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Teacher Interview

1) What are the student's strengths/weaknesses? a. The student has little motivation to do class work. Compared to the other students

in class, he is extremely slow. b.

If the student stopped and really took the time to do his work he is able to do it correctly.

2) Why is there difficulty? a. This student has a hard time gasping ideas. In terms ofclass time, the student has

an extremely short attention span. 3) Does the student have a Learning Disability? a. The student has an IEP for a Learning Disability. He also participates in the Title

1 groups. He is in need ofextra instruction and extra time. 4) Are there family issues?

a. The students family has been contacted and seems to care, but has not come through YET.

Student Observation

The setting for this observation is in an urban fifth grade classroom. There were about twenty students in the room, and the student being observed, "Danny", sits on the far right side of the room in the second to last row. Danny has nothing on his desk but a pencil, and he is slouched down in his chair. While other students continually talk to each other, Danny sits quietly. I would expect this student to be more aware of his surroundings and more social with his peers. This class has a general issue with behavior and talking out of turn. Many times it is a domino effect. If one student gets out of line everyone else in the class follows. This student is not an outcast from the rest of the class, and he seems to have a lot of friends. I would expect the student to partake in the chatter, rather than sitting there calmly. Also, I would expect that the student, being in school, would be more attentive. There are lessons being taught in the room, students talking, worksheets being passed out, and I would expect the student to be more responsive to the world around him. Due to his inattentiveness, I would expect the student to have a lot of issues completing his work. During this observation there was a math lesson being taught. After the teacher gives out the worksheet on angles, the student pays little attention to the teacher's direction. He begins to play with his pencil under his desk and then gets in trouble for it. Following this reprimand, the class was instructed to turn over their worksheet and begin. The student slouched deeper into his seat, held onto his pencil and put his head on his desk. After a few minutes of rest, Danny picks his head up and starts to write down the notes off the board. One can see that he is squinting;

therefore, the board may be hard for him to read. After writing down some notes the student becomes unaware again. He stops writing and stares at the class around him. He tries to start the worksheet again. In frustration Danny hits his desk with his fist and then throws his head back. As the teacher sees the students not working she goes to the board and explains how to do each problem. Danny does finish the worksheet and gets up to tum it in. While all of this is going on, the other students in the class are chaotic. Eventually, the teacher shuts the lights off and starts to yell. Everyone must put away all of their things and sit in silence. Danny falls into his seat and puts his hand over his face. The children sit with their heads down and are reprimanded for the remaining ten minutes. This student does and does not match my expectations. I am surprised that he did not participate in all of the class chaos. It seems to be a snowball effect, and he never got caught in it. Something that matched my expectations was the students lack of work ethic. He had little motivation to write down the notes and listen to the teacher when she was explaining the worksheet. As a result he was very frustrated. With the many worksheets the students complete daily, I would expect that the students are not engaged in their learning. As a result of this disengagement the student gets frustrated and the class gets out of control. I would expect the child to behave differently if the situation were to be different. The routine of the explanation of a worksheet, pass out the worksheet, complete the worksheet, get yelled at, and sit with your head down is a common occurrence in this classroom. If the student were to be more engaged in the lesson, his behavior would follow in suit. In terms of development, the student seems to be at the same developmental levels as many of the other students in the class, but overall the class is below grade level. The type of work that the student is completing is not meaningful. The student is not developing his thinking

skills, and more than likely does not retain most of what he should in class. The student does not have the ability to scaffold his thinking. Seeing that the students are not allowed to talk in class, Danny has not developed his ability to solve problems and interact with other students. Instead of speaking up about his frustrations, Danny simply shuts down. Generally speaking, Danny is not at the developmental level of other students his age. Overall I feel that this observation was not the best look at this students skill. There were many interruptions and the student was not fully invested in his work. While I think that a new situation would be beneficial to see how the student works in a math lesson, this observation was very true to the classroom environment which the student is a part of. There are constant interruptions and yelling. To some extent I feel that the students work is reflective of this environment. The student is in need of some individual math help

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Survey Test: Testing Conditions

The survey test was administered in the workroom of the elementary school. The student had a spot which was cleared off for him to work. The student sat on one side of the table with only his pencil and test. I sat on the other side of the table. The door was closed so that there would be little interruption while the student was taking the test. Upon starting the test the student was told that this test would not be taken for a grade and to try his hardest. I also said thatl I would not be able to answer any questions. If the student did not know how to complete the question he was told to try his best to solve the problem, but not to be upset if he did not know all of the answers. While taking the test, the student stopped often to look out of the window. There would be times when he would set his pencil down and simply stare. Other times he would stare at the paper and not write anything for a few minutes. The beginning of the test seemed to be easier for the student, and it become more difficult as the test went on. Often during the testing the student would ask me how to do a problem or ask ifhis answer was correct. The fact that I could not answer these questions for him was frustrating for him. He often chose to just write down a random answer. Towards the end of the test it was getting close to lunch time and the student asked me how long he had until lunch. He was concerned with the time and paid less and less attention to the test itself. When completed he put his head down and waited for me to take the test. He then left the room and went down to lunch.

Survey Test: Results After the student left the room I sat down and graded the survey test. On a side note, after speaking with the students teacher and being given academic information about the student, the student was given a survey test at a fourth grade level, although he is enrolled in the fifth grade. This served to be a good choice because most of the areas on the test did not meet the mastery level. The student excelled at operations, number concepts, and data. All other mathematical areas on the test did not meet mastery level. The two areas which will be chosen for the probes are time and fractions/decimals. In the following three weeks the student will be administered three sets of probes.

Survey Test: Fractions and Decimals Facts: Automatic and accurate computation.

Yes

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No

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Survey Test: Time

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Fractions and Decimals

(4.2.10) Use a standard algorithm to add and subtract decimals with fewer than 4 errors out of 17 questions given weekly. (4.1.7) Name and write mixed numbers as improper fractions using objects or pictures with fewer than 4 errors out of 17 questions given weekly.

Time

(4.5.9) Add time intervals involving hours and minutes with fewer than 7 errors out of20 questions given weekly.

