Strong Start Math Project. Learning Trajectories

Transcription

1 Strong Start Math Project Learning Trajectories Counting Subitizing Composing and Decomposing Comparing Numbers Addition and Subtraction Understanding the Equal Sign

2 Learning Trajectory Developmental Levels for Counting The ability to count with confidence develops over the course of several years. Beginning in infancy, children show signs of understanding number. With instruction and number experience, most children can count fluently by age 8, with much progress in counting occurring in kindergarten and first grade. Most children follow a natural developmental progression in learning to count with recognizable stages or levels. This developmental path can be described as part of a learning trajectory. Level Level Name Description Notes 1 Pre-Counter A child names some number words in no apparent order and without meaning. Chanter 3 Reciter Reciter (10) A child sing-songs numbers often in some order, but it is a song and without meaning of quantity or counting. A child verbally recites number names as separate words with the intention to count, but does not necessarily recite the correct order. A child verbally counts to 10 with some correspondence with objects. The child may point to objects to count a few items but then often loses track Corresponde r Counter (Small Numbers) Producer Counter To (Small Numbers) A child can keep one-to-one correspondence between counting words and objects at least for small groups of objects laid in a line. When asked how many, the child often recounts the objects starting over with one each time. A child begins to count meaningfully. The child accurately counts a given set of objects to 5 and answers the how many question with the last number counted without needing to recount the objects. When asked to show a specific number of objects, a child can accurately produce or make a set of objects up to 5. 8 Counter (10) A child accurately counts structured arrangements of objects to 10. He or she may be able to draw representations for quantities up to 10. The child can also find the number just after or just before another number, but only by counting up from 1. 9 Counter and Producer Counter to (10+) Child accurately counts and produces sets to 10 and beyond to 30, keeping track of objects that have and have not been counted. Child draws representations to 10, then to 0 and 30, and can find the next number to 0 or 30. Child recognizes errors in others counting and can eliminate most errors in one s own counting.

3 10 11 Counter Backward from 1 Counter from N (N+1, N-1) The child is able to count backwards from 10. The child begins to count on from numbers other than one, either in verbal counts or with objects. The child can determine the number just before or just after another number quickly without having to start counting back at one Skip- Counting by 10s to 100 Counter to 100 Counter on using patterns The child can count by tens to 100. The child can count by ones through 100, including knowing the decade transitions from 39 to 0, 9 to 50, and so on, starting at any number. The child keeps track of counting acts by using numerical patterns or movements, such as tapping as he or she counts. 15 Skip Counter The child can count by five and twos with understanding Counter of Imagined Items Counter On Keeping Track Counter of Quantitative Units Counter to 00 Number Conserver The child can count mental images of hidden objects. The child can keep track of counting acts numerically with the ability to count on (one to four counts) from a given number. The child can count unusual units such as wholes when shown combinations of wholes and parts. For example when shown three whole plastic eggs and four halves, a child at this level will say there are five whole eggs. The child counts accurately to 00 and beyond, recognizing the patterns of ones, tens, and hundreds. The child demonstrates the ability to conserve number. She or he understands that a number is unchanged even if a group of objects is rearranged. For example, if there is a row of ten buttons, the child understands there are still ten without recounting, even if they are rearranged in a long row or a circle. Source: Sarama, J., & Clements, D. H. (009). Early childhood mathematics education research: Learning trajectories for young children. Routledge.

