David Chard’s (and Carnine) example of area-The article talks about how we teach area by unique formulas and they argue that all computations of area can be derived by using one formula of base x height. Teaching with the Big Ideas makes lesson planning more difficult and requires teachers to step out of the box and embed the big ideas into all concepts. Research says that students will become more proficient and as we use the Big Ideas to teach we become better teachers---we must understand the mathematics rather than just teach procedures. IN order to support teachers with the big ideas, we have created the Components of Number Sense model. This model helps you to understand what to look for in your lessons to make sure that you are communicating big ideas. Where did they come from? Reinforcing the Big Ideas in lessons is critical. What are the most important big ideas that you should be driving home in all of your lessons. Use these components of the big ideas to connect new concepts to those already learned---often teachers try to build lessons around one of the big ideas – when in fact, we should build our lessons around ALL of the Big Ideas for each concept.
Number and Operations, Measurement, Geometry, Data Analysis and Probability, and Algebra It is as if the teacher is thinking that she needs to teach five separate subjects---when in fact all of the components of number sense connect all the strands. What NCTM is trying to achieve in the strands curriculum----the number sense components will connect the strands into the rope of mathematical understanding
Number sense is the DNA of mathematics. Math without number sense is just ransom atoms.
We will be discussing these idea.
The physical reality of the mathematics that we model with symbols and the number line (how much, how far, how big, how bright, etc.) This component is foundational to every lesson---am I communicating the quantity and magnitude of the math concept? For instance---often times we will teach number as memorization rather than students deeply understanding sets. Can our students only rote count to 100 rather than understanding the quantity and magnitude first. Division of fractions is a key example—it is important to teach that concept with a deep understanding of the quantity. Keep in mind that “estimation” is a part of every component of number sense.
Naming number and connecting with one to one correspondence. It is the word, symbol, and the visualization. Thomas says that the numeration system allows us to allocate words for numbers---This connection allows for the sense making of quantity. Always use one-to-one correspondence with what you are doing at the time— Example 3 x = x + x + x Connect factoring to the distributive property. X(x+3) = x squared + 3x Bring it back as deeply as we can so the student can make connections. Hook them into what they do understand. An analogy to reading is when helping a student to understand the word rapid---take them back to what they do understand ---fast. Just as a strong teacher in teaching vocabulary, do not be afraid to take them back to explain a new concept as it attaches to a deeply understood and previously learned concept. This is called anchoring instruction. The first three components of number sense are anchors for building a rich understanding of math. What teachers tend to do is think of math as a discipline where you build on the skills, but their instruction does not tend to be recursive. In other words, whereas in teaching reading, we would have no problem talking about rapid in terms of fast. In math, we don’t think about going back to one-to-one correspondence to discuss the new concept. Would we ever consider not mentioning fast when discussing rapid? “ Oh my goodness---I can’t say fast---cause that is a baby word!”
The state of being of the same value, A=B. Children need to connect to equality physically and representationally and eventually symbolically. Equality is an incredibly powerful tool that permeates all of mathematics. Students need to recognize the many ways we can represent equality and inequality. What is equality? Are a and b the same thing? How are they equal? How are they not equal? The essence of equality is not sameness---it is equal value---equal weight---equal number of sides---congruent in size--- What is the difference between congruency and equality. There are more things in place for congruence---must be exactly the same. What is wrong with saying things are the same? The students understand what same is---but how can we connect understanding of equal through mathematical language? Same what????
Our number system is a Base Ten system. Because of this, knowledge of what that means and how that works is critical. The system defines how we build and construct number. We use 0-9 to represent all rational numbers. Base ten does not go away and continues to support our mathematical understanding. We need to have students think beyond units and with powers of ten.
This is probably pretty familiar to you in terms of standard notation. It does expand upon the powers of ten. Secondly, the chart should be constantly in the forefront of discussions involving number. It is not one unit that is taught in Grade 5, but rather permeates all of the development of the math concepts to follow and the discussions that we have with students. Warm-ups with students ---How many bricks are there in the school building? How many hairs are on your head? If you can estimate within a power of ten, you have a great deal of mathematical power. We don’t have estimation as a wedge as it is part of each component of number sense. What is the difference between guessing and estimating? If kids are thinking is base ten, then you are half way to having your students estimate sensibly! The number in the blue box is 6.5452.
