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Using Math Manipulatives- conference notes
I am adding some notes to this...see the bottom of this entry.
We have been on the run since March... Saint Patty's in Savannah, a week later Charleston South Carolina as we said good- bye to Steve's mom. Four days camping out under the stars at Shakori Hills music festival...nine days at Vogel State Park in Georgia and we just returned from nine days at Dreher Island in South Carolina. I can't say it is letting up anytime soon...we leave for 9 days to tube in the mountains of North Carolina in two weeks...then onto Floydfest in Virginia and we finish with camping on the Ocean at Hunting Island. I did the math and I figured this year we will be spending 60 nights on vacation in our pop up. After Europe we decided that is how we would spend vacations in 2008. The children love it.
I hope to share the memories of these trips... until then I come to the family blog to share notes that I promised to share from my pre-conference experience during MY vacation two weeks ago at the four day Charlotte Mason conference here in NC. Steve worked from home for 3 1/2 days and took the kiddos on to Dreher Island to allow me to run away and to soak up the sunshine of why I homeschool. He then took the kiddos camping where I joined them two days later. They kept us wonderfully busy at the conference, if I thought I was going to get some rest I was wrong, but each and every minute was life changing .... I arrived a day early to attend a workshop on using math manipulatives in the UPPER elementary grade levels. My boys are all really gifted in Math. Reichen in his last semester of 2nd grade tested at a Algebra Level in math, the scores urged placing him in the algebra curriculum but I decided to start him in the pre-algebra curriculum to make things more concrete. I don't use computers, websites or any "computer games" with any of the children so all of his teaching have come from manipulatives, thoughts and discussions. I loved this conference in that I really got a chance to see math with manipulatives at the higher grade levels.
Here are my quick notes to those who I promised them to...
Bare with me, I typed this quickly for my homeschool group. If I
waited till I had a moment to type it all pretty with thoughts in a
row it would never get done. So I thought I would share what I have.
I spent the extra $80 to go to the conference a day early to attend
the pre-conference workshop on using manipulatives in the upper
elementary grades. I love math and I love manipulatives and this
workshop was life changing in how I will teach math.
If anyone wants maybe before or after the kids co-op I can spend some
time going over this with them.
Here are the quick notes I typed and shared with my HS group.
I hope to get my notes down better...but wanted to share some things
from a Math Seminar I went to. I spent five days at a Charlotte
Mason conference and paid an additional amount to attend a pre-
conference on using math manipulatives with upper elementary
students. We of course covered using them with the lower level to.
The Pre-conference was taught by Dr. Milton Uecker and honestly was
life changing in ways to help students view math. I thought his
ideas were perfect with our current strong core knowledge curriculum.
Some quick things that we did and that he shared.
We really need to work on children understanding math.
Sure a child can add 4 plus 1 and get 5. Sure they can divide 16 by
4 and get 4...but can they easily tell you a story about that
problem.
I would have assumed yes that they easily could...but was pretty
shocked when I just sat my boys down and asked them to tell me a
story giving 61 divided by 13 as an example. I got a lot of "ummms"
before they came up with a story. Our goal should be that they can
do this easily that examples pour from their mouths.
He said "speak Math"!!!!!!!! Math is a language!!!!!!
Have interactive dialogue...assess both the understanding and the
ability to solve problems.
Do a pictorial bridge.
We can use manipulatives that we get in the higher level
maths. He also had us (the parent students) draw our base ten blocks.
a square= 100
a line =10
dots =1
This was huge drawing our base ten blocks!!!!!!!!
So we had to draw them in addition to playing and manipulating them.
Multiplication: The rectangular model
When you see the problem ask
"what is the question"
"What is known"
"What is the length of the sides (factors)
"What is unknown"
"What is the area (the answer)"
****He made us solve 14x3= by drawing the picture with lines dots.
He then made us show "the proof" of it.
I immediatly put 14 x3 down on the paper and multiplied but he
prompted us to look at the drawing and see the 3x10 and the 3x4.
****He then made us show 15x12 and then to prove it "the proof" we
had to show the sub products. Show the kiddos (using the drawing)
how you have 5x2 and 10x2 and 5x10 and 10 x10 those added give you
the 180.
