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Counting Principles 1-1 Correspondence: one word tag to each object Stable Order: order of word tags invariant across sets Cardinality: final word tag = quantity of set Abstraction: objects of any kind can be collected and counted Order-Irrelevance: items in set can be tagged in any sequence Recall the basics of counting. Counting is keeping track of quantity.
This is a critical concept that we tend to lose in our procedural instruction. Yet it is critical. Consider what it means to add. We can only add things that have a defined commonality. In life we can talk about this concept and say, I am going to add all of the people and see how many I have, or I can say I am going to add all of the things with legs. So now I can include the tables and chairs. With mathematics we are tied to the idea of the UNIT when we add. We need to have LIKE UNITS in order to add things. Remember the 1/6 + 1/6 situation. The issue here is that our unit is 6ths and we are seeing “How many 6ths”. The addition is driven by the like units. If we had 1/6 + 1/5 we need to change the form so that we can get like units. Again the unit drives the addition. Format discussion like teacher in the Sims article “Look Who’s Talking” in the Trainer Notes (page 123 bottom of blue box). Engage students as this teacher engages her students. Make sure participants see that the issue of Same Unit is critical when we consider quantity.
Note to trainer—Have a participant read this slide.
This is a visual representation linking magnitude to number lines. We have discussed this slide in a previous unit. Further information is provided in Unit 5 Trainer Notes Slide 18. Questions that will instill quantity are listed below: How many? Who has more or less? What did you roll? How many spaces did you move? How high are you ? How far did you move around the dial? What was the question we asked in trashketball? (Which team scored the highest number of correct free throws? Which team had the most opportunity for improvement?)
When young students are told a story and asked to retell a story there is often difficulty. This issue relates to math when a student is asked to pattern colors, example red, blue, blue, red, blue, blue, etc. and ordinal counting. With respect to visual spatial issues, a student will not always be able to complete a pattern. Students are also unable to reverse patterns. 1, 2, 3, 4, 5, 6, etc. Looking at q quantity and naming it with a number.
See Trainer notes Unit 5 Slide 11 Dialogue.
The Preliminary item - This item is generally mastered around the age of 3 years. It is included in this test as a warm-up item, to orient children to the nature of the test and to give them a successful experience at the start. Level 0 items - These items assess children's ability to count and to quantify small sets, when concrete objects are available and can be touched and manipulated. Teacher says out loud everything that is not in parentheses. Everything in parentheses are directions for the teacher. Trainer note: Make sure to communicate stopping point in the testing process for this level is 3 questions correct. Stress that In order to have an Item that has an a and b, both a and b must be correct to be counted in the score.
Level 1 items - There are two classes of items at this level. Items 2, 3, 4, 5, 6 and 9 assess children's knowledge of the number sequence. Items 1, 7, and 8 assess children's ability to handle simple arithmetic problems. To solve items at this level, children need to rely on something like a &quot;mental counting line&quot; inside their heads.
Level 2 items - There are two classes of items at this level as well: those that assess knowledge of the number sequence and those that assess knowledge of arithmetic. The primary distinction between items at level 1 and level 2 is that items at this level require children to deal with double-digit numbers (i.e., tens and ones) and/or depend on the use of two mental number lines for successful solution.
Level 3 items - Like the earlier levels, there are two classes of items at this level: those that assess knowledge of the number sequence and those that assess knowledge of arithmetic. The primary distinction between items at level 2 and level 3 is that items at this level require children to deal with triple-digit numbers and/or to solve more complex problems involving double-digit numbers (e.g. addition and subtraction problems that require re-grouping). Have participants in pairs give the NKT to each other, taking turns administering and being the test taker. Assign each participant to give the Number Knowledge Test between Day 3 and 4. Give them information for on how to score the test. Bring back the results to Day 4 meeting. Guidelines from the Number Worlds Website. http://clarku.edu/numberworlds
See Trainer Notes Unit 5 Slide 18 Dialogue.
Trainer responsible to read and review this article. “Subitizing” is in the Trainer Notes Refer to Subitizing: What is it and why should we teach it. Doug Clements Make available for teachers to read on their own. Discuss how subitizing is the ability to ‘see suddenly’ the amount. Subitizing is a pre-skill for students to be able to see number as composed of other numbers.
If we only present numbers through manipulatives we are not supporting students’ ability to subitize well. They then live in “ones world’ and do not see how numbers can be composed. If 7 is presented like this, the tendency is to count each dot.
If we present 7 as a 3 and a 4, we begin to encourage students to see numbers as compositions of other numbers. This is a critical element of quantity. Recommend that elementary teachers, particularly early elementary teachers read and re-read the Clements article.
Notice how the picture land supports students in developing their ability to see number quickly, or to subitize. The repeated presentation of number through the standard arrangements in dice, cards and dominos is an excellent way to support kids in seeing number without counting.
Take for instance—a lesson on percentages. Later on in this Foundations training, we will build on student’s fundamental understanding of quantity and magnitude to explore, develop and understand percentages. We believe that this is a more powerful way for us to communicate percentages, ratio, division, fractions and rate than through procedural and oftentimes disconnected methods emphasizing multiplication and “moving it over two”.
Algebra is so concrete there are many examples of algebra in everyday life. Take your run of the mill speeding ticket, this can be turned into an exciting and motivating lesson for young high school about to get there license. Algebra can help them find the answer to that all important question of “where is the best place to speed”? For instance in the symbolic section here we have a state that calculates how many miles over the 65 MPH speed limit you were (s-65) multiplies that number by four and then charges you a $10.00 constant fee for court cost. How much would a speeding ticket cost in this state if you were clocked at 80 MPH. So algebra allows us to create a line that will tell us our exact cost for every possible speed.
This lesson maps Hands On Equations which is an excellent program that teaches algebraic equations using manipulatives. Consider how most of us were taught algebra particularly solving equations—it is taught in a highly symbolic fashion that is completely removed from quantity and magnitude. Let’s give it a try – Note to trainer: Each participant will need 1 Hands On Equation Packet.
