Source:
National Research Council of the National Academies. (Ed.). (2005). How Students Learn. Washington D.C.: The National Academies Press. (Chapters Introduction, 5-8).
Chapter 7 - Pipes, Tubes, and Breakers: New Approaches to Teaching the Rational Number System (Joan Moss)
1. How do the principles of How People Learn apply to the development of Rational Numbers?
2. How do we teach it?
3. Is it effective?
1. Difficulties of the Rational Number System
1.1. Extensive research shows that many people, adults and children alike, having difficulty with the rational number system (fractions, decimals, etc.).
1.2. Introduced in early elementary school, students must reformulate the way they think about numbers in a major way. They must go beyond whole-numbers ideas, which are expressed in fixed quantities, to understand numbers expressed in relationship to other numbers.
1.3. New relationships are grounded in multiplicative reasoning very different from additive reasoning that characterizes whole numbers.
1.4. The majority doesn’t make the transition smoothly, and become disenchanted and frustrated with mathematics.
1.5. A common misunderstanding is continued use of whole-number reasoning in situations where it does not apply. Examples:
1.5.1. Children say .059 > .2
1.5.2. Children say 1/3 + ½ = 1/5 (even many 5th and 6th graders)
1.5.3. Children say 1/8 > 1/6
1.6. Rational number system introduces a major stumbling block in children’s mathematical development.
1.7. It marks a time where students no longer know what is going on in the math classroom, and it is disheartening to both students and teachers.
1.8. It is a cause for concern because rational numbers are an underlying basis of advanced mathematics.
1.8.1. Students cannot succeed in advanced mathematics (ex. algebra) without rational numbers.
1.8.2. The rational number system is also a major part of our daily adult lives. (p. 310)
2. Rational-Number Learning and the Principles of How People Learn
2.1. The Knowledge Network: New Concepts of Numbers and New Applications (Principle 2) (p. 312)
2.1.1. Rational numbers is a new knowledge network – it’s filled with new and intertwined concepts, new facts, and new symbols.
2.1.2. Based on multiplicative instead of whole-number relations.
2.1.3. Operations involved may appear less intuitive and at odds with earlier understandings.
2.1.4. New Symbols, New Meanings, New Representations
2.1.4.1.A specific rational number can take several many forms (i.e. fraction, decimal, percent)
2.1.4.2.A rational-number quantity can be represented by an infinite number of equivalent common and decimal fractions. (1/4, 2/8, 4/16, ….25, .250,
2.1.4.3.For many students, flow between representations does not come easily (p.313).
2.1.4.4.Most traditional instruction of decimals, fractions, and percents is taught separately as distinct topics, which makes it confusing and more difficult for students to realize that a single quantity can have many representations.
2.1.5. Subconstructs
2.1.5.1.A single rational number can have several conceptually distinct meanings.
2.1.5.2.Example = ¾
2.1.5.2.1. Part-whole relationship, as in 3 of 4 equal size shares.
2.1.5.2.2. Quotient interpretation, as in 4 kids sharing 3 pies. (p. 314)
2.1.5.2.3. Ratio, as in 3 red cars for every 4 green cars, not to be confused by the part-whole relationship of 3/7, where 3 out of the 7 cars are red.
2.1.5.2.4. Measure, as in ¾ of an inch.
2.1.5.2.5. Multiplicative operator, behaving as an operation that reduces or enlarges the size of another quantity, as in ¾ of the original size.
2.1.5.3.Coordinating these different interpretations requires a deep understanding of concepts and interrelations among them.
2.1.6. Reconceptualizing the Unit Operations
2.1.6.1.Rational numbers are dense – we can find an infinity of other numbers between any 2 rational numbers.
2.1.6.2.Rational numbers are also implied, as opposed to explicit. (p. 315)
2.1.7. New Conceptualizations
2.1.7.1.Numbers must be understood in multiplicative relationship.
2.1.7.2.Students must understand that numbers are no longer independent, and understand a fraction as a new kind of quantity that is defined multiplicatively by relative amount conveyed by symbols.
2.1.7.3.Looking at students prior conceptions and relevant understandings can help support the conceptual change.
2.2. Students Errors and Misconceptions Based on Previous Learning (Principle 1)
2.2.1. Conceptions of numbers are grounded in whole-number learning, which leads them to confusion in rational numbers. (p. 316)
2.2.2. For example, many students think that 2/3 and ¾ are equal because “one piece” is missing. They are thinking in additive, absolute terms instead of relative or proportional (multiplicative) terms. (p. 317)
2.3. Metacognition and the Rational Number (Principle 3)
2.3.1. Students must be actively engaged in sense making to solve problems competently.
2.3.2. Most middle school children do create appropriate meanings for fractions, decimals, and percents; instead they rely on memorized rules for symbol manipulation.