This is a great way to show math and not just fall back on
the "carry".
****He had us do 21x23 and break it down to the "sub products"
3x1 and 3x20 and 1x20 and 20 x20. Have them draw it...Have them use
their base ten blocks then have them physically show you the "sub
products"....then have them tell you a story about it, describe it,
talk it and then go back to the "carry" multipication so they can
see how they relate and how we got our number.
He had us multiply the 22 x13 with the carrying.
He then showed us to multiply it by looking at it as
"3x2" and "3 x20" and "10x2" and "10 x20"
This is hard to describe but if you write it down you will see it.
It was in this simple math problem that he showed us to "look for
factors in the edges"...."answer the area".
It was in this example where he made us look at the "square"
(exponent) in the problem from our picture AND the base ten blocks.
Division the Rectangular Model
Again on this sheet he made us ask
"What is the question"
"What is known"
the answer to "what is known" is "the area and one side (factor)".
"what is unknown" and that would be "the other factor or side".
He had us figure out 182 OVER 14 (or 182 divided by 14)
Tell the students that the numerator is "the area"
the denominatior is "how many times can I can 14 out of 182".
Use your base ten blocks to do this. You then are showing the
students how to break down the number is "subs"
Model it!!!!!!
Talk it!!!!
Some of my notes that I wrote are....
Define "what is one". If you use manipulatives you need to define
was it "one". For your one could be the smallest block or if doing
decimals one could be your hundreds block.
To think math you must "talk math".
Bring thoughts out in the open access.
Manipulate math, understand it, ask yourself "does the answer make
sense?. "Does the answer make sense" is a biggie!!!!!!!!!!
Build a model to show you got the answer right.
Give something beyond the answers.
You can spend 10 minutes after breakfast showing children the
difference between "squared" and "cubed" and they will probably
retain it forever.
Model "squared" with a 100 base ten block.
It is the same distance over and up.
Then have them show "squared" with any numbers you can think of.
Model "cubed" with a 1,000 block
the same distance over and up...and height.
While doing the most simple math problem as "what were you thinking
about while doing it". Teacher listens to description....AND if
needed the teacher supplies the right word (math term) for what they
were thinking. For examples "6 cookies shared with 3 people" and we
introduce the word "divide".
Teach our children that we EXPECT them to "prove it".
Hey that's fun!!!!!!!!!!!!
Teach them when we multiply we are really counting.
Express a math problem in as many ways as possible...make them see
the relationships with their base ten blocks and drawings.
When dealing with fractions
"think about them"
"think about the answer"
"does your answer make sense"
When doing algebra Illustrate the "=" sign as a "scale"
Illustrate it as a scale.
Make a visual scale on your kitchen table. We all know how I use my kitchen table in math.
Tape an "equal" sign to your kitchen table.
Tape two lines on the opposite side...make it look like a scale.
Cut out an "x" from a piece of paper...
Now show the problem.
Teach children that the = sign is "Same Value As"
In summary he wants children to see the edges and the rectangles in
their math problems. It all really comes down to that...see the
edges and see the rectangles in your manipulatives.
While we take a break from typical school year math over our summer
what a great way to show children math for a few minutes a day.
It is hard to explain but I did a quick search and found this from
him.
Go directly to pages 24 and 25
http://www.childlightusa.org/review/Winter2007_Review.pdf
Sorry so rushed but laundry to do...and kiddos to enjoy...
Charlotte Mason Pre-conference notes Teaching Mathematics: A shift in Paradigm
Speaker: Dr. Milton Uecker
This seminar focused on teaching math to train students to begin to think and speak mathematically. To think mathematically, children need to be engaged in concrete, activity-centered instruction that requires children to see and describe relationships through the language of mathematics.
Our goal…mathematical thinking, understanding, problem solving/application.
Role of the teacher…provide materials and experiences to think about and talk about.