We represent the same values in many different forms. As mathematicians we choose the form that works best for us, or describes best for us that part of the story. The concept of Form of the Number, begins in pre-kindergarten skills and continues through trigonometry, physics, etc. We need to develop in students’ minds that these different systems for evaluating values are interconnected (For instance 0.5 is a different form of the value ½; also, graphic representations are a different form of the same information we might also look at with an equation, or a chart.) These are all different forms of the same values/numbers/mathematical information. i.e. T-Chart, Cartesian Coordinate plane, concrete reality of the mathematical model, equation. Think about the Fraction domination game we discussed. For students to see that 4/6 + 1/6 is just a different form of the value 5/6 allows them to better access the idea of equal values looking different. Also it connects the idea of composing and decomposing numbers and allows them to have a framework for manipulating numbers without changing their values because they are thinking about “different forms of the same number”
Consider Liping Ma’s Article- Discuss article “Arithmetic in American Mathematics Education” by Liping Ma. -Discuss –What is Lei Shi? What are some ways to represent the number 12? Lie Shi is a mathematical concept that Ma had difficulty finding an equivalent for in American mathematics classes. Lie Shi, Ma said that is missing in math instruction in the U.S. She found the definition of what she is talking about in Sheldon’s Complete Arithmetic written in 1886. Lie Shi is a mathematical expression---various forms of the number. i.e. 4 + 4 is the same as 5 + 3 and can be represent by lines that make up a drawing of a house. Roof is 4 plus the other parts of the frame of the house is 4 and so is five lines for the roof frame plus 3 vertical lines that represent the structure lines Consider Liping Ma’s Article- Discuss article “Arithmetic in American Mathematics Education” by Liping Ma. -Discuss –What is Lei Shi? What are some ways to represent the number 12? Lie Shi is a mathematical concept that Ma had difficulty finding an equivalent for in American mathematics classes. Lie Shi, Ma said that is missing in math instruction in the U.S. She found the definition of what she is talking about in Sheldon’s Complete Arithmetic written in 1886. Lie Shi is a mathematical expression---various forms of the number. i.e. 4 + 4 is the same as 5 + 3 and can be represent by lines that make up a drawing of a house. Roof is 4 plus the other parts of the frame of the house is 4 and so is five lines for the roof frame plus 3 vertical lines that represent the structure lines
Beginning to think of number in a dynamic unit so that you are now investigating not the number itself, but it’s relationship to other numbers. This also begins early on—understanding cost of items and figuring a “unit rate”. Also, consider that a percentage is not really a “number” it is telling you what rate something is happening. So if I say I made 75% of my foul shots this year, I may have made 3 shots or 300 shots, but I did it at that dynamic ratio of 3:4. “ We view proportional reasoning as a pivotal concept. On the one hand, it is the capstone of children’s elementary school arithmetic; on the other hand, it is the cornerstone of all that is to follow.” (Lesh, R., Post, T. , & Behr, M. (1988).
How are things related physically, what information can we derive from patterns, how are the numeric patterns and the physical world related. Once internalized these concepts can be used to investigate both the physical and the theoretical universe. It is not a coincidence that Geometric and Algebraic Thinking are right next to Quantity and Magnitude----Algebraic and Geometric concepts describe, explain and predict quantity and magnitude in the real world.
As participants work the next two problems, they should think about procedural versus conceptual perception in building mathematical power.
Compare the two stories. You are coming up with a combined value of the two amounts. The values are constant. You will come back to reflect on this story later in the presentation.
Did you use the same algorithm for both situations? Should you have? What algorithms did you use? Why? STOP!!!! You will discuss the stories in a later slide.
Just say NO to just doing it! It is all about the understanding!! Picture drawing and visual imagery help students to fully comprehend a mathematical problem. Students need time to develop diagram fluency.
