Trajectorias Aprendizagem Clemets&Sarama 2009

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Learning Trajectories in Early Mathematics Sequences of

Acquisition and Teaching

Written by: Douglas H. Clements, Ph.D. and Julie Sarama, Ph.D., Graduate School ofEducation, University at Buffalo, the State University of New York at Buffalo

Introduction

Children follow natural developmental progressions in learning and development. As asimple example, children first learn to crawl, which is followed by walking, running,skipping, and jumping with increased speed and dexterity. Similarly, they follow naturaldevelopmental progressions in learning math; they learn mathematical ideas and skillsin their own way. When educators understand these developmental progressions, andsequence activities based on them, they can build mathematically enriched learningenvironments that are developmentally appropriate and effective. These developmentalpaths are a main component of a learning trajectory.

Key Research Questions

Learning trajectories help us answer several questions.

1. What objectives should we establish?2. Where do we start?

3. How do we know where to go next?4. How do we get there?

Recent Research Results

Recently, researchers have come to a basic agreement on the nature of learningtrajectories (Clements & Sarama, 2004). Learning trajectories have three parts: a) amathematical goal; b) a developmental path along which children develop to reach thatgoal; and c) a set of instructional activities, or tasks, matched to each of the levels ofthinking in that path that help children develop higher levels of thinking. Let's examineeach of these three parts.

Goals: The Big Ideas of MathematicsThe first part of a learning trajectory is a mathematical goal. Our goals are the big ideasof mathematics clusters of concepts and skills that are mathematically central andcoherent, consistent with childrens thinking, and generative of future learning. Thesebig ideas come from several large projects, including those from the National Council ofTeachers of Mathematics and the National Math Panel (Clements & ConferenceWorking Group, 2004; NCTM, 2006; NMP, 2008). For example, one big idea is that

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counting can be used to find out how many are in a collection. Another would be,geometric shapes can be described, analyzed, transformed and composed anddecomposed into other shapes. It is important to realize that there are several such bigideas and learning trajectories, depending on how you classify them, there are about12.

Development Progressions: The Paths of LearningThe second part of a learning trajectory consists of levels of thinking; each moresophisticated than the last, which lead to achieving the mathematical goal. That is, thedevelopmental progression describes a typical path children follow in developingunderstanding and skill about that mathematical topic. Development of mathematicsabilities begins when life begins. Young children have certain mathematical-likecompetencies in number, spatial sense, and patterns from birth (Clements & Sarama,2007a, Sarama & Clements, 2009).

However, young children's ideas and their interpretations of situations are uniquely

different from those of adults. For this reason, good early childhood teachers are carefulnot to assume that children see situations, problems, or solutions as adults do.Instead, good teachers interpret what the child is doing and thinking; they attempt to seethe situation from the childs point of view. Similarly, when these teachers interact withthe child, they also consider the instructional tasks and their own actions from the childspoint of view. This makes early childhood teaching both demanding and rewarding.

The learning trajectories we created as part of the Building Blocks(http://ubbuildingblocks.org) and TRIAD (http://www.ubtriad.org) projects provide simplelabels for each level of thinking in every learning trajectory. Figure 1 illustrates a part ofthe learning trajectory for counting. The Developmental Progression column providesboth a label and description for each level, along with an example of children's thinkingand behavior. It is important to note that the ages in the first column are approximate.Without experience, some children can be years behind this average age. With high-quality education, children can far exceed these averages. As an illustration, 4-year-oldsin our Building Blocks curriculum meet or surpass the 5-year-old level in most learningtrajectories, including counting. (For complete learning trajectories for all topics inmathematics, see Clements & Sarama, 2009; Sarama & Clements, 2009. These worksalso review the extensive research work on which all the learning trajectories arebased.)

Instructional Tasks: The Paths of TeachingThe third part of a learning trajectory consists of set of instructional tasks, matched toeach of the levels of thinking in the developmental progression. These tasks aredesigned to help children learn the ideas and skills needed to achieve that level ofthinking. That is, as teachers, we can use these tasks to promote children's growth fromone level to the next. The third column in Figure 1 provides example tasks. (Again, thecomplete learning trajectory in Clements & Sarama, 2009a and 2009b includes not onlyall the developmental levels, but several instructional tasks for each level.)

http://ubbuildingblocks.org/http://www.ubtriad.org/http://www.ubtriad.org/http://ubbuildingblocks.org/


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Table 1. Samples from the Learning Trajectory for Counting (all examples fromClements & Sarama, 2007b, Clements & Sarama, 2009, Sarama & Clements, 2009).

Age Developmental Progression Instructional Tasks

1

yearPre-CounterVerbalNo verbal counting.

Chanter VerbalChants sing-song orsometimes-indistinguishable numberwords.

Associate number words with quantitiesand as components of the countingsequence.