Hands-on-Equations shows us how even solving abstract algebraic equations has a fundamental quantity component that students can HOLD ON TO. This is a very Physical presentation of algebra which is NOT a gimmick this IS the real math that is going on in Equations. Note also that this approach fits in nicely with the research not only of Griffin but also of Cecil Mercer whose focus is special education. He has found consistently that the struggling learner shows improvement when they are given direct instruction in moving from the concrete to the REPRESENTATIONAL to the abstract. Presenter may chose to click on the website and show one or two of the videos to see the language that students use with this approach to algebraic equations.
These are the types of equations that students will be able to solve readily after just a few days to weeks of work with Hands-on-equations. They learn to solve equations in a way that connects to what they already understand both the physical and the idea of reacting to stimuli sort of like they do with a video game, for example. The engagement is really amazing.
Get these materials out
As you know, another one of the Components of Number Sense is Equality. This mode of teaching equations not only emphasizes quantity but also the Equality behind equations.
At this stage of Hands On Equations, students solve by trial and error. Set up a blue pawn on the your right hand side of the balance and a number cube of 3 on the left hand side. (Participants should view the equation with the pawn(s) and number cubes as shown above. Your number cubes should face out, but participant number cubes should face up so they can be seen more easily. Model your thinking. Hmmm—if my system is balanced, then the blue pawn must be equal in value to three. Also, if I try the value of 1 for the blue pawn in the second example, it will create an unbalanced system, since 1 + 1 + 2 = 4 and not 8 + 1. I need to try some other values, higher values. Let me try 2 + 2 + 2 = 6 and 8 +2 = 10, so I must need a higher value for my pawn. Let me try 6. Yes! It works since 6 + 6 + 2 = 8 + 6 (Both sides of my system has a value of 14, so my scale balances. Let the students know that the pawn has a special name “x”. A check of the equation is part of the process so if x =6, then 14= 14 with a check above the equals sign.
Lesson centers around equations, variables, and unknowns on both sides of a setup---but still at the intuitive stage. The word variable is not used at all. Students acquire concepts through work with the materials. Show participants how to set up the equation physically. Try x = 3 When I try this solution, I have 7 on one side of my balance but 9 on the other side. Let me try 4. When x = 4, I have 8 on once side and 11 on the other side of my balance. My numbers on each side of the equation are now further apart than when I tried 3, so let me try 2. When x = 2, I have 6 on one side of the equation and 7 on the other side. Hmm—my numbers on both sides are getting closer together. Let me try x = 1. Yes---that works to balance my system. Set up the check by setting up the equation physically again and substituting in 1 for the blue pawn. The check is 5 = 5 so x does equal 1.
A legal move is when the same value is removed from both sides simultaneously. Show how when you physically set up 4x + 2 = 3 x + 9, you can remove 1 blue pawn from both sides, then another blue pawn from both sides and finally a third blue pawn from both sides. You now have (1 blue pawn) X+ 2 = 9. Students will readily see the solution for x is 7. Set up the original equations and check. 30=30 so x does indeed equal 7. We have a balanced system when x = 7.
Have participants try this equation by removing pawns from both sides. 2x + 2 = 10 X = 4 works since 4 + 4 + 2 = 10 Students do not remove number cube values from both sides at this point.
Note what happens in algebra. Up until now, we have almost always maintained a value. In algebra, you actually change values as long as you maintain equality, this is okay. Here is where the “soul of mathematics” the equals sign becomes the most important consideration/concept. Note that if I add 4 to both sides, I am changing my value on both sides. Why is that okay now? It is okay because our purpose has changed. We are no longer just simplifying numbers or writing the same value in different forms, now we are using the power of an equation to chop away at things until we find the hidden value of a variable.
Participants can now be shown about subtracting the same number-cube value from each side of a balance setup. This removal of the same value will leave the setup in balance. Four can be removed from both sides of the balance. Some participants may want to decompose 10 into 4 + 6. Our equation now reads 2x = 6 so students will readily see that x = 3. Checking with the original equations 10=10, so x is equal to 3.
Show participants how to remove the pawns first. Three blue pawns (3x) can be removed from both sides Try removing one from one side and two from the other side and ask if our system remains in balance. Click---t6 show our new equation. The value of 2 can be removed from both sides. This one is a little more difficult to decompose 14 since there are not enough of the correctly numbered cubes, but some participants may try 10 and 4 and then take 2 away from 4 to get 2. Adding back in the 10, we have a value of 12.
Have participants try this equation. They should set up the equation and keep the system balanced by first removing 2 blue pawns from both sides and then the value of 5 from both sides. 13 may be decomposed into 5 + 8 or 10 + 3 to help in the solution process. X = 4 in the check process.
Participants take away pawns as a part of the set-up process. 5 blue pawns are placed on the balance and then 3 are removed. Participants use legal moves to solve the equation. 1 pawn from each side and a value of 2 from each side. Click to show each stage of the symbol with the hands on.
Participants should set up the equation on the balance. Three blue pawns and a 4 number-cube on left side and five pawns on the right side, then remove three pawns from the right side of the balance to show the subtraction, then place a 9 number-cube on the right hand side. Allow participants to use legal moves to solve for x.
Participants learn that what is on the inside of the parentheses is doubled as the number outside the parentheses are distributed over each value. The set-up is pawn plus 3 two times. Have students display the doubled portion in two rows on the mat. On the teacher set-up the doubling in seen next to each other. Subtract one x from each side and six from each side, the value of x is 2. 10 = 10 so it checks.
Students can begin to transfer their concrete, hands-on experience in solving algebraic linear equations to a representation with pencil and paper.
Have participants use the concrete to model the equation, represent the equation with paper and pencil and show their solution. Click on each step of the solution above and on the next slide.
