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Table S1. Description of Subtests within Each Clinical Interview Domain Measure

Domain Subtest Description* ExampleCounting Forwards by Ones Students count as high as he/she can, starting from 1. 1, 2, 3…

Backwards by Ones Kindergarten students are asked to count backwards from nine to one. In first grade, students should be able to count backwards from 20 to 1.

9, 8, 7…

Forwards by Tens Students are asked to count forwards by tens up to 100. Now I want you to count by tens starting with 10.

How Many Students are asked to enumerate sets of objects. Tell me how many chips are here. [6,11]

Addition Small Numbers Students are asked to solve problems involving digits less than ten. How much is 4 plus 6?

Zero Principle Students are asked a relatively simple problem involving zero, then are pressed to generalize the principle to a problem involving larger numbers.

How much is 6+0? How much is 0+23?

Order Principle The order of numbers in addition problems does not affect the sum (a + b = b + a). Students are asked to solve a simple problem involving single digits, then are pressed to generalize the principle to a problem involving large numbers.

How much is 3+4? How much is 4+3? 23 plus 12 equals 35. How much is 12 and 23 altogether?

Mental Calculation Solving problems without the aid of manipulatives is a critical step in mathematical learning. Successful students should be able to use base ten concepts and other mental strategies to decompose and solve complex problems mentally.

How much is 11+7?

Subtraction Small Numbers Students are asked to solve problems involving digits less than ten. How much is 6 minus 2?


How much is 7 minus 0?

Same Number Principle

Any number subtracted from itself results in zero (a – a = 0). Students are asked to solve a simple problem involving single digits, then are pressed to generalize the principle to a problem involving large numbers.

How much is 6 take away 6?

Inverse Principle Subtraction combinations can be solved by recalling a complimentary addition combination, and vice versa. If a – b = c, then b + c = a. Students are asked to solve a simple problem involving single digits, then are pressed to generalize the principle to a problem involving large numbers.

7 minus 5 is 2. How much is 2 plus 5?


How much is 13 take away 9?

Multiplication Small Numbers Students are asked to solve problems involving digits less than ten, and the difficulty of the problems vary by grade.

How much is 3 times 5?


How much is 0 times 5? How much is 17 times 0?

Identity Principle Any number multiplied by one equals that number (n × 1 = n). Students are asked to solve a simple problem involving single digits, then are pressed to generalize the principle to a problem involving large numbers.


Order Principle The order of numbers in multiplication problems does not affect the answer (a × b = b × a). Students solved a simple problem involving single digits, then are pressed to generalize the principle to a problem involving large numbers.

3 times 4 is 12. Now you figure out the next problem. How much is 4 times 3?



Written Number Writing Numbers Students are asked to write numerals. Write down two hundred five.

Setup Problem Students are asked vertically align addition, subtraction, and/or multiplication problems.

Write down four thousand one hundred thirty-three plus eight hundred thirty seven.

Place Value Students are asked to explain what specific numerals represent in the context of a written number and to identify how many of a given unit there are.

What does the five mean in the number 257? What is the place value of the number five? How many tens are there in the number 257?

Computation Addition

Students are asked to addition problems some of which involve carrying.

Here is 38 + 24.

Computation Subtraction

Students are asked to solve subtraction problems some of which involve borrowing.

Here is 53 – 26.

Computation Multiplication

Students are asked to multiplication problems some of which involve carrying.

Here is 96 × 3.

* Note: Problem difficulty within each subtest varied by grade. For example, first graders were asked to solve 11 + 7 within Addition Mental Calculation whereas third graders were asked to solve 25 + 34.