See Trainer Notes Unit 4 Slide 23.
Math is a mystery- but it is one we can solve through both inductive (parts to whole—if I have some of the parts of a triangle I can figure out the other parts) and deductive (whole to part-if I know that two things are equal, I can then figure out the values of variables within that equation) reasoning, just like - Sherlock Holmes. For example we teach addition and when the child gets this “skill” we move on to another “skill”, as if every skill is it’s own story, rather than all being connected through structure of mathematics. When you tell a first / second grader that you “Can’t take a big number from a smaller number” you are giving this student false information- so that – when students reach a latter stage in the “novel”, i.e. integers, they must debug themselves. For each time a student does a skill wrong – it takes six times of doing it right to learn it correctly.
Speaking of language—let’s see how Peter understands the mathematical term expand---
He’s thinking---he is trying to make a connection to something---
And—oh—he is on to something
He’s getting there—is it working???
ETC??? He is seeing a pattern
Note that Peter’s mathematical understanding of the homograph “Expand” is not understood. Like Tracy, he is doing the best he can with his understanding of the situation. Developing mathematical language is more than memorization.
You DO NOT RESOLVE all of these question yet. We are going to resolve them in the next few slides. You are THROWING THESE QUESTION OUT FOR THOUGHT AT THIS POINT! Did you use the same algorithm for both situations? Should you have? What algorithms did you use? Why? So why did you “add the denominators”? You did not really “add” in this situation, you adjusted your proportion based on total amount of shots taken. Why is it not okay to create equivalent ratios and add them in the basketball situation: While you would be maintaining the correct ratio for that player, you would be messing up the ratio for the team for that moment in time. You would be doubling the impact of the one player and would be saying that the team went 9/20? Or 9/40? Is this right? The reality is that as you combine the shot making of different team members it is not the same as adding fractions. Why? Because the proportions themselves change. You are talking about “out of” a different number of shots. You are using the same mathematical symbols to tell different stories. You have to know/understand your story to understand how to manipulate the symbols. We will confuse kids if we do not separate the stories into two very different problems in the math.
Let’s see how strong instruction might resolve these questions in students minds and enable them to approach these situations with meaning. We have covered the components of number sense and how important language is, What does this look like in the IMPLEMENTATION of a lesson. Remember we know from the international studies that our instruction is only as good as the IMPLEMENTATION choices we make. How can we use the framework of the Comp. of Number Sense Model to improve our implementation choices? Give credit to Faulkner/Wake County see copyright. Training available through NCDPI
We are going to play Trashektball
Agreement on placement of foul line and backboard availability (2 meters is about the right distance) or get participant suggestions that seem reasonable. Free throw averages are based on the number of successful shots from the foul line out of the number of shots that you take. Have participants determine factors that will effect the outcome of the trashketball. (i.e. Athletic ability, distance from goal from where the ball is shot, size of ball, size of the basket, backboard availability, height of player, length of arms, correct conversion procedure, etc.) Decimals, fractions (ratios) and percents Percents do not tell us how many shots we actually took. Good free throw percentage is 78% for the pros. Maybe 48% for our teams. Statistics help us to predict, but don’t guarantee that the same rate will be achieved. The same number out of may not stay static!
Review the procedures. Demo with the next two slides and show how to record the statistics in the various forms of the number. Have participants break into teams of 3-5 people. They will use the chart (next slide) to track their performance.
Have copies of this available. 1 per 3-5 participants (1 per team). Each team will take a total of 20 shots.
Need a copy of this slide (or 100 grid). One for each team. Teams should have already filled this out. Pick one team to work with and fill out for entire group. Use either a large graph paper or on overhead or Elmo OR project onto a whiteboard and have them use this grid to outline their work. Pick a team at random and demonstrate what their Percentage (per 100) is using the 100 grid.
