NEW TEACHING APPROACHES

NEW TEACHING APPROACHES

Moss and Case identified three different proposals on approaches to teaching of fractions that address the above mentioned problems in various ways and then propose a new curricular approach which they tested themselves in a study involving fifth and sixth grade students. The first of the older studies conducted by Hiebert and Warne (as cited in Moss & Case (1999)) was judged to have addressed primarily the syntactic and notational problems mentioned above and placed a great deal of emphasis on the use of base 10 blocks. In the second study Kieran (as cited in Moss & Case (1999)) was seen to address the syntactic and representational issues and, among other innovations, used paper folding to represent fractions in preference to pie charts. The third of the studies, conducted by Streefland (as cited in Moss & Case (1999)) attempted to address all four concerns and was based on using real-life situations to develop children's understanding of rational numbers.

Moss and Case's (1999) own approach was designed to address all four of the identified problems and was characterized by several qualities distinguishing it from previous approaches. They started with beakers filled with various levels of water and asked students to label beakers from 1 to 100 based on their fullness or emptiness. They emphasized two main strategies: halving (100 -> 50 -> 25) and composition (50 + 25 =75) in determining appropriate levels. Refining this approach they developed the notion of two place decimals with five full beakers and one three-quarter full beaker making 5.75 beakers. Four place decimals were then introduced with 5.2525 (initially, spontaneously denoted as 5.25.25 by the students) characterized as lying one quarter of the way between 5.25 and 5.26. Students eventually went on to work on exercises where fractions, decimals and percentages were used interchangeably. Moss and Case found that this approach produced deeper, more proportionally based, understanding of rational numbers. They see their approach as having four distinctive advantages over traditional approaches: (a) a greater emphasis on meaning (semantics) over procedures, (b) a greater emphasis on the proportional nature of fractions highlighting differences between the integers and the rational numbers, (c) a greater emphasis on children's natural ways of solving problems, and (d) use of alternative forms of visual representation as a mediator between proportional quantities and numerical representations (i. e. an alternative to the use of pie charts).

Source: http://www.homeedsa.com/

MISTAKES TEACHERS MAKE

MISTAKES TEACHERS MAKE

Based on previous research Moss and Case (1999) identified four major problems with current teaching methods in the area of fractions. The first is a syntactic rather than a semantic emphasis, which is to say that researchers have identified that teachers often emphasize technical procedures in doing fraction arithmetic at the expense of developing a strong sense in children of the meaning of rational numbers. The second problem identified is that teachers often take an adult-centered rather than a child-centered approach, emphasizing fully formed adult conceptions of rational numbers. As a result teachers often do not take advantage of students "prefractional knowledge" and their informal knowledge about fractions thus denying children a spontaneous "in" to their formal study of fractions. A third issue is the problem of teachers using representations in which rational and whole numbers are easily confused e.g. students count the number of shaded parts of a figure and the total number of parts so that each part is regarded as an independent entity or amount (Kieran cited in Moss & Case (1999). Finally, researchers have identified considerable problems in use of notation that can act as a hindrance to student development. These problems center around teachers' perceptions that the notation used for rational numbers is transparent while this has been shown not to be the case, especially with regard to decimal fractions (Hiebert, cited in Moss & Case (1999)). Tirosh (2000) conducted a study on teacher knowledge in teaching of fractions and concluded that teachers needed to pay considerably more to analysis of student errors.

Source: http://www.homeedsa.com

Fraction Teaching Stategies

Materials

• M & Ms of different colors
• Hershey Candy Bars
• Magnetic fraction bars

Activity

I start by passing out a paper candy bar to everyone that has 12 parts.

We talk about how a candy bar is the whole and they they cut it in half so they see that 6 pieces=1 half.  I model with a real candy bar.

We talk about them splitting this with a friend so they would have EQUAL PARTS. Then I tell them 2 other friends come over so they have to split each half into equal parts again. As we do each split, I draw a picture of the fraction and label it.

Once we have divided the candy bar into twelve pieces and I have shown them how we went from 1/2 to 1/12, I use magnetic fraction bars to show them briefly how 1/12 is much smaller than 1/2.

Once they get this down and have enjoyed a real candy bar, I then take small bags of M & Ms and pass them out to pairs of kids. We divide our candy into groups by color and then figure out which fraction of the different colors everyone has. For example, if they have 24 total candies and 8 of them are red, I show them that 8/24 are red. Those that are ready always seem to tell me that this is 1/3. (I divide candy in advance so there is always an even number in bags, but not always the same number so some groups may get 24, others 30, others 36, etc.).