2.3.3. Mistakes are made by students’ failure to monitor their operations and judge the reasonableness of their responses.
2.3.4. Classroom teaching must support students in developing metacognitive skills (explaining their answers, strategies, interpretations, etc.) or math may no longer make sense to students, particularly at the start of rational numbers.
3. Instruction in Rational Number
3.1. Why does instruction so often fail to change students’ whole-number perceptions?
3.1.1. HSL principles are routinely violated.
3.1.2. Topics in rational numbers are typically covered quickly and superficially, but it takes time for students to master it thoroughly.
3.1.3. Not enough time is dedicated to teaching conceptual meaning behind rational numbers, instead procedures receive greater emphasis.
3.1.4. Procedural competence is important, but it must be anchored by conceptual understanding, and this is not the case for a great number of students.
3.1.5. Other aspects, such as teaching the notation system for decimals, are short-changed in this way as well.
3.1.6. Operations tend to be taught in isolation, (p. 319) divorced from meaning.
3.1.7. Almost no time is spent relating various representations – fractions, decimals, percents – to each other.
3.1.8. Researchers agree that textbooks fail to provide a grounding for the major conceptual shift to multiplicative reasoning that is essential to mastering rational number.
3.1.9. Examples of instructional approaches
3.1.9.1.Pie Charts and a Part-Whole Interpretation of Rational Numbers
3.1.9.1.1. Typically a pie or cake is used to introduce fractions. The object is divided into equal pieces, with a certain number of pieces shaded. Students count the number of pieces which indicates the denominator, then count the shaded pieces, which becomes the numerator. (p. 320)
3.1.9.1.2. This instruction is an obvious approach, but it is an additive approach and does not introduce children to the difficult conceptual shift of multiplicative reasoning.
3.1.9.2.Alternative Instructional Approaches: Ratio and Sharing
3.1.9.2.1. Multiplicative operations of splitting.
3.1.9.2.1.1.Folding paper rather than pie charts. (Kieren)
3.1.9.2.1.2.Contexts of ratios using cooking, shadows, gears, and ramps. (Confrey)
3.1.9.2.1.3.Contexts of ratios using equal shares and quotients by having children use realistic situations like sharing pancakes or chocolate bars and the children devise a system themselves. (Streefland) (p. 321)
3.2. Cirriculum developed using a different approach to rational number, alongside Robbie Case, shown through controlled experiment trials to be effective in helping students in the 4th, 5th, and 6th grades gain a strong initial grounding in the number system, highlights multiplicative understanding, with an additional focus on interrelations among fractions, decimals, and percents.
3.2.1. Pipes, Tubes, and Breakers: A New Approach to Rational Number Learning
3.2.1.1.Percents as a stating point
3.2.1.1.1. Start with percents first because it only involves fractions of the base 100.
3.2.1.1.2. Done through everyday understanding of students.
3.2.1.1.3. Students are challenged to consider relative lengths of different quantities.
3.2.1.1.4. Initial activities direct students’ attention to ideas of relative amount and proportion from the very beginning of their learning of rational number. (p. 322)
3.2.1.1.5. Ex: 2 different size beakers both 50% full.
3.2.1.1.6. Ideas of percents and proportion serve as the anchoring concept for subsequent learning of decimals and fractions, then for understanding of the number system as a whole.
3.2.1.2.Starting Point: Visial Proportional Estimation and Halving and Doubling
3.2.1.2.1. From prior experiences and understandings, students at this age have generally developed the ability to estimate proportions, such as halves, visually, and have the ability to work with successive halving. (p.323)
3.2.1.2.2. Both abilities have grounding in multiplicative operations.
3.2.1.2.3. Idea is to merge these separate understandings to construct conceptual grounding for rational numbers.
3.2.1.2.4. Strategy was to develop a “bridging context” to help students first access their knowledge of visual proportions, then integrate it with their knowledge of halving.
3.2.1.2.5. Context is having students work with percents and linear measurement, allowing students to access initial kind of understanding and integrate them in a natural fashion.
3.2.1.3.Why Percent as a Starting Point?
3.2.1.3.1. Students always working with the denominator of 100, thus postponing problems that arise when students must compare or manipulate ratios with different denominators.
3.2.1.3.2. Allows students to concentrate on working with their own procedures rather than struggling to master complex algorithms or procedures.
3.2.1.3.3. All percentages have a corresponding decimal or fractional equivalent that can be relatively easy to determine, which helping in starting to develop the understanding of how the 3 representations are related.
3.2.1.3.4. Children know a good deal already about percents from their everyday experiences. (p. 324)
3.2.2. Curriculum Overview
3.2.2.1.Divided into roughly 3 parts.
