Let’ assume that a = b. Then using the Multiplication Property of Equality : (States that when both sides of an equation are multiplied by the same number, the remaining expressions are still equal.) we can multiply both sides of the equations by a. a x a could be represented as a x b. (Multiplicative Property of Equality) Using the Addition Property of Equality: (Adding the same number to each side of an equation produces an equivalent expression.) we can add a squared to both sides. We can simplify both sides of the equation combining like terms and using the commutative property
Subtract 2ab from both sides of the equation using the subtraction property of equality. Simplify the right side of the equation by combining like terms. Using the division property of equality, divide both sides of the equation by a squared - ab
Cancelling out like terms leaves you with 1 = 2. Why?
Since a = b, you end up dividing by zero in the final step when you divided by (a squared –ab). You may want to do the proof with whole number to allow participants to see the division with zero.
In sign language, the representations for 11 and 12, just as in English are very different in the fact that they don’t follow the standard pattern. When we count, we say 6, 7, 8, 9 and then 10. 11 and 12 are the odd man out as we return to the pattern with 13, 14, etc. In sign language, the mechanical representations for 10, 11 and 12 are also represented in irregular patterns. Ten is a “thumb up, shaken”, Eleven is “index finger, bent back and forth” and twelve is “index and middle finger bent, back and forth”.
Have participants read this slide.
Handout: Math Student Profile checklist is in the handouts folder.
Site Page 23 in Griffin Article “The Development of Math Competence in the Preschool and Early School Years” in Royer’s Math Cognition . Note to trainer: read example 2 page 23 from Griffin’s Article and discuss how instead of the children thinking of place value as designated spot in a procedural system rather than understanding the true value of the number. We must tap into student understandings and misunderstandings. 8 year old structure from Griffin article. Students to not make connections between two number lines at once until they are about 8 years old. With this more complicated structure, students are able to understand place value. More concrete experiences will help develop readiness. Return to the six year old stage for remediation.
Piaget’s developmental model supports both Griffin’s and Mercer’s ideas of moving from the concrete to the more symbolic and formal reasoning.
Vigotski is a constructivist (the idea that knowledge builds on prior knowledge). He agrees with Piaget’s model, however, he says that we can move student through Piaget’s stages more quickly depending on the types of activities we do with them. He also speaks to the Zone of Proximal Development and states that children must be taught at the level they are ready to learn.
In the next few slides, we will pose some scenarios of student weaknesses. Get together in pairs and consider the research we have presented so far, the readings that you have done and your own understanding of students and the curriculum and suggest insights and appropriate interventions for each case.
Possible interventions: Use a hundreds board and repeatedly add and subtract ten using rods (or snap cubes) on top of the hundreds board. Have the student discuss the concept of the pattern that is occurring as the student is adding each ten rod. This reinforces both the concrete and its connection to the symbolic. Get students to think in base ten by subtracting 20 from 108 to get 88 and then talking through the quantity that they have subtracted one more than they needed to. How do I get that one back again? I have to add it back into my difference to get 89. This activity will reinforce the “mental number line” that children have.
Possible Interventions: Use base ten blocks and the concrete multiplication mat to have students see that there are 3 groups of 20. Compare what happens with 2 x 3 and 20 x 3. Encourage students to see that it is 3 x 2 ten rods which is 2 x 3 x 10. Have student compose 108 using base ten blocks. Begin subtracting out 9 since division is repeated subtraction until they can explain to you that the answer cannot possibly be 2. Ask student how many tens there are in 100. Then consider how many 10s there are in a 110 then how many 9s are in 108 in order to come up with a good estimate. After a sound understanding and an estimate is established, consider reviewing the algorithm for long division.
