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Spring 2010
Melissa L. Morgan
It was difficult to choose what to pick for this paper. I enjoyed the class, and I feel that I have grown, both in my understanding of teaching mathematics and in mathematics methods. I especially enjoyed creating lesson plans, and enjoyed sharing them with students.
However, as my goal is to learn to mentor homeschoolers and evaluate portfolios, I have included my Math Methods Observation.
I also included the Fractions and Decimals paper, below. Parents generally have little difficulty teaching counting and the basic operations in elementary school math. They start to experience difficulty when they encounter fractions, and become intimidated with needlessly complex education jargon that they often encounter.
I would like to use some of what I learned to explain and simplify more advanced and confusing concepts and teaching methods. Elementary school mathematics, including fractions doesn’t have to be so hard! For instance, I wrote:
“Of course, the text is correct that young children intuitively develop many concepts of fractions on their own, and have many uses for fractions. It doesn’t necessarily follow, however, that large amounts of time (or any) should be spent on fraction instruction in the classroom, before fourth grade, especially if children still struggle with understanding basic math concepts such as place value. However, if a teacher must teach according to national standards, the text’s recommendation--to limit early fractions to “informal but mathematically correct development”—makes sense. Teachers might disagree, however, with how to define “informal.”
I question the text’s idea that young children, at ages of five and above, have to be “taught” what the words “one-half” mean. One wonders, if true, if these kids have been living totally isolated from any older children or adults to conversationally model these concepts in normal everyday living situations. A normal kid knows when his brother got more than one-half of the cookie! I also questioned how the text’s examples of early lessons on fractions could be called “informal.” I consider informal activities as those that are incorporated into daily living, not arbitrary activities with an agenda.
Fractions can be used to represent division, and the “ability to convert a fraction to an equivalent form is a prerequisite for all but the most simple computation involving fractions.” Real world activities involving comparing which fractional form is larger can give children a mental picture of fractions and equivalencies. They can also compare fractions on number lines, use fraction charts, and in pictures.”
Mrs. Bird, thank you so much for the time and care that you put into teaching this course. I would appreciate any more suggestions and ideas that you might have for me, on what I should take next. May God bless you and your teaching.
Fractions and Decimals
Melissa Morgan
Chapter 8, Fractions, starts with a brief history lesson on fractions, explaining that they were introduced when people began to measure. It is important to learn to use “models to relate fractions and decimals and to find equivalent fractions,” apply fractions to problems, and use mental computation. In fractions, the top number is called the numerator and the bottom is the denominator. The denominator cannot be zero, because that would be meaningless.
Of course, the text is correct that young children intuitively develop many concepts of fractions on their own, and have many uses for fractions. It doesn’t necessarily follow, however, that large amounts of time (or any) should be spent on fraction instruction in the classroom, before fourth grade, especially if children still struggle with understanding basic math concepts such as place value. However, if a teacher must teach according to national standards, the text’s recommendation--to limit early fractions to “informal but mathematically correct development”—makes sense. Teachers might disagree, however, with how to define “informal.”
I question the text’s idea that young children, at ages of five and above, have to be “taught” what the words “one-half” mean. One wonders, if true, if these kids have been living totally isolated from any older children or adults to conversationally model these concepts in normal everyday living situations. A normal kid knows when his brother got more than one-half of the cookie! I also questioned how the text’s examples of early lessons on fractions could be called “informal.” I consider informal activities as those that are incorporated into daily living, not arbitrary activities with an agenda.
Fractions can be used to represent division, and the “ability to convert a fraction to an equivalent form is a prerequisite for all but the most simple computation involving fractions.” Real world activities involving comparing which fractional form is larger can give children a mental picture of fractions and equivalencies. They can also compare fractions on number lines, use fraction charts, and in pictures.
I was interested that “only 24 percent of 13-year olds in the United States were able to correctly estimate 12/13 = 7/8.” They had no mental picture of what they were adding, apparently, or they would have recognized that both fractions were almost one whole number. I agree with the book, that “pupils can best develop understanding if they can begin by dealing with physical-world problems situations.” I only differ with the book to the extreme it appears to advocate that teachers must not give children direct instruction.
“In cases where one of the denominators is a multiple of the other, children will be able to determine this by dividing the larger denominator by the smaller denominator. However, when neither denominator is a multiple of the other, a common multiple of both denominators must be found. …the least common multiple (LCM)” Regrouping subtraction problems can cause difficulties that can be avoided by using models such as Cuisenaire rods or other manipulative device. Multiplication is “the simplest of the fundamental operations with rational numbers…”
I wonder if the book is perhaps over-analyzing multiplication when it says that multiplication of fractions is not a series of additions, but is closely related to partition division. It seems to be a series of additions, if you consider it this way (the way I think of it):
½ X 4=1/2+1/2+1/2+1/2=2
How is that not a series of additions? A real world example would be to receive 4 half dollars, or 4 halves of a pie. I think they are just making something simple into something needlessly complicated.