Next use the numbers from Story One and demonstrate what that would look like. How is our money story like a percentage problem? (get someone to say that they are both based on 100 cents. Make the language connection.) Let’s look at what this would look like on our 100 grid. What is our unit whole? Yes. 100. Just like in a percentage problem where we are looking at how many in 100, a money problem revolves around 100 cents or one unit whole dollar. 1 dime plus 7 nickels 1/10 plus 7 nickels 7/20. How would we draw in our money? How is the money problem different from the basketball problem? You know how to “do it”---but why? If it is hard for you to understand---how do you communicate the understanding to students. Shade in 10 units for the one dime. Shade in 5 units for each nickel. 45/100 should be shaded.
What would the Story Two stats look like?
What would Story Two look like here? Just do 1 out of ten and just 7 out of 20 do NOT convert to percentage.
See Trainer Notes for Slide 42 Dialogue.
Remember the Prototype for Lesson Construction. We need to make sure that our instruction supports students in making important mathematical connections so that they can function in the world of the symbolic. How do we do this?
Use the Components of Number Sense to guide your questioning! This will improve your implementation choices, improve they types of mathematics discussions you have with students and improve student understanding of mathematical structures! Review the handout on questions to ask for Trashketball. Give participants 4-5 minutes to review hand-out and discuss briefly with table.
You have a class that reports the following statistics. Discuss fractions first. Make sure the students are grounded in the quantity of the fractions using Benchmark fractions. Before you estimate---what must you understand? Use of benchmark fractions---especially ½. For starters, we can look to see if anyone did better or worse than making Half of their shots! Do this with the group. The coach will give out a prize –who gets the trophy? We don’t know yet, but we have narrowed it down to two. There two groups that did better than 1/2. Who did the best? Why is it difficult to tell who did the best in fractional form? How can we determine who did the best? What form of this ratio/value is the best form for us to figure out the winner? Unlike denominators can make it hard to see who did the best. What if we convert to percentages. Do you see that converting to percentages shows us how each person would do if they made 100 shots? We are essentially making all of these fractions have the same denominator of 100! Then it will be easy to see. Use next slide to show this process. 5/8 > ½ = 0.625 = 62.5% 7/14 = ½ = 0.5 = 50 % 7/11 > ½ = about 0.64 (the .63 repeats as a decimal) = about 64% 7/16 < ½ = 0.4375 = about 44 %
You can use the grids to record ratios of shots made out of shots taken. Have participants think and talk through the representations of their ratios. Consider how you would do this with your class so that they can see what it means to represent the fractions in decimal/percentage form. Notice that you are really just creating the like denominator of 100 so that you can compare the fractions that have unlike denominators. That is one of the reasons to utilize percentages!
Need Bobcats Stat sheet. Have participants work in groups. Pretend you are the assistant coach and have to make a recommendation to the Head Coach. Note that there are several players who would be legitimate choices. It is important to see how, once you understand the different forms of the information that sometimes we want one form, sometimes the other and sometimes we need both! Consider a player who has made 3/4 foul shots all year versus one who has made 150/200. Both have same percentage, but are both performing in the same way? The stats can be projected for the participants on the next slide.
Do you get it? You must tell the story and connect all the previous chapters to the the rest of the novel. Sometime it is our job to help the child construct / write the chapters that have been previously taught in a “just do it” fashion.
Unit4 components
1.
Components of Number Sense Overview Unit 4
2.
What’s the “Big Idea”? <ul><li>Multiple authors recommend that teachers utilize the “Big Ideas” in their instruction. </li></ul><ul><li>Generally authors present the “Big Ideas” as lesson specific. </li></ul><ul><li>Often lessons are disconnected from each other and / or focus on one “Big Idea” for that lesson. </li></ul><ul><li>Big Ideas Revisited </li></ul><ul><li>We use the “Components of Number Sense” to connect every part of the curriculum in almost every lesson, thus, making our lessons more powerful and more efficient. </li></ul><ul><li>Outcome equals stronger teachers and stronger math students . </li></ul>
3.
How does this fit with the NCTM? <ul><li>The NCTM utilizes a strands curriculum </li></ul><ul><li>The Components of Number Sense are not in conflict with the NCTM strands. </li></ul><ul><li>The Components enable teachers to build true understanding of the mathematical strands. </li></ul><ul><li>This helps maintain and build connections for the way students think about mathematics. </li></ul>
4.
