Times Are A Changing!

It is interesting to me how many times I have heard voiced over the last couple of years the following words:  “That’s not how I learned math!”  As a parent, I have said these words myself (many times).  I have heard other parents say them.  I have heard fellow students say them during discussions in my current math classes.  I have even heard other teachers say them, when I have been subbing at the local elementary school.  Something has changed about math.

Gone are the days…

Math has come a long way from the days when we just plunked a math book down in front of a student and taught them the one way to do their math problems!  Gone are the days when students sat at their desks for the whole math lesson and listened to the teacher impart knowledge (whether they got it or not).  The days when all students were expected to reach a solution to the problem in exactly the same way.  The days when if you didn’t get it, you had to conclude that you were just not good at math. 

That was me.  I always thought that I just really stank at math!  Hours of trying to memorize multiplication tables and days of being pulled in during recess to work on math when I was in elementary school only did one thing for me…it made me really hate math!  Now, I have to wonder if I had been shown more than one way of doing it, if it would have made a difference in how I viewed math. 

We aren’t all the same…

 Not all students are the same.  Not all students look at math in the same way.  Not all students learn the same way.  So, why would we expect them all to “get it” if we only show it to them in one way?    According to Howard Gardner there are Multiple Intelligences .  Students usually have an inclination towards learning based on which intelligence comes more naturally to them.  Here is a short video to describe the types of intelligences and how they impact learning.

What’s different?

Math has changed to focus on giving every student the opportunity to learn in a way that works for them.  Students are introduced to many different methods of solving problems and are able to choose a method that works well for them.  In addition, they are not just given the how but also focus on the why.  Math has become more interactive.  Students may be working in pairs or groups on problems.  They are using manipulatives to help them in problem solving.  They are using technology to help prepare them for their future. 

Here are a couple of articles about the changes that have taken place in math class:

How Has Math Changed Since You Were In School?
What Does Effective Mathematics Instruction Look Like?

Here is a slide show to help dispel some of the myths surrounding these changes.

Why Has Mathematics Instruction Changed? Myths & Facts
 
There are many things that are not done the same way anymore as when I first learned them.  This does not mean they are being done wrong today, just that they are being done differently or have advanced.  “This is not the way that I learned math!”  However, when I think about the strides that have been taken in technology and the greater resources available to teachers today, I have to wonder why we would ever expect math class not to change.

What's So Rational about Fractions?

When I think about the word “rational”, I have to tell you that “fractions” does not automatically pop into my head.  Nevertheless, fractions are considered to be “rational numbers.”  I have to admit that I have always sort of thought of the rational numbers as the ones they threw into the system to mess us all up!  However, when I really look at fractions and how much we use them in our day to day lives, I would have to say that, yes, they are rational.  Another word for rational, according to my Thesaurus, is “balanced”.  This is a good way to look at fractions.  They help to balance out the system.  What if there was no such thing as a fraction?  Well, some math operations might be easier, but how would we break wholes into parts?  Having fractions to represent parts of whole numbers helps to keep the number system all in balance!  That is very rational.

How do I feel about fractions?  Well, it’s complicated…

The word rational may mean balanced, but it definitely does not mean simple!  If you are not a person who has to worry about doing operations with fractions  on a regular basis, then it sure is easy to forget exactly what they are.  When I haven’t had to deal with a fraction for a while, I really have to think about how to add, subtract, multiply or divide them. 

“Do the denominators have to be the same for this one?”  “Am I just working with the numerator?”  “Do I work this one across?”  “Is this the one where I make an x?”  “Which ones do I have to flip?”    

Wow!  This seems so complicated! 

Making complicated relevant…

How do you teach students all of these complicated concepts so that they will stay with them?  I think you have to make it relevant to their lives.  We use fractions every day when we measure, use money, cook, or divide anything up into parts.  Using some of these things to help teach fractions will make it more meaningful to students. 

Another thing that can help to make fractions more relevant is to make learning operations that are complicated more fun by using manipulatives, games, videos, songs, jokes or rhymes (all things that kids enjoy).  Here are some fun things I found for teaching fractions…

Fraction Games - take your pick!

Classroom Games/Activities

A Fraction Mystery Story

And a song...



And a wacky but memorable way of doing fractions...

 

Fretful about Factorization!