Click to show the set up of the equation and the subsequent steps. Demonstrate that the balance now illustrates x + x = 8. Discuss that at this stage students will use their intuitive knowledge as they did in the first lessons to go to 2 x = 8 to x = 4. Most students will not be thinking of these equations in terms of procedures at this point.
Remember Quantity and Magnitude have many dimensions
Note Trainer: Make sure you are comfortable with the Mayer reading for the later parts of this section. Review both these slides and the reading to develop your comfort level.
Student need to understand fraction from different angles. It is important that we develop students ability to use benchmark fractions to understand quantity at the rational number level. The magnitude model of the number line is an excellent way to do this. Play from minutes 18:00 to 24:40
Explain that this is an adaptation of the problem used by Mayer in his article.
Ask the participants what type of knowledge is used in translating. Depends on both semantic (knowledge of facts and objects—knowing butter comes in sticks and there are 100 cents in a dollar) and linguistic knowledge (language used to present the problem) i.e. Per stick means each stick.
Problem solving must select the relevant information and organize it into a coherent mental representation such as a mental number line with Food Lion at 65 and Lowes at two steps to the right. Involves schematic knowledge. (Seeing the problem as a total cost= Unit cost x number of units)
Strategic knowledge is needed. Breaking a solution plan into its component parts. Monitoring and reflecting as you work through the plan is a part of planning.
Mayer’s stage of Planning. While we are planning we need to make sure that we are training kids to think about these types of issues as they approach problems. When you ask kids to organize a set of problems by similarity, students in the U.S. will organize them by their surface similarity---car problems, food problems, recipe problem whereas students from higher performing countries are far more likely to organize the problems by their structural similarity---in other words—problems that can be solved by algebraic slope would be organized together, and ratio problems would be organized together. What we tend to seem to do in the classrooms in the US, is that we work through each problem trying to get an answer for that problem rather than using physical quantity to connect to mathematical structure. Consider our work so far. We now know that the money problem fits in with the basketball percentage problem Structurally. Before our work, we may have organized all of the basketball problems together and not noticed which ones were related to 100 as the unit whole like a money problem. Just as Sharon Griffin found that quantity is more than just the quantity itself, but rather deeply attaching it to the structure of the number line and magnitude, we need to make sure that whenever we are using quantity (real-world problems) that we are at the same time connecting this to the underlying mathematical structure.
Procedural knowledge is needed. Symbols represent the quantity. Strategy or component error can be determined. Sense-making comes into play here.
Documentation Mayer article
One last note on subtraction. As you do word problems and discuss the math remember that there are 3 basic types of things that can happen in life that we can model and solve utilizing the operation of subtraction. This is NOT a key word strategy. It is designed to help students understand the underlying structure of the mathematics. As you will learn through foundations training, key words don’t work. Teaching underlying structure and teaching students how to represent these structures to make sense of the mathematics does work and develops students sense-making skills.
Repeated Addition: I am going to get 3 dollars each day I dog sit for my neighbors and they will be out of town for 4 days. Array/Row-Column: Area Questions such as My garden is 4 feet by 3 feet, how many square feet is it? Or questions such as, I need to set up the chairs in 4 rows of 3 chairs, how many chairs do I need? Counting Principle: I have 3 types of cones and 4 flavors of ice cream, how many unique one-scoop cones can I make? Also works for others with same structure like 3 tops and for pairs of shorts, how many different outfits, etc. These are the three types of things that happen in the world that we can model and then solve using multiplication. Students should be explicitly taught these structures and then use them to understand word problems and to model the problems at the representational level.
These are examples of assignment sentences – the meaning here can be picked up form a sentence without contextualized interpretation. Consider that key word strategies assume that every math word problem is made up of only assignment sentences. Often times teachers seem to convince themselves that key word strategies work by creating problem sets that are so controlled that they only contain assignment sentences. In the beginning stages of problem solving key words may work. This is an opportunity to begin to teach children structure that will last a lifetime; not key words that are sentencing them to “walking the plank” they will fall off if key words are their problem solving strategy.
Here is another place where the “plank” ends! Just as our elementary students struggled with comparison subtraction problems middle school and upper elementary faced with relational sentences stumble and fall into the sea of misunderstanding
Have participants translate these relational sentences. They have already seen this same type of problem with the students and principals in Unit 2, Slide 50. Extensive notes on this slide are in Unit 2 Trainer notes that will match the dialogue you may need to have again in this slide. What you can expect is the following equation 8 R= D because we read left to right—we translate left to right. You may also get R=D+8 In order to translate the sentence into R=D+8 the translator is not thinking proportionally. Proportional reasoning is not additive but rather multiplicative. Let’s consider the R=D+8, is it an accurate representation of the relational sentence. If there are 8 raccoons how many deer are there? 8= 0+8. Does this match our intuitive feeling about quantity? Another common misrepresentation of relational sentences is to understand that it is a proportional sentence and that the operation of multiplication is appropriate. The translator, therefore, invokes multiplication but perhaps because of earlier practice using key words and assignment sentences they literally translate from left to right. Incorrectly translate to 8R=D. Does this work? What is this sentence saying it means 8 multiplied by R = D. Why does this not work? Note to Trainer: This is algebra – take time to write / draw out / diagram what the sentence is saying. Make sure to have the participants use a number line for Zelda and Yolanda. Ask the question--Who is older? Mark that on the line. So where does Zelda go? What do we know?
Concrete and Quantity help the students to translate. Representational/Structural and Verbal help the student to Integrate and Plan The abstract symbolic level is where the students execute the plan.
Read through this slide with participants so that they get an idea of what is involved in numeration.
If you are working off the SCOS you may not recognize that there are foundational skills necessary for proper understanding as per the developmental sequence. Look at the standard course of study and find out how often a particular concept fits into your math curriculum. Trainer: access the website and glance at the NCSCOS, and other resources for K-12 math Proposed changes can be found to the new curriculum. Make certain that you show the participants the SCOS links, the Indicators by grade level, Week by Week Essentials, classroom strategies and black line masters that are in the resources to the right on the page.
