Tag / conceptual understanding

by Beyond Traditional Math

What Exactly is a Fraction Anyway?

When I was a student in elementary school, I dreaded learning about fractions. It was a very tough concept for me. All I remember is shading in boxes and finding common denominators. I never understood what I was doing.

I decided as a teacher that my mission was to help fractions make sense to my students.  So I introduce the concept very slowly and very carefully.  Because this is so abstract for students, it must be connected to the real world the whole way through the unit.

We started learning about fractions by trying to figure out what a fraction actually is.  I know that sounds obvious, but I need to find out what my students know. So I posted the question, What is a fraction?

I gave a large piece of paper to small groups and asked them to write everything they know. They all pretty much came back with something along these lines.  There were a lot of I don’t knows, and a lot of blank stares. The people who did write something just wrote symbols or numbers.

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Not one student could tell me what a fraction really was. So I tried to clarify it for them with a simple drawing.

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Now that the definition is out of the way, maybe we can move into the conceptual understanding part! I make it a point to say those words daily as we talk about what we learned the day before. The emphasis in the Common Core State Standards for fractions in third grade is on parts of a whole, so that is what we’ll focus on!

How do you help your students know the meaning of math vocabulary?

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by Beyond Traditional Math

You Call That a Multiplication Table? THIS is a Multiplication Table

Multiplication Table to Scale

Have you even seen this yet?! Click on the image to go straight to the PDF of this amazing multiplication table by David Millar of thegriddle.net. It is free to educators to print and use in the classroom. Even better if you go to the educator section, there is a black and white version that is both with and without numbers.

My mind is churning with the possibilities for using this in the classroom! I think it is the best multiplication table I’ve certainly ever seen. Talk about helping students conceptually understand multiplication, connect it to arrays as well as the concept of area.

Outstanding!

by Beyond Traditional Math

Inquiry Learning: YES!

A little while back I wrote about letting students discover math patterns and connections on their own.  Inquiry learning truly does help students do the majority of the learning.

Well, check out this blogger from The Research Based Classroom. I love her post this morning citing Piaget:

The Research Based Classroom

“Each time one prematurely teaches a child something he could have discovered himself, that child is kept from inventing it and consequently from understanding it completely.”   Piaget

Right on Brandi!

How do you all help your students discover concepts on their own?

by Beyond Traditional Math

Even and Odd: The Importance of Conceptual Understanding

I can’t tell you how many times I learn through the eyes of my five year old.  She asks amazing questions, especially when it comes to numbers.  While we were doing some daily math play today, I thought about introducing her to the concept of even and odd numbers.  I was taught even and odd by being told to memorize the numbers: 2,4,6,8.  As a tiny child I don’t think I truly understood why a number was even vs. odd.

So we built the numbers 1-10 (in hindsight I wish I had included zero) with unifix cubes. Then I told her to put them in pairs. We put the pairs together, and lined everything up. During this process she was uncomfortable whenever one was left over, and wanted to pair it up with one from another number set! This is how it looked once we had it all situated.

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Then I asked her what she noticed.

“There is ‘ones’ all by itself!”

We pointed out which numbers had an “odd one out” as I put it. That was when I introduced the words odd and even.  I think it is essential to introduce the vocabulary to kids after they have explored the concept, so that they have a way to name what it is they are seeing.

She then noticed that every other number was odd.  So we looked at and extended the pattern: odd, even, odd, even….until we got to the 10. I then asked her what she thought 11 would be, odd or even?  She shouted out “odd!” without even thinking about it.

I still have a few third graders that are struggling with this concept (can you even believe that?). I am going to try to see if this could help them understand this very basic concept.

 

by Beyond Traditional Math

Purposeful Play: Explore Concepts Before You Name Them

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In the beginning of my teaching days, I was terrified of manipulatives. I thought that students would mess around with them, blocks and chips would be flying through the air and no one would be learning. I’ve learned the last few years that the opposite is true. Manipulatives have become a way for students to both play and discover math concepts in my classroom.

Today the learning target was: I can discover addition rules. 

In the middle of the floor I put out cups and two color counting chips. I showed them two cups with a few chips in each one and asked them how I could find out how many chips I had. I had 4 chips in one cup and 5 in the other. Many students scoffed (exclaimed “this is easy!”) and told me to count them. So to get at the properties of addition naturally, I asked how many different ways I could count them.  I asked them to write some number sentences in their math journals. The students came up with:

4 + 5

5 + 4

(3 + 2) +4

9 + 0 (They made me pour all the chips in one cup!)

It was awesome to see how they were thinking.  In just a minute or two, they figured out the commutative, associative and identity properties of addition. I walked around to each student, dropped counters into their cups, and asked them to record as many number sentences as they could with their chips.   I was able to differentiate easily by dropping a small number in the student’s cups who were struggling, and larger numbers in those gifted student’s cups. Some of them asked for 3 cups, others stuck with two. They were sorting and dumping the chips, writing number sentences as they went along.

After about 5 minutes I asked them to share some addition rules that they discovered.  They reported to me (in kid language):

  • In every number sentence I always had the same sum.
  • No matter how many ways you break the numbers you still get the same sum. (This is the associative property!)
  • It didn’t matter which cup I added first, I always got the same sum. (This is the commutative property!)
  • If I had a cup with zero in it, the other cup was the sum! (This is the identity property!)

Tomorrow I will name these things for them. We will learn that those exact “rules” they came up with are actually properties of addition, which can then translate to multiplication later in the year.

I now know that purposeful play is an important part of the learning process.  My goal is to integrate this more and more each day!