NCTM Strands Without Number Sense Number & Operations Measurement Geometry Data Analysis and Probability Algebra
5.
NCTM Strands With Number Sense Number & Operations Measurement Geometry Data Analysis and Probability Algebra
6.
<ul><li>What are the Components of Number Sense? </li></ul><ul><li>Language Connections </li></ul><ul><li>Classroom Implementation of the Components of Number Sense </li></ul>Components of Number Sense
10.
Reflection <ul><li>Write down individually: </li></ul><ul><ul><li>Reinforced </li></ul></ul><ul><ul><li>Learned </li></ul></ul><ul><ul><li>Want to Know </li></ul></ul><ul><li>Share with a partner or a small group </li></ul><ul><li>Complete this task in five minutes. </li></ul>
12.
* Table taken from Binary Number Systems, 2007 Total 654.52 Total 654.52 0.02 0.5 4 50 600 Weighted Value 2 5 4 5 6 Digits 10 -2 10 -1 10 0 10 1 10 2 Weights One hundredth One tenth Units Tens Hundreds
14.
“ Arithmetic in American Mathematics Education” <ul><li>What are the steps for solving a mathematical problem in China? </li></ul><ul><li>What is the American equivalent to Lie Shi? </li></ul><ul><li>Using 10 sticks model a figure and then talk about the mathematical expressions that exist within the visual. </li></ul><ul><li>What are the key points that Ma makes in this article? </li></ul>
17.
Reflection <ul><li>Write down individually: </li></ul><ul><ul><li>Reinforced </li></ul></ul><ul><ul><li>Learned </li></ul></ul><ul><ul><li>Want to Know </li></ul></ul><ul><li>Share with a partner or a small group </li></ul><ul><li>Complete this task in five minutes. </li></ul>
18.
Numeration Quantity/ Magnitude Base Ten Equality Form of a Number Proportional Reasoning Algebraic and Geometric Thinking Components of Number Sense
19.
Analyze the Lesson! <ul><li>How is that each component of the Components of Number Sense gets expressed in the teaching of the lesson as we connect the language and build true mathematical understanding? </li></ul>
20.
Story One <ul><li>You have a dime (1/10 of a dollar) and seven nickels (7/20 of a dollar). </li></ul><ul><li>In fractional form show how much money you have all together. </li></ul>
21.
Story Two <ul><li>A team has two team members </li></ul><ul><ul><li>3-point shots 1/10 and 7/20. </li></ul></ul><ul><li>How did the two of them perform as a team ? </li></ul>
23.
Language and Math Language <ul><li>Is math-language language? </li></ul><ul><li>How can we use some of what we do in reading instruction to improve our mathematics instruction? </li></ul>
25.
We have to teach math like we teach reading! <ul><li>How is math exactly like a mystery novel? </li></ul><ul><li>Do we really try to see the BIG picture? </li></ul><ul><li>How does our chapter in the novel fit the whole story? </li></ul>
32.
Two stories <ul><li>You have a dime (1/10 of a dollar) and seven nickels (7/20 of a dollar). </li></ul><ul><ul><li>In fractional form show how much money you have all together. </li></ul></ul><ul><li>Two team members record their 3-point shots as 1/10 and 7/20. </li></ul><ul><ul><li>How did they do as a team? </li></ul></ul>
34.
Trashketball The Power to Understanding Percents, Fractions and Decimals
35.
Let’s Try It! <ul><li>Agreements </li></ul><ul><ul><li>How far will the foul line be from the basket? </li></ul></ul><ul><ul><li>Are backboards allowable? </li></ul></ul><ul><ul><li>What else? </li></ul></ul><ul><li>Demo with two players </li></ul><ul><ul><li>How hot is their shot as a team? </li></ul></ul><ul><ul><li>Representing their stats </li></ul></ul>
36.