Strategies The two weak areas which have been chosen are fractions/decimals and time. After analyzing the survey test and probes, one can see that this student does not understand the underlying concepts of these subjects. In terms of fractions and decimals the student places fractions in a random order, decimals are placed in order of largest to smallest despite their place value, and sometimes the student re-writes the problem given. In terms of time, the student does know how to tell time on an analogue clock, but does not understand time lapse. Often the student is inconsistent with his answers and has difficulty reading. These difficulties are hindering the student in moving forward in his mathematical studies, but there are many strategies which can be used to aid the student in comprehension of various mathematical areas. Strategy One The strategy which will be discussed is decimals, and the important of place value. This is a particular interest for the student because when placing in order decimals he ignores place value. The article, "Place Value as the Key to Teaching Decimal Operations" was published in Teaching Children Mathematics, a monthly journal. The article first states that many students are weak in decimal knowledge due to their lack of knowledge in place value. Students who do not have a flexible understanding of place value often confuse two things. First the students see the decimal portion of a number as a whole number. An example of this is 1.16 is larger than 1.8 because 16 is a larger integer than 8. The second misconception is the more digits there are, the smaller the number. An example of this is thinking that 12.94 and 12.32 are both smaller than 12.6 because they have more digits. These misconceptions are easily changed if the students practice understanding and expanded their knowledge of place value.

In order to expand on place value knowledge, students must understand the "dth's". Many students understand the idea of tens place, hundreds place, etc. but they may not see to the right of the decimal as tenths, hundredths, thousandths, etc. In order to work with decimals the student needs to be able to make that distinction. In order to help the student understand this concept manipulatives are often used. With the base ten blocks, teachers can change the value of what is "one unit" and make it possible for a single cube to be representative of a hundredth or tenth. With this strategy, students will learn to be flexible with decimals. The ultimate goal of this strategy is to have students see the right side of the decimal as an extension of the place value system, and to see the decimal as one quantity. If students are flexible and comprehend decimal place values, the two misconceptions mentioned above are no longer applicable. With ample practice, the student will be able to apply their knowledge to number sense, decimals and their close relative of fractions. The second part of fractions/decimals is working with fractions. Knowing that often students are confused by fractions, it is great to find ways to integrate and visualize these concepts. In the article "Painting Watercolor Fractions" the main goal for this elementary school teacher is for her students to get hands on work with fractions. The students start with a large piece of paper and cut it into two pieces, and then four pieces, and then each of the four sections get cut into an additional eight pieces. After the students have cut their paper they have seen the process of making 1/2, 1/4, 1/8, 1/16, and 1/32. From there, the students are told to paint 2/8 of one section blue, or 1/4 of the entire piece red. In order to follow the directions and make the art grid correctly, the students must understand fractions. Furthermore, throughout the art making process students are being questioned about their fractional pieces. These questions keep students thinking about the mathematical concepts as the primary part of the lesson and the art

making as the secondary part. This project is helpful for students who need the visual support of lessons. This project would be helpful to this specific student because he does not have an understanding of what fractions are larger than or smaller than others. By physically cutting out the pieces of the art grid the student would be able to see that although 8 pieces is more then 4 pieces, 1/8 is smaller than 1/4. Also, when the student is being questioned on what is bigger or smaller, he has tangible fractions to work with. The art section of this project allows not only for a visual representation, but it also alleviates some of the math pressure that this student feels. Often the student gets very frustrated with math worksheets and problems. If the student is doing a math assignment which is integrated with art he may be less likely to feel that pressure and become frustrated. Strategy Two For the third strategy the student will be concentrating on time. Often, if a student understands fractions and decimals they have a better understanding of time. The analogue clock works with fractions. While the knowledge of fractions may help a student learn about time, this student's difficulty is time lapse. Looking at the probes and survey test, it seems as if the student does not understand how time "adds up". Although many of the number systems taught in school are based on the idea of ten, time is not one of them. In the article "Time out for Time" this is a topic which is discussed. Time is based on numbers which can be divided by 60, 24 and 365. This is an abstract thought for many students. A strategy which this article suggests is to teach the idea of time and the lapse of time though an informal unit of measurement. The specific example is the use of a pendulum, but any consistent measure would work. Using the example of a pendulum, the students are asked to see how many swings it takes for the students to complete

a task. This task could be singing a song or cleaning up after an activity. The number of swings is the students unit of time. Once the students have this understanding of an informal unit of measurement, the teacher can introduce its relationship to time. Students will begin to understand what a minute really is, and how different measures of time relate to one another. Understanding the relation of time will help this student understand time lapse. If the student understands that once sixty minutes has passed it turns into one hour, the student will be less likely to answer a question with one hour and seventy-five minutes. The student would then realize that the answer would be two hours and fifteen minutes. This student would be able to understand time intervals and their relation to how the clock is read. Time intervals are important to understand how time repeats and how time moves throughout the day. In the article "It's About Time", the author suggests various activities for a class to do frequently. One of the suggestions it makes is for each student to create a clock and create pictures of various activities which are done throughout the day. Examples of these activities would be eating breakfast or lunch, taking the bus or going to soccer practice. The students are instructed to place the events on clock where they would occur during their day. Once they have done that the teacher can observe and ask them questions. "How much time is between breakfast and lunch?" "If soccer practice starts at 3:45 pm and school ends at 3:15 pm, how many minutes do you have to get to soccer?" The students would have a clock to look at in order to figure out how much time there is between these activities. This will support the students learning of time lapse and time intervals. This will be a worthwhile strategy for this specific student for multiple reasons. This activity can be done quickly, often, and it is able to be modified for many students. Seeing that this student has a hard time paying attention, the fact that this is done quickly is effective. The

student gets a review of time lapse and working with the clock. He is not overloaded with information and is able to work with his own clock at his own desk. In terms of modification, each student may have as many or as few events on their clock. Where one student may have five things to do that day, another may only have two. This allows every level oflearning to participate in the same lesson. Conclusion The main goal for this student is to find strategies that will help him learn the mathematical concepts, as well as accommodating to the students specific needs. The strategies given above are all interactive and hands on. Seeing that the student is enrolled in a classroom that is primarily worksheet driven and the student is not finding success, the worksheets are not an effective route. Looking at the research, the strategies provided allow the student to work with materials that are real and substantial, rather than trying to conceptualize abstract ideas. Moreover, knowing that the student has difficulty with attention span and reading, the activities alleviate some of that stress. These strategies should help the student comprehend the math and be able to apply the knowledge elsewhere.