Prime factorization!  Those two words have always caused me some anxiety and a bit of nail biting.  So, when I saw that we were covering this topic in my Math for Elementary Teachers class, I was a little nervous.  After all, it has been quite a while since I really looked at factoring numbers.  However, I am happy to say that I made it through with my nails still intact and a much better understanding of prime factorization.

What’s so bad about prime factorization?

I had to ask myself this question as I started looking at prime factorization again.  Why all the anxiety over a few little numbers?  At some point in my past I must have had a bad experience with prime numbers.  Or, maybe it is just the word, “factorization”, that seems so daunting to me.  Either way, I found out that it is not as complicated as it sounds!  As a matter of fact, I almost would go so far as to say…I actually enjoyed it.

How does it work?

Prime factorization is basically breaking a number down to its prime factors.  A prime number is one that only has two factors, 1 and itself.  For example, the number 7 only has factors of 1 and 7.  No other numbers can be multiplied together to make 7. So, 7 is a prime number.  The first five prime numbers are 2, 3, 5, 7 and 11. 

If I am going to find the prime factorization of the number 84, I start by dividing it by the lowest prime number that can go into it.  If I am not sure of what the lowest prime is that can go into 84, then I can use the rules of divisibility to help me find it.  I know that 84 can be divided by 2 and that 2x42=84.  Now I move on to find the lowest prime number that can go into 42.  This would be 2 again and 2x21=42.  Now I need to find the lowest prime number that goes into 21.  This would be 3 and 3x7=21.  Since 3 and 7 are both prime numbers, I can divide no further.   I end up with the prime factorization for 84 being 2x2x3x7.   Here is a video which shows how to use a factor tree  to find the prime factorization of 84.

Here is a math manipulative that can be used to practice making factor trees and finding the prime factorization of numbers:

 Factor Tree Manipulative

Why is prime factorization necessary?

Prime factorization is a great resource to use when we need to find the greatest common factor (GCF)  or least common multiple (LCM) of two numbers.  It works especially well if we need to find these things when we are working with large numbers.  Sounds like a topic for another post!  Here are a couple of links to show how we find the GCF and LCM with prime factorization: 

Using prime factorization to find the GCF

Using prime factorization to find the LCM

Is Common Better?

I have been reading about the Common Core State Standards Initiative  this week.  What exactly is a Common Core Standard?  Well, it is the effort being led by states to have a single set of educational standards across the United States.  The standards apply to students in Kindergarten through 12^th Grade and currently pertain to Math and English Language Arts.  Previously every state has had its own set of unique standards for K-12 education.  Currently 45 states and Washington D.C. have adopted these standards.  The remaining five are Alaska, Minnesota, Nebraska, Virginia and Texas (at the time of this post).   

The basic premise behind having a common set of standards for the whole country is that it will provide consistency in education that will better prepare students for success in their future.  This will in turn contribute to the success of our nation’s future.  This sounds like a really good plan when you first look at it.  However, I have to wonder if making education common is really making it better.

A Few Questions I Am Pondering…

• What about the cost?  Implementing new educational standards in Math and Language Arts makes old curriculum obsolete.  This means that every school will have to purchase new curriculum.  With the lack of funding to some schools, this is not an easy undertaking.
• What if the new standards are more rigorous for some states than their old ones?  Students may be starting out behind with the implementation of new standards.  Catching up may be very difficult and confusing.  Causing education to be frustrating for them and not a good experience.
• What if the new standards are less rigorous for some states than their old ones?  Students may find themselves bored and less engaged with learning.
• What effect will this transition have on teachers?  Teaching is a very demanding and high-stress profession.  They will have to be the ones to deal with re-vamping their lesson plans, aligning to new standards and finding and incorporating ways to teach the new standards effectively.  This sounds like a daunting task in a field that already has high rates of burn-out.
• Who is really in control?  The responsibility of education has been a duty of the state.  The Common Core State Standards initiative is said to be led by the states.  However, with the push and incentives offered by federal government to sign up for the initiative, it makes me wonder how much control the states will retain once this goes into effect.

Here is some information that was helpful to me in looking at the pros and cons of Common Core…

Pros And Cons Of The Common Core Standards
Prezi - Pros and Cons of Common Core
Common Core - Arguments for and Against

 

Does Common = Ordinary?

