Algebraic Thinking Framework 1997

8/11/2019 Algebraic Thinking Framework 1997

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One teacher made up this story to introduce

her sixth-grade students to Crossing theRiver, one of the lessons in a unit called Patternsin Numbers and Shapes.

You and your partner have gone on a long hikewi th 8 adult s. You are very hot and t ir ed . Yourgroup gets to a wide river that you must cross toget home. No one in your group knows how toswim. On the riverbank is a small boat, which canonly hold 2 children, or 1 adult, or 1 child. Howmany one-way trips does it take to get all the peo-ple in your group across the river?

After a brief discussion of the story by the class,

groups of students sought different tools to solvethe problem. Some groups began diagramming thestory on paper, others acted it out, and others usedcubes of different colors to track the trips acrossthe river. Whatever their strategy, students in eachgroup were following an investigative process thatthey had learned for finding and generalizing pat-terns. At the same time, they were building a foun-dation in algebraic thinking.

The Patterns in Numbers and Shapes unit wasdeveloped as part of the National Science Founda-

tion-funded curriculum MathScape: Seeing andThinking Mathematically and was field-tested inclassrooms across the country. In each lesson, stu-dents encounter a problem presented in a context,and they work in pairs or small groups to act outthe story, whether kinesthetically, visually by draw-ing pictures, or manipulatively by modeling the sit-uation with physical objects. They engage in an in-

vestigative process to solve the problem: (1) theyseek out a pattern in the story, (2) they recognizethe pattern and describe it using different methods,and (3) they generalize the pattern and relate it to

the story. The Patterns in Numbers and Shapesunit and all the algebra-focused units in the cur-riculum are intended to provide a new image ofhow students can develop algebraic-thinking skills.

A broad view of algebra ic th inking i s t aken toshow students the real-life uses and relevance ofalgebra. Algebraic thinking is using mathematicalsymbols and tools to analyze different situations by(1) extracting information from a situation, such as

Patterns as Tools

for AlgebraicReasoning

Kristen Herbert and Rebecca H. Brown

Kristen Herbert has been a curriculum devel-oper, with particular interests in algebra andperformance assessment, at the Education De-velopment Center (EDC) for five years. She iscurrently teaching middle school mathematics.Rebecca Brown has been a classroom researcher,with particular interests in algebra and technol-ogy, at EDC for the past three years. She plans tocontinue classroom research in mathematics.The authors are very grateful to the many teach-ers and students who field-tested this unit and

would like to thank their colleagues at the Edu-cation Development Center, who helped createthis unit, and Charles Lovitt, who provided theinitial drafts of many of the problems in the unit.

This material is based on work supported bythe National Science Foundation under grantno. ESI-9054677. The views expressed in thisarticle do not necessarily reflect those of theNational Science Foundation.

Reprinted from Barbara Moses,ed.,Algebraic Thinking, Grades K12:Readings from NCTMs School-Based Journals and Other Publications (Reston,

Va.: National Council of Teachers of Mathematics,2000),pp. 12328.Originally appeared in Teaching Children Mathematics 3 (February 1997):34045.

1997 by NCTM.


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the one described in Crossing the River; (2) rep-resenting that information mathematically in words,diagrams, tables, graphs, and equations; and (3) in-terpreting and applying mathematical findings,such as solving for unknowns, testing conjectures,and identifying functional relationships, to thesame situation and to new, related situations. Theinvestigative process used in the Patterns in Num-bers and Shapes unit is an initial, informal exampleof this three-part framework (see fig. 1).

Using this investigative process to solve contextual-ized problems gives students the kind of informalexploration of algebra discussed in the Curriculumand Evaluation Standards for School Mathematics

(NCTM 1989, 102). It states, It is thus essential thatin grades 58, students explore algebraic conceptsin an informal way to build a foundation for thesubsequent formal study of algebra. Rather thanpush students into formal symbolic algebra, the

unit emphasizes algebraic thinking by leading stu-dents to communicate their toughts in their own

words or thei r own symbols. In addition, thefive-week unit gives students the opportunities andsupport to develop pattern-seeking skills and anability to generalize patterns from concrete situa-tions. The approach is also intended to increasestudents confidence in themselves as being capa-ble algebra students.

The Investigative Process toSolve Problems

The investigative process consists of three phases:(l) pattern seeking, (2) pattern recognition, and (3)generalization. These three phases are specificcomponents of the general framework shown infigure 1; pattern seeking is extracting information,

pattern recognition is mathematical analysis, andgeneralization is interpreting and applying what

was lea rned. S tudents fo ll ow the invest iga tiveprocess for all twelve lessons in the unit. Crossingthe River, one lesson from the unit, will next beused to illustrate the investigative process that stu-dents follow. Students are guided through the les-son by the student page shown in figure 2. In ateachers guide, suggestions are made for the useof manipulatives, physical activity, and drawings tohelp students solve the problems.