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SOURCE: Teaching Children Mathematics v3 p448-53 April '97

The magazine publisher is the copyright holder of this article and it is reproduced with permission. Further reproduction of this article in violation of the copyright is prohibited. Some years ago 1 examined several middle school students' understanding of numbers (Threadgill-Sowder 1984). The answers that students gave me during that study showed me that their understanding, developed largely through experiences in the elementary grades, was fuzzy and led me to undertake a decade of research on children's number sense in the elementary and middle school grades. 1 will set the stage for this article by sharing two of the questions 1 gave the students during that study and some of the responses 1 received.

QUESTION 1 The sum of 148.72 and 51.351 is approximately how much? One student said, "Two hundred point one zero zero. Because the sum of 72 and 35 is about 100, and then 148 and 51 is about 200." (Note: 1 have used words for numerals where there is confusion about how the students read the numbers.) Another said, "One hundred fifty point four seven zero, because one hundred forty-eight point seven two rounds to one hundred point seven and fifty-one point three five one rounds to fifty point four zero zero. Add those." Fewer than half the students gave 200 as an estimate of this sum. The others saw a decimal number as two numbers separated by a point and considered rounding rules to be inflexible.

QUESTION 2 789 x 0.52 is approximately how much? One response was "789. 1 rounded point five two up to 1 and multiplied." A second student said, "Zero. This (789) is a whole number, and this (0.52) is not. It (0.52) is a number, but it is very small. You round 789 to 800, times zero is zero." Only 19 percent of the students rounded 0.52 to 0.5 or V2 or 50 percent. Several of them said that answering this question without paper and pencil was impossible and refused to continue. The majority of students had little idea of the size of a decimal fraction and applied standard rounding rules that were inappropriate for this estimation. Others who have studied elementary school children's understanding of decimal numbers have found that when

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. . . . . . . . . . . .(p. 161). These rules worked just often enough that students did not recognize that they were in error. (1 suspect that many teachers will recognize them.)

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More recent research on decimal-number understanding confirms that man1a1, . . .t";~}~1:~f~J,1Pin~,

of;d.IWI..um~lFor a summary of this work, see Hiebert (1992). The children in these studies were primarily' f frO'm classes where the introduction to decimal numbers was brief so that sufficient time would remain for the more difficult work of learning the algorithms for operating on decimal numbers. But time spent on developing students' understanding of the decimal notation is not time wasted. Teachers with whom I have worked claim that much less instructional time is needed later for operating on decimal numbers if students first understand decimal notation and its roots in the decimal-place-value system we use. In the remainder of this article I will discuss decimal notation and how we can help students construct meaning for decimal number.

GIVING MEANING TO DECIMAL SYMBOLS The system of decimal numbers is an extension of the system of whole numbers and, as such, contains the set of whole numbers. For the sake of convenience, this article refers to decimal numbers as those numbers whose numerals contain a decimal point. Decimal numbers, like whole numbers, are symbolized within a place-value syste~:i

'";O"""_«"<""f~"_.U'_'i~j;rhUS,children are taught that the 7 in 7200 i~"iii'th';;Ttf\~~l\,,!t<&ftce, the 2 is in the unarMs place, a (j Is in the tens place, and a 0 is in the ones place. But when asked how many $100 bills could be obtained from a bank account with $7200 in it, or how many boxes of 10 golf balls could be packed into a container holding 7200 balls, children almost always do long division, dividing by 100 or 10. They do not read the numbers as 7200 ones or 720 tens, or 72 hundreds, and certainly not as 7.2 thousands. But why not? These _<~.I~a(~~!!in ....
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question and the golf-ball question, and we need to practice reading numbers in different ways. Problems that require working with

powers and multiples of 10, both mentally and on paper, give students a flexibility useful with whole numbers, and

this flexibility makes it easier to extend instruction to decimal numbers.

'·"liif>_It~.The place-value name for 0.642 is six hundred forty-two is form with 642'''Y.he~e we simply say six hundred forty-two, not 642 ones~! IW_(thousandths, hundredths) or nths (tenths) with deci~i?F ana the use ot d (tnousanc. hundred') ~.~ n (ten) with whole numbers. The additional digits in the whole number with a similar name is another source of confusion. Whereas 0.642 is read 642 thousandths, 642 000 is read 642 thousand, meaning 642 thousand ones. In a number containing a decimal point, the units place, not the decimal point, is the focal point of the number, as shown in figure 1. The decimal point identifies where the units, or ones, place is located; it is the first place to the left of the decimal point. The decimal point also tells us that to the right the unit one is broken up into tenths, hundredths, thousandths, and so on. So really, 0.642 is 642 thousandths of 1. Put another way, 0.6 is six-tenths of ,ut lust as 0.6 is six-tenths of 1, 6 is six-tenths of 10,60 is six1, whereas 6 is 6 ones and . 6 te tenths of 100, and so on. < n in figure 2. Similarly, starting with the smaller numbers, 0.006 is six-tenths of 0.01, whereas 0.06 is six-tenths of 0.1. Moving in the opposite direction, 6000 is 6 hundreds 600 is 60 tens 60 i 0 ones! 6 is 60 tenths, 0.6 is 60 hundredths, 0.06 is 60 thousandths, and soon. '.:::~~ ~' . . . These issues are discussed more fully in Sowder (1995). Students who t2:...t:.2.!J:la~.sense of mathematics must become very confused when thev are told to aring numbers such as 0.45 and 0.6. _ • .. .,,,, . . . ''''~''!!'I .... , .• ,. r nstead of annexing zeros ,0 <.. ,_', ""'.•, ;,'"". ".'0 ,<~"., ,', . 0

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iil-leibert (1992) discusses research

showing that if students do not have a sound understanding of place value when they learn to add and subtract

decimal numbers, they make many errors that are very difficult to overcome because they are reluctant to relearn

how to operate on decimal numbers in a meaningful way.