How do we know what the students do know? When we look at a broad objective such as 1.01 in Grade 7, it is imperative to find out what the students do know in order for that student to master that objective. The student may have a simple misconception that is getting in the way of mathematical success. Once we know what the student does know, we can address what they don’t know. Note to trainer: This is just one example of how NCSCOS fits these probes. See handout folder. These probes will be used in Unit 6 also.
The arithmetic sub-test on the WISC factors in on the verbal section. For example y=x+5. This statement just says that y is the same value as x, just x has five more.
Teacher models counting sequence, emphasizing the new part. 2. Teacher models and test students on the new part of the sequence until students are able to say the new part correctly three times in a row. 3. Tell the students what number they will be ending with. 4. Test students from the beginning through the new part. 5. Frequently repeat in multiple settings in one to two minute sessions. NOTE: When the child is counting DO NOT interject corrections as the child is counting.
Often children get the multiplication facts dealing with 2, 5, 10’s. Why is this? It is because we have taught them to skip count earlier in the SCOS. Consider using basic numeration skills to support students with their multiplication tables. Why would you teach counting in this order? Because you can connect to student’s prior knowledge of skip counting by 10’s. 9’s (using the distributive property) is just one less than 10. 12 is (11+1) therefore 12 x 2 is (11 x 2 + 2) and 12 x 3 is (11 x 3 + 3) and etc. The language piece is very important. Ask the student, “How do you do that?”
Procedurally, how do we figure these problems out? Base Ten---trust it!! Discuss the importance of these questions as an opening activity regarding ten percent. Possible answers to the questions. Connecting to prior math knowledge Pointing out patterns Connecting to benchmark fractions Reinforcing the importance of benchmark fractions Creating opportunity for students to discuss and help make sense of the concept Creating opportunity for students to connect to what they already know Getting smaller Less Less Ask where 10 % falls on a number line. Closer to 0 or closer to 1? (0) Not very good since it is less than ½ of his/her shots It means that for every $10 you pay—you get $1 off or you pay 90 % which is $ 9 for every $10.00 Stress here that what you pay and your discount equal 100%.
Practice, practice, practice, but what are your practicing? Procedurally, how do we do this? How do you teach this without just practicing the procedure of “moving it over one”? How do you get kids to visualize and deeply understand ten percent? What is ten percent? It means, for instance, if someone has money, you get ten dollars our of every 100 dollars. So, if they have $100, you get $10. But what if they have $400, how much do you get? You could teach this by saying, let’s move it over one, but particular for the struggling leaner it is important that you connect through counting/numeration and not just a procedural trick that we know from experience that doesn’t build the conceptual understanding of 10%, nor does it work long term as trick for this type of student. So what you are practicing for the kid for $400, “ So, let’s think what this means, let’s see if he has $400, that mean s we get 10 for the first $100, 10 for the next $100, then 10 for next 100, and ten for the last $100. Have I counted for $400? Okay, so how much do I get? 10, 10,10 and 10 more makes 10, 20, 30, 40. So, I get 10% of $400 and that is 10 for every 100, which is $40.00!” “Remember, that is what ‘per cent’ means. So in this case, I get 10 per cent which means 10 per hundred, which we figured out to be $40 per $400. A link to the hundreds board is appropriate as a visual for students. Example 1: What is 10% of 50. Well 50 is ½ of a hundreds grid. If I had 10% of a hundreds grid, then I would have ten, but I only have ½ of a hundreds grid so I must have 5. Example 2: What is 10 % of 400 Ask participants what 10 % of 100 would be. (10) But I have 400 which is 4 x 100 so I must have 4 x 10 which is 40. Proportional reasoning is stressed here as well in this though process.
Thinking about hundreds grids again could be very helpful as you guide participants through this discussion. 20% of 800 10 % of a hundreds grid is 10, so 20 % of a hundreds grid is 20. I don’t have just one hundreds grid, I have 8, so 8 x 20 = 160. 20 % of 800 is 160 Find 15% of 800 10% + 5% means that I have 10% of 8 100s grids---so I have 80, but then I have another ½ of that amount since 5% is ½ of 10%. I have 80 + 40 which is 120. Also, think about complements. We know that if we have to find 20% of something, the complement to that is finding 80% of that amount. If I go to the mall to buy a pair of pants and they are have a 40% off sale, then I know I will be paying 60% of the original price. For example, the cost of the pants is $30.00. 40 % off would be 3 + 3 + 3 + 3 = 12 off (10% + 10% + 10% + 10%) or we could think of it as 3 + 3 + 3 + 3 + 3 + 3= 18 that we will pay (10% + 10% + 10% + 10% + 10% + 10%). Students will be empowered with these connections to base ten and equality.
Fifty percent of what is important to know as we measure quantity.
Bullet 1 Keep it, change it, switch it. What does this mean???? We teach students to keep the sign of the first integer the same, change the operation sign from subtraction to addition and then change the sign of the second integer to its opposite. Are we teaching the understanding of why this is so or are we just teaching them to memorize the trick. Bullet 2 When good things happens to bad guys, that is bad, right? ( + times a – is a -) When bad things happen to good guys, that is bad, right? (- times a + is -) When a good thing happens to a good guy, that is good. ( + time + is +) When a bad thing happens to a bad guy, that is good. (- times – is +) Bullet 3 When two socks that match are pulled out of drawer---that is good (+). (+ times + = + and – times – is +) When two socks that don’t match are pulled out of drawer, that is bad(-). (- times + = - and + times – is -). What might confuse our students here is that putting socks together implies adding them not multiplying serving to only make things more difficult in terms of understanding. Bullet 4 What does the word “larger” imply? Is -8 larger than 3? No, we need to communicate to students the meaning of absolute value and how we find difference between two absolute values and then use the sign of the number with the greater absolute value.