When we have figured out the fraction of each color (I have them draw it out on paper with crayons and label the pictures), I let them eat their fractions. My kids always love this since it involves food and they always get fractions too!

Teaching Fractions with Tortillas

Submitted by Debora from Naples, Forida

Materials

• 10 inch flour tortillas (2 for each student)
• Markers and marker board (one large one for all responses or individual ones for each student)

Activity

1. Each student gets 2 tortillas and puts one on top of the other.

2. The teacher explains that one tortilla equals one whole.

3. Each student carefully folds the flour tortilla in half (as the teacher models it).

4. The teacher asks students questions about what each has now and asks for math statements about what they have. Students have to concretely show their statement as well as state it.

5. Students reply and teacher writes down their responses - examples 1/2 + 1/2 = 1, 1/2 = 1/2, 1>1/2, 1/2<1

6. The students then carefully fold one half into two pieces and the teacher asks the questions again.

7. This continues until the student has one tortilla in eighths and the whole one stays the same so the students can compare and make many statements.

8. This exercise leads to many statements that can lead to establishing rules about fractions. For example - 2/2 = 1, 4/4 = 1, 8/8 = 1 If the teacher continues to ask questions and does not give the students answers, the students will discover the rules on their own. I believe this is vital for the students to "own" their learning.

Paper Folding Fractions

Submitted by Cathy from Holly Springs, NC

Materials

• Colored Paper (various shades)
• Crayons/colored pencils
• Overhead (optional)
• Base-ten blocks

Activity

I start by talking about the vocabulary that they'll encounter with fractions and decimals (numerator, denominator, fraction, decimal).  I usually put the word and definition on the overhead for everyone to see.

Then I have the students make fraction pieces with whole sheets of paper.

I first have them get a piece of white paper (8.5 x 11) and then they write "Whole" on the middle.  Then we take another colored sheet (any color) and fold that sheet in 1/2 and we write 1/2 on the sheet.  We do this for all the fractions and then I introduce fraction strips. 

I have them show me various fractions with their fraction pieces or strips. We also learn about decimals at this time. We talk about tenths and hundredths using base-ten blocks.

I use the overhead to show fraction pieces and base-ten blocks and how they all work together.

I usually end the lesson with having random students come to the front of the room and we talk about what fraction has brown hair, is wearing red, is a boy, etc. and then we connect the fraction to the decimal.

Fractions with Real Pies!  Yum!

Submitted by Leslie from Ohio

Materials

• Pies, and/or cakes (All the same size!)
• cutting utensil

Activity

I divide the class into groups, each group with a different number of students.  Then, I explain that each group will share a pie (or cake). We talk about being fair and giving each member of the group the same size piece. No one is permitted to eat anything until everyone in the class has a piece.

After the pies (or cakes)are divided, we talk about the size of each groups pieces (halves are going to be much larger than the fourths, etc.).  After all the discussion and observations, we eat our lesson. Yes, I do try at the end to give similar amounts to each student by recutting.

After thought: I usually do this activity after a measurement activity where we actually make the pies or cakes.

Fun Data Collection Project that Teaches Fractions

Submitted by Jessica from Seattle, Washington

Materials

• Just the kids at first!
• Graph paper and lined paper (or you can make your own worksheets)

Activity

I begin with the introduction of surveys. We do this early in the year as part of the get to know you activities. I have the kids create a survey about favorite things, allowing for 4 categories. So someone could survey favorite sports and offer the choices: Soccer, Football, Baseball, or Swimming.

They begin by taking a tally and getting answers from everyone in the class. Then they make a graph. We analyze the graphs and we begin to discuss how 9 out of 25 like baseball, and 4 out of 25 like swimming.

We continue with this activity as part of our daily morning meeting activity we do a short survey with a raise of hands. We report the results in fraction form on the board. 3/25 had pancakes for breakfast, 22/25 did not have pancakes. Something simple. I try to vary it enough so they can see lots of examples. Some results in which it is almost everyone, some which it may be nearly half and some which would be very few.  Sometimes as a visual the kids also stand up and we can see how many out of the whole like one thing or another.