AN INSTRUCTIONAL UNIT ON DECIMAL NUMBERS

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The unit summarized here was developed for a research study (Markovits and Sowder 1994) and resulted in students' performing much better on later decimal topics in their textbook. This unit has also been used by teachers who asked me for a way to teach decimals meaningfully. These teachers later told me that they thought the students who completed this instructional unit had a much better grasp of decimal numbers than did their students in prevJ.9YS l'~prS'
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which can be ordered from most catalogs of mathematical aids. The materials consist of individual centimeter cubes, long blocks that are marked to look as though ten cubes have been glued in a row; flat blocks that are marked to look as though ten longs have been glued that are marked to look as though ten flat blocks have been glued to form a into a ten-by-ten block, and lar e. . blocks .. " _ -- ._-- --.-.­ ten-by~ten-by-ten cube. .r&.~;:'·"'-'-··

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Students must play with the blocks and learn relationships to answer such questions as the following: * How many longs are in a flat? * How many small blocks are in 3 longs? * Where do you think there will be more longs, in 3 flats or in 1 big block? * I have 6 longs and 3 small blocks. What do I have to add in order to have a flat? * Which is bigger, that is, has ~_Iats or 48 longs? In the next lesson we begin to"" " ' ',:';1<' " - : : The small blocks are used to represent the

number 1. Students then are asked what numbers are represented by various sets of blocks: two big blocks, three

flats, and 2 little blocks; one flat and 2 longs; and so on. They must also represent numbers with blocks; for

example, they show 404 with blocks. Two-dimensional drawings can later be used for the blocks, and these drawings

can be used on assignments for problems like the one in figure 3.

AlternativelY,' stu,den,ts can be asked t(),,~~o~ WitQ);lI,9s's,~,!h.e 1~r,9'~,rL"~Jt",Q",,,.9,f!,,1J9)U~,l',W, P,i,~Ro,fth",e ""um,,b,e,rs 204 and 258 end so OfblOCk'"{ _d~(~:nutnbers when the small cube represents 1: .' * Can you represent 46 321 with the blocks you have? Why or why not?

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* Can you represent 8 V2 with the blocks you have? ~~~,""~~,, The next lessons should focus on changing the unit. _~~_(tudents

can then be asked to represent 76. (They would do so With seven flats and SIX longs.) they can again be asked, "Can you represent 8 V2 with the blocks you have? Why or why not?" (Yes, with eight longs

and five small cubes.) It is then worthwhile to ask a few questions--remaining in the whole-number system--where

the flat represents one unit.

Aff~'m'any'ilu~stions,

It is then natural to begin decimal instructton.. , , ' " < . "~'~.~~"_I'.'l!"~/"";l It is obvlouslv less than 1. What part of 1 is it? Since ten longs are in a flat, one long represents O.L Several " questions should follow: * How would you represent 0.3? 4.3? (See fig. 4.) * How many tenths are in four wholes? * What do you have to add to 0.9 to have one whole? * 4.5 is _ ones and _ tenths, or _ tenths. * Which of the followlnq are equivalent to one flat and four longs: 14? 1.4? 140? 14 longs? 41 longs? 41?

ukewise, children can come to understand that a small block in this context represents one hundredth, and many

questions similar to the previous questions can be asked. Teachers can also present such problems as the following:

In 6.40 are _ _ tens and _ _ ones and _ _ tenths and _ _ hundredths, In 6.4 are _ _ tens and

ones and _ _ tenths and

ones and _ _ tenths and _ _ hundredths. In 6.04 are _ _ tens and _ _ hundredths. Are any of these numbers the same? Why?

A great deal of practice is needed in each of the lessons described here; the questions indicated areonly a small

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{\sk hundred-thousandth. When students feel very secure with the blocks, with changing units, and with problems

involVing decimals, it is time to SWitch to another representation. A day or two spent with money--dollars and cents-­

will work well. Finally, a lesson or two should focus directly on decimal numbers without using another representation

(although many children will naturally answer in terms of "blocks" or "wood"). Questions like the followlnq can be

asked:

* Is 0.1 closer to 0 or to 1? * Is 1.72 closer to 1 or to 2? * I am a number. I am bigger than 0.5 and smaller than 0.6. Who am I?

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Page 40f5

Full text

* Are there decimals between 0.3 and 0.4? How many do you think there are?

* Are

there decimals between 0.35 and 0.36? How many?

* Are there decimal numbers between 0.357 and 0.358? How many? Draw baskets and label them "Numbers smaller than 0.5," "Numbers bigger than 0.5 but smaller than 1," "Numbers between 1 and 3," and "Numbers bigger than 3." Then give the students the following numbers and ask them to place each number in the appropriate basket: 0, 0.03, 1.01,5.08,2.63,0.49,0.93,0.60, 1.19, and so on. This type of problem can be made more difficult with baskets labeled "Numbers between 0.4 and 0.5," "Numbers between 0.7 and 0.8," and "All other numbers." If desired, these lessons could be interrupted before decimal numbers are introduced, and addition and subtraction of whole numbers could be introduced using the blocks. But when addition and subtraction of decimal .,---.--.......

numbers are introduced, ~-,-

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,··_c,~~~,·'!Br.J4k.gerfl _,_.____ uch students also develop a good feeling for th'e~sries of edina I numbers and can compare them with one another. It did not occur to any of the students who received this instruction to round 0.52 to 0 or to 1 when estimating a product--0.52 was simply seen as "about a half." When students understand what they are doing, they tend to enjoy doing mathematics. It is worth the time needed to build strong foundations. The time will be easily made up in future lessons, and students are much more likely to be successful.

ACTION RESEARCH IDEAS 1. For each of the following pairs of decimal numbers, ask students to tell which is smaller. Then analyze their answers to see if any are making the rule-lor rule-2 errors identified in the Sacker-Grisvard and Leonard (1985) study. Number Pair Use of Rule 1 Use of Rule 2 Correct 3.17 or 3.4 3.4 3.17 3.17 14.285 or 14.19 14.19 14.285 14.19 6.43 or 6.7216.436.7216.4311.01 or 11.002 11.01 11.002 11.0029.642 or 9.999.999.6429.642 15.134 or 15.12 15.12 15.134 15.12 156.1 or 156.012 156.1 156.012 156.012 If you find evidence of systematic rule-lor rule-2 errors, use some of the instructional ideas in this article and then reassess your students to determine whether their understanding of place value has improved. In addition to rule-lor rule-2 errors, look for other systematic errors that students make. What are the misconceptions that underlie these errors? 2. Assess your students' understanding of place value by asking such questions as the following. (a) The Sweet Candy Company places 10 pecan clusters in each box they sell. The cook just made 262 pecan clusters. How many boxes can be filled with the fresh pecan clusters? (b) There is $2148 in the bank, ready to be used for prizes for the state science fair. If each prize is $100, how many prizes can be given? Students with good place-value understanding will not need to do any division. Some students may solve (a) and (b) by using division. Some may not solve them at all. In either case try numbers like 260 or $2100 to see if easier numbers allow them to use their more limited place-value knowledge. If you find some students making large numbers of errors, use some of the instructional ideas in this article. Then reassess them using similar questions to determine whether their knowledge of place value has improved. Added material Judith Sowder is professor of mathematical sciences at San Diego State University, San Diego, CA 92182. She is the editor of the Journal for Research in Mathematics Education, published by NCTM. She has had a long-standing interest in number sense and has published several research articles and book chapters on this topic. Edited by Donald Chambers, National Institute for Science Education, Wisconsin Center for Education Research, University of Wisconsin--Madison, Madison, WI 53706. Readers are encouraged to send manuscripts appropriate for this section to the editor. FIGURE 1 Ones as the focal point of the decimal system