Teachers need to ask…what is one? Can you prove it? What is the question? Can you state that another way? Can you tell me a story that requires this type of solution? Teachers must promote thought through interactive dialogue
Virtual Manipulative and Internet/Web Resources
National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html
NCTM Illuminations http://illuminations.nctm.org
Multiplying/Dividing Whole Numbers
- Rectangle Multiplication-visualize the two numbers as an area with manipulatives
- Rectangle Multiplication of Integers-visualize and practice using an area representation.
Factoring/Multiplying Variables
- Algebra Tiles
Multiplying/Dividing of Fractions
- Fractions-Rectangle Multiplication-visualize and practice multiplying fractions suing an area representation.
Fractions/Ratios/Percents
- Percentages - Discover relationships between fractions, decimals and percents
- Fraction Model I – Explores fractions using adjustable numerators and denominators. You can see decimal and percent equivalents, as well as a model that represents the fraction http://illuminations.nctm.org/
Balancing Equations
- Algebra Balance Scales- Solve simple linear equations using a balance beam representation.
- Algebra Balance Scales-Negatives- Solve simple linear equations using a balance beam representation.
- Can you balance? Add blocks to make the scale balance
- http://teams.lacoe.edu/documentation/classrooms/linda/algebra/activities/balance/balance.html
Patterns and Functions
- Color Patterns- Arrange colors to complete a pattern.
- Set Game- recognize similarities/differences www.setgame.com
- Function machine – Explore the concept of functions by putting values into this machine and observing output.
- Grapher – A tool for graphing and exploring functions.
What is one? Is it a small ant, is it a congregation of people, is it a country? Is it 100, is it 1.0?
What is one? This is important as you start to study and do fractions/decimals “parts” of one.
You can use your base ten blocks and you can use your “100 block” as one and show parts of one. We were given plastic overhead projector plastic to place over our 100 block to see fractions within a 100 block=1.
To think math talk math.
Bring thoughts out in open access.
Manipulate math, understand it, does the answer make sense?
Think about the answer you expect…does your answer make sense?
Build a model to show you got the answer right.
Do something that is beyond just giving an answer.
“square” explain it…same distance over and back. This is why area is squared.
“cubed” explain it as the same distance over, back and up. This is why volume is cubed.
A line at this time show lines as a distance. This is why perimeter has not square or cube.
Ask children, “tell me a story?” This is very important. Tell a story no matter how simple. Give a word problem that shows an understanding of the math. “Tell me a story and show me an example of this math problem.” Build a problem…go beyond the answer.
Seek balance.
Get to know the problem.
Consolidate and interact with grids.
Ask, “what were you thinking as you worked the problem” (if right and if wrong).
The teacher can listen to the description and then supply the correct word needed.
You want your child to say, “My teacher expects me to prove my answer.”
When we multiply we are simply counting.
Express multiplication in as many ways as possible.
Fractions- Think about them, think about your answer, what are they asking, what example can you think of…is your answer small or large, is it greater than or less than certain numbers.
Think of adding parts and not kinds (denominator)
Balancing equations- look at it like a balancing scale.
Illustrate with a scale. We have one supplied from seminar.
Make sure children understand that an “=” equal sign means “value is the same”/”same as” (value).
A book recommended for all “Teaching Math to Students with Special Needs” –Thornton.
Dr. Uecker recommends this book even if not teaching a special needs child.
An equal sign is “the same value as”.
When you multiply fractions like 1/3 of 5/8 you go over 1/3 and go up 5/8 (just like with whole numbers). When you multiply you go over and up.
Look at the edges and rectangles!!!!!!!!!!!!!!!!!!
Be careful with your young children as they learn to add and multiply. We are asking them to do math from the ones place, then tens place etc… so they are writing numbers backwards. We have to be careful that we are not training their little minds to look at numbers backwards. I am working at preventing this by adding another lesson to their math on these days. I am doing a reciting of numbers and then asking my children to write the number I said. This could be an exercise to help them write out numbers in the correct way and not backwards. By doing this I hope to diffuse any damage (if any) caused from writing numbers backwards all of the time. It was suggested by someone in the class that dyslexia could be related to have our children look at common numbers backwards at such a young age. I want to counter act that by reciting numbers and having them write what is the correct way of writing them.