in everything: clothes sales, gas prices, etc. Linking these items to as much stuff as possible lets kids know that it is something that is important in their life.
9. Take the opportunity to interview several students one on one on the kinds of tasks discussed in this article to gain awareness of their thinking and strategies--Always keep communication open with your students, not only with math strategies (new and old), but with learning strategies for every subject.
10. Look for examples and activities that can engage students in thinking about fractions in particular and rational number ideas in general--Keep it interesting, but, informational.

Manipulatives

How kids can use manipulatives.

https://www.teachingchannel.org/videos/teaching-fractions

Take a look at this video above. Fran Dickinson does a good job explaining how beneficial manipulatives can be. Manipulatives can be used to break down a whole and see the little parts that make it up.

Dickinson taught his lesson in a very confident format, which helped his students have confidence. Fractions are a topic that most people like to run from, including teachers. When we have this aura about us on a certain topic students can sense it. We need to be comfortable with teaching this category so that we can exude that comfort onto the students.

Technology Manipulatives

• ThinkCentral

Why Use Manipulatives?

Why should we have kids use manipulatives?

We do this because it allows students to VISUALIZE a problem and see multiple ways that the problem can be solved. If a student is confused about how fractions work, manipulatives break down the fraction so that they can see all the parts. Fractions are part of a whole, manipulatives can show students all the part that make it up.

13 Rules that Expire

Expiring Words?

The deeper I get into the education field, the more I realize that I have more to learn. Learning how to teach new math concepts is actually a little confusing for me. I grew up with it one way. Now that I'm an adult and about to teach myself, I'm learning that they way I learned is actually not the best. Education is an ever growing and ever changing field.

13 Rules

A breakdown of the 13 Rules:

1. When you multiply a number by ten, just add a zero to the end of the number---while this is true quite often, and can simplify many problems, this particular rule is talking about exceptions, like with fractions/decimals. This is definitely not a rule we can follow when addressing these problems.
2. Use keywords to solve word problems---students need to be sure to look at the entire problems. Key words can be helpful, but they must pay attention to the entire problem.
3. You cannot take a bigger number from a smaller number---kids learn later in their education that this is actually true. So while we teach whole number subtraction in elementary school, we cannot make a blanket statement like this.
4. Addition and multiplication make numbers bigger---This is another thing that coincides with decimals and fractions. Multiplication and addition may just do the reverse.
5. Subtraction and division make numbers smaller---This is not true when dividing negatives, and sometimes with fractions. Numbers can actually come out quite larger.
6. You always divide the larger number by the smaller number---This is just completely false, and students don't even realize they do this regularly because it is not presented as a mathematical problem.
7. Two negatives make a positive---integers can throw this rule through the loop.
8. Multiply everything inside the parentheses by the number outside the parentheses---PEMDAS. If the numbers within the parentheses are being added then this statement is true. However, when the numbers are being multiplied in the parentheses, then they must be multiplied first with each other, and then multiplied by the outside number.
9. Improper fractions should always be written as a mixed number---"This rule can certainly help students understand that positive mixed numbers can represent a value greater than one whole, but it can be troublesome when students are working within a specific mathematical context or real-world situation that requires them to use improper fractions."
10. The number you say first in counting is always less than the number that comes next---When we put a relationship with that number, it could be quite the opposite, like "three dozen eggs is more than eight eggs, and three feet is more than eight inches."
11. The longer the number, the larger the number---0.12345 is much smaller that 0.6. Negative numbers, even in the thousands, will always be smaller than 1. The length of the number has no real bearing on it's size.
12. Please Excuse My Dear Aunt Sally---In a way, order matters. Parentheses, exponents, multiplication, division, addition, and subtraction. However, multiplication and division hold the same weight, as does addition and subtraction. There are problems where both are present and you can take different routes to anser WITHOUT it effecting the outcome. For example, 3^2 – 4(2 + 7) + 8 ÷ 4. This problem can be addressed first by solving the exponents, the addition problem within the parentheses, or simplifying the division section. This problem cannot continue, though, until 2+7 has been solved and that has been multiplied by 4.
13. The equal sign means Find the answer or Write the answer---The equal sign means that both sides are equivalent. 3+4=7 means that 3+4 is the same as 7.

Expired Language

Reasoning About Division

The Old Way

Every week I continue to learn just how much everything has changed since I was a kid in grade school. Growing up, we did math in the standard formats. For me, there was only one way that a problem could be solved. Luckily, that method worked for me and I never questioned it. However, now there are several approaches to take for solving a division problem. The first three that will be talked about are similar. There are only slight changes among them.