FIGURE 2 Alternative number names and representations when a long represents one unit

FIGURE 3 Substituting a two-dimensional drawing for blocks

FIGURE 4 Representing numbers with base-ten blocks

REFERENCES Hiebert, James. "Mathematical, Cognitive, and Instructional Analyses of Decimal Fractions." In Analysis of

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Arithmetic for Mathematics Teaching, edited by Gaea Leinhardt, Ralph Putnam, and Rosemary A. Hattrup, 283-322. Hillsdale, N. J.: Lawrence Erlbaum Assoc., 1992. Markovits, Zvia, and Judith T. Sowder. "Developing Number Sense: An Intervention Study in Grade 7." Journal for Research in Mathematics Education 25 (January 1994): 4-29. Sackur-Grisvard, Catherine, and Fran cots Leonard. "Intermediate Cognitive Organizations in the Process of Learning a Mathematical Concept: The Order of Positive Decimal Numbers." Cognition and Instruction 2 (1985): 157­ 74. Sowder, Judith T. "Instructing for Rational Number Sense." In Providing a Foundation for Teaching Mathematics in the Middle Grades, edited by Judith T. Sowder and Bonnie P. Schappelle, 15-29. Albany, N. Y.: SUNY Press, 1995. Threadgill-Sowder, J. "Computational Estimation Procedures of School Children." Journal of Educational Research 77 (July-August 1984): 332-36. J

WBN: 9709100445006

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4/19/2008

INTEGRATING

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the curriculum

second-grade teacher at our school had A asked for my help preparing his students for the standardized math test they would soon in

be taking. There m:e.fractipDs OD tbe,le§t and he "~ _fJ,(Our previous math/art lesson on symmetry' had produced excellent results on last year's test (see "Sym­ metrical Aliens," Arts & ActivitiesOct. 2004). This would be another great opportunity to cre­ ate a new lesson, combining two of my favorite subjects: art and math. The title of our new lesson opened eyes and produced comments from "Oh good, I love math," to "Oh no, not fractions!" There was mostly intrigue at the thought of bringing math into the art room and curiosity at how we would go about paint­ ing a fraction. To proceed with our lesson, we needed to review our color wheel in order to iden­ tify warm, cool and neutral colors. From an "Elements and Principles of Design" poster we learned that rhythm is the rep­ etition of colors. as well as shapes, lines, values, forms, spaces and

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< Luke Evans textures. It is what we can use to make our paintings active and exciting. We looked at reproductions of paintings that were divid­ ed into grids and observed how the artists used repetitions of shapes and colors to create beautiful art. (Spectral Sq %Wres, by Richard Anuszkiewicz, Spectrum Colors Arranged by Chance by Ellsworth Kelly, and Flora on the Sand by Paul Klee.) We began by folding a 12" x IS" sheet of SO-lb. drawing aoer in half both vertically and ho~zonta1ly-"" ~; I placer~~, crayons at the tables. Workin t02:eth~r, we used a red cray­ on to divide th

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if asked to paint 3Al of the blue rectangles a warm color? This was the format of the math test. We used the same format to reinforce the learning experience and connect the mathematical and artistic processes. I direct­ ed the activity. We worked on the entire

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r, be able to identify positive and negative shapes.

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12" X 18" heavy white drawing or watercolor paper '0 Watercolor sets containing 16 half-pans of warm, cool and neutral color> Q Yellow, blue, red and black crayons for each student

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let the students choose what colors to paint the remaining rectangles, which allowed all those who needed it, time to catch up. There was lots of interaction and sharing of knowledge during the process: "Was that 3"s of the blue section?" "Is magenta a warm color? Yes ... no '" look at the color wheell" My students enjoyed both the process and the great range of beautiful colors from which they had to choose. Because of the nature of watercolor paint, we made use of our "accidents" by creating colors with feathery blends and tie-dyed swirls. Part II of our lesson inVOJv=~'oUt 12Jl x :, olding our paint­ mgs horizontally, we cut them in half. One half of our painting was kept intact. The other half was cut into 16 col­ orful rectangles, Then out came the paper punches in a variety of shapes: snowflakes, stars, moons and ovals, extra large leaves, a musical note, a hand ... And, the Fiskars" Paper Edgers": Victorian, Seagull. Scallop, Wave .... I put six or seven of these tools on each of six tables see FRACTIONS 011 page 42

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and the stu­ dents shared. (The one rule I asked the students to follow was to only cut and punch shapes from the smaller rectangles, leaving the gil x 12 11 paper whole.) We referred back to our Elements of Design Poster on Rhythm and found the following quote. "Move­ ment and rhythm work together to create the visual equivalent of a musical beat." There were 16 little rectangles from which to create designs. In addition to repetitions of shapes and colors, we kept in mind the visual effects of contrasting col­ ors and, most importantly, positive and negative spaces. After a yellow paper was used to produce many lit­ tle positive shape stars, the negative star spaces became positive again when the paper was glued onto an orange rectangle. We overlapped lines to create plaids and created little landscapes. Some of our rectangles had radial symmetry, while others achieved asymmetrical balance. At one table, a group of drag­ ons appeared! Because our lessons last just 45 minutes, we made folders from 1211 x 1811 sheets of manila paper. At the end of one session, our collages were placed on the drying rack and our remaining colors of paper were stored in our folders. My students were very happy with the results. Our collages were dis­ played at our school's annual art show and received many compli­ ments. A beautiful part of these col­ lages is the quality of the colors, which were painted by the students as they were learning about fractions. Math is very often an integral part of artistic creation. It is gratifying when art can be used to teach and reinforce mathematical concepts and help our students to succeed. •

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10rdo No. 235 "" ARTS & AcnVll1fS ReaderSorvico e;.,rdl

42

e r t s & activities

I

april 2005

Michael

i'~aylor on

INTEGRA TING

Math IN YOUR CLASSROOM

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Building Decimal Success.