If yellows are positive and reds are negative, what is the value of the quantity? ( 0 ) What happens when you add two numbers that are opposites, like 5 and -5? In a sense, they &quot;cancel&quot; each other, resulting in zero. In other words, the sum of a number and its additive inverse is zero. What is the absolute value of the red counters? (3) How many units is that from zero? What is the absolute value of the yellow counters? (3) How many units is that from zero? What is the opposite of the red counters? (Yellow counters) What is the value of the red counters. (-3)
This is model for vocabulary that helps students to develop linguistic understanding of mathematical concepts. It’s power is in the use of non-examples. Non-examples should be introduced after students have been exposed and understanding examples of the concept.
There is a difference between a tool, such as a calculator, which does the thinking and supplies the answer and a concrete model that helps students explore concepts. Model Integers on a number line. Students perform operations with integers on number lines. Two color chips can provide a visual that can then be represented as a tool to provide meaning for integers and their operations.
Think of all of the math that can be communicated with this mathematical tool.
If a number is to the left of a number on the number line, it is less than the other number. If it is to the right then it is greater than that number.
Discuss how -3 can be modeled on the number line. How would we add the following expressions on the number line? 2 + 2 -2 + 2 1 + -2 -1 + 2 Subtraction -1 - -2 In order to subtract, I must add the opposite, so, I begin at -1 and then I move to the right 2, since I must reverse directions in order to add the opposite. What about multiplication? It is just repeated addition! -1 x 3 = -3 (I have three groups of -1) Since I can commute numbers, 3 x -1 is also -3. Division is the opposite operation of multiplication so if we begin with -12, how many groups of -3 can we make with -12? (4 groups)
See Slide 89 Dialogue in Trainer Notes.
We talk about distance above and below sea level. (Sea level is at 0) Stocks rise and fall from one day to the next so this rise and fall is measured as gains and loses with respect to particular benchmark---the day before, the month before, etc. Temperature rises and falls, again with respect to a particular benchmark---last year at the same time, etc. Golf is played according to par—over or under. In this case of quantity--the lower score wins the game since you were able to play the course in fewer strokes.
Connecting the Components of Number Sense of Quantity, Magnitude and Numeration therefore is more than just working with computations. A student must be able to not only be accurate but understand the math well enough to use different solution methods and make decisions about which solution method is best for a given situation. Toni Meyer and Elizabeth Murray---- Adding It Up
Presentation 5 quantity magnitude and numeration january 2
Unit 5: Quantity / Magnitude and Numeration
Quantity / Magnitude Sharon Griffin Richard Mayer Russell Gersten, et al
Quantity / Magnitude Defining the Concept Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
Geary and Hoard, Learning D isabilities in Basic Mathematics from Mathematical Cognition , Royer, Ed. <ul><li>1-1 Correspondence </li></ul><ul><li>Stable Order </li></ul><ul><li>Cardinality </li></ul><ul><li>Abstraction </li></ul><ul><li>Order-Irrelevance </li></ul>Gellman and Gallistel’s (1978) Counting Principles
Abstraction <ul><li>3X and 2X </li></ul><ul><li>3Y and 2Y </li></ul><ul><li>3X and 2Y </li></ul><ul><li>3 ones and 2 ones </li></ul><ul><li>3 tens and 2 tens </li></ul><ul><li>3 tens and 2 ones </li></ul><ul><li>3/6 and 2/6 </li></ul><ul><li>3/6 and 2/5 </li></ul>
Use of the Number Line <ul><li>Quantity and Magnitude: </li></ul><ul><li>The physical amount of something. In mathematics this is displayed in many forms. For example sets, symbols, growth patterns, measurement, weight and relationships are all ways to demonstrate and quantify magnitude. </li></ul><ul><li>It is particularly important for the student to associate magnitude to the NUMBER LINE. The idea that moving in a certain direction implies an increase in quantity. (Griffin) </li></ul>
Different Forms of a Number-- Linking to Magnitude to Number Lines Number Worlds Griffin
Diagnosis Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
Early Indicators of Problems with Quantity/Magnitude: <ul><li>Early sequencing skills are delayed </li></ul><ul><li>Visual spatial issues </li></ul><ul><li>Child has difficulty rote counting </li></ul><ul><li>Child has difficulty with rational counting (1 to 1 correspondence) </li></ul><ul><li>Adapted from Harding & Harding </li></ul>
Number Knowledge Test http:// clarku.edu/numberworlds
Name__________________________ Score _____ Date________ Preliminary Let’s see if you can count from 1 to 10. Go ahead. Number Knowledge Test Level 0 (4-year-old level) ____ 1. Can you count these Counters and tell me how many there are? (Place 3 Counters in a row in front of the child.) ____ 2a. (Show stacks of counters, 5 vs. 2, same color) Which pile has more? ____ 2b. (Show stacks of counters, 3 vs. 7, same color) Which pile has more? ____ 3a. This time, I’m going to ask you which pile has less. (Show stacks of counters, 2 vs. 6, same color.) Which pile has less? ____ 3b. (Show stacks of Counters, 8 vs. 3, same color.) Which pile has less? ____ 4. I’m going to show you some Counters. (Show a line of 3 blue and 4 white Counters in a row, as follows: BWBWBWW.) Count just the white Counters and tell me how many there are. ____ 5. (Pick up all the Counters from the previous question. Show a mixed array—not a row—of 7 white and 8 blue Counters) Here are some more Counters. Count just the blue counters and tell me how many there are. *Go to Level 1 if 3 or more correct In order to receive credit an item that has an a and b, both a and b must be correct to be counted in the score.