Possible interventions: Give the gal a calculator!!! Reinforce concepts of powers of ten so that she has a sense of whether her calculator answers make sense. If students have a strong number sense for base ten and understand that a reasonable answer for 4281 x 27 is 4000 x 30 which is 4 x 3 x 10 x10 x 10 x 10 (120,000). We did not say “move it over or add four zero”, instead we developed the concept of times ten utilizing base ten blocks and money. This way the student can actually visualize powers of ten and not just “add zeros”. Only then can a student “use a calculator”.
These strategies can be credited to Dr. John Woodward and are used throughout his Transitional Math program. If time permits, show Dr. Woodward’s website. He has word problems that promote number sense included on his website. Go to downloads and links.
After instruction with base ten blocks, students can begin to create a representation that builds on student understanding through the concrete. Demonstrate on Overhead using base ten blocks and talk about the mathematics and how it transfers to the symbolic. Use a representation system for the base ten blocks. Example: Use four lines to represent 40 (4 tens) and 3 dots to represent the 3 ones, etc. Make an overhead of the Concrete Multiplication Mat for this purpose. You will need overhead base ten blocks to illustrate the concrete of this model. Use the mat to illustrate the representations as well.
Demonstrate on Overhead using base ten blocks and communicating the mathematics of the symbolic. Use a representation system for the base ten blocks. In this example be sure to show how 1 ten is composed with the 9 ones and 1 more from the 5 ones. Use the grids provided in the trainer notes to explain the symbolic of the concrete.
Have participants work in pairs and use base ten blocks to model the addition problem. Talk through the quantity as it relates to the symbolic representation. Presenter note: Note that John Woodward writes answer. You might want to consider simplified form instead of answer because it more deeply communicates to the student that this is an equal representation---it is in a different form. Some students will still struggle adding 49 + 37 without the concrete blocks in front of them. Consider our synthesis of the Griffin and Mercer model and remember that these students may need more support at the representational level. Consider teaching these students explicitly how to draw four tens and nine one using sticks and dots, for example to help them to transition from the concrete to the symbolic.
Model this problem with base ten blocks and transfer to representational grid. 76 decomposes into 70 (7 tens) and 6 ones. Twenty-nine decomposes into 20 (2 tens) and 9 ones. We only have 9 and cannot subtract 9 from 6, so must decompose one ten. This will leave us with 6 tens and 10 ones and 6 ones. We combine the ones 10 + 6 which equals 16 ones. We subtract 16 -9 to get 7 ones. We subtract 60 (six tens) minus 20 (2 tens) to get 40 (4 tens), which equals 47. If our answer is 47, does that make sense? Estimate 80-30 = 50—pretty close and checking with a calculator—it is the correct solution.
Have participants complete this problem in their pairs and discuss. Use the Subtraction grid on the overhead to show the process. Participants should model 80 and 50 in the tens column and 1 and 2 in the ones column. Decompose 80 into 70 + 10. Then decompose the ten into 10 ones and combine with the 1 one to get 11 ones as in the previous example. Show the subtraction stage (11-2) = 9 ones and 70-50 is 20 which is 29. Estimate: 80 – 50 = 30 so we are close Calculator: 29—we got it! So the Answer is 29.
A reminder that this is not a base ten chart, but another way to decompose the quantity. It is 5 ones X 3 ones and 90 ones x 3 ones which equals 285 ones.
Participants should decompose the numbers as in the previous slide. 80 tens and 3 ones are multiplied by 7 ones. Use the multiplication grid in the trainers notes of Unit 6. 21 ones is recorded. 560 ones is recorded. 581 is the total amount of ones.
Estimation is valuable as a tool to gain insight into accuracy of our solutions.
With division we use a number line and extended facts to help estimate. We will see this in the following slides.
It takes me three times of moving 9 units to reach 27.
Dr. Woodward’s research shows the power of using powers of tens and near facts to have students make sense of their answers. Problem 1: We know that 48 divided by 3 is 16, but having students thinking about 48 as 50, the nearest power of ten and finding the near fact of 5 x 10 = 50 gives students a “ballpark” answer. This prevents kids from getting an answer like 30 divided by 48 and got 0.625 and accepting it as fact since the “calculator told them” that was the answer.