3.2.2.1.1. Percent – uses concrete props that highlight linear measurement.
3.2.2.1.2. Two-place decimal – introduced like the percent of the way between 2 whole numbers.
3.2.2.1.3. Activities that promote comparing and ordering rational numbers and moving along decimals and percents. Fractions are also taught at this stage in relation to percents and decimals.
3.2.2.2.Lessons
3.2.2.2.1. Lessons Part 1: Introduction to Percents
3.2.2.2.1.1.Percents in Everyday Life – discussion to elicit what students know about the topic.
3.2.2.2.1.2.Pipes and Tubes: A Representation of Fullness
3.2.2.2.1.2.1. Props include black drainage pipes and white venting tubes that can be raised or lowered simulating (p. 325) the action of water filling them to different levels.
3.2.2.2.1.2.2. To get a better idea of the students’ understanding, they are asked how they would teach percent to another child using these props.
3.2.2.2.1.2.3. Children are asked to use the props to generate ideas about percents above 100.
3.2.2.2.1.3.Percents on Number Lines: More Estimations
3.2.2.2.1.3.1. Included activities with laminated meter-long number lines calibrated in centimeters to provide students with another way of visualizing percent. (p. 327)
3.2.2.2.1.3.2. Activities included using the line as a sidewalk where students were asked by peers to walk some %.
3.2.2.2.1.3.3. Activities are used to consolidate percent understanding and extend linear measurement context.
3.2.2.2.1.4.Computing with Percent
3.2.2.2.1.4.1. Estimate the fullness of various beakers of water.
3.2.2.2.1.4.2. Leads naturally to a focus on computation and measurement using halving strategies.
3.2.2.2.1.5.Invented Procedures – From the previous activity, students used their halving strategy as the basis for their procedure of calculating certain percents (e.g. 75%).
3.2.2.2.1.6.String Challenges: Guessing Mystery Objects
3.2.2.2.1.6.1. String measurement activities as a way of considering percent quantities and calculating percentages using benchmarks.
3.2.2.2.1.6.2. “The Mystery Object Challenge” – teacher held up a piece of string that was cut to the percent of the length of a certain object in the room. (p. 329) Eventually, the students were invited to pick the object to challenge the rest of the class.
3.2.2.2.1.6.3. Students then created “percent family” posters of strings using their mystery object as the base, which provided another opportunity for calculating percents, as well as reinforced the idea of proportion. “Our string lengths are different even though all our percents are the same.”
3.2.2.2.1.7.Summary of Lessons Part 1
3.2.2.2.1.7.1. Estimations then calculations of percent quantities.
3.2.2.2.1.7.2. Presented in the context of linear measurement of pipes, tubes, beakers, string, and number lines.
3.2.2.2.1.7.3. Students were not given formal instruction, but naturally employed procedures of their own.
3.2.2.2.1.7.4. While percent was the only form of rational numbers introduced, students often referred to fractions when working these initial activities. (p. 331)
3.2.2.2.1.7.5. Students were told that 12 ½ % is 1/8, as they new 25% is ¼.
3.2.2.2.1.7.6. Activities continually helped students integrate their sense of visual proportion with their ability to do repeated halving and help build the foundation for further learning of this number system.
3.2.2.2.2. Lessons Part 2: Introduction of Decimals
3.2.2.2.2.1.Two-place decimal.
3.2.2.2.2.2.Initial lessons also had strong focus on measurement and proportion.
3.2.2.2.2.3.Research has shown that a solid conceptual grounding in decimal numbers is hard for a student to achieve.
3.2.2.2.2.4.Makes a direct link from percents to decimals.
3.2.2.2.2.5.Show students that a two-place decimal number represents a percentage of the way between 2 adjacent whole numbers, or an intermediate distance between 2 numbers.
3.2.2.2.2.6.Decimals and Stopwatches
3.2.2.2.2.6.1. Props for students were LCD stopwatches displaying seconds and hundredths of seconds.
3.2.2.2.2.6.2. Students were asked to consider what the 2 “smaller numbers” might mean.
3.2.2.2.2.6.2.1.Students noted that there were 100 of these time units.
3.2.2.2.2.6.2.2.“It’s like they are percents of a second.” (p. 332)
3.2.2.2.2.6.3. Asked to name the time units and it was termed as the quantity they understood to mean time that had passed between any 2 whole seconds.
3.2.2.2.2.6.4. Work continued with the stopwatches and decimals with a focus on ordering numbers.
3.2.2.2.2.7.Magnitude and Order in Decimal Numbers
3.2.2.2.2.7.1. To illuminate the difficult concepts of magnitude and order, activities were designed to help students with ordering decimals.
3.2.2.2.2.7.2. “Stop-Start Challenge”
3.2.2.2.2.7.2.1.Students attempted to start and stop the stopwatch as quickly as possible, several times in succession.
3.2.2.2.2.7.2.2.After discussion, they learned to record their times as decimals.