Name__________________________ Score _____ Date________ Number Knowledge Test Level 1(6-year-old level) __ 1. If you had 4 chocolates and someone gave you 3 more, how many chocolates would you have altogether? __ 2.What number comes right after 7? __ 3.What number comes two numbers after 7? __ 4a Which is bigger: 5 or 4 ? 4bWhich is bigger: 7 or 9 ? __ 5a This time, I’m going to ask you about smaller numbers Which is smaller: 8 or 6? 5b Which is smaller: 5 or 7? __ 6a Which number is closer to 5: 6 or 2? (show visual array after asking) 6b Which number is closer to 7: 4 or 9? (show visual array after asking) __ 7. How much is 2 + 4 ? (okay to use fingers) ___ 8. How much is 8 take away 6 ? (okay to use fingers) ___ 9a When you are counting, which of these numbers do you say first? (show visual array 8526-ask child to point and say each numeral) 9b When you are counting, which of these numbers do you say last? *Go to level 2 if 5 or more correct In order to receive credit an item that has an a and b, both a and b must be correct to be counted in the score. .
Name__________________________ Score _____ Date________ Number Knowledge Test Level 2 (8-year-old level) _____ 1.What number comes 5 numbers after 49? _____ 2.What number comes 4 numbers before 60? _____ 3a.Which is bigger: 69 or 71 ? 3b.Which is bigger: 32 or 28 ? _____ 4a.This time I’m going to ask you about smaller numbers. Which is smaller: 27 or 32 ? 4b Which is smaller: 51 or 39 ? ____ 5a Which number is closer to 21: 25 or 18? (Show visual array after asking the question) 5b Which number is closer to 28: 31 or 24? (Show visual array after asking the question) ____ 6. How many numbers are there in between 2 and 6? (Accept either 3 or 4) ____ 7. How many numbers are there in between 7 and 9? (Accept 1 or 2) ____8. (Show visual array 12 54.) How much is 12+54? ____9. (Show visual array 47 21.) How much is 47-21? (use term take away) *Go to Level 3 if 5 or more correct In order to have an Item that has a and b, both a and be must be correct to be counted in the score.
Name__________________________ Score_____ Date________ Number Knowledge Test Level 3 (10-year-old level ____ 1. What number comes 10 numbers after 99? _____ 2. What number comes 9 numbers after 99? _____ 3a. Which difference is bigger, the difference between 9 and 6 or the difference between 8 and 3? _____ 3b. Which difference is bigger, the difference between 6 and 2 or the difference between 8 and 5? _____ 4a. Which difference is smaller, the difference between 99 and 92 or the difference between 25 and 11? _____ 4b. Which difference is smaller, the difference between 48 and 36 or the difference between 84 and 73? _____ 5. How much is 13 + 39? (Show visual array of 13 and 39.) _____ 6. How much is 36 – 18? (Show visual array of 36 and 18.) _____ 7. How much is 301 take away 7? In order to receive credit an item that has an a and b, both a and b must be correct to be counted in the score.
Developmental Level Comparison Chart Raw Score Developmental Level Score C.A. Equivalents 1-3 -0.5 2-3 years 4-6 0.0 3-4 years 7-8 0.5 4-5 years 9-14 1.0 5-6 years 15-19 1.5 6-7 years 20-25 2.0 7-8 years 26-28 2.5 8-9 years 29-30 3.0 9-10 years
Where Research Meets the Road Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
Different Forms of a Number-- Linking to Magnitude to Number Lines Number Worlds Griffin
Doug Clements, Julie Sarama <ul><li>Subitizing </li></ul><ul><li>This is a critical skill and may lay underneath early math number sense difficulties with addition and subtraction. </li></ul>
“Subitizing” Doug Clements <ul><li>What is subitizing? </li></ul><ul><li>What is the difference between perceptual and conceptual subitizing? </li></ul><ul><li>What factors influence the difficulty level for students in subitizing? </li></ul><ul><li>What are the implications for teaching? </li></ul><ul><li>What are some strategies that teachers can use to promote subitizing? </li></ul>
Different Forms of a Number-- Linking to Magnitude to Number Lines Number Worlds Griffin
Example: Percentages Predictions, Box scores, % vs. # 70/100 = 70 % 7/10 = .7 = 70% 35/50 = .7 = 70% 1 2 Recording the information Sense-making Touchable or visual
Even Algebra can fit this mold! How does Speed affect $; What is the constant? 1 2 4(s-65) + 10 Faulkner adapting Leinwald, Griffin Quantity: Concrete display of concept Verbal: Mathematical Structure Symbols: Simply record keeping!