So, does making it common make it better?  I don’t have an answer to that question!  However, I can say that the word common does not conjure up images of individuality and diversity that we have worked so hard to cultivate in America.  To me, the first thing I think about when I hear the word “common” is ordinary.  As a teacher, I am hoping that the education the students receive in my classroom is as individual as they are and encourages them to go beyond the ordinary to reach their full potential as individuals.  This is my goal, no matter what the standards. 

Math in My Head!

Taking math classes really makes a person think about…well, math!  I am currently taking two math classes and one thing that I have noticed is how often I reach for the calculator.  Is this a good thing or is this a not so good thing?  Hmmm…

This left me thinking, “shouldn’t I be able to do more of this math in my head?”  If a person tries to Google “mental math strategies”, they come up with a pretty extensive list of methods for doing math in their head.  It seems a little overwhelming and my brain is already protesting not using a calculator.  So, here are just a couple of methods I learned about in math class.  These seem pretty straightforward and a good place to start.

Not just people are compatible…numbers are too!

Compatible numbers are numbers that easily go together to help solve a problem.  For starters, it can be helpful to look for numbers that make ten in a problem or multiples of ten.  These are easy to calculate.  Here is an example with addition:

7+9+3=

 I know that 7+3=10.  If I add those together first, then it is easy to add 10+9=19.

How about multiplication?

                (2x3)x(5x4)=
I know that 2x5=10.  This leaves me with 3x4=12.  Finally, I can solve the problem with 10x12=120.

OR I know that 5x4=20.  This leaves me with 2x3=6.  Finally, I can solve the problem with 20x6=120.

Here is a video, showing how this method could be helpful in teaching elementary students to solve problems mentally that require adding a long list of numbers.

 

Once students have mastered looking for 10’s, they can move on to looking for other number combinations that are compatible to help solve problems mentally.

If it looks too difficult to do in your head…just compensate!

Compensation is another technique which uses a form of compatible numbers.  It simply means that if there is not anything in the problem that looks compatible, we just substitute (or compensate) to make it compatible.  Here is an example with addition:

35+48=
I see that 48 is pretty close to 50.  So, I change it to 50 in my head.  Then it is easy to add 35+50=85.  Now, since I added 2 to the 48 to make 50, I simply take away 2 from my answer.  85-2=83.

How about multiplication?

9x14=

I see that 9 is close to 10.  So, I change it to 10 and multiply 10x14=140.  Now, since I multiplied by 10 instead of 9 (10 sets of 14 instead of 9 sets of 14), I subtract the extra set of 14 from my answer.  140-14=126.

There are many other techniques for doing math in our head and teaching elementary students to do math in their heads too.  I found this List  to be helpful.

Should I toss out my calculator?

Probably not!  It sure comes in handy for that math that I can’t do in my head…  

Hooray for the Array!

While studying sets and whole number operations and properties this week in my summer Math for Elementary Teachers class, the topic of the rectangular array was broached.  Arrays can be used starting in the early elementary years to help students understand a variety of number concepts and operations.  As a teacher of elementary students, it is beneficial to understand different ways that arrays can be used to help teach math.  The possibilities are endless.  Only limited by how creative you are willing to be.   

Aren’t arrays just for teaching multiplication?
Of course not!  How boring would that be?  Before students learn multiplication, arrays can be used to teach many other number concepts.  A few of them are…

• Equal /Not Equal
• Odd/Even
• Sorting
• Skip Counting

Some of these concepts are related to multiplication but younger students will probably not notice this connection until later.  Arrays can be used to teach many things and exposing students to them early in their education will help to build links to prior knowledge for future learning.

How can arrays be used for multiplication?
Arrays are an excellent way to help students understand multiplication.  Breaking number sentences into rows and columns helps to give them a visual representation of the problem.  Besides this arrays can be used to explore and demonstrate other concepts that go along with multiplication.

• Factors/Primes/Squares
• Properties of Multiplication
• InverseOperation of Multiplication - Division 

How can we get creative with arrays?
Even though graph paper and a pencil work just fine for making arrays, that does not mean that we are limited to using only this method.  Arrays are all around us (muffin pans, legos, tiles on a floor, and eggs in a carton) and students should be encouraged to explore them in many different forms.  Students can be given grids and various types of materials (buttons, blocks, candies, and coins) to use in making arrays.  Arrays can be a great method of visual and hands-on learning. 

Hooray for the array!   

More information on arrays can be found at:

More about arrays!