Pattern Seeking

To begin the Crossing the River problem, stu-dents model the situation presented in the first stepof the task (see step 1 in fig. 2) to look for pat-terns. As suggested in the teachers guide, teachersgive students some type of counters to help solvethe task; two counters of one color represent thechildren, and eight of another color represent theadults (see fig. 3).

As students begin to move the boat back and forth

across the river, they begin to make comments,such as, Someone has to row the boat backacross the river, maybe a kid can do it and Thearms on these kids are going to be really tired afterall this rowing!

Students collect and record numerical data as theywork through the situation. They develop manydifferent methods of recording the data. Some stu-

Fig. 1. A framework for algebraic thinking in context

Situation

Extracting Interpreting

and applying

Mathematicalrepresentation

Mathematicalfindings

Mathematicalanalyses

Fig. 2. Student sheet for Crossing the River

1. A group of 8 adults and 2 children need to cross a

river. A small boat is available that can hold 1 adult,

or 1 child, or 2 children.Everyone can row the boat.

How many one-way trips does it take for them all to

cross the river?

2. What if there were different numbers of adults?

How many trips would be required in the following

situations?

6 adults and 2 children?




3. How would you find the number of trips needed for

2 children and any number of adults? Describe the

method you would use to solve the problem.


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dents record the number of trips either as tallymarks or in a list. Others draw a picture or dia-gram that shows each trip across the river (fig. 4).

While gathering and recording their data, studentsbegin to notice recurring patterns. Some studentsnotice a pattern while working with the manipula-tives and use their recorded data to check theiridea of the pattern. In one class, for example, twogirls quickly exclaimed to their teacher, We foundthe pattern. Using the manipulatives, they showedtheir teacher and explained, First you send twochildren over, next you bring one child back, then

you send an adult over, and the other child bringsthe boat back. It doesnt matter how many adults

you have, you just need two children. You can getany number of adults across even if it takes allday! To explain how many total trips occurred for8 adults, however, the two girls had to go throughthe process again, record all the trips, and thencount the total number of trips. Other studentslook for the pattern in their recorded data (2 chil-dren across, 1 child back, 1 adult across, the sec-

ond child back, 2 children across, 1 child back,and so on) and recognize that it would take fourtrips to get 1 adult across the river.

Pattern Recognition

Students try additional cases of the problem withdifferent numbers of adults (step 2 in fig. 2). Forsome students, solving additional cases serves as a

way to test and refine their understanding of thepattern (fig. 5), and for others, it provides addi-tional opportunities to recognize a pattern (fig. 6).

In the last problem of this step, students are askedto explore how many trips it would take to get 100adults across the river. At this point, some students

Fig. 3. Students work through the problem using ma-

nipulatives and a river drawn on paper.

Fig. 4. One student recorded the trips across the riveras a picture.

Fig. 5. One student explains how she and her partnerused the pattern to solve the problem and generate aformula.


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do not use the manipulatives to get a solution; in-stead these students substitute 100 into their under-standing of the pattern to calculate a solution (fig.7). However, other students have been using themanipulatives to solve all the cases up to thispoint. One students dilemma is captured in hiscomment, On the paper it said what would be theanswer if there were 100 grown-ups, and I thought

we had to do the problem [for 100 grown-ups]. Ihope we dont have to do that problem! Askingstudents to solve the prob lem for a very largenumber of adults motivates them to look for a gen-eralized method for solving the situation on thebasis of patterns they saw in simpler cases ratherthan act out a very large case.

Students use multiple representations of the data intheir search for a generalized statement of the pat-tern. If students are stuck, teachers may suggestthat they represent the data in a table or a graph.

Students have learned to read data from tables andgraphs in introductory lessons in the unit; thus, therepresentations can help them find the patterns.

Generalizing

In the final step of the task (step 3 in fig. 2), stu-dents develop a generalized method for figuringout how many trips it would take to get everyoneacross the river, given any number of adults. Stu-dents articulate a generalized rule from their pat-tern in a way that they feel most comfortableby

using words, diagrams, their own invented sym-bols, or equationsand explain it by relating it tothe original situation (figs. 8 and 9).