Table S2. Profile Definitions

Small Numbers

Mental Calculation

Question 1 Question 2 Strategy Profile

Question 1 Question 2 Strategy Profile

Correct Correct

Memory (Expressed Strategy) + Sensible (Alternate Strategy) ExpertSensible (Expressed Strategy)Memory (Expressed Strategy) MechanicalOther (Expressed Strategy)

Ordinary

Correct(Incorrect)

Incorrect(Correct)

Memory (Expressed Strategy) + Sensible (Alternate Strategy)Sensible (Expressed Strategy)Memory (Expressed Strategy) MechanicalOther (Expressed Strategy)

Struggling

Incorrect Incorrect

Memory (Expressed Strategy) + Sensible (Alternate Strategy)Sensible (Expressed Strategy)Memory (Expressed Strategy) LostOther (Expressed Strategy)

Correct CorrectSensible Expert

OtherOrdinary

Correct(Incorrect)

Incorrect(Correct)

Sensible

OtherStruggling

Incorrect IncorrectSensible

Other Lost

Principles

Expert All questions correct using the principle

Ordinary Most questions correct using the principle

Promising A few questions correct using the principle

Lost Almost no questions correct using the principle

Counting

Kindergarten: Forwards By One

Grade 1: Forwards By One

Backwards By One

Forwards by Ten How Many

Expert Up to 43 (no errors) Up to 103 (no error) Expert Down all the way to 1 Up to 70-100 Expert Correct on

both

Competent Up to 40-43 (with errors)

Up to 100-103 (with errors) Struggling All else All else Promising Correct on 1

Sloppy Up to 20-39 Up to 44-99 Struggling

Incorrect on both

Struggling Up to 0-19 Up to 0-43

Written Number

Alignment and Place Value Computation

Expert All correct Conventional Correct with standard algorithm

Promising At least 1 correct Flexible Correct with NO standard algorithm

Lost All incorrect Promising Incorrect with bug

Struggling Incorrect with no bug

Table S3. Comparison of results to past research

Our findings (see Tables 9, 10, and 11) Research Literature

Counting

Forwards and Backwards by

Ones, Forwards by

Tens

54% of the Kindergarteners were able to count from one to 43 without making any errors and 61% were able to count up to 43 with three or less errors.

44% of the Grade 1 students were able to count up to 103 without making any errors whereas 68% were able to count up to that same number with two or fewer errors.

Grade 1 students encountered some difficulty count-ing up to 103 without any errors.

Both the Kindergarten (79%) and the Grade 1 stu-dents (70%) were able to correctly count backwards by ones from 9 and 20, respectively.

90% of Grade 1 students accurately counted forward by tens.

Children between Kindergarten and Grade 1 are learning to count by ones from one to 100 (Clements, Sarama, & DiBiase, 2004).

The average age for children to acquire counting fluently without error is 7-years (PBS, 2011).

Kindergarten and Grade 1 students success-fully count backwards from 10 and 20, respec-tively (Clements et al., 2004).

Typically Kindergarteners can skip count and skip count by tens (Clements et al., 2004; PBS, 2011).

(Table continues)

How Many

74% of the Kindergarten and 87% of the Grade 1 (87%) students were able to successfully produce sets of objects that ranged from six to twelve for the younger students and eight to seventeen for the older ones.

The dominant strategy used by Kindergarten stu-dents to produce a set of objects was pointing to each object as it is counted followed by pushing aside each object as it is counted.

Though Kindergarten students relied on pointing, using this strategy was about equally likely to produce a correct answer as an incorrect one (41% compared to 37%).

The dominant strategy used by Grade 1 students to correctly enumerate sets of objects was pushing aside followed by pointing and visually counting.

In Grade 1, the use of counting and pointing resulted in incorrect answers about equally often (39% and 38%, respectively), whereas the use of pushing side was least likely to result in an incorrect answer (i.e., 15%).

In Grade 1, the strategy least likely to produce incor-rect answers was pushing aside.

Typically, Kindergarten students can success-fully enumerate and produce collections of up to 20 objects and Grade 1 students can do the same with sets containing up to 100 objects (Clements et al., 2004; PBS, 2011).