These are some of the questions that we need to ask ourselves regarding the students understanding. We should not assume anything we must “know” that the student understands the concepts at hand. If we do not know if the student has the concept or how strong the understanding of concept is- how do we find out? We can use multiple types of assessments including curriculum based evaluation and / or informal probes
Special education research on math has tended to focus on procedural metacognitive strategies and developing automaticity and fluency. General education math research has focused on international comparisons and trying to understand why our students are not performing as well as other countries. This research has been largely qualitative in nature. In other words, they are reviewing videotape and trying to understand the difference in implementation. Deborah Ball’s work fits in nicely with this in that she has been able to show that teacher understanding has an impact on student success. Much of what we can say regarding these topics is inferred from the research. In other words, what are these stronger implementation choices and how do teachers communicate a stronger understanding of the mathematics. Also, what types of metacognitive strategies truly communicate the math and not just the procedures of the math. Considering Liping Ma’s work, we begin to see that equality and forms of a number may be a large piece of this equation. In your reading you will remember, particularly in the subtraction section, that teachers consistently referred to teaching the idea of different forms of the same value and their sense of equivalent exchange as critical components in the lessons of subtraction. We in the United States did not refer to these concepts when teaching subtraction at all.
You have seen these next two slides earlier, but we need to revisit them. Is 1 equal to 1 cubed? Talk amongst yourselves. What does that mean? Make a sketch—one is a linear measurement and 1 cubed is a volume measurement. They are not the same thing. What does the form of the number stand for? (linear vs. cubic) Does this mean that they are still equal. We can’t say that they are the “same thing”, because they are not! But they are equal in numeric value! Equal in number dimension but not in the physical dimension.
Reinforce the information from the previous discussion. Language and accurately and consistently communicating a common language is a huge part of implementation choices. Are these equal? In what ways are they equal? Are there ways in which they are “not the same”? These two are equal in number value, but they are not the same thing when we consider their physical dimensions. This is what they found in the TIMSS studies. The countries that are producing stronger math students are the ones where the teachers are making these kinds of habitual implementation choices. They are very aware of how language communicates math accurately for all time. Are we laying down accurate mathematical language that will facilitate future learning and understanding of math as opposed to facilitating getting the right answer in this class at this minute. You are laying down synapses in the student’s brains that will last a lifetime.
What ever 3x is then y is the opposite.
Wow---all of these are forms of a number!!!
Give each table a different concept. Have them create a short role play that illustrates the student error. Address student strengths and weaknesses. Address how the implementation of a specific strategy might remediate the problem. In the role play, address the language that would be used in the process of instruction.
Have each group review via this slide the error and explain their remediation and use of language. The error is that the student does not understand the decomposing. Is the student seeing what we see “One hundred thirteen or are they seeing one ten and 13 ones. Possible remediation would include use of base ten blocks and use of John Woodward’s representational grid along with communicating the value of the digits and their place.
In all of these problems they are regrouping whether they need to or not. They have no concept of place value. 34-2 They are breaking it down into 2 tens and 14 ones subtract 2 ones = 2 ten and 12 ones = 212 56-51=4 tens and 16 ones subtract 5 tens and one one. They subtract the one from the 16 ones and end up with 15 ones and since they can't subtract 5 tens from 4 tens they say problem and reverse the order and end up with 1 ten = 1 ten and 15 ones = 115. 854 -60 = 7814 (Student borrowed from the tens place to exchange for 14 ones when he didn’t need to do so. 14 tens minus 6 tens is really 8 tens, but as a result of the previous error, the 8 is in the hundreds place. Is this still a correct digit? (yes) Having borrowed from the 100s place the 800 became 700 and 7 is a correct digit.