4 Recognizing Number and Subitizing The ability to recognize number values develops over the course of several years and is a foundational part of number sense. Beginning at about age, children begin to name groups of objects. The ability to instantly know how many are in a group, called subitizing, begins at about age 3. By age 8, with instruction and number experience, most children can identify groups of items and use place values and multiplication skills to count them. Most children follow a natural developmental progression in recognizable levels. This developmental path can be described as a learning trajectory. Level Level Name Description Notes 1 Small Collection Namer The first sign of a child s ability to subitize occurs when the child can name groups of one to two, sometimes three. For example, when shown a pair of shoes, this young child says, Two shoes Nonverbal Subitizier Maker of Small Collections Perceptual Subitizer to Perceptual Subitizer to 5 Conceptual Subitizer to 5+ Conceptual Subitizer to 10 Conceptual Subitizer to 0 Conceptual Subitizer with Place Value and Skip Counting The child can name the value of a small collection (one to four objects) only briefly, the child can put out a matching group nonverbally, but cannot necessarily give the number name telling how many. For example, when four objects are shown for only two seconds, then hidden, child makes a set of four objects to match. The child can nonverbally make a small collection (no more than five, usually one to three) with the same number as another collection. For example, when shown a collection of three, makes another collection of three. Progress is made when a child instantly recognizes collections up to four when briefly shown and verbally names the number of items. For example, when shown four objects briefly, says four. The child instantly recognizes briefly shown collections up to five and verbally names the number of items. For example, when shown five objects briefly, says five. The child can verbally label all arrangements to five shown only briefly. For example, a child at this level would say, I saw and and so I saw. The child can verbally label most briefly shown arrangements to six, then up to ten, using groups. For example, a child at this level might say, In my mind, I made two groups of 3 and one more, so 7. The child can verbally label structured arrangements up to twenty, shown only briefly, using groups. For example, the child may say, I saw three 5s, so 5, 10, 15. The child is able to use skip counting and place value to verbally label structured arrangements shown only briefly. For example, the child may say, I saw groups of tens and twos, so 10, 0, 30, 0,,, Conceptual Subitizer with Place Value and Multiplication The child can use groups, multiplication, and place value to verbally label structured arrangements shown only briefly. At this level a child may say, I saw groups of tens and threes, so I thought, five tens is 50 and four 3s is 1, so 6 in all. Source: Sarama, J., & Clements, D. H. (009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge.

5 Learning Trajectory Developmental Levels for Composing and Decomposing Number Composing and decomposing are combining and separating operations that allow children to build concepts of parts and wholes. Most prekindergartners can see that two items and one item make three items. Later, children learn to separate a group into parts in various ways and then to count to produce all of the number partners of a given number. Eventually children think of a number and know the different combinations of quantities that make that number. Most children follow a natural developmental progression in learning to compose and decompose numbers with recognizable stages or levels. This developmental path can be described as part of a learning trajectory. Level Level Name Description Notes Pre-Part- Whole Recognizer Inexact Part- Whole Recognizer Composer to, then 5 Composer to 7 Composer to 10 At the earliest levels of composing a child only nonverbally recognizes parts and wholes. For example, when shown four red blocks and two blue blocks, a young child may intuitively understand that all the blocks include both the red and blue blocks, but when asked how many there are in all, may respond with a random number word. A sign of development in composing is that the child understands that a whole is bigger than parts, but does not accurately quantify the set of objects. The child may be able to subitize quantities of 1,, or 3, but can not compose larger quantitmies. For example, when shown three red blocks and two blue blocks and asked how many there are in all, the child names a big number from the child s perspective, such as 5 or 10. The child can produce number combinations to and then to 5. A child at this level names parts of any whole (decomposing), or the whole given the parts (composing). For example, the child can show different ways to decompose four when asked, Show me four fingers using both of your hands. The child can use five as an anchor for the numbers within five. The child can produce number combinations to seven. A child at this level names parts of any whole to seven (decomposing), or the whole given parts (composing). For example, when playing hiding cubes, if seven cubes are shown, then some are hidden (e,g., behind the teacher s back), initially the child can determine how many are hidden and eventually can just name the hidden amount. The child can now also anchor the quantities six and seven to five (e.g., 7 is 5 and more). The child can produce number combinations within ten. A child at this level names parts of any whole (decomposing), or the whole given parts (composing). The child initially determines and then becomes solid in using ten as an anchor for any of the numbers zero through ten. For example, a child can find all the combinations for problems such as, There are ten flowers in my garden. Some are red and some are purple. How many are red and how many are purple? Adapted from: Sarama, J., & Clements, D. H. (009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge.