The most important aspect of this step is for stu-dents to describe how their generalization relatesto the physical situation; why do you multiplythe number of adults by 4 and add 1 for the lasttrip? When students can answer these questions,they have gained an important understanding ofhow to use algebraic thinking to model a con-crete situation.

From generalizing the pattern, students come tounderstand the power of algebraic thinking. Asone student commented, There was a pattern.Every four trips were the same. You can figure outhow many trips it took by using a formula. To fig-ure it out fast, you could use a formula. Anotherstudent wrote, I think we try to find formulas soit will be easier to get the problems done. Formu-las make problems easy to solve. Its very helpful.

A third student wrote, Sometimes when you thinkyou found an equat ion, what you really found is

the steps you took to get the answer in a short-ened way. Seeing the power of algebraic thinkingmotivates students to engage in this kind of think-ing when presented with similar situations.

Benefits of a Whole Unit

Dedicated to this Approach

As in the Crossing the River example, the otherlessons in the unit employ the same approach:students are given a contextualized problem,

which they then solve through an investigativeprocess. For example, in Painted Rods studentsare told of a company that produces painted rodsof varying lengths by using a paint-stamping ma-chine (see fig. 10). The paint stamp marks exactlyone square of the rod each time. Students areasked, How many paint stamps does it take topaint all the faces of different lengths of rods?

Fig. 6. Some students begin by writing things out butthen recognize the pattern.

Fig. 7. One student used a generalized understandingof the pattern to solve the problem for 100 adults.


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Following the investigative process, students usemanipulatives to construct models of the rods. Asingle cube serves as the paint stamp as studentseither physically emulate or visualize painting rods

of lengths 1 to 10. As students solve the problemsfor these rods, they collect numerical data, orga-nize them in a table, and use their table to searchfor a pattern in the numbers. Next, students findthe number of stamps required to paint rods oflength 12, 25, and 100. Solving such large cases asa rod of length 100 motivates students to find thegeneral relationship between the length of a rodand the number of stamps required to paint it. Stu-dents then use their method of choicewords, di-agrams, or their own invented symbolsto repre-sent their generalized understanding of the pattern

as a rule. Finally students explain their rule interms of the original situation.

To extend their thinking, students consider whatthe input may have been for a given output. Forexample, in Painted Rods students are asked touse their understanding of the pattern to explorethe question If it takes 86 stamps to paint therod, how long is the rod? Another way that stu-dents thinking is extended is by asking them tosolve a similar, but more challenging, problem.

After completing Crossing the River, for exam-ple, students are asked to solve the same problem

with a different number of children. What hap-pens to the pattern if we have a different numberof children? (a) 8 adults and 3 children or (b) 2adults and 5 children? Similarly, after PaintedRods, students are asked to solve the problemfor rods of double width.

Together the lessons in the Patterns in Numbersand Shapes unit afford students multiple opportu-

nities to learn the investigative process of patternseeking, pattern recognition, and generalization.Students engage in the process in each lesson,and, consequently, over the course of the unitthey internalize the approach to solving situationsinvolving patterns. Students understanding of andability to use the process is evident in the strate-gies and skills for pattern seeking that they ex-hibit; the tools of tables, graphs, and verbal rulesthat they use for describing patterns; and the gen-eralized rules they articulate. Thus, in the future

when students are presented with a problem in-

volving patterns, they have strategies for ap-proaching the problem.

The unit also has a positive effect on students per-ception of their ability to generalize a rule from aconcrete situation, that is, to think algebraically. Asstudents follow the process, they are able to find apattern and express it as a generalized rule. Onestudent wrote, Finding formulas is exciting. Its

Fig. 9. Another student drew the generalized rule andexplained how it relates to the original situation.

Fig. 10. Drawing of problem in Painted Rods

Fig. 8. This student articulated the generalized rulein symbols and explained how it relates to the origi-nal situation.


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like artifacts to an archaeologist. Finding a rulemakes students feel successful, and, consequently,they begin to see themselves as being capable al-gebra students. Teachers who have taught the unit[in a field test] reported, [Students] are confi-dent. They think they are good at it. Theythink they are smart. They know that their b igbrothers and sisters do algebra. Students in-creased confidence in their ability to engage in al-gebraic thinking ultimately contributes to a positiveattitude about algebra. Their positive attitude in

conjunction with the use of an investigativeprocess and their new understanding of the powerof algebraic thinking give sixth-grade students asolid foundation on which to build a formal under-standing of algebra throughout the middle grades.

Reference

National Council of Teachers of Mathematics. Cur-riculum and Evaluation Standards for School

Mathematics. Reston, Va.: The Council, 1989.

Documents

Algebraic Thinking Framework 1997