The book makes sense, however, in the recommendation that division “of a whole number by a fraction is conceptually close to division of a whole number by a whole number. Therefore, this is a good type of problem to use an introduction to division with fractions…Because the use of multiple solutions to problems and exercises is a valuable learning procedure, we suggest that neither the common-denominator method nor the reciprocal method be discarded, both can be used in developing the process of division.” (However, I don’t think a lot of time trying to lead children to “discover” procedures is necessary…)
Question number 1: (Duplicated from lesson) What whole-number concepts that children have learned will be likely to confuse them when they first start learning about fractions? They may be confused about place value with fractions, if they do not do sufficient work to understand pictorial fraction comparison, as well as understanding “one-whole” in different fraction forms. Children may over generalize what they learned about subtraction whole numbers with regrouping.
Decimals, chapter 9, began with the history of decimals, describing the development of decimals from the Babylonians, through the Middle Ages, and into present times. I found it interesting that France and Scandinavia used a comma instead of a decimal point. That seems confusing to me. The text stated that decimals “are used extensively in real-world situations. Thus, it is fairly easy to develop good, truly useful situations for introducing them.” Yet the text also said that the ‘child’s familiarity with dollars and cents may obscure his or her mathematical thinking concerning decimals.” How is this so? The text didn’t explain. I attempted to research on this internet, also, and found no documentation for this statement.
I have used some of the ideas for teaching decimals, such as using Cuisenaire rods, which measure from 1 cm to 10 cm, and considering the white rod as one tenth, etc. I have also used squared paper. Place value charts, as shown in the text, are also useful. The text states that if “the students are accustomed to using and only with decimals, they will interpret verbal numbers correctly and consistently.” However, when I use verbal decimals, I make a “point” to use the word point, not the word “and”—which clarifies things, I think. It is, as the book says, ‘helpful to place a zero before the decimal point for decimals with a value less than 1…Emphasis should be placed on the ones place as central to decimal notation.”
With addition and subtraction with decimals, like in whole numbers, you must remember to keep place value in mind. The book says that the “major difficulty in teaching multiplication with decimals is determining the number of decimal places in the product. Traditional mathematics programs simply taught the following rule: “Count the number of decimal places in the factors.” The book claims that teaching this “at best…produced only computational proficiency and left pupils wondering why a computation such as 2X.3=N produced a product less than 1.”
The author of the book must assume that a student that is simply told this fact has had no real world experience with decimals. A student who has had real world experience with decimals would know that .3 means the same as .30, or 30 cents, and that if I got 30 cents three times, I would only have 90 cents, which is still less than one whole dollar. Somehow, however, this experience is thought to be confusing to children, and should be avoided?
Anyway, after a long tedious process, the kids investigated and “invented” the same rule anyway, that the “product will have as many decimal places as there are in both factors.” So I assume the author of the book is simply saying they think that children will only understand something if they have discovered it themselves without being taught—“discovery learning.” In further discovery learning investigations involving multiplication, the “most common solution involved writing the division as a fraction and reducing the fraction…” The students said that “When you divide tenths, the answer will be expressed in tenths. Or you can use place value. Place the decimal point in the quotient directly above the decimal point in the divisor.” The book recommended memorizing common decimal equivalents, such as 1/8=.125 and ¾=.75. Finally, the book discussed expanded notation.
Question number 4: Develop a set of activity cards for each of the four operations with decimals, using a variety of approaches.
I would make activities using the following methods:
Addition with decimals: Use place value mats with
decimals, decimal square manipulatives and number lines with
decimals. Use flash cards, each with a decimal printed on it. Play a
game to see who has the highest decimal, adding two together. Play
store, writing prices and adding up what two items will
cost.
Subtraction with decimals: Use place value mats with
decimals, decimal square manipulatives and number lines with
decimals. Play the same game with decimal flash cards, only this
time, see who has the lowest number, after subtraction. Play store,
subtracting how much money you have left after you buy and
item.
Multiplication with decimals: Use place value mats with
decimals, decimal square manipulatives--figuring out sq. km. of a
tract of land). How much will it cost, if you buy the same item two
times? Three times? Four times? Five times? For instance, each
eraser costs $.85. How much for 6 erasers? Also use activities such
as figuring how many miles from one city to another city, using a
legend on a map.
Division with decimals: Teach division of
decimals as the opposite of multiplication. Use the division of
fractions. (Writing the division of decimals as a fraction and then
reducing it.) Use place value mats with decimals, decimal square
manipulatives and number lines with decimals. For additional
practice, use in practicing all operations, at Mr, Martini’s
classroom,
http://www.thegreatmartinicompany.com/decimals/decimals-home.html
Melissa L. Morgan
3-31-10
I observed Mrs. Huston, a homeschool teacher and Ohio certified teacher. Mrs. Huston holds a Masters in Education from OSU, and a Bachelors from the College of Mount St. Joseph. She has been teaching since 1982, and for the last 20 years, has been responsible for teaching and evaluating students of all grade levels and abilities (K-12), including children with special needs.