Example: Algebra—Solving Equations 1 2 Sense-making 8 Equations Maintaining Balance X+X = 8 2X = 8 X = 4 Adapted from Borenson: Hands on Equations Touchable or visual Recording the information
Research <ul><li>Mercer’s research supports the idea of teaching students with disabilities with </li></ul><ul><li>C-R-A approach </li></ul><ul><li>Concrete </li></ul><ul><li>Representational </li></ul><ul><li>Abstract </li></ul><ul><li>http:// www.borenson.com/html/video_demo.html </li></ul><ul><li> Hands-on Equations provides excellent Concrete/Representational support for students. </li></ul>
Level #1: Lessons #1-#7 <ul><li>Solve equations such as: </li></ul><ul><li>2X + X + X + X + 2 = 2X + 10 </li></ul><ul><li>and </li></ul><ul><li>2(X+4) + X = X + 16 </li></ul>
Materials for level #1 <ul><li> Eight Blue Pawns </li></ul><ul><li> Two Red Cubes , Numbered 0-5 </li></ul><ul><li> Two Red Cubes , Numbered 5-10 </li></ul><ul><li> A paper drawing of Balance Scale </li></ul>
Demonstrate Equality <ul><li> Using “Human Scale” demonstrate the idea behind EQUALITY </li></ul><ul><li> This is a Big Idea in algebra and needs to be reinforced again and again </li></ul>
Solve by Trial and Error beginning with Modeling and Think Aloud <ul><ul><ul><ul><ul><li>3 2 8 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Lesson #1 </li></ul></ul></ul></ul></ul>
Blue Pawn is named “X” <ul><li>X + 4 = 2X + 3 </li></ul><ul><li> Physically Represent Equation </li></ul><ul><li>Solve by Trial and Error: for us that will generally mean Model how you would solve by trial and error. Think Aloud. </li></ul><ul><li>Lesson #2 </li></ul>
Legal Move and Equality <ul><li>Introduce the Concept of a “ Legal Move ” </li></ul><ul><li>Review the idea of Equality , show that it is a “legal move” for a mathematical reason </li></ul><ul><li>4X + 2 = 3X + 9 </li></ul><ul><li>Lesson #3 </li></ul>
Pictorial Notation <ul><li>4x + 3 = 3x + 9 </li></ul><ul><li>6 </li></ul><ul><li>x x x x 3 x x x 9 </li></ul>X = 6 Check 27 = 27 Lesson # 7
Solve Using Representation <ul><li>2 (x + 4) + x = x + 16 </li></ul><ul><li>4 4 16 </li></ul>
<ul><li>2 (x + 4) + x = x + 16 </li></ul><ul><li>8 </li></ul><ul><li>x 4 x 4 x x 16 </li></ul>2x = 8 so x = 4 Check 20 = 20
Quantity/Magnitude <ul><ul><li>Size </li></ul></ul><ul><ul><li>Weight </li></ul></ul><ul><ul><li>Number in a set </li></ul></ul><ul><ul><li>Length </li></ul></ul><ul><ul><li>Capacity of a container </li></ul></ul><ul><ul><li>Value of an object </li></ul></ul><ul><ul><li>Area </li></ul></ul><ul><ul><li>Time </li></ul></ul><ul><ul><li>Likelihood of an occurrence </li></ul></ul>
Classroom Applications Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
Quantity/Magnitude in action Utilizing the Number Line to develop understanding of fractions <ul><li>http:// www.centerforcsri.org /webcasts/fractions/ </li></ul>
Mathematical Problem <ul><li>At Food Lion, butter costs 65 cents per stick. </li></ul><ul><li>This amount is 2 cents less per stick, than butter at Lowes. </li></ul><ul><li>If you need to buy 4 sticks of butter, how much will you pay at Lowes? </li></ul><ul><li>(adapted from Hegarty, Mayer, Green, 1992) </li></ul>
Translating <ul><li>Converting a sentence into a mental representation </li></ul><ul><li>At Food Lion, butter costs 65 cents per stick. </li></ul><ul><li>So --- Food Lion = 0.65 </li></ul>65
Integrating <ul><li>Building a mental model of the problem situation </li></ul><ul><li>Butter problem seen as “total” cost </li></ul><ul><li>(Food Lion Butter + 2) x 4 = Cost at Lowes </li></ul>FOOD LION LOWES 65
Planning <ul><li>Devising a plan for how to solve the problem. </li></ul><ul><ul><li>First, add 2 cents to 65 cents and then multiply the result by 4. </li></ul></ul>
Planning <ul><ul><li>Structural similarity vs. Surface similarity </li></ul></ul><ul><ul><ul><li>Structurally similar </li></ul></ul></ul><ul><ul><ul><ul><li>Money problem (100) and basketball percentage (100) </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Speed of a car (slope) and salary problem (slope) </li></ul></ul></ul></ul><ul><ul><ul><li>Surface similarity </li></ul></ul></ul><ul><ul><ul><ul><li>All basketball problems together </li></ul></ul></ul></ul><ul><ul><ul><ul><li>All transportation problems together </li></ul></ul></ul></ul>
Executing <ul><li>Carrying out the plan </li></ul><ul><li>65 + 2 = 67 </li></ul><ul><li>67 x 4 = 268 </li></ul><ul><li>268 cents = $2.68 </li></ul>
<ul><li>Executing </li></ul><ul><li>Those who learned with a </li></ul><ul><li>multi-representational approach were able to generalize the skill better. </li></ul>
Structures of Addition <ul><li>Join and Part-Part Whole </li></ul><ul><ul><ul><li>There is something, and you get more of it? </li></ul></ul></ul><ul><ul><ul><li>There are two kinds, how many all together? </li></ul></ul></ul><ul><ul><li>Start Unknown </li></ul></ul><ul><ul><ul><li>Some are given away, some are left, how many were there to start? </li></ul></ul></ul><ul><ul><li>Compare--total unknown </li></ul></ul><ul><ul><li>- I know one amount and I have some amount more than that--how many do I have? </li></ul></ul>? ? ? Addition types Adapted from Carpenter, Fennema, Franke, Levi and Empson, 1999, p. 12 in Adding it Up, NRC 2001. Taken Left What did I start with? How many altogether? How many do I have?
Structure 3 types of subtraction <ul><li>Ask yourself if a problem is a subtraction problem— </li></ul><ul><li> Does it fit one of these three types?: </li></ul><ul><ul><li>The Classic “Take away” </li></ul></ul><ul><ul><ul><ul><li>(how many left?) </li></ul></ul></ul></ul><ul><ul><li>Comparison </li></ul></ul><ul><ul><ul><ul><li>(difference between? who has more?) </li></ul></ul></ul></ul><ul><ul><li>Deficit/Missing amount </li></ul></ul><ul><ul><ul><ul><li>(what’s missing?) </li></ul></ul></ul></ul>You’ve got some amount and “take away” from it: What’s left? You compare to see: Who has more or less? You need some more to get where you want to be: What is the missing amount? ? ? ?
STRUCTURE: 3 Types of Multiplication: 4 x 3 <ul><li> </li></ul><ul><li>Repeated Addition </li></ul><ul><li> Array/row-column </li></ul><ul><li>Counting Principle </li></ul>
STRUCTURE: 3 Types of Division: <ul><li>Measurement/Repeated Subtraction </li></ul><ul><li>“ How many 2s can I get out of 10? ” </li></ul><ul><li>Partitive/Unitizing/Fair Shares </li></ul><ul><li>“ How many would 1 person get? </li></ul><ul><li>Or “What would that mean in relation </li></ul><ul><li>to 1?” </li></ul><ul><li>Product/Factor “If I have an area of 10 and one side is 2, how long is the other side?” </li></ul>If I have 10 cups of beans and I give out 2 cup portions, how many servings will that provide? 10 10 If 2 people find $10 how much will each person get ? 10 2 ? ? ? 2 2 2 2 2 ?