Students’ ability to successfully enumerate sets of objects in part depends on set size (i.e., smaller sets are easier to count) and the type of counting strategy employed. For example, managing a set of 30 objects may be cumber-some. Further, strategies such as pushing aside that clearly separate the counted objects from those that remain to be counted reduce memory strain and may result in greater success (Gins-burg, 1989).

Kindergarten and Grade 1 students typically rely on concrete counting strategies such as pushing aside and pointing when enumerating sets of objects (Ginsburg, 1989). These types of strategies consist of labeling each object in the set with one and only one number word un-til all the objects have been counted (Ginsburg, 1989; PBS, 2011).

(Table continues)

Small Number Addition

Kindergarteners relied heavily on sensible strategies such as counting all with chips and memory (34% and 24%, re-spectively) and were often correct when using these meth-ods, (30% and 19%, respectively).

The dominant method used by the Grade 1 students to solve these types of problems was sensible with a slight in-crease in derived strategies and decrease in memory.

Grade 1 students were more often correct when using sensible strategies followed by memory and derived, 49%, 17%, and 10%, respectively.

In Grade 2, derived facts and sensible methods were equally used to solve small number addition problems. The use of these strategies was also about equally likely to re-sult in a correct answer (34% and 38%, respectively).

Memory use decreased from Grade 1 to Grade 2 and in-creased from Grade 2 to Grade 3.

The use of derived facts also increased from Grade 2 to Grade 3 as sensible strategies decreased.

When memory was used by Grade 3 students, it was more likely to be correct (19%) than incorrect (17%).

The dominant strategy correctly used by Grade 3 students was derived (40%) followed by sensible (28%). Grade 3 students rarely generated incorrect answers to these types of problems.

Kindergarten and Grade 1 students typically rely on strategies that directly model the num-bers in the problem by using fingers or coun-ters to represent the addends and ultimately the sum (e.g., Carpenter & Moser, 1984; Carpenter et al., 1993; PBS, 2011).

Toward the middle to end of Grade 1 and within Grade 2 students tend to use more ad-vanced counting strategies such as counting-on from the larger addend and doubles to solve problems involving single-digit numbers (Clements et al., 2004; PBS, 2011).

There is an increased use of more abstract strategies such as derived facts, making tens as well as retrieval from Kindergarten through Grade 3 that are not directly tied the structure of the problem (Ginsburg, Klein, & Starkey, 1998; Siegler, 1996) and to accuracy (Siegler, 1998).

(Table continues)

Subtraction

The kindergarteners relied heavily on sensible strategies, though when using these strategies students were equally likely to answer correctly (14%) and incorrectly (14%). Note that Kindergarteners have not yet learned subtraction in school.

Grade 1 students relied on sensible strategies with a slight increase in derived and decrease in memory and were most accurate when relying on sensible strategies to solve these types of problems (46% compared to 12% and 8%, respec-tively).

Only a small proportion of the Grade 1 students re-sponded incorrectly when using sensible strategies (9%).

In Grade 2, derived facts and sensible methods were equally used to solve small number subtraction problems though students were slightly more accurate when using sensible strategies when compared to derived (41% com-pared to 33%).

There was also an increase in derived facts and memory for small number subtraction problems from Grade 2 to 3 and a decrease in sensible strategies.

Grade 3 students were typically correct when solving these types of problems, though the method most often yielding the correct answer was derived followed by sensi-ble and memory, 38%, 19%, and 32%, respectively.

Kindergarten and Grade 1 students typically rely on strategies that directly model the num-bers in the problem by using fingers or coun-ters to represent the minuend, subtrahend, and ultimately difference (e.g., Carpenter & Moser, 1984; Carpenter et al., 1993; PBS, 2011).

The use of counting strategies such as count-ing-down and counting-up are used a bit to-ward the end of Grade 1, but mostly in Grade 2. It is not until the end of Grade 2 and within Grade 3 that students’ flexibly shift between counting-down and counting-up strategies de-pending on the numbers in the problem (PBS, 2011).

Woods, Resnick, and Groen (1975) also found that a substantial portion of the Grade 2 students in the sample relied on counting down and up when solving single digit subtraction problems.