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. The sixth installment of our series on fractions takes '8 slightly different point of view - a decimal point, to be exact ,

~ecimal-Fractioi1

Match

Grades 3-5 Keepconnecting decimalideas to fraction ideas and your students will learn that, there's nothing.mysterious about either. Here are several trusted activities that will help kids build decimal success.

_II

rovide a list of deci­ mafs~~ctions which are close Comparing Fractions to t~ those decimals, Ask ;your sn;dents to Tentha(Gradea 2-5) . .. 'In the "N,,",".t _ • •~cJass cu"~~~;slOn, nave ~aems expiam Using a fraction circle, theirreasons for. the matches they made. squaresor othermanipulatives, have your Here's a sample list: students set up 10 tenths to make a unit. 115­ 0.732 Then have them build various fractions 1/2 0.490 and compare them to the tenths. Students 0.67 3/4 should feel free to estimate fractions 0.201 2/3 which-don't match exactly, "One half is 1.252 7/8 the same as five-tenths." "Three-quarters 0.811 119 is betweenseven-tenths and eight-tenths." 0.12 3/4 + 1/2

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Spot (Grades 3-5) Have your students cut out a circle the size ofa half-dollar and draw a smiley face on it. The face should be looking up and to the left. This guy's name is "Spot," and he looks at the pieceswhich are units, or ones.

Manipulatives can help students build fraction sense, encouraging them to make estimations such as, "One-third is slightly morethan three-tenths."

i I

I

I Michael Naylor is a professor of math education at

WesternWashington University, Bellingham, WA and -a Teaching Editor of Teaching K-8. E-mail: mnay )[email protected]

26

CIRCLE 12 ON READER SERVICE CARD

Spot keeps his eye on base ten pieces.

MARCH

2003 • www.TeachingK-8.com

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Have-yourstudentsset upa whole num- ~ul:tiplying With CWecimals Place the Point (~rades 3~t

berWithbaselOpieces- say, 156- and (~racles 5-8) Give your studentsdecimal number sen­ , ask them to place Spotso he's looking ~tWhen'multiplying two decimalnumbers, tenceswherethe decn:nal has been leftout

the ones. , suchas30.7x4.35,weusuallyignorethe . of the answer.Askthem.to use.number

Now move Spotso he's lookingat the decimal places atfirst and compute307x . sensetodetermine where the point goes. five long pieces. They were five tens, but 435 = :133545. Students learn a rulefor Here~ a few examples:

now Spot.says they're five .oneY. What where to place the decimal point- count . .

number do the pieces represent now? . the total number of places to the right of 14.5.08 + 3.799 7.] 8307

]'01.35 - 35,]01 ;= 66249

the decimal point 60.15 x 1.55=932325'.

and be sure the 84.7-:-1.75= 484

product has the [] 0 of same number 'D 0 decimal 'places. " Oecimal .~umber Line '1:1 [J instead of teach­ (GJrades 3-8) ,~ " ing the rule, have One problem children have with deci- . ~ kidsestimate what mals is tbinking5.l2 is .greater than 5.8, " the answer Will be since J2 is .greater than S.,You can help Now that Spot's looking at the tens, they become ones. This students see values of decimalswith a means tfJe single blocks-that.had been ones ,havebecore tenths. and'place the dec­ imalpoint accord­ number line. Draw a number line and The smallestpiecesare one-tenth the size .ingly. Students may reason this way: 30 label a few points for reference: Write .of the ones, so they'are tenths, and-the- - x 4=,120; so 30.7:x4.35should be.alit­ decimal numbers, on sticky-notes and large square is the -tens. The .nnmber 'is, . tie more.They'll have no.trouble placing, have studentsstickthem on the line. This , read as 15 aridsixtenths and written15.6., ,the decimalpoint: 133:545. Furthermore, . activity can be donein spare moments­ Students Can model other decimal num- theylll be ~timating and doing mental just be sure you have time for a discus­ I bers.writingthem in base 10 notation.•. ' matlt·. sion-afterwards. .'

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,Join Todi'f! 800·933·ASCD (2723) • www.ascd.org/joinascd.htrnl www.TeachingK-8.com·.MARCH2003

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1703 North Beauregard Stfeel Alexandria, VA U31N7UI

27

. MichaelNaylor on

Teaching Math

Time Out for Time

"Saxon bas enhanced our reaJi1"K and math programs, Tick-tack activities foreven the youngest children. and our scores reflect tbe $IICCe$S! we certainly have improved 011 ITBS, bllt we i also have other data to

You can also use a pendulum to time mirror the achievement.

how long it takesthe class to clean up ,Snxonprovides an easy-to­ from an activity or get readyto use mttllwl complete with go to recess. !ess(m plans a7ldsttuJent Using .a pendulum sheets ill an orgallked, also presents ~ oppor­ tt!acher-friendly manner. ..• tunityfor a science con­ Hats offto a wonderfUl nection: the amount of ~'" program that helps align time it takes for a pen­ Here are activ­ our clirriculmN to . ,'~ l dulum to swingbackand itiesthat makegooduse assessments 4"d our state:' 1.-1 forth depends only on the of classtime whileensur­ and local QCC. v :' ing a good time for all! length of the .string - not ~e ':§:.;..

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Hanna Fowler, PrinciP~ )

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Maxwell Thomson, ElementaryGeor Scho? , j'.,'.

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FIRST GllADE

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Understanding Time : (Grades ~·S)

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Try t~ching your st~dents ho.w to read time by removing the mmute hand from a clock and asking them to approximate the time. The above clock shows that it is "between 2 0:" clock and 3 o'clock."

how far you pull the pendulum to the side before releasing it. If you make all the pen­ dulums the same length, they'll swing at the same rate. The longerthe string, the longerthe periodof the swing.