6 Learning Trajectory Developmental Levels for Comparing Numbers Comparing sets is a critical skill for children as they determine whether one set is larger than another. Prekindergartners can learn to use matching to compare sets or to create equivalent sets. The ability to compare sets with fluency develops over the course of several years. With instruction and number experience, most children develop foundational understanding of number relationships of more, less, and the same. Most children follow a natural developmental progression in learning to compare numbers with recognizable levels. This developmental path can be described as part of a learning trajectory. Level Level Name Description 1 Object Corresponder At this early level a child puts objects into one-toone correspondence, but with only intuitive understanding of equivalence. The child does not attend to cardinality of the sets. For example, a child may know that each carton has a straw, but does not necessarily know there are the same number of straws and cartons. Notes (e.g., What language would you use with your students? ) Perceptual of Similar Items by Subitizing of Dissimilar Items Matching Counting within 5 The child can compare sets that are quite different in size (e.g., one group is at least twice the size of the other group) and know that one set has more objects than the other set. The child can compare sets of 1- items by subitizing (i.e., just by looking at the sets), but the items in each set are the same or similar in size and type. For example, compare 3 bears and bears. The child can match small, same-sized sets of dissimilar items, such as comparing bears and chairs, and show that they are the same amount. The child compares same or different-sized sets of 1-6 objects by matching and can tell If the result is the same if there are no unmatched items or more if there are extra items. For example, a child gives one toy bone to every dog and determines if there are the same number of dogs and bones, or if there are extra dogs or bones. The child begins to compare sets by counting. Initially, the child is not always accurate when comparing different-sized objects, for example a child might think that 3 soccer balls are more than 5 golf balls. The child can make accurate comparisons, but only when objects are about the same size and the sets are small (about 1 5 objects). The child is solid at this level when he/she can accurately count to compare two sets and say which is larger even if the sets contain dissimilar items.

7 8 Counting within 10 The child compares sets within 10 by counting, the objects do not need to be the same size within 0 by Using Ten Place Value The child compares sets of objects within 0 and uses ten as a benchmark. The also begins to compare sets of objects by coordinating quantity with the relative position of numbers on a mental number path. For example, when comparing 15 pillows and 8 children, the child reasons that 15 is more than 8 because 15 is more than ten and 8 is less than ten. Another example; when comparing 13 pennies and 17 pennies, the child realizes that both numbers are composed of 10 ones and some extra ones, and reasons that 17 is greater because 7 ones is more than ones. As numbers get bigger, the child starts to anchor to the decades and can compare numbers using place value understanding. The child also extends his/her mental number path to compare larger quantities. For example, a child at this level can explain that 63 is more than 59 because six tens is more than five tens, while simultaneously knowing that they do not need to compare the ones. A child might also reason that 63 is greater than 59 because it is more than 60 and further along on the number path. Adapted from: Sarama, J., & Clements, D. H. (009). Early childhood mathematics education research: Learning trajectories for young children. Routledge.

8 Adding and Subtracting Learning single-digit addition and subtraction is generally characterized as learning math facts. It is assumed that children must memorize their facts, yet research has shown that addition and subtraction have their roots in counting, counting on, number sense, the ability to compose and decompose numbers, and place value. Research has shown that learning methods for adding and subtracting with understanding is much more effective than rote memorization of seemingly isolated facts. Most children follow an observable developmental progression in learning to add and subtract numbers with recognizable stages or levels. This developmental path can be described as part of a learning trajectory. Level Level Name Description Notes 1 3 Pre-Explicit +/- Nonverbal +/- Small Number + Find the Result The child shows no sign of being able to add or subtract. The first inkling of development is when a child can add and subtract very small collections nonverbally. For example, when shown two objects which are then covered with a cup, and then shown one more object going under the cup, the child identifies or makes a set of three objects to match the hidden amount. The child can find the total amount for joining situations (i.e., word or story problems) up to 3 + by counting all with objects. When asked, You have bears and get 1 more bear. How many do you have now? The child counts out, then counts out 1 more object, then counts all three objects. The child counts, one, two, three and then exclaims, three! Addition (finding the total amount): The child can find total amounts by directly modeling and counting all objects for joining word problem situations (e.g., You had 3 apples and then get 3 more apples, how many apples do you have now?) and part-part-whole situations (e.g., There are 6 girls and 5 boys on the playground, how many children are on the playground?). When asked, You have red balls and 3 blue balls. How many balls do you have? the child counts out red objects, then counts out 3 blue objects, then counts all 5 objects. Subtraction (finding the missing part): The child can solve take-away problems by separating with objects. For example, when asked, You have 5 bears and then you give bears to Kyle. How many bears do you have now? the child counts out 5 bears, then removes bears, and then counts the remaining 3 bears. 5 Make It N A significant advancement occurs when a child is able to count on in problem situations. This child can add on objects without needing to count from one. When asked, This puppet has balls but should have 6. Make it 6, child puts up fingers, immediately counts up from, saying, Five, six. The child does not necessarily represent how many were added and is not expected to tell the change amount. 6 Find the Change or Difference The child can find the missing addend in join, change unknown, word problem situations by adding on objects using fingers. When asked, You have 5 fish and then get some more. Now you have 7 fish in all. How many fish did you get? the child counts out five cubes, then counts those five again starting at one and adds on more objects as needed to get to the total number while counting six, seven, and finally counts the balls added to find the answer of two. The child initially may have to recount to separate the start amount from the change amount, but eventually keeps track of the separate parts. The child compares two sets by matching in simple problem situations that have a natural matching setting and that use informal language (won t get, extra). For example, when asked, Here are 6 dogs and balls. If we give a ball to each dog, how many dogs won t get a ball? the child counts out 6 dogs, matches balls to four of the dogs, then counts the two dogs that have no ball.