During Grade 2 students begin to achieve flu-ency with subtraction facts thereby resulting in increased accuracy and use of retrieval (PBS, 2011).

(Table continues)

Multiplication

The most common type of strategies used by the Grades 2 and 3 students were sensible, such as repeated addition and skip counting, 32% and 42%, respectively. This approach was also the most accurate for both Grades 2 and 3, 22% and 24%, respectively.

Though derived and memory were also used, their frequency and accuracy were low.

Using sensible strategies, Grade 3 students were about equally likely to produce correct (24%) and incorrect (18%) answers.

The Grades 2 and 3 students in our sample did not use memory often (i.e., 4% and 6% respec-tively) or accurately (i.e., 2% and 1% respec-tively), though this is not an unexpected finding as our sample of students did not receive instruc-tion on multiplication until the middle of Grade 3.

We begin to see an increase in derived and sen-sible strategy use from Grade 2 (45%) to Grade 3 (53%).

There are three types of strategies Grades 2 and 3 students used to solve small number multiplication problems: retrieval from memory; repeated addition where one of the multiplicands is represented the num-ber of times indicated by the other (e.g., 2×3=3+3=6); and representing one or more of the multiplicands on paper and counting the objects that correspond to the problem (for example, a student solves 2×3 by writing 3 sets of 4 tally marks on a piece of paper and counting the tallies) (PBS, 2011; Siegler, 1996).

Lemaire and Siegler (1995) also found that students only used retrieval on familiar multiplication number facts and relied on repeated addition or counting strate-gies on ones deemed difficult.

Steel and Funnell (2001) found that the younger ele-mentary students in the sample (e.g., Grade 3 students) did not often use retrieval on small number multiplica-tion problems. However, these researchers found that there was a shift away from less effective and toward more efficient strategies such as partitioning as well as retrieval in Grades 4 through 6.

Clark (1996) found that Grade 2 students were capa-ble of solving small number multiplication problems and displaying multiplicative thinking. However, the results from this study suggest that educators should not expect mastery of this material until Grade 5 and beyond.

(Table continues)

Zero Principle

Addition

The kindergarteners were struggling with the zero principle whereas almost all of the students in Grades 1, 2, and 3 were able to apply their knowledge of this principle on both familiar (e.g., 4+0=4) and unfamiliar (e.g., 0+23=23) type problems (32 percent compared to 74, 92, 92 %, respectively).

As early as Kindergarten, students begin ap-plying existing knowledge and reasoning strate-gies to sensibly determine unknown sums up to 18 (Clements et al., 2004). However, the ability to use this type of knowledge improves through-out Grade 1 and onward (PBS, 2011).

Subtraction

Kindergarten students struggled with the zero prin-ciple in subtraction, though most students in Grades 1 (75%), 2 (90%), and 3 (91%) applied their knowledge of this principle on both familiar (e.g., 6-0=6) and un-familiar (e.g., 37-0=37) type problems.

As early as Kindergarten, students begin ap-plying existing knowledge and reasoning strate-gies to sensibly determine unknown differences (Clements et al., 2004). However, the ability to use this type of knowledge improves throughout Grade 1 and onward (PBS, 2011).

Multiplication

The zero principle in multiplication was very chal-lenging for Grade 2 students though not all Grade 3 students applied this principle (42% and 71%, respec-tively).

Students typically consistently and accurately apply this principle in Grade 3 (PBS, 2011).

Identity Principle

Subtraction

There is a dramatic increase in the use of the Iden-tity Principle from Kindergarten to Grade 1 (9% to 60%), Grade 1 to Grade 2 (60% to 83%) and Grade 2 to Grade 3 (83% to 92%).

Research suggests that by the end of Grade 1, students can apply their knowledge of the Iden-tify (Negation) Principle to subtraction prob-lems containing single-digit numbers (e.g., 6-6) (PBS, 2011).