Trashketball Procedures <ul><li>Team members rotate shots. </li></ul><ul><li>After you shoot, you record your result. </li></ul><ul><li>Return to end of line for your next shot </li></ul><ul><li>Team Members stay positive with team members: “Nice shot!” “Nice try.” “You’re getting better.” “We’re doing great.” </li></ul><ul><li>Stay humble with other teams: “Great percentage!”, “Thanks”. </li></ul><ul><li>In teams of four, play Trashketball. </li></ul>
37.
Trashketball Stats Group Name _____________ Group Members: 1._________,(2)________,(3) _________(4) __________ <ul><li> </li></ul><ul><li> </li></ul> Shots taken: Make or Miss: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total made/Total taken _______/20 Decimal _____ Percentage_____
38.
What was our percent? <ul><li>How do we determine this ratio? </li></ul>___ / ___
39.
How Do We Represent the Money Problem Story One? <ul><li>How do we determine this ratio? </li></ul>___ / ___ 1 Dime 7 Nickels
40.
Trashketball Stats for Story Two Group Name _____________ Group Members: 1._________,(2)________,(3) _________(4) __________ <ul><li> </li></ul><ul><li> </li></ul> <ul><li>Shots taken: Make or Miss: </li></ul><ul><li>1 2 3 4 5 6 7 8 9 10 </li></ul><ul><li>Total made/Total taken 1/10 Decimal ___.10__ Percentage__10%_ </li></ul><ul><li>Shots taken: Make or Miss: </li></ul><ul><li>1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 </li></ul><ul><li>Total made/Total taken 7/20 Decimal __.35___ Percentage_35%___ </li></ul>
41.
100s Grids for Story Two <ul><li>How can our ratios be represented? </li></ul>___ / 10 ___ / 20
42.
How Did the Team Do Altogether Story Two? <ul><li>How do we determine this ratio? </li></ul>___ / ___
43.
How can the Quantity and its connections to the Math Structure provide understanding for the Symbolic ?
44.
What Do We Ask? <ul><li>Quantity/Magnitude </li></ul><ul><li>Numeration </li></ul><ul><li>Equality </li></ul><ul><li>Base Ten </li></ul><ul><li>Form of a Number </li></ul><ul><li>Proportional Reasoning </li></ul><ul><li>Algebraic and Geometric Thinking </li></ul>
45.
Trashketball Stats Group Name _____________ Team Members: 1._______,(2)_______,(3)________,(4)________ <ul><li> </li></ul><ul><li> </li></ul> <ul><li>Shots taken: Make or Miss: </li></ul><ul><li>1 2 3 4 5 6 7 8 </li></ul><ul><li>Total made/Total taken _______5/8 Decimal _____ Percentage_____ </li></ul><ul><li>Shots taken: Make or Miss: </li></ul><ul><li>1 2 3 4 5 6 7 8 9 10 11 12 13 14 </li></ul><ul><li>Total made/Total taken _______7/14 Decimal _____ Percentage_____ </li></ul><ul><li>Shots taken: Make or Miss: </li></ul><ul><li>1 2 3 4 5 6 7 8 9 10 11 </li></ul><ul><li>Total made/Total taken _______7/11 Decimal _____ Percentage_____ </li></ul><ul><li>Shots taken: Make or Miss: </li></ul><ul><li>1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 </li></ul><ul><li>Total made/Total taken ______7/16 Decimal _____ Percentage_____ </li></ul>
47.
Bobcats Stats! A coach can choose any one player to shoot “technical fouls.” If you were the assistant coach of the Bobcats and your team had to shoot technical fouls, who would you suggest to the head coach to shoot the foul shots and why?
50.
It is all about the language--- <ul><li>“I have heard many People say, ‘Give me the Ideas. It is no matter what Words you put them into.’” </li></ul><ul><li>To this he replies, “Ideas cannot be Given but in their minutely Appropriate Words.” </li></ul><ul><li>William Blake (quoted by J. Newman, The World of Mathematics , 1956) </li></ul>
51.
Assignments for Units 3 and 4 <ul><li>Due Day 3 </li></ul><ul><li>Learning Task 4 </li></ul><ul><li>(Components of Number Sense Questions) </li></ul>
Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.
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