I have my students work in teams of four to come up with a survey question they will ask 100 people (25 each.) Groups of 5 will ask 20 people each and I have a separate form in this case. They create a survey and get the results, breaking up and going to visit other classes, the library, the office, and so forth. We get the answer, the person's age and gender. We sort and classify the results, and create graphs. We create a fraction to show each answer as well.

With the number 100, I also briefly introduce decimals and percentages, though my third graders are just beginning to understand them. They do wonder about them though, and this is a good way to show them how a fraction, decimal and percent are connected.

Along with these lessons in data, I also do hands on activities with fraction puzzles, pie slices, the pizza game and some worksheets.

We do paper folding and create fraction books with common fractions represented such as half, fourths, thirds and so on. However, the most memorable part of the fraction unit is the data collection project.

Our curriculum is great, and it introduces fractions as brownies, and cookies, which the kids love and I do those lessons throughout the unit as well, though am not bound to the curriculum and can use what I like from it.

Spelling Word Fractions

Submitted by Jennifer from Lawton, Oklahoma

Materials

• Paper with three columns
• pencil

Activity

Word fractions Write our spelling words in first column.

Second column is fraction of vowels.

Third column is fraction of consonants.

example: determine   4/9    5/9

Fraction Pizzas

Submitted by Jennifer from Georgia

Materials

• Construction paper cirlces (pizzas)
• Index cards with fractions
• linking cubes, bingo markers, or any small items in classroom to use as toppings

Activity

The circles (pizzas) are already divided into halves, thirds, forths, sixths, and eighths (depending on level of students).

As a whole class the pizzas are displayed where everyone can see them. Each student will take turns and pick an index card with a fraction.

The student must decide which pizza they must put thier topping on.

Example. If the student draws 2/3, they must determine that they have to use the pizza that is divided into thirds to put thier topping on. The student will then cover 2/3 of the right pizza with (pepperoni, cheese) whatever they choose for their topping.

After this lesson the materials may be placed for center time or free choice play.

Teaching Division of Fractions

Submitted by Matt from Cincinatti, Ohio and Taiwan

Materials

• Fraction Skittles (Not Skittles like you eat.  Skittles are a manipulative teaching tool.  See a picture here.)  This material has 4 "skittles." The first one is a whole one and cannot be separated. The 2nd one is cut in half. The 3rd in thirds and the 4th in fourths.

• Fraction insets (See a picture here.)  These are 10 circles.  The first one is not cut at all (it is a whole circle). The 2nd one is cut in halves, the 3rd in thirds, and so on all the way up to ten.

Activity

There are many things that can be done with these from just general exploration to high concepts of math. I am going to explain the division of fractions. It is not clear to many adults why we flip the 2nd fraction and multiply.  With these two tools, it becomes obvious.

Let's assume the problem is 2/9 divided by 1/3. We take the 1/3 of the fraction skittle and set it down. We then have to give this 2/9, so we take 2 of the 9th pieces and set them down.  The question in division always becomes "how much does each one get?" We know that 1/3 is not 1. So we investigate what we need to do to make it one. The obvious answer is to get the other two skittle pieces. The child can put them together to confirm that it makes one, then set them next to each other. "If he has 2/9 (pointing to the original one), what does he need? (pointing to one of the new ones)" The child will put down 2/9 under each one. "Now, let's put them together to see how much we have." When the child puts them together, thus creating one of the skittles, then puts the insets together, thus creating 6/9. By this time, the child has already worked with these materials enough to recognize that shape as 2/3. If they do not, it is ok at first, as there is plenty of time to work with that later. But it should be as simple at this point of just saying, "What can we change that for?" since they've already done an exchange work. They will put 3 of the 9 insets away and get 1 of the 3 insets. They will do this until they are left with its simplified version of 2/3

Eating Your Way Through Fractions

Submitted by Leslie from Sunbury, Ohio

Materials

• Graham Crackers
• Chocolate Bars
• Mini Hershey Kisses
• Marshmallows

Activity

We make smores to eat, but first make fraction parts with the chocolates and marshmallows...for equal fractions: If Katie handed out 6 marshmallows to Brad how many of them would she eat if she ate 1/3 of the marshmellows.

We make fraction story problems with the chocolates, marshmallows and graham crackers eating along the way until we made fraction parts: give half of the graham cracker to a friend: now who us 1/4 of the new graham cracker, etc..kids love this..you can also use mini marshmallows to make it cheaper and mini Hershey bars too!

SOURCE: http://www.superteacherworksheets.com