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weight of the object hanging from it or

What's a Minute? (Grades C~·~) Keeping your eye on a clockwhich dis­ plays seconds, have the class clap with ,."lWfiiI',....".. .. -- ..',.I"a' our students should first record an estimate of you once each second Havethem count thenumber of swingsthependuium will with you from 1 to 60,and explain that make while they complete a task.'" 60 seconds make one minute. Do activi­ ~~ ties suchas thesefor exactly one minute: have the students closetheir eyesand sit quietly; have them stand on one foot; read them a story; have them write counting numbers. Have students rest Michael Naylor is a professor of math education at WesternWashington University, Bellingham. WAand their heads on their desks 'and raise their a Teaching Editor of Teaching K-8. E-mail: ronay Continued on page 24 [email protected] ~'

-1o_'1atIoiBalcSU••

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CIRCLE 45 ON READER SERVICE CARD

APRIL 2002 • www.TeachingK-8.com

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Teachers and students agree­ Saxon phonics programs produce results.

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R~vlew-

SAXON

PUBLISHERS

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For more information or free samples, . call (600) 284·7019. .

Time out for time.;. continuedfrom page 22

rove Strategic Thinking and Test-Taking Skills

hands when the

passed._

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Reading 'rime (Grades 1-3) Childrencan learn to read a digitalclock easily- it's simplya matter ofbeingable to read two-digit numbers, Learning to read an analog clock is another matter. _ _U1l1.R.emovethe minute hand and periodically ask stu­ dents to approximate the time. Is it al­ most 11 o'clock?Is it halfwaybetween2 o'clock and 3 o'clock?

't 'If Encourage thinking about quarters and twel:fths ofan hour- you'll be teach­ ing fraction conceptsat the same time! '.

'We know "it's .important to you to help your students practice more thanjust test-taking skills, You want your students to develop skills thatcanmake them successful throughout theiracademic careers AN.D beyond. The multilevel books in the Mathematical Thinking series give you thetools to help students develop creative thinking skills-so they not only gettherightanswer, they understand how they got it. Six levels of math problems help you tailor yoW: instruction to yo~ students' abilities: CTP2613 Level A (Gr.1-2)

CTP2616 Level D (Gr.3-4)

ISBN 1-57471-784-7

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CTP2614 Level B (Gr.1-2)

CTP2617 Level E (Gr.5-6)

ISBN 1-57471-'"785-5

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Crp 2618 Level F (Gr.5-6)

ISBN 1-57471-789-8

Tolocate the educational supply storenearest you, call 800-287-8879, or look one up on theWeb at www.creativeteaching.com.

24

CIRClE 19 ONREADER SERVICE CARD

"~';.)'.,;::,

., tr It's easy for students to figure out how many seconds are in a given period .of time. It's more difficult for them to convertin the other direction - for exam­ ple, how many days are in a millionsec­ onds? To convert from a million sec­ onds, divide onemillionby 60 to find minutes, then by 60 again 10find hours and so on. Have students compare one million seconds (about 10.5 days), on~ billion seconds (almost 32 years) and one trillion seconds (32,000 years!). ; (lo";,'''''lr~('?;'''' ~<:."":~'""~":~,~"""7""" --."" ~~>~';"~.N·

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APRIL 2002

• www.TeachingK.-8.com

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It's about Time

he "Math by the Month" activi­ ties are designed to appeal

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directly to students. Students may work on the activities individually or in small groups. No solutions are suggested so that students will look to themselves

as the

mathematical author­ ity, thereby develop­

• .uwer. ing the confidence to validate their work. This month's activities focus on investigating and exploring questions related to time, calen­ x:;:;::'::.::;:::'.'

dars, and the New Year.

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CURRICULUM-BASED MEASUREMENT DATA CHART

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Appendix C-2

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Appendix C-2

CBM Data

Probe 1 This probe was administered on February 28, 2008. This probe was administered in the workroom with the door closed. After the student sat down at the table he began to work on the fractions and decimals probe first. He asked multiple questions and was frustrated that I could not answer them. He continued to work and methodically went through the test. He started at the first question and went in order until the end. When he did not know the answer he started to circle things and write down random answers. Many of his answers were inconsistent and showed little strategy. When he was done with the fractions and decimal test he gave it to me and I handed him the time test. He sighed and put his head on the desk. After a moment of sitting there he picked his head up and began to take the time test. Throughout this test he asked fewer questions since he knew there was little chance of me answering them. He finished quickly and wanted to leave the room. In the middle of his test a teacher's aide came into the workroom and began singing. She stated that she eats her lunch in here every day at this time and that there is nowhere else she could eat. She saw that he was testing but made little effort to be quiet. She made her lunch in the microwave, photocopied, sang a little song and tried to have a conversation with me while my student was taking his test. It was very distracting to him, but in the middle of the probes I did not want to get up and make him move to a new location. I mentioned to her that he was taking a test but it did not seem to change her behavior. I do not know if this occurrence changed his test scores or if it just made the testing longer.

Fractions and Decimals Probe 1 Facts: Automatic and accurate computation.

Yes

Operations: Error patterns

No

Problem Solving: Incorrect! inconsistent strategies used in word problems

• • •

Place value error Random assignment of fractions No knowledge of mixed numbers

Not Applicable

Concepts: The underlying concept of the operation

No

Strategies: What strategies does the student bring to the computation?

No

• • •

Student circles the largest number despite place value Student randomly assigns answers to fractions Student rewrites the problem

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4) Write the fraction for 0.32 -2. :' . )

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5) Order this group of decimals from greatest to least \

0.368

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7) Order this group of fractions from greatest to least

1/4

1;2

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Order this group of fractions from least to greatest

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1/3

1/2

1/5

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Write this improper fraction as a whole or mixed number

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6/5

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Write this mixed number as an improper fraction

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Time Probe 1 Facts: Automatic and accurate computation.

Yes

Operations: Error patterns

No

Problem Solving: Incorrect! inconsistent strategies used in word problems

No

Concepts: The underlying concept of the operation

No

Strategies: What strategies does the student bring to the computation?

No

•

Reading clock

•

Time lapse

• •

Inconsistent strategy Reading difficulty

• •

Time lapse Does not understand 60 minutes is the same as 1 hour

• • •

Random answers Finger counting Leaving answers blank

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Time

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1) What time does this clock say?

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2) What time does this clock say?

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)(\ 3) In 15 minutes what time will it be?

4) What time was it 20 minutes ago? ..... \

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5) Leslie got on her bike at 2:30 pm. When she got home it was 3:45 pm. How

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long did Leslie ride her bike?

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7) How much time passes between 12:00 pm and 4:30 pm?