Assignment Sentences <ul><li>Food Lion = 65 cents </li></ul><ul><li>Let x = 5 </li></ul><ul><li>Laura’s Age = 21 </li></ul><ul><li>The 727 travels at 250 mph </li></ul><ul><li>1 pound = 16 ounces </li></ul>
Relationship Sentences <ul><li>The price of Food Lion’s butter is 2 cents less than the price of Lowe’s butter per stick. </li></ul><ul><li>y = x - 1 </li></ul><ul><li>Laura is 2 years younger than Valerie. </li></ul><ul><li>The 757 travels 56 mph faster than the 727. </li></ul><ul><li>The package weighs 5 ounces less than the pound of hamburger. </li></ul>
<ul><li>Translate these Relational Sentences </li></ul><ul><ul><ul><li>There are 8 times as many raccoons as deer </li></ul></ul></ul><ul><ul><ul><li>Zelda is four years younger than Yolanda </li></ul></ul></ul>
Numeration Defining the Concept Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
Numeration: Defined <ul><li>numeration noun. the action or process of calculating or assigning a number to something. a method or process of numbering, counting, or computing.ORIGIN late Middle English : from Latin numeratio(n-) payment (in late Latin numbering ), from the verb numerare to number. </li></ul>
Numeration Concepts <ul><li>Set – collection of elements </li></ul><ul><li>Cardinal Number – identifies number of set </li></ul><ul><li>Numeral – symbol representing number </li></ul><ul><li>Rote counting – sequence of numbers </li></ul><ul><li>Rational counting – one-to-one correspondence </li></ul><ul><li>Ordinal numbers – first, second, etc. </li></ul><ul><li>Manipulatives- concrete objects </li></ul><ul><li>Skip counting – 2, 4, 6, 8 </li></ul>
Craven County 7th Grade Probes <ul><li>Goal 1 Number and Operation </li></ul><ul><li>1.01 Develop and use ratios, proportions and percents to solve problems </li></ul><ul><li>Decimal Representation </li></ul><ul><li>What decimal is represented by each picture? </li></ul>
Diagnosis Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
Indicators: <ul><li>Child can’t rote count </li></ul><ul><li>Rapid naming (number identification) </li></ul><ul><li>Language piece (has issues talking about numeration) </li></ul><ul><li>(Arithmetic on IQ test IQ/ Verbal side) </li></ul>
Where the research meets the road Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
Introduction of Skills <ul><li>Rote counting- </li></ul><ul><ul><li>Determine how high they can count. </li></ul></ul><ul><ul><li>Stop student as soon as you hear a mistake just say “stop” and start over do not say the missed number out of sequence. </li></ul></ul><ul><ul><li>Spread out practice. </li></ul></ul><ul><ul><li>10 min. is too long </li></ul></ul>
Introduction of Skills <ul><li>Skip counting </li></ul><ul><ul><li>Pre-skills </li></ul></ul><ul><ul><li>Rote count ten numbers for each series </li></ul></ul><ul><ul><li>Order of skills - counting by 10’s, 2,5, 9, 4, 25,3,8,7,6 </li></ul></ul><ul><ul><li>Model-lead-test </li></ul></ul><ul><ul><li>Add new numbers each day with at least 2 of the old numbers </li></ul></ul><ul><ul><li>Practice a number set </li></ul></ul>
Classroom Application: Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
Set 2 WCPSS Middle School Remedial Warm-ups Wake County Public School System/Valerie Faulkner email@example.com
Numeration across the NCSCOS <ul><li>Percents </li></ul><ul><li>Ratios </li></ul><ul><li>Decimals </li></ul><ul><li>Fractions </li></ul>
<ul><li>10% of 100 10% of 200 </li></ul><ul><li>10% of 50 10% of 150 </li></ul><ul><li>10% of 23.28 </li></ul><ul><li>a) What is happening here? The numbers are ______________. </li></ul><ul><li>b) When you figure 10% of something, will your result be more or less than what you began with? ______________ </li></ul><ul><li>c) Is 10% more than ½ or less than ½ of something?_______. </li></ul><ul><li>d) If a basketball player made 10% of their foul shots, would that be very good, or not good at all? ________________. Why?__________________________________________. </li></ul><ul><li>e) If a store item is on sale for 10% off, what does that mean? </li></ul><ul><li>________________________________________________. </li></ul>
Tricks We Use <ul><li>Same-Change-Switch </li></ul><ul><li>Bad Guys vs. Good Guys </li></ul><ul><li>Matching Socks </li></ul><ul><li>Just Subtract The Smaller From The Larger And Keep The Sign Of The Larger </li></ul><ul><ul><li>( - 8 + 3 ) </li></ul></ul>
Prototype for Lesson Construction Additive Inverse Opposites Absolute Value 1 2 Symbols Simply record keeping! Verbal: Discussion of the concrete Quantity: Concrete display of concept (-5) + (-6) = (-11) -5<5 8 – 6 = 8 + (-6) V. Faulkner and DPI Task Force adapted from Griffin
Frayer Model for Linguistics Characteristics Definition (in own words) Non-Examples Examples Integers Whole numbers and their opposites Thermometer (-1 degrees) Stock Market (Down 2) Elevation (45 feet high) <ul><li>Positive, negative and zero </li></ul>0.56
Concrete Display Of Integers <ul><li>Number Line </li></ul><ul><li>Two Color Chips </li></ul><ul><li>Four Pan Balance </li></ul><ul><li>Thermometer </li></ul>
The Number Line Source: http://www.learningwave.com/chapters/integers/numline.html