9 7 Counting Strategies 8 Part-Whole 9 Numbersin- Numbers 10 Deriver 11 Problem Solver 1 Multi-digit Addition: The child can find the total amount for joining and part-part whole situations by counting on orally, often using finger patterns, objects, or rhythmic movements to track the quantity being added on. When asked How much is bears and 3 more bears? the child answers,... 5, 6, 7, using finger patterns and then exclaims, Seven! The child may rearrange addends to start count from larger number. Subtraction: The child can solve missing addend problem situations or compare problem situations by counting on. When asked, You have 6 bears. How many more bears would you need to have 8 bears? the child says, Six then puts up first finger and says, Seven and then puts up a second finger and says Seven and then answers, Two! most often by subitizing the quantity of fingers she held up. The child has initial part-whole understanding and can solve all previous types of word problem situations using flexible strategies, and may use some derived facts. When asked, How many total bears do you have if you have 5 green bears and 6 blue bears? The child reasons that 5 bears and 5 bears is 10 bears, so 5 bears and 6 bears would be one more, so the total is 11 bears. Sometimes the child can solve start unknown problem situations, but only by trial and error. When asked, You had some bears. Then you get 6 more bears. Now you have 11 bears. How many bears did you start with? the child might lay out 6 objects, then set out 3 more objects, then count them all and get 9. Initially, when the child realizes not enough objects were added the child often starts over, but eventually reasons he/she can just put out more objects until the total amount is reached, but then must determine the quantity of the part added, often with several recounts, and eventually says, Five! The child recognizes that a number is part of a whole and can solve problem situations when the start is unknown with counting strategies. When asked, You have some bears, then you get more bears, now you have 9 bears. How many bears did you have to start with? The child counts, putting up fingers, and says, 5, 6, 7, 8, 9. Looks at fingers, and says, Five! The child can use flexible strategies and derived combinations to solve all types of problem situations within 0. When asked, If you had 5 fish and then bought 7 more fish, how many fish would you have now? the child reasons, is the same amount as and more, which is 10 plus, so it s 1. The child can simultaneously decompose a number and move part of a number to another number, aware of the increase in one and the decrease in the other. The child can solve simple multi-digit addition word problems by incrementing. When asked, Ms. Robinson s class has 0 students and Ms. Brown s class has 8 students, how many total students is that? the child uses connecting cubes in stacks of ten to count 10, 0, 30, 0, plus 8 more is 8 students. The child can solve all types of problem situations by using flexible strategies and many known combinations. For example, when asked, If I have 13 bears and you have 9 bears, how could we have the same number? this child says, 9 and 1 is 10, then 3 more to make 13; one and three is four; so I need four more! The child can solve most multi-digit addition and subtraction word problems by incrementing or combining tens and ones. When asked, Ms. Evan has 8 students and Ms. Adams has 35 students, how many total students is that? one child reasons that 0 and 30 is 50, then adds 8 more to get to 58, then adds more to get to 60, and finally adds 3 more to get to 63. Another child reasons that 0 plus 30 is 50 students; that 8 and 5 is the same amount as adding 8 and and then 3 more which is 13; then adds to get 63 total students. Adapted from: Sarama, J., & Clements, D. H. (009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge.