Standard Division

This method is what I grew up with. To me, it's common knowledge. Now that I have heard what my teacher has said I can understand why some of my students in the afterschool program I used to work for struggled with this. We teach our students in addition, subtraction, and multiplication to work from the right to left. When it comes to long division, we change direction; we move left to right. Again, this worked for me, and still does. I never questioned with it, and it is a little hard for me to SEE the problems, although I can understand the thought processes behind them.

Greenwood Method

This method seems a little easier to work with. The students do not need to necessarily work the problem out exactly. They make close estimations and work the problems out that way.

Pyramid Model

This is very similar to the previous model, except that the numbers are stacked on top and added together, rather than on the side.

Know, know, know

As is usual, when it comes to teaching, you need to know yourself and your students. Knowing your students is the most important factor in all of this. You need to know how to teach multiple methods of learning because there will be a few students who need a different method. You just have to be comfortable with teaching these methods.

Multiplication Song

2x Table Rap

• Antenna Rap

  I like this video because students love rap. This kind of music always gets kids up dancing and involved. The catchy tune can be easily memorized and they also match the numbers with something that helps to keep it memorized, "jumping 4," "bowling 10," etc.

3x Table Jam

• Jammin' Song

  I like the way this song says the equation, "two 3's is 6." It says that for each part, so students can better understand and visualize what 2x3 actually is.

5x Table Pop

5x Table Song

• This song is made similarly to a singer that kids love. While her actions and songs may not always be the most appropriate for young ears, Miley Cyrus is VERY popular among the little ones. This song, however, introduces more skip counting. They begin the song by including multiplication, but it quickly turns into skip counting by 5's. This is not a song to play at the beginning of the learning segment, however, later it can be introduced and students can have some fun.

Times Tables

Each of these videos has something interesting and unique to the way that they can reach kids and teach multiplication. You have to know your students, though; you have to know where they are in the learning sequence and be careful about what you show.

Multiplication

Why know multiplication and division over addition and subtraction?

Addition and subtraction definitely have their advantages over multiplication and division. Working with smaller numbers, it is not quite as important to know the fact families and have them ready to go at all times. However, once the numbers start growing, fact families become much more important. Students who do not know their fact families are at a disadvantage. Without knowing these groups, it takes much longer to determine the answer to larger problems. Ann H. Wallace and Susan P. Gurganus' article, "Teaching for Mastery of Multiplication," breaks down this area of math and goes over several different categories of teaching multiplication.

Types of Multiplication

Reading through this article, I had no idea there were so many categories of multiplication. I am sure I learned this information years ago, when multiplication was new. However, everything is engrained, now, so it is just something I now know.

1. Repeated Addition- the groups exist simultaneously. One factor describes the number of items in each group, while the other factor describes the number of groups.
2. Scalar Model- the scalar multiple expresses a relationship between the original quantity and the product, but the scalar multiple is not a visible quantity. I.E.: Marcus has eight marbles. His brother has three times as many marbles. How many marbles does Marcus’s brother have?
3. Rate Model- the product is the total value or distance associated with all the units, usually represented in a number line.
4. Cartesian Product Model- two disjoint sets exist and the size of each set is known. The sets are paired and the product is represented by the number of pairings.
5. Area- a region is defined in terms of units along its length and width. The product is the number of square units in the region.

Repeated Addition Model

Scaler Model

Rate Model

Cartesian Model

How Should We Teach Multiplication?

In the past, teachers have taught by following the book.That is likely how most of us were taught: in whatever order the book went. Actually, it wasn't until college that I had teachers go out of order of the texts, because it made more sense that way sometimes. Recent studies have shown that the most effective course of instruction should be:

1. Introducing the concepts through problem situations and linking new concepts to prior knowledge
2. Providing concrete experiences and semiconcrete representations prior to purely symbolic notations
3. Teaching rules explicitly
4. Providing mixed practice

Instruction should be incorporated with realistic problems with hands on materials. For me, hands on activities is the way to go. I learn and understand most things kinesthetically over audio or visual. Allowing students to learn fact families by their own drawing is also beneficial. This not only lets them start understanding fact families, but lets them practice it by also skip counting.

How to Use Technology in Elementary Math

What types of technology?

The most prominent piece of technology that is becoming prevalent in classrooms is some sort of Smart Board. This is something that can not only work as a regular, old fashioned, presentation piece, but it can also be used as a great interactive device. This tool gets kids up and participating in class, rather than sitting back and filling out papers.

Another new piece is iPads. These are becoming popular as well. In some of the classes that I have had to observe and substitute, I have seen iPads more and more. These can be used in centers or whole-class, depending on how many iPads your school has been given.

Computers and laptops are also becoming more popular. I see these most often as a group of five or so, but some schools are lucky enough for every student to now have their own laptop. These laptops that are given out are completely intrusted with the student. While the computers are likely monitored on school premises, the students have full control outside of school.

Why use it?

What is the point of incorporating technology more into the classroom?

Technology is growing. Everyone has a smartphone. Everyone has a tablet. Children are using these devices at younger ages every year. I think bringing these devices into the schools can be a really good idea because we can teach our children how to use these devices. We can teach our students to make good decisions while growing up in a world that is only growing more and more.

Addition and Subtraction

Number Sense

With learning how to use numbers in math, aside from counting, allows children to really understand numbers and their representation to one another. When we just teach children to count (i.e. 1, 2, 3, 4, 5...) those numbers do not mean anything. Children can say those numbers in order, or in reverse order. However, math is what helps children understand numbers, their true purpose, and how to use them.

How can we teach math?

We always start with simple problems. One easy task, for more visual or kinesthetic students, is using Domino's or popsicle sticks (or some other easily countable object).

Domino's give the numbers out directly. So we can have the students add the two sides of the Domino and see what the whole piece is.

Popsicle sticks allows children to physically manipulate a set and see how those "addends" equal the sum.

Unifix Cubes Activities to Teach Addition, Subtraction, Patterns & Sorting

Variety

We must be conscious of what we teach our children and how. We must teach to strengths and interests. We must show our students the variety that life has to hold and how to think differently.

"Education is not the learning of the facts, but the training of the mind to think."--Albert Einstein

Counting

Counting Efficiency

I watched a video of a teacher, Laretha Todd, who had a great method of allowing her students to determine, on their own, the best way to record and count a large group of something. Her students worked in partners and each group had a different baggie full of objects. Mrs. Todd's class was a 3rd grade room.

The first thing Todd did was involve the whole class in reading the objectives. I like that she did this rather than reading the objectives to the students. The students started the lesson being involved and alert.

Next, Todd modeled what was expected. The model did not only include her, but she also called on students to give their ideas of how the assignment was to be completed.

Third, the students split up into their groups and gathered their supplies.

Fourth, Todd lets the students take over. The students determine the best method for grouping, counting, and recording. She walks around and talks with each group. She asks questions, but in the long run, she lets them do what they think is best.

Fifth, Todd comes together with the class and shares students' strategies. She also revisits the objectives with this.

I think Todd did a great job with this lesson.

https://www.teachingchannel.org/videos/counting-collections-lesson 

Number Sense

What happened to number sense?

Technology.

While technology is amazing, and we are only progressing into this category, it could be a factor in stumping our children's education. While I was a kid we didn't have a computer, there were five channels on the television, and imagination was a very powerful tool. Nowadays, every adult has a computer of their own, there are smart phones and tablets. Answers are at our fingertips now. I remember when I was a kid and my mom had us in the car, when she would go through a fast food restaurant she would tell us to come up with the change before she got to the window. I don't know many people that can do math like that anymore, even though it is simple, because of technology. I truly believe that this is hurting our youth, at least to a degree.

What is Number Sense?

Transitioning into a Differentiated Classroom

Our Childhood

How many people remember being taught with the whole group? I do. For my elementary experience, I only had two subjects in those six years that provided some sort of differentiation. In third grade, our teacher split my classroom into two groups for our weekly spelling tests. In fifth grade, our teachers swapped classes for math. In both subjects, I knew where I sat: I was advanced in reading/spelling and slower in math.

Why Differentiate?

When starting to differentiate your classroom, it can be difficult and you might change things periodically if things don't seem to be quite right. This strategy is very beneficial to both you and your students. Differentiation allows you to target all the different needs and abilities of your students. It can benefit you directly by also allowing to see information in a different light and explain information in various ways.

"If you cannot explain it simply, then you don't understand it well enough."--Albert Einstein

7 Steps to High-End

Katherine Gavin and Karen Moylan have summarized seven steps to transforming your class in their article "7 Steps to High-End Learning."

1. Select an appropriate task: make sure that what you teach is worthy of being taught.
2. Increase expectations for all students: just because your students cannot complete a task in the same way does not mean that they are unable to complete a task. Do not lower expectations because some students may struggle with a certain format.
3. Facilitate class discussions about the concepts: class discussions can allow students to see different ideas or different ways to come to the same answer.
4. Encourage all students to communicate their thinking in writing: writing allows students to try explaining their ideas and feelings in ways that others can better understand.
5. Offer additional support: this article talks specifically about hint cards, but there are so many more ways to incorporate additional help. Help can be tutoring, a friend in the classroom, posters around the room, etc.
6. Provide extended challenges: these allow students an opportunity to practice more on a specific task.
7. Use formative assessment to inform instruction: this allows teachers to modify their instruction throughout the learning process.