In all of these problems, the student is regrouping before they multiply the ones times the tens value. Example: 1 The student multiplies 5 ones times 7 ones, puts down the 5 ones, combines the three tens with the one ten to get 4 tens before multiplying by the 5 ones. The error continues. Possible remediation: Use concrete multiplication mats and look at the partial products strategy of multiplication. Transfer to Woodward grids as a representational step in the process and then to the symbolic.
The student knows their basic facts. The student’s error is conceptual as the student doesn’t understand the meaning of the digits that she is placing in the quotient. She does not understand the concept of division, but can do the procedure. The student is confusing the procedures for division with procedures that are involved in addition and subtraction. Possible remediation. Have the student use base ten blocks. Example 1: Have the student model 86 with base ten blocks. Have her divide the 8 tens into two equal groups and the 6 ones into two equal groups. Help the student to understand
The student does know their addition facts. The student seems to have no sense of place value. The student understands the procedure. For example: 9 hundredths is being combined with 3 tenths and the problem persists from that point with no regard to the value of the quantity. Possible remediation would include reinforcement with base ten blocks and emphasis on powers of ten. Transfer again to the representational stage is important prior to the symbolic stage.
Consider the following corrective dialogues Nice work. Yes, 13-5 is 8. You did that all in ones form, right? You put your 13 in ones form and then subtracted 5 ones? Good. Remember, though, that mathematicians like to start in Standard Form and then evaluate the form. So what is 13 in standard form? How would we say 13 in the Chinese language? Good-1 ten and 3 ones. That’s standard form. Go ahead and put 13 up in standard form. Okay now, evaluate that form and see what needs to happen. Yes, you need to make and equal exchange and turn it into ones form. I know it seems like an extra step, but remember we are not just learning how to “get the right answer” we are also learning how the number system works and how to think like a mathematician. So we will start in standard form and then evaluate the form and see if we need to make equal exchange. Okay, what do you mean by “the same problem”? (student will likely say that it is the same answer). Now that is true that these two problems are the same in simplified form, but do you see that they are decomposed differently here? .8 does equal .8, but let’s think of a story problem for these two situations. What if we said that you had 50 cents and than your sister gave you 30 cents so you could buy a bottle of apple juice. And in the second problem it says you had 40 cents and your sister gave you 40 cents for a bottle of apple juice. Is that the same thing? In one you needed your sister to give you more money to get the apple juice. It’s not the same thing! But, the apple juice did cost the same in both of these problems, this is true. Both problems are equal and therefore are the same in simplified form. (have student explain to you how the problems are different (different situations, different decompositions) and what aspect of them are the same (simplified form) Let’s look at this. I know you may have heard teachers say that these are “the same” but really they are not. What is the same? The “answer” right—they are the same in simplified form. But remember, that ‘s now the only thing to consider in a math problem. Let’s think of a story problem here. Let’s say you have 3 chickens and then you give 2 away to your neighbor. That models the phrase 3-2 right? You have 3 things and then you “take away” 2. Okay, but look at the second situation, in this situation you are Adding something. What if you had 3 chickens but then an infectious disease comes in and is added to the mix. It Kills 2 of your chickens. How many chickens do you have now? Yes, in both cases you end up with one chicken. But did it happen the “same” way? No. I like the fact that you can see that these things are the same in simplified form, but these are not exactly the same problem. Excellent. You are right, they are not the same thing. How are they different? (one is .50 and one is 50) okay good. But let’s look at this. What does .50 mean? It means you have one half of a whole, right. Let’s look at it on a number line. Let’s see, .10, .20, .30, .40, .50. Do you see it is right in the middle between 0 and 1? It is exactly one half. Now let’s look at 50 percent. Remember percent means “out of 100” so let’s look at this on a number line. 10, 20 , 30 , 40, 50. Do you see that 50 is exactly halfway between 0 and 100? So 50% is also one half. So these are both the same in fractional form, but you are right that they are not exactly the same thing. Often times, though we use decimals and percentages to talk about the Rate at which something happens. In this case, I could use either one of these expressions to say that someone was doing something half of the time. If I wanted to say that a basketball player made half of their shots I could say that in decimal form as .50 or in percentage form as 50%.
1. Base Ten, Equality and Form of a Number Unit 6
2. 1 = 2 ?
3. <ul><li>Let a = b </li></ul><ul><li>a 2 = ab Multiplicative Property of Equality </li></ul><ul><li>a 2 + a 2 = ab + a 2 Addition Property of Equality </li></ul><ul><li>2a 2 = a 2 + ab Simplify </li></ul>
4. <ul><li>2a 2 – 2ab = a 2 + ab – 2ab Subtraction Property of Equality </li></ul><ul><li>2a 2 – 2ab = a 2 – ab Simplify </li></ul><ul><li>2(a 2 – ab) = 1(a 2 – ab) Factor </li></ul><ul><li>2(a 2 – ab) = 1(a 2 – ab) Division </li></ul><ul><li>(a 2 - ab) (a 2 – ab) Property of Equality </li></ul>
5. <ul><li>Therefore 1 = 2 Simplify </li></ul>
6. <ul><li>Can you find the flaw in the proof? </li></ul>
9. Base Ten Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
10. Base Ten-Defined <ul><li>Base ten digits are often used with a decimal separator and includes the start of a fractional point and positive and negative numbers. </li></ul><ul><li>Linguistically, when we use English, the structure of the number words shows a base of ten, at least at the outset. When we write numbers, the structure of our number symbols also shows base ten. </li></ul><ul><li>Sign language mimics our language not our number system. </li></ul><ul><li>Walter S. Sizer Base and Subbase in a Number System </li></ul>
11. Equality Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
12. <ul><li>Equality is a mathematical statement of equivalence of two quantities and nothing more. </li></ul><ul><li>Cain, Faulkner, Hale 2007 </li></ul>
13. Form of a Number Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
14. Form of a number <ul><li>Form of the number can be defined as multiple representations of quantity, ratios, and mathematical information. </li></ul>
16. Diagnosis Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
17. Base Ten <ul><li>Has the student been exposed to base ten concepts? </li></ul><ul><li>What are the early signs of base ten issues? </li></ul><ul><li>Math Student Profile Checklist </li></ul>
18. Where Research Meets the Road Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
19. <ul><li>Sharon Griffin’s developmental nature of children using two number lines. </li></ul><ul><ul><li>6 year old structure </li></ul></ul><ul><ul><li>8 year old structure </li></ul></ul><ul><ul><li>10 year old structure </li></ul></ul>
21. <ul><ul><li>Zone of proximal development –Vigotski </li></ul></ul><ul><ul><ul><li>Constructivist </li></ul></ul></ul><ul><ul><ul><li>We can move students through Piaget’s stages more quickly depending on the types of activities in which engage them. </li></ul></ul></ul><ul><ul><ul><li>Learning happens just above the mastery level. (ZPD) </li></ul></ul></ul>
22. Classroom Application: Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
23. Base Ten <ul><li>Sharon Griffin </li></ul><ul><li>Liping Ma </li></ul><ul><li>John Woodward </li></ul>
24. Base Ten <ul><li>My student can’t do 42 plus 10. How do you remediate this student? </li></ul><ul><li>My student subtracts 108 – 19 and arrives at an answer of 89, but cannot explain their answer. Does that tell you that they understand base ten? How would you know? </li></ul>
25. Base Ten <ul><li>My student is multiplying 20 x 3 and writing the entire problem out and solving it procedurally and getting it correct. How do you respond? </li></ul><ul><li>My student divides 108 by 9 and arrives at 2 for the quotient? How do you respond? </li></ul>
26. Base Ten <ul><li>You have a student who struggles with memory and clearly cannot do 2-digit by 3-digit multiplication. She is a seventh grader and must be able to perform computations with decimals, fractions and percents. </li></ul><ul><li>How do we help her to use base ten to allow her access these SCOS skills? </li></ul>