10 Understanding the Equal Sign as a Symbol of Mathematical Equality Level Level Name Description Core Equation Structures 1 Rigid Operational Students can solve equations or evaluate true-false statements successfully that only have operations on the left side of the equal sign. Equations with operations on left: a + b = c + = 7 T or F: 3 + = = T or F: 3 + = 8 + = 7 T or F: 5 + = 8 Equations with operations on right: c = a + b Flexible Operational Students can successfully solve equations with operations on the right side of the equal sign or interpret statements that have no operations. = 3 + T or F: 8 = = + T or F: 7 = 3 + Equations with no operations: a = a 7 = T or F: 7 = 7 = n T or F: n = n 3 Basic Relational Students can successfully solve or evaluate statements with operations on both sides of the equal sign, and begin to recognize or explain a relational understanding of the equal sign. Equations with operations on both sides: a + b = c + d a + b c = d + e = x = + 0 7m + 3m + 5 = m Comparative Relational Students can successfully use shortcuts (e.g., compensation strategies) and properties of the operations to solve equations or evaluate statements; and can consistently explain and generate a relational understanding of the equal sign. Equations that can be most efficiently solved by applying simplify transformations: Use a shortcut to tell if the equation = is true or false? Figure out the value of m in "3 x 7 = 60 + m" without fully multiplying out 3 x 7. Adapted from: Matthews, P., Rittle-Johnson, B., McEldoon, K., & Taylor, R. (01). Measure for measure: What combining diverse measures reveals about children s understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 3(3),

11 Developmental Levels for Multiplying and Dividing Multiplication and division builds on addition and subtraction understandings and is dependent upon counting and place value concepts. As children begin to learn to multiply they make equal groups and count them all. They then learn skip counting and derive related products from products they know. Finding and using patterns aids in learning multiplication and division facts with understanding. Children typically follow an observable developmental progression in learning to multiply and divide numbers with recognizable stages or levels. This developmental path can be described as part of a learning trajectory. Level Level Name Description Notes Non- Multiplication and division concepts begin very early with the problem of 1 Quantitive sharing. Earlyevidenceoftheseconceptscanbeobservedwhenachild Sharer dumps out blocks and gives some (not an equal number) to each person. 3 5 Beginning Grouper and Distributive Sharer Grouper and Distributive Sharer Concrete Modeler x/ Parts and Wholes x/ Progression to the next level can be observed when a child is able to make small groups (fewer than 5). This child can share by dealing out, but often only between two people, although he or she may not appreciate the numerical result. For example, to share four blocks, this child gives each person a block, checks each person has one, and repeats this. The next level occurs when a child makes small equal groups (fewer than 6). This child can deal out equally between two or more recipients, but may not understand that equal quantities are produced. For example, the child shares 6 blocks by dealing out blocks to herself and a friend 1 at a time. As children develop, they are able to solve small-number multiplying problems by grouping making each group and counting all. At this level a child can solve division/sharing problems with informal strategies, using concrete objects up to twenty objects and two to five people although the child may not understand equivalence of groups. For example, the child distributes twenty objects by dealing out two blocks to each of five people, then one to each, until blocks are gone. A new level is evidenced when the child understands the inverse relation between divisor and quotient. For example, this child understands If you share with more people, each person gets fewer Skip Counter x/ Deriver x/ Problem Solver Partitive Divisor Multi-digit x/ Uses repeated adding, additive doubling, or skip counting to solve multiplication and for measurement division (finding out how many groups) problems. Uses trial and error for partitive division (finding out how many in each group). Measurement division example: given twenty blocks, four to each person, and asked how many people, the child skip counts by, holding up one finger for each count of. Partitive division example: A child at this level also uses trial and error for partitive division (finding out how many in each group). For example, given twenty blocks, five people, and asked how many should each get, this child gives three to each, then one more, then one more. Child uses strategies, patterns, and de/composition (1 x = 10 x x ) and derived combinations, such as finding 7 x 8 from 7 x Solves multi-digit problems by operating on tens and ones separately. Child solves many types of multiplicative problems, with flexible strategies and known combinations. Multi-digit may be solved using combinations separately on ones and tens. Example: Each of these packages has 5 pieces of gum. How many pieces of gum are there in 7 packages? 35, because 7 times 5 is 35. The next level can be observed when a child is able to figure out how many are in each group. May first repeatedly add a divisor until the dividend is reached. For example, given twenty blocks, five people, and asked how many should each get, a child at this level says four, because 5 groups of is 0. Child uses compensating to paper-and-pencil procedures. For example, a child becoming fluent in multiplication might explain that 19 times 5 is 95, because twenty 5s is 100, and one less 5 is 95. Adapted from: Sarama, J., & Clements, D. H. (009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge