07. Pipes, Tubes, and Beakers: New Approaches to Teaching the Rational-Number System

        The rational number system, which includes fractions, is notorious for causing students agitation.  It requires reformulating the whole number system, which is taught previously.  It is grounded in multiplicative reasoning that is much different than the additive reasoning needed for the whole number system.  This switch from additive to multiplicative reasoning is the source of many problems.  This includes decimals, where the same type of reasoning is needed (4 + .3 does not equal .7).  This failure in reasoning carries significant academic consequences, not only for future studies, but also for many practical tasks, such as reading maps, preparing budgets, and calculating discounts. 


The Knowledge Network: New Concepts of Numbers and New Applications (Principle 2)

        There is a network of inteconnected ideas, including symbols, concepts, and facts to move to the rational number system.  They are also less intuitive.  First, a number like 4 can be represented in many ways, for example 4/1 and 4.0 and the student must be able to move from one to another effortlessly.  Second, there are multiple interpretations for rational numbers.  For example, 3/4 could mean 3 of 4 equal shares (part-whole), 4 children dividing 3 pies (quotient), 3 red cars for every 4 green cars (ratio), etc.    Coordinating these interpretations requires deep understanding of the concepts.  These interpretations form the network that make up the knowledge network. 
        Unit and operations must be reconceptualized too.  Density and continuity, meaning that there are an infinite number of numbers between any two rationals, is an example.  A unit of 1 is newly implied too.  With whole numbers, 6 is 6 units.  With fractions, 1/2 means 1/2 of the implied unit 1. 
Students' Errors and Misconceptions Based on Previous Learning (Principle 1)

        There is a necessity to look at student preconceptions, which make clear the type of faulty reasoning students are making. 

Metacognition and Rational Number (Principle 3)

        A metacognitive approach to get students to monitor their own learning is needed, especially as students must actively make sense of the mathematical concepts.  Yet, most school children do not create appropriate meanings.  Classroom teaching that does not support this can have serious consequences, namely that students can stop expecting math to make sense. 


        Analysis of common curriculum shows that the 4 learning principles are often violated in traditional curriculum.  While truly understanding the concepts takes a great deal of time, in contrast to whole-number learning, rational-number learning is covered quickly and superficially.  Procedures for manipulation are emphasized much more than making sense.  Virtually no time is spent teaching how the representations - decimals, fractions, percents - relate to one another. 
        In traditional practice, fractions usually begin with shapes partially shaded, with students counting to understand the numerator and denominator.  This introduction, however, is grounded in additive thinking.  It reinforces the concept that must, in fact, be changed to truly understand rational numbers. 
        Alternative approaches have been investigated, such as beginning with ratios and proportions, and these have been very successful and changing the students' schemas.  A program called Pipes, tubes, and beakers has a similar approach. 

Pipes, Tubes, and Beakers: A New Approach to Rational-Number Learning

        Based on principle 1, the program begins with a bridging program between proportional and halving understandings, based on what the students already understand.  The proportional reasoning is applied with percents, and these are really just "priveledged" fractions, always based on base 100.  Instead of confusing students with many bases, this gives students a chance to develop their conceptual understandings. 
        The curriculum has three parts.  First, students are introduced to percents using concrete props that highlight linear measurement.  After, the 2 place decimal is introduced, as an in-between percentage points.  Finally, activities comparing and ordering rational numbers are introduced, along with fractions.  Many metacognitive opportunities are included as well to allow students to monitor their work. 

Lesson Part 1
        Percents in everyday life:  Students were prompted to say how they were familiar with percents in every day life, and what they thought percents meant. 
        Pipes and tubes:  Black drainage pipes with white venting tubes were used to simulate water in a beaker.  Students were asked to show how the pipes could show percents to younger children. 
        Laminated meter-long lines calibrated in centimeters were put along the classroom, and students were challenged to walk a certain distance (e.g. walk 70 percent of the room)
        Next, beakers with water with varying fullness were introduced.  Students had to estimate what percent of each beaker was full.  This led naturally to calculation, as students who saw the water was 4cm out of 8cm needed to simplify.  Yet, the students were not given any standard rules. 
        Then, strings were cut.  Students were told they represented a certain percent of a mystery object in the room.  They also had to make their own strings of certain percentages compared with a piece of cardboard. 
        These activities all focused on linear measurement and were designed to extend students informal, previous understandings. 

Lesson Part 2: Introduction of Decimals

        Students, now being told they were percent experts, moved on to become decimal experts.  To begin, students were given stopwatches and asked to figure out what the two smallest digits (tenths and hundredths) represented.  They had to come up with names. 
        Next, several activities were devised to help students order decimals.  The first was called "Stop-Start-Challenge."  They had to start and stop the stop watch as quick as possible 5 times, then challenge their partner to do it faster.  Then they had to order their intervals from smallest to largest. 
        Then, they played "Stop the Watch Between", where they had to stop the watch between .45 and .50, for example.  After, they were asked to put decimals on a number line. 

Lesson Part 3: Fractions and Mixed representations of Rational Numbers

        First, students were asked to represent fractions in as many ways as they could.  They were also asked to compose word problems that included fractions.  Also, they had to complete and compose string problems with fractions, such as 1/8 + 1/2 + ? equals 1.  With time, they were also asked to substitute some of the fractions with decimals, such as 1/8 + .5 + ? equals 1. 
        Students were given stopwatches again and played "Crack the Code."  They were given 2/5, for example, and had to stop the watch as close as possible to the decimal equivalent .4. 
        They then played card games, with fractions, decimals, and percents.  The students had to sequence the cards in order, which led to a great deal of debate and discussion among the kids.  This included some harder fractions which the students had yet to encounter.  The game ended with students reaching consensus.
        Then students played war, with the larger number being added to their total.  By this point, students were already comfortable exchanging between the decimals, fractions, and percents. 


        This model, although modified slightly, was implemented in four other experimental programs.  These were compared with control groups.  The results showed significant improvement with the experimental group.  Not only could they answer more questions than the "traditional" students, they made many more references to proportional concepts.  Many post-experiment interviews illustrated the new kinds of connections and understandings that students developed. 

        Principle #1 - Changing the focus to multiplicative understanding instead of additive, only by looking at students understanding, was one way this was implemented.  Furthermore, building on students' prior understandings by engaging their prior knowledge was helpful. 
        Principle #2 - The learning sequence was designed around mastery of multiplicative reasoning, such as proportion. 
        Principle #3 - Though this was not so much detailed in the chapter, students were regularly engaged in whole-group discussions where they could explain their reasoning and share invented procedures. 
        Allowing the students to build on their knowledge made them also feel confident. 


CONNECT:How are the ideas and information presented 
CONNECTED to what you already knew?
EXTEND:What new ideas did you get that EXTENDED or pushed your thinking in new directions?
CHALLENGE:What is still CHALLENGING or confusing for you to get your mind around? What questions, wonderings or puzzles do you now have?
1.  Connect: I knew the principles very well and could connected them to what was described in the lessons.  

2.  Extend:  That if you analyze the type of reasoning behind new concepts and the required change, the curriculum can be re-ordered.  For example, by focusing on percents first, instead of pie charts, additive reasoning was not reinforced and instead students immediately had to switch to multiplicative. 

3.  Challenge:  The 4th principle of community was not described, and I wonder how it could be applied.  Furthermore, I am puzzled by the role of emotions in this entire process.  Although they refer to their role on several occasions (such as allowing students to be confident in the sense-making process), they did not explicitly refer to emotions.