3.2.2.2.2.7.2.3.Students then compared their reaction times with their peers and ordered their times.
3.2.2.2.2.7.2.4.Students can learn from their experience in trying to get the quickest time that .09 is small than .10.
3.2.2.2.2.7.3. “Stop the Watch Between” is another game to engage students in the issue of magnitude. (p. 333)
3.2.2.2.2.7.4. Laminated number lines where students were asked to indicate parts of 100 using decimal representations.
3.2.2.2.2.8.Summary of Lessons Part 2
3.2.2.2.2.8.1. First understand how decimals and percents are related, then learn how to represent decimals symbolically.
3.2.2.2.2.8.2. Activities proceeded for students to consider and reflect on magnitude.
3.2.2.2.2.8.3. Finally students engaged in comparing and ordering decimals.
3.2.2.2.2.8.4. First step in students’ learning to translate among the representations of rational number and gain fluency with different kinds of operations.
3.2.2.2.3. Lessons Part 3: Fractions and Mixed Representations of Rational Numbers
3.2.2.2.3.1.Fractions First: Equivalencies
3.2.2.2.3.1.1. Give students a chance to work with fractions more formally and then provide them with opportunities to translate flexibly among fractions, decimals, and percents.
3.2.2.2.3.1.2. Students were asked to represent a fraction in as many ways as they could. (Ex: 3/4 they would represent as 6/8, 75/100).
3.2.2.2.3.1.3. They were also asked to incorporate fractions into word problems they created and other students solved.
3.2.2.2.3.1.4. Students also created equations with questions for other students to solve. (Ex: How many to make a whole for 1/8 + ½ + ¼?) (p. 334)
3.2.2.2.3.1.5. Students initially used fractions in the equations, but then started adding percents and decimals on their own.
3.2.2.2.3.2.Crack the Code
3.2.2.2.3.2.1. Further work with LCD stopwatches.
3.2.2.2.3.2.2. Students moved between representations of rational numbers as they were challenged to stop the watch at the decimal equivalent of different fractions and percents.
3.2.2.2.3.2.3. This allowed them to increase their understanding of the possibility of fluid movement between representations.
3.2.2.2.3.3.Card Games
3.2.2.2.3.3.1. Specially designed set of cards depicting various representations of fractions, decimals, and percents.
3.2.2.2.3.3.2. Students used the cards to design games tat challenged their peers to make comparisons among and between representations. (p. 335)
3.2.2.2.3.4.Summary of Lessons Part 3
3.2.2.2.3.4.1. Focused primarily on students’ uses of mixed representations.
3.2.2.2.3.4.2. Started with some formal activities with fractions and equivalencies, then had students make up own games and challenges for more practice in flexible movement from one operation to another.
3.2.2.2.3.4.3. One of the primary goals was to provide students with habits of mind regarding multiple representations that will be with them throughout their learning and lay the foundation for their ability to solve mathematical problems.
3.2.2.3.Results from Studies
3.2.2.3.1. Needed to look at the improvement made by individual students at the end of the experimental intervention.
3.2.2.3.2. Also interested to see the performance of students in experimental group compared to those with traditional classroom instruction.
3.2.2.3.3. Thus, assessed the experimental students on a variety of task before and after course instruction and administered these same tasks (p. 337) to students from classrooms with standard textbook instruction.
3.2.2.3.4. Students in experimental group improved significantly, often higher than those students receiving traditional classroom instruction, who were often also older.
3.2.2.3.5. Quality of answers was also better from the experimental group, and also made more frequent reference to proportional concepts. (p. 338)
3.2.2.3.6. Knowledge Network
3.2.2.3.6.1.Students gained the following understandings toward developing their knowledge network:
3.2.2.3.6.1.1. An overall understanding of the number system, as evidenced by their ability to use representations of decimals, fractions, and percents interchangeably.
3.2.2.3.6.1.2. Appreciation of the magnitude of rational numbers, as seen in their ability to compare and order numbers.
3.2.2.3.6.1.3. Understanding of the proportional- and ratio-based constructs of this domain, demonstrated by equivalencies.
3.2.2.3.6.1.4. Understanding of percent as an operator, as demonstrated by their ability to invent a variety of solution strategies for calculating with these numbers.
3.2.2.3.6.1.5. General confidence and fluency in their ability to think about the domain using the benchmark values they had learned, which is a hallmark of number sense.
3.2.2.3.6.2.Research is still in an early stage.
3.2.2.3.6.3. Research will continue to pursue questions, including the potential limitations of continued halving as a way of operating with rational numbers, downplaying the importance of quotient subconstruct, as well as a limited view with fractions.
3.2.2.3.6.4.Research will also continue to learn more about how children learning this way should proceed with their learning of mathematics. (p. 340)
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