Repeated experience with the countingsequence in varied contexts.

2ReciterVerbalVerbally counts with

separate words, not necessarily in thecorrect order.

Provide repeated, frequent experiencewith the counting sequence in variedcontexts.

Count and Race Children verbally countalong with the computer (up to 50) byadding cars to a racetrack one at atime.

3Reciter (10)VerbalVerbally counts to

ten, with somecorrespondence withobjects.

Count and MoveHave all children countfrom 1-10 or an appropriate number,making motions with each count. Forexample, say, one [touch head],two [touch shoulders], three [touchhead], and so forth.

Corresponder Keeps one-to-onecorrespondence between countingwords and objects (one word for eachobject), at least for small groups ofobjects laid in a line.

Kitchen Counter At the computer,children click on objects one at a timewhile the numbers from one to ten arecounted aloud. For example, theyclick on pieces of food and a bite istaken out of each as it is counted.

4Counter (Small Numbers) Accurately

counts objects in a line to 5 andanswers the how many question

with the last number counted.

Cubes in the BoxHave the child count asmall set of cubes. Put them in thebox and close the lid. Then ask the

child how many cubes you are hiding.If the child is ready, have him/herwrite the numeral. Dump them outand count together to check.

Pizza Pizzazz 2Children count items upto 5, putting toppings on a pizza tomatch a target amount.



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Producer (Small Numbers) Counts out

objects to 5. Recognizes that countingis relevant to situations in which acertain number must be placed.

Count Motions While waiting duringtransitions, have children count howmany times you jump or clap, or someother motion. Then have them dothose motions the same number of

times. Initially, count the actions withchildren.

Pizza Pizzazz 3Children add toppings toa pretend pizza (up to 5), to matchtarget numerals.

5Counter and Producer (10+) Counts

and counts out objects accurately to10, then beyond (to about 30). Hasexplicit understanding of cardinality

(how numbers tell how many).Keeps track of objects that have andhave not been counted, even indifferent arrangements.

Counting Towers (Beyond 10) To allowchildren to count to 20 and beyond,have them make towers with otherobjects such as coins. Children build

a tower as high as they can, placingmore coins, but not straighteningcoins already in the tower. The goal isto estimate and then count to find outhow many coins are in your tallesttower.

Dino Shop 2Children add dinosaurs to abox to match target numerals.

In summary, learning trajectories describe the goals of learning, the thinking and

learning processes of children at various levels, and the learning activities in which theymight engage. People often have several questions about learning trajectories.

Future Directions

Although learning trajectories have proven to be effective for early mathematicscurricula and professional development (Clements & Sarama, 2008; Sarama, Clements,Starkey, Klein, & Wakeley, 2008), there have been too few studies that have comparedvarious ways of implementing them. Thus, their exact role remains to be studied. Also,in the early years, several learning trajectories are based on considerable research,such as those for counting and arithmetic. However, others, such as patterning and

measurement, have a smaller research base. Further, there are few guidelines for manymore sophisticated math topics for teaching older students. These remain challenges tothe field.

Conclusions

Learning trajectories hold promise for improving professional development and teachingin the area of early mathematics. For example, the few teachers that actually led in-depth discussions in reform mathematics classrooms saw themselves not as moving



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through a curriculum, but as helping students move through levels of understanding(Fuson, Carroll, & Drueck, 2000). Further, researchers suggest that professionaldevelopment focused on learning trajectories increases not only teachers professionalknowledge but also their students motivation and achievement (Clarke, 2004; Fennemaet al., 1996; Wright, Martland, Stafford, & Stanger, 2002). Thus, learning trajectories

can facilitate developmentally appropriate teaching and learning for all children.

Author Note:This paper was based upon work supported in part by the National Science Foundationunder Grant No. ESI-9730804 to D. H. Clements and J. Sarama Building BlocksFoundations for Mathematical Thinking, Pre-Kindergarten to Grade 2: Research-basedMaterials Development and in small part by the Institute of Educational Sciences (U.S.Department of Education, under the Interagency Educational Research Initiative, orIERI, a collaboration of the IES, NSF, and NICHHD) under Grant No. R305K05157 to D.

H. Clements, J. Sarama, and J. Lee, Scaling Up TRIAD: Teaching Early Mathematicsfor Understanding with Trajectories and Technologies. Any opinions, findings,conclusions, or recommendations expressed in this material are those of the authorsand do not necessarily reflect the views of the funding agencies. The curriculumevaluated in this research has since been published by the authors, who thus have avested interest in the results. An external auditor oversaw the research design, datacollection, and analysis, and five researchers independently confirmed findings andprocedures. The authors, listed alphabetically, contributed equally to the research.

Date Posted Online: 2009-04-30 10:02:54



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Trajectorias Aprendizagem Clemets&Sarama 2009

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