Reflection for September 22nd Class

1. A mathematical idea from the session that is new or important.
   This session validated how important it is to teach students different strategies when adding and subtracting.  After the previous session I left class thinking to myself, "Why on earth would I teach students compensation?  It is SO confusing!"  This class reminded me how important it is to teach the strategies, and then students will pick what strategy works for them.  When I was in the classroom, I was constantly having my kids come to board and solve the problems, but not just solve them but talk the rest of the class through what they did to get to the answer.  It is also nice to hear how the other teachers are connecting what we are doing in out math endorsement class to what they are doing with their students.  I feel that I actually have a disadvantage being an Ed Tech.  I don't get to practice what I am learning on students, but at least I can learn from my peers.
2. A question I have.
   I don't have a question right now, but I am excited for the next session when we get into multiplication and division.
3. An application of the mathematical idea for my classroom.
   My answer to this is the same as last week.

September 22, 2010

LookUsing a Ten Structure to Solve an Addition Problem

 

pg 52 of Beckman activity manualing at examples of students solving an addition problem using Ten Structure pictures

• Students #1 and 2 get it.  They understand how to group numbers and "carry" to the next place value.
• Student #3 - may not fully understand place value, and doesn't understand "changing out"  will struggle when moves on to subtraction.

Subtraction

How can we represent subtraction so students understand?

• 2+__=7 - student will subtract 2 from 7.....or count up
• Students need to understand fact families

• Number bond idea  

A misconception that a lot of student have is that addition and multiplication ALWAYS make numbers bigger.  NOT TRUE....not when you are adding or multiplying negative numbers.

Many students would say that addition is easier because they can "move" the numbers around, where they can't be moved in subtraction.

Pictorial Models for Subtracton:

Solving the same problem using Compensation

Another strategy to solve a subtraction problem  (I think its the Austrian strategy)

Pg. 54 in the Beckman activity manual

Pg. 221 of the Van de Wall book

• examples/strategies that students have used to solve the subtraction problem 73-46, using a number line.
• Add tens to get close, then ones
• Add tens to overshoot, then come back (it's good for kids to know that they can go backwards on a number line)
• Add ones to make a ten, then tens and ones
• Take tens from tens, then subtract ones
• Take away tens, then ones
• Take extra tens, then add back (compensation examples)
• Add to the whole if necessary (compensation examples)

Reading Assignment

• Van de Wall pg. 157 - 161
• Parker 1.5 pgs. 25-30

Reflection - September 15

1.  A mathematical idea from the session that is new or important.

• I have always thought it was important to have students explain how they solved a problem.  By letting students explain their process, other students learn that there is more than one way to solve a math problem.  I can see why it is important to teach students about the different properities such as commutative and associative.

 2.  A question I have.

• Why and how would I teach students about compensation without confusing them.  I had a difficult time with the compensation activity in class, and left with the conclusion I wouldn't teach my students that strategy.   

3.  An application of the mathematical idea for my classroom.

• Like I said earlier, I like having students share how they solve different problems.  When students are at the whiteboard solving and explaining their problem, I need to make sure I point out the use of different properities.  For example, if I can see that a student used the associative property, I need to point that out to the rest of the class.  I think that would make the learning more meaningful. 

September 15, 2010

Introduces different systems of counting including the Egyptian base ten, the Babylonian base sixty, the Hindu system using three symbols, and the abacus and computer systems.

How Does Explicit Instruction relate to Constructivist Teaching?

• A teacher should use both strategies.  It is good to let students explore in their learning, but sometimes students can explore forever, but never come to the correct conclusion.  It is good to let students explore, but the teacher should guide their exploration.  Students can explore and come to the wrong conclusions.  Some students can't come up with strategies on their own.

Addition

• Addition is combining two numbers.
• Kids start adding by counting.  For example, if a students has 8 unix cubes.  The students will count each cube individually until they count to 8.

When should students be able to do these addition strategies?

• Count all - students should know this by Kindergarten
• Count on - students should be able to do this by 1st grade
• Count on from larger number -  students should know something about addition, so students in 1st and 2nd grade should be able to do this.
• Make 10 model - 2nd graders
• Make 10 from larger - 2nds graders
• Double +1 - 2nd grades
• The first three years of teaching addition is crucial.  Students should master addition strategies by 2nd grade. 

Parker Book, pgs 14 - 23

• pg. 14 Addition in relationship to set and measurement.  An issue that can come up with using set, students learn to manipulate objects rather than manipulate numbers.  
• Properties of Addition:
• Identity - anything plus zero remains that same number.  Adding something to zero, gets the identical number.  Don't say, "Adding a number to zero doesn't do anything."
• Commutative - order doesn't matter.  4+5 = 5+4.  This can fuel fact families.  Stressing this property can help students see the relationship with numbers.  Build on this, build the foundation to help students get the automaticity.  
• Associative - grouping doesn't matter.  (4+5)+6 = 4+(5+6).  Students will use associative property so the numbers are "more friendly" to add together.  For example: 3+5+7.  Group the 3 and 7 to get 10, then add the 5 to get 15.
• Compensation: 6+9 = take one from the 6 and add it to the 9 to get 10 then add the remaining 5 to get 15. (bottom of pg 17)
• pg. 18 - shows student getting the right answer, but using the equal sign incorrectly. It is important to help students understand how to use the equal sign.  Math is a language. Students need to understand and learn to use this math language to communicate.   

