# Making Fractions Real: An RME Task

Recently, I’ve been investigating Realistic Math Education (RME).  I like the idea of building on what is real to the student.

Because I spend a lot of time thinking about fractions, I wanted to know how RME approaches them.  Luckily, Streefland wrote about his three-year teaching experiment in the Netherlands.

In the experiment, students were introduced to the “Fractured Family.”  The family encounters many experiences that require fractional thinking and proportional reasoning.  For example, they need to divide an omelet at lunch or share apples after school or bake cookies from a recipe.

It’s not necessary that students have experienced these situations themselves, but rather that they can imagine the adults and children in the family doing them.

Here’s one of the initial tasks students encounter.

When Anja and Monica Fractured come home from school they may have an apple each.  But what do you do about such a difference in size?

Here’s what I like about the task:

• It’s a great context for introducing fair sharing.  Students know that it’s not fair if one child gets the big apple and one gets the small apple.
• It’s a great context for talking about the unit.  Is one-half of the small apple the same size as one-half of the big apple?  Why are they both called one-half?
• Using an imaginary family allows students to connect sharing with specific people they can imagine as opposed to the more abstract idea of sharing with unnamed people.
• Students are encouraged to draw how they would share the apples.  At first students create very detailed drawings–drawing leaves and stems for the apples. But as they do more tasks, they move away from detailed drawings and use circles or rectangles to represent items.   The drawings become representations or models of the situation.  Eventually they become a mental model.

I’m not a big fan of calling the family fractured, but you can adapt this task and name the family anything you want.

I’m still making my way through the book, but I’ll be sure to share any other tasks that I find interesting.  I’d also love to know if anyone has any experience using RME tasks in their classes.

Want to read along with me?

Streefland, L. (Ed.). (1991). Fractions in realistic mathematics education: A paradigm of developmental research (Vol. 8). Springer.

# Realistic Math Education: Defining the “real” in real world

Dan Meyer recently posted about how students aren’t easily fooled by attempts to make make tasks “real world” by placing a photo next to them.

I found myself nodding along at what he described.  I once naively asked a fourth grade class to write real world problems about fractions and received the following response:

The Knicks scored 12 ½ points and the Nets scored 13 ¾ points.  How many points did they score all together?

When I asked the student how it was possible to score ¾ of a point or why you would want to know the combined score of two opposing teams, he looked at me like I was crazy and said it didn’t matter–“you told me to make a problem with fractions so I put fractions in it.”

Even at young ages, students learn there is a game teachers play where they make “real world” problems.  Kids pick up pretty quickly that these aren’t about the world they live in–they are a special type of problem in math.  So they skim the word problems.  They pull out the numbers and assume they should try some mathematical operations they recently did in class.

So I agree with Dan that spreading the “real world” over a task doesn’t fool students–even our younger ones.

I do believe in building off students’ experiences . Although math tasks don’t have to be real world, they need to build on what students know.  An activity they can do.  An experience they had.  An operation they already understand mathematically.

As I was reading the post, I thought about Realistic Mathematics Education (RME), developed in the Netherlands over thirty years ago.  RME is strongly based on Hans Freudenthal’s philosophy that students should be guided in a process that allows them to “mathematize” the world around them.   They should be provided with tasks that allow them to use mathematics to organize and solve a problem.

What’s interesting is that many people incorrectly assumed that this meant real world problems of the kind Dan showed.  However, realistic in RME referred to situations students could imagine, not necessarily something that would happen in real life.  As a result, “the fantasy world of fairy tales and even the formal world of mathematics can be very suitable contexts for a problem, as long as they are real in the student’s mind.”

Making it clear that real is about what is real in the student’s mind can be helpful when thinking about how to select appropriate tasks for students.  Unfortunately, it’s much harder than simply throwing a real world picture next to the problem.

Want to know more?

Check out the Freudenthal Institute’s page:  http://www.fi.uu.nl/en/rme/

Read about Freudenthal and RME: Gravemeijer, K. & Terwel, J. (2000): Hans Freudenthal: A mathematician on didactics and curriculum theory, Journal of Curriculum Studies, 32(6), 777-796.

Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational studies in mathematics, 39(1-3), 111-129.

# Another way to introduce ratios: develop relative thinking

One of the most important things students need to understand to reason proportionally is the difference between comparing two quantities in relative (multiplicative) versus absolute (additive) terms.   Students often struggle with making the move to thinking multiplicatively.

How can we begin to help them make this transition?

The research suggests one way:

Make the two ways to compare quantities explicit.

What does this look like?

Here’s an example of how ratios are introduced in Chinese classrooms (Cai, 2004; Cai & Sun 2002):

Miller Middle school has 16 sixth-grade students and 12 of them said that they are basketball fans.  The remaining students are not basketball fans.  How could we describe the relationship between the students who are basketball fans and those who are not?

Once students determined that there are 12 fans and 4 non-fans, they described the relationship in the following ways:

• There are 8 more fans than non-fans.
• There are three times as many fans as non-fans.
• For every three fans, there is one non-fan.

The first description is an additive comparison, while the next two are multiplicative comparisons.  Through continued exposure to the different ways to compare quantities, students begin to move flexibly between additive and multiplicative reasoning.

Of course that’s only the beginning of developing proportional reasoning.   But being aware of the different ways to compare the situations is an important first step.  Knowing why one comparison might be more useful than another is the next step.

Cai, J. (2004). Developing algebraic thinking in the earlier grades: A case study of the Chinese elementary school curriculum. The Mathematics Educator 8(1) 107-130.

Cai, J., & Sun, W. (2002). Developing students’ proportional reasoning: A Chinese perspective. In B. H. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios and proportions: National Council of Teachers of Mathematics 2002 Yearbook (pp. 195-205). Reston, VA: NCTM

# Proportional Reasoning: Absolute vs. Relative Thinking

As I’ve mentioned before, proportional reasoning is complicated.  Researchers refer to it as a “watershed” concept because of its role as both the capstone of K-8 mathematics and the cornerstone of high school mathematics.

How do we begin to tackle such a complex concept?

One of the most important things students need to understand in order to begin to reason proportionally is the difference between comparing two quantities in relative (or multiplicative) terms versus in absolute (or additive) terms.

For example, say there is a group of 6 boys and 3 girls.  You can compare the number of boys and girls in various ways.    When you say there are 3 more boys than girls, you are using absolute or additive thinking.  When you say there are twice as many boys as girls, you are using relative or multiplicative thinking.    Understanding the difference between these two comparisons and what each tells you is an essential building block in being able to reason proportionally.

We know from the research that students often struggle with moving from thinking additively about situations to thinking multiplicatively.    In a classic study, Hart (1982) found that two-thirds of 13-15 year olds answered the following task incorrectly

When measured with paperclips, Mr. Short is 6 paperclips tall.  Mr. Short has a friend Mr. Tall.  When you measure their heights with matchsticks, Mr. Short’s height is 4 matchsticks and Mr. Tall’s height is 6 matchsticks.  What would be Mr. Tall’s height if you measured it in paperclips?

The majority of students who answered incorrectly said 8.  They reasoned about the situation using absolute or additive thinking–adding 2 to 6.   They did not see the multiplicative relationship in the situation–that Mr. Tall was one and a half times as tall as Mr. Short.

How can we begin to encourage students to think multiplicatively about situations?  I’ll talk more about what the research says next time, but I’d love to hear your thoughts.  If you want to know more now, check out the resources below.

Hart, K. M. (1984). Ratio: Children’s strategies and errors. A report of the strategies and errors in secondary mathematics project. London: NFER-Nelson

Karplus, R., Karplus, E., Formisano, M., & Paulson, A., (1979) Proportional Reasoning and control of variables in seven countries in J. Lochhead & J. Clements (eds), Cognitive process instruction. The Franklin Institute Press, Philadelphia.

Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning, (pp. 629-666). Reston, VA: National Council of Teachers of Mathematics.

Lesh, R., Post, T., & Behr, M. (1988). Proportional Reasoning. In J. Hiebert & M. Behr (Eds.) Number Concepts and Operations in the Middle Grades (pp. 93-118). Reston, VA: Lawrence Erlbaum & National Council of Teachers of Mathematics.