Mrs. Huston has evaluated portfolios for three of my children, for at least the last seven years. I have observed her methods in math, as well as other subjects, for much more than 100 minutes. Mrs. Huston also performs portfolio evaluations for many other homeschool families, and I have been present for some of those evaluations also. Mrs. Huston evaluates homeschool portfolios for homeschooling children and parents under the written narrative option, reviewing education plans and methods as per the Ohio Administrative Code 3301-34 Department of Education, Ohio Administrative Code, Chapter 3301-34 Excuses from Compulsory Attendance for Home Education, Promulgated pursuant to Ohio Revised Code, Chapter 119 (See http://www.ohiohomeeducators.net/--copy of notification form, below).
How are the children learning Math? Do they use physical or picture models for lessons, or math manipulatives? If so, how and for what purposes are these used?
Mrs. Huston teaches with and recommends using Saxon Math materials. Saxon math is a traditional math program which involves teaching a new mathematical concept every day and consistently reviewing old concepts. She uses manipulatives infrequently if children struggle with understanding a new concept, but frequently utilizes picture models for contrast, review, and practice in elementary mathematics instruction. Huston is open to using various methods to match modalities of learning strengths.
How much instruction is given before students are asked to complete math tasks on their own?
Generally students participate in practice exercises first, where any mistakes or misunderstandings are checked and addressed. Mrs. Huston gives a short verbal introduction along with guided practice before letting the students practice on their own. If needed, supplementary lessons are given. After a child demonstrates understanding, independent learning is stressed, and children are encouraged to read and do lessons independently. Mrs. Huston has found the Error-less teaching technique of Opportunities To Respond (OTR) very helpful in teaching problem solving. She recommends keeping instruction simple, and to the point; “then give a lot of opportunities to do the math and demonstrate understanding.”
In regard to their math assignments, how much is involved in problem solving? communication? reasoning? Connections?
Saxon math connects math concepts to previous learning. Approximately one-third of assignments are focused on problem solving, and two-thirds apply to concepts, reasoning and calculations. Communication occurs both in initial instruction and in reviewing assignments.
In homeschool evaluations, Mrs. Huston reviews portfolios of student work, employing Socratic questioning and observation to determine compliance with all Ohio education standards (see form, below.)
OAC 3301-34-02 Statement of Purpose:
The purpose of the
rules in this chapter is to prescribe conditions governing the
issuance of excuse from school attendance under section 3321.04 of
the Revised Code, to provide for the consistent application thereof
throughout the state by superintendents, and to safeguard the primary
right of parents to provide the education for their child(ren). Home
education must be in accordance with law.
OAC 3301-34-03
Notification
(A) A parent who elects to provide home education
shall supply the following information to the superintendent:
(1)
School year for which notification is made: 20_________
(2)
Name of parent:
__________________________________________________
Address:
____________________________________________________________
___________________________________________________________________
Telephone
number (optional): (______)___________________________________
(3)
Name, address, and telephone number (telephone number is optional) of
person(s) who will be teaching the child the subjects set forth in
Paragraph (A) (5) of this rule, if other than the parent.
Name:
_____________________________________________________________
Address:
___________________________________________________________
__________________________________________________________________
Telephone
number (optional): (______)__________________________________
(4)
Full name and birth date of child(ren) to be educated at
home:
_________________________________________
____________________
_________________________________________
____________________
_________________________________________
____________________
_________________________________________
____________________
_________________________________________
____________________
(5) _____ Assurance that home education will
include the following, except that home education shall not be
required to include any concept, topic, or practice that is in
conflict with the sincerely held religious beliefs of the parent:
(a)
language, reading, spelling, and writing;
(b) geography, history
of the United States and Ohio, and national, state, and local
government;
(c) mathematics;
(d) science;
(e) health;
(f)
physical education;
(g) fine arts, including music; and
(h)
first aid, safety, and fire prevention.
(6) ___ Brief outline of
the intended curriculum for the current year. Such outline is for
informational purposes only. (attached)
(7) List of
_____
textbooks,
_____ correspondence courses,
_____ commercial
curricula, or
_____ other basic teaching materials that the
parent intends to use for home education. Such list is for
informational purposes only. (attached)
(8) ____ Assurance that the child will be provided a minimum of nine hundred hours of home education each school year.
(9) ____ Assurance
that the home education teacher has one of the following
qualifications:
(a) a high school diploma; or
(b) the
certificate of high school equivalence; or
(c) standardized test
scores that demonstrate high school equivalence; or
(d) other
equivalent credential found appropriate by the superintendent; or
(e) lacking the above, the home teacher must work under the
direction of a person holding a baccalaureate degree from a
recognized college until the child's or children's test results
demonstrate reasonable proficiency or until the home teacher obtains
a high school diploma or the certificate of high school equivalence.
(10) The parent(s) shall affirm the information supplied with his or her signature prior to providing it to the superintendent.
_____________________________________________________________________
Signature and Date
_____________________________________________________________________
Signature and Date
*Form Optional