Hundred Chart

• fractions
• adding
• rounding
• an easy way for students find the "nice" numbers
• prime number
• skip counting

Assignments for next week:

• Reflection
• no readings
• Share a difficulty.....

 

  

Sept. 8th Class Reflection

Egyptian Number System

 A mathematical idea that was new/important to me is the importance of teaching student the history of numbers.  This isn't a new concept to me, but for some reason something clicked on how important it is for students to know this.  By teaching students how the Egyptians used to write numbers, how they used those characters to add and subtract can be a great way to teach students place value.  Place value has always been an important concept for students to learn and understand, and as a teacher I am constantly trying to think of ways to teach it to my students.  Place value is HUGE, and that is another thing that really clicked for me today.  Students need to understand what the 3 really represents in the number 3,245.  I am hopeful that throughout the course of this math endorsement class I will learn different teaching strategies to reach the different learners that I will encounter in the classroom.

Sept. 8, 2010

Earlier number systems can be used to help students understand the importance of place value.

Part Part Whole relationships

• The most fundamental concept you can teach to kids
• If students can understand how to break up numbers math will be easy/easier
  
• This is HUGE for students to undertand.
  
• There are different ways to represent numbers
• Example: 23 = 10 + 10 +3, 23 is the whole, 10, 10 and 3 are the parts that equal the whole
  

Ah-Ha Moment -

• I never realized how important it was for students to understand part part whole relationships, but that makes sense. Break up the numbers. Doing math is MUCH easier when you break up the numbers. Thinking about what I do when solving math problems, I break up the numbers. When I do that it's easier and much quicker for me. Every student is different and that means that they will have different approaches to solving math problems. When they use part part whole relationships, they are going to go about it differently.

Developing Whole-Number Place-Value Concepts pg 187 - 193

• Why is "base-ten language" important?
  5 tens + 3 ones is and example of "base-ten language"
  It is the base of our number system.
  Being consistent w/vocabulary and reinforces the vocabulary
  It helps students understand the concept of part part whole
  
  
• Explain three ways one can count a set of objects and how these methods of counting can be used to coordinate concepts and oral and written names for numbers.
  
  
  
  
  1. Counting by Ones -
     
  2. Counting by Groups and Singles - Example: "One, two, three, four, five bunches of ten and one, two, three singles." This method does not tell directly how many items there are. This counting must be coordinated with a count by ones before it can be a means of telling "how many."
  3. Counting by Tens and Ones -
     

• Describe the three types of physical models for base-ten concepts. What is the significance of the differences among these models?
  
  
  
  
  1. Base-Ten Models - a good base-ten model for ones, tens and hundreds is one that is proportional.
     
  2. Groupable Models - clearly reflect the relationships of ones, tens, and hundreds are those for which the ten can actually be make or grouped from the singles.
     
  3. PreGrouped or Trading Models - are commonly shown in textbooks are are commonly used in instructional activities.
     
  4. Nonproportional Models - these models can be used when students don't need to understand how ten units make a "ten" or by students who need to return to place-value concepts as they struggle w/more advance computations.

Reflection - How does the knowledge you have gained about place value impact your teaching of place value in your classroom?

• I've always known place value is important, but as I have learned more and more about place value I have found how important it is to teach place value using different approaches. The students in my classroom are all different and have different learning styles. Helping students understand place value can be difficult. Some students totally get it, some students kind of get it, and some students will pretend to get it. As an educator I try to teach place value in concrete, representatonal and abstract ways.

Draw 3 representations of 463

Set Models and Measurement Models

• Set - The answer must be a whole number. Example: counting discrete objects, groups, singles.
• Measurement - The answer is often not a whole number. Example: number line, distance, height, length.

Assignment for Next Week

• Read Van de Walle - pgs 167 - 170 - Helping Children Master the Basic Facts.
• Read Beckman - pgs 100 - 104
• Reflection of class - view reflection question on pg 5 of syllabus

Math Endorsement - Day 1

Vocabulary for Course

• Explicit vs Implicit
• Engagement
  
• Concrete - using tangible items
  
• Representational - using a diagram
  
• Abstract - using a number sentence

Example - Liam has 5 trucks. How many wheels are on all 5 trucks?

• Concrete - students solve problem looking at 5 toy trucks and counting the wheels on each truck.
• Representational - students solve problem by drawing a picture of 5 trucks that each have 5 wheels and then counting the wheels.
• Abstract - students solve problem using the number sentence 4+4+4+4+4=20 or 5x4=20

How do you teach students what you want them to learn?

• Know where they are starting
• Where do you (the teacher) want them to be?
• What do you (the teacher) want them to know?
• Model math practices
• Analyze student thinking errors
• Make use of student background
• You have to know where your students are going - it doesn't end in your class, math builds on each other. Example: adding and subtracting fractions leads to algebraic thinking.
• 
  

Assignment for next class:

• Parker - 1.1 and 1.2
• Van de Walle - pgs 134 - 138 (Part/Whole relationships)