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9) Jacki has to be at school by 8 am. It is 6:45 am right now. How long does she have to get to school?

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Probe 2 This probe was administered on March 6, 2008. Considering what happened the week before in the workroom, I decided to change our location and go to a table in the library. This worked out well, even though there was a class in the library. My student finished quicker, which could have been result of fewer distractions around him. Also, this test was done after lunch, rather than before. This made a difference because the student was not overly concerned with time. He was not asking what time it was or when lunch was going to start. This time the student asked for the time probe first instead of the fractions and decimals. Throughout this test the student asked few questions. While watching him it seemed as of this may have gotten easier for him to complete due to the speed which he was finishing the probe. When grading the probe that statement did not stand true. In many of the questions he did not answer completely or correctly. Often in this time probe he would only give answers in hours without considering minutes at all. He rushed through it and did not take the time to really work on the problems. After completing the time probe he had asked to use the bathroom. I gave him a pass and he did not come back for a few minutes. I went to go look for him and he was found talking to other students in the hall way. He stated that he did not want to take another test, but reluctantly came back to the library. During the fractions and decimals probe, the student worked quickly and complained often. He wanted to go to the bathroom, the nurse and back to class. He randomly assigned numbers, left answers blank and rewrote what was in the question. Due to the behaviors during testing I do not know if this probe is a good representation of his knowledge in fractions and decimals. He was extremely happy to be completed that day.

Fractions and Decimals Probe 2

Facts: Automatic and accurate computation.

Yes

Operations: Error patterns

No

Problem Solving: Incorrect! inconsistent strategies used in word problems

• • •

Place value error Random assignment of fractions No knowledge of mixed numbers

Not Applicable

Concepts: The underlying concept of the operation

No

Strategies: What strategies does the student bring to the computation?

No

• •

•

Student circles the largest number despite place value Student randomly assigns answers to fractions Student leaves problem blank

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4) Write the fraction for 0.68

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5) Order this group of decimals from greatest to least

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0.23

0.087

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7) Order t.hiS group of fr.1J... " ns fromgreatest to least

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1 / 4 < 1/7

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11/9

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Write this mixed number as an improper fraction

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Time Probe 2 Facts: Automatic and accurate

computation.

Yes

Operations: Error patterns

No

Problem Solving: Incorrect! inconsistent strategies used in word problems

No

Concepts: The underlying concept of the operation

No

Strategies: What strategies does the student bring to the computation?

No

•

Reading clock

•

Time lapse

• • •

Inconsistent strategy Reading difficulty Incomplete answers

• •

Time lapse Does not understand 60 minutes is the same as I hour

• • •

Random answers Finger counting Answers in hours without minutes

Probe 3 This probe was administered on March 20,2008. This test set of probes was given during morning class activities. The student had a doctor's appointment in the afternoon; therefore, he was unable to take the probe later in the day. For this set of probes I took the student back to the library. There seemed to be more success in there than there was in the workroom. For this last set of probes the student chose to take the time probe first and then the fractions and decimals probe second. When taking the time probe he seemed to be more comfortable with the material. This was also evident in the fact that this was his highest scoring time probe. In the library he started to actually look at the clock on the board and the blank clocks on the probe to help him solve the problem. The visual support really helped this student. Also, knowing I could not help him, the student did not ask me any questions during this probe. This helped with his frustration level. Since he was not asking any questions, he was not getting upset with the lack of answers. He had come to terms with trying each question on his own. The student turned in the probe and asked for the second one. With the fractions and decimals probe there seemed to be a disconnection. Problems which came with ease on the first two probes he did not know how to do. Eventually he worked through them, but this probe did not have the success that the time probe did. By the end of this probe the student still did not have any knowledge of mixed numbers or improper factions. He randomly listed the order of fractions, and listed the order of decimals from greatest to least despite their place value.

Fractions and Decimals Probe 3 Facts: Automatic and accurate computation.

Yes

Operations: Error patterns

No

•

• •

Problem Solving: IncorrectJ inconsistent strategies used in word problems

Place value error Random assignment of fractions No knowledge of mixed numbers

Not Applicable

Concepts: The underlying concept of the operation

No

Strategies: What strategies does the student bring to the computation?

No

• • •

Student circles the largest number despite place value Student randomly assigns answers to fractions Student guesses

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Facts: Automatic and accurate

computation.

Yes

Operations: Error patterns

No

Problem Solving: Incorrect! inconsistent strategies used in word problems

No

Concepts: The underlying concept of the

operation

No

Strategies: What strategies does the student bring to the computation?

No

•

Reading clock

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• •

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•

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• •

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3) In 45 minutes what time will it be?

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CBA Reflection

The CBA process was a challenge for me. When giving the survey test I could already see that this was going to be a struggle. Personally, it was very hard for me to watch one of my students sit and be frustrated and not be able to help them at all. When my student would ask questions and for four weeks get problems wrong, it was hard to sit there and not help him. He did not mean to do poorly. When looking at all of the data together I feel that the things my student did poorly on where mathematical skills that he was never taught. If someone were to take the time to teach him these processes, he would do well. The idea of mathematics not being taught well in the classroom is disheartening. I know that this student is able to complete his work, but without instruction this student is blinded. When looking at the math observation, I wonder how the students learn anything in this environment. There is little instruction, no creativity, constant frustration and utter chaos. With all of the distractions that are in the classroom there is little learning that goes on. At this point I am unsure if the poor instruction is the fault of the teacher, the system, or a combination of both. If there were simple changes made to the environment itself I think that the entire class would benefit from the modification. Modify and accommodate are the two words which would be most important for this student. Ifthere were small changes made to the curriculum and the instruction this student would find more success. Right now he is traveling down an academic path full of struggle and frustration. If he is this disengaged in the fifth grade a change needs to be made. If this continues the student will continue to have difficulty in his academic future and adulthood.

If I were to do this type of assessment again I would take more time to plan things out. I feel that I did not explain what he was doing well enough, and I did not plan well enough. I should have had a set location and a set time to administer my probes so that the testing conditions were as consistent as possible. Also, I do not think I would have chosen the same subjects to test. If I would have taken more time looking at the survey test and deciding where the student had difficulty, I may have made a more informed decision. In terms of expectations, I did not really have any coming into this. I was not sure what I was getting myself into when administering and analyzing all of this information. I feel that CBA's are a challenge, but are worthwhile.