27. Transitional Math John Woodward’s Strategies Building Number Sense http://www2.ups.edu/faculty/woodward/home.htm
32. You Try It! 81 - 52 Estimate: Calculator: Answer:
33. 90 5 x 3 15 270 285
34. Try It! <ul><li>83 x 7 </li></ul>x
35. What about Larger Numbers? <ul><li>389 x 78 = 400 x 80 = </li></ul><ul><li>Sense making of the numbers. </li></ul><ul><li>In the real world we use a calculator to at least check our work. </li></ul>Use Estimation to get a ballpark number. 4 x 100 x 8 x 10= 4 x 8 x 100 x 10= 32 x 1000=32,000 297 x 31 = 300 x 30 = 9000
36. Division <ul><li>Use a Number Line </li></ul><ul><li>Use Extended Facts </li></ul><ul><li>Use Estimation </li></ul>
37. <ul><li>Division </li></ul><ul><li>(Works with facts and Conceptually Guided Operations.) </li></ul><ul><li>Use of the Number line </li></ul>0 9 18 27 1 2 3
38. Division <ul><li>Estimation </li></ul><ul><ul><li>Promotes number sense </li></ul></ul><ul><ul><li>Gives students an explicit strategy to check solutions to problems they work on a calculator. </li></ul></ul><ul><ul><li>Uses the strategy of near fact. </li></ul></ul><ul><ul><li>Helps with two digit multiplication by removing power of tens. </li></ul></ul>
40. Diagnosis Equality/Forms of a Number Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
41. Diagnosis Strategies <ul><li>What kinds of experiences have students had with understanding and exploring equality/forms of a number? </li></ul><ul><li>Can the student tell you what equality is? </li></ul><ul><li>Can they recognize different forms of a number? </li></ul><ul><li>Does the student have an internal sense of a balance scale? </li></ul><ul><li>Can students explain what it means that these representations are different forms of equal values? </li></ul><ul><ul><li>Example .45 = 45/100 = 9/20=45% </li></ul></ul><ul><li>Assessments </li></ul><ul><li>Number Knowledge Test </li></ul><ul><li>Informal probes </li></ul>
42. Where Research Meets the Road Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
43. Research Equality / Form of a Number <ul><li>What does the research say about using equality and form of a number </li></ul><ul><li>TIMSS (1999, 2003) </li></ul><ul><li>Ball, 2006 </li></ul><ul><li>Ma, 1999 </li></ul>
44. Classroom Application: Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
45. Are these equal?
46. Are these the same? Are two cars equal to two cars? Is one elephant equal to two cars? Are these the same?
47. What about this? <ul><li>X = Y on a balance scale </li></ul><ul><li>3x + Y = 0 on balance scale </li></ul><ul><li>What else could you put on the left hand side the balance? </li></ul>
48. Forms of a Number 2x = y sin u = 1/csc u 45 = 40 + 5 X 2 = 0 x y 1 2 3 2 4 6
49. <ul><li>Discuss in small groups base ten concepts for addition, subtraction, multiplication, division, decimals and percents. </li></ul><ul><ul><li>Assignments by tables of concept </li></ul></ul><ul><ul><li>Report out by group </li></ul></ul><ul><ul><li>Now try it! </li></ul></ul><ul><ul><ul><li>Change the way we talk to kids </li></ul></ul></ul><ul><ul><ul><li>Role playing with student mistakes. </li></ul></ul></ul>
55. <ul><li>You say 13-5, and the student responds by placing 13 ones on the base ten mat and then takes away 5. How should the teacher respond? </li></ul><ul><li>A student says that 0.5 + 0.3 and 0.4 + 0.4 is the same problem. How do you respond? </li></ul><ul><li>A student says 3 – 2 is the same as 3 + -2. How do you show the student that these statements are not the same, but they have the same result? </li></ul><ul><li>A ask the student if 0.50 and 50% is the same thing and they say no. How do you respond? </li></ul>