Multiplication

Grade 2 students appear to struggle with the use of the Identity Principle on multiplication problems whereas a good portion of Grade 3 students use their knowledge of this principle when problem solving, 37% and 79%, respectively.

Some Grade 2 students possess and apply knowledge of the Identity Principle on small number multiplication problems, but this is more common toward the end of Grade 3 (PBS, 2011).

(Table continues)

Commutativity Principle

Addition

45% of the Grade 1 students, 66% of the Grade 2 stu-dents, and 70% of the Grade 3 students used their knowledge of this principle when solving addition commutativity problems.

By the end of Grade 1, students can apply their knowledge of the Additive Commutativity Principle to addition problems containing sin-gle-digit numbers (e.g., 2+3) (Clements et al., 2004; PBS, 2011).

Multiplication

Even though most Grade 2 students had not received any multiplication instruction and Grade 3 students were just learning multiplication, a good portion of these students possessed and applied knowledge of the Commutativity Principle on appropriate problems (48% and 64%, respectively).

Students ability to apply the multiplication commutativity principle develops within Grade 3 (Baroody, 1999).

Inverse Principle Subtraction

The Inverse Principle posed the greatest challenge to the students, especially for Grades 1 and 2.

Even within Grade 3, only 50% of the students used their knowledge of this principle while problem solv-ing.

On average, the ability to apply knowledge of addition complements begins in Grade 1 and continues through elementary school (Clements et al., 2004; PBS, 2011).

Research suggests that students who are ex-periencing mathematics difficulty in Grades 1 and 2 perform poorly on problems involving the inverse principle (Jordan & Hanich, 2003).

(Table continues)

Mental Calculation Addition

The Grade 1 students were more likely to rely on sensi-ble strategies compared to Grades 2 and 3 students, 39% compared to 15% and 14%, respectively.

Grade 1 students were also mostly likely to be accurate when using sensible strategies compared to derived and partitioning, 24% compared 6% and 2%.

Students in Grade 2 used partitioning and derived strategies equally often (22.1%) though Grade 3 students used derived (27%) a bit more than partitioning (24%).

As expected, students’ use of memory was infrequent regardless of accuracy.

Grade 2 students were equally likely to be accurate when using these two methods (i.e., 17% and 16%, re-spectively) as were Grade 3 students’ (i.e., 22% accuracy when applying partitioning or derived).

As students progress through Grades 1 through 3, there is a shift away from sensible strategies and toward more abstract problem solving methods. There was also a significant improvement in the accu-racy of large number mental calculation of addition prob-lems from Grade 2 to Grade 3 (53% to 75%).

Usually during Grade 2 students reach a de-gree of proficiency with small number prob-lems and learn to apply this knowledge to multi-digit problems either by executing an al-gorithm or mental computation (Siegler, 1998).

Mental calculation relies on students’ base ten /place value knowledge and ability to par-tition large numbers into small parts (Clements et al., 2004).

Typically Grade 2 students use their knowl-edge of the base ten system to convert ad-dends into tens such as 7+4 as 7+3+1 = 10+1 = 11 (PBS, 2011).

Grade 2 students are exposed to double-digit problems using manipulatives that represent the base-ten system such as unifix cubes or stern blocks (PBS, 2011).

Grades 2 to 4 students show a transition from manipulative strategies to more mental strategies (Carr & Alexeev, 2011)

(Table continues)

Subtraction

Grades 1, 2, and 3 students relied more heavily on sensible strategies (28%, 25%, and 24%, respectively) though Grade 3 students were more accurate when using derived (27%) com-pared to sensible (18%).

The accuracy of Grade 1 students was higher when using sensible strategies compared to derived (11% compared to 3%) whereas Grade 2 students were about equally like to be accurate when using these methods (13% compared to 15%).

Student accuracy was low, 11% for grade 1, 13% for grade 2, and 18% for Grade 3.

There was very limited use of partitioning in Grades 1, 2, and 3, 1%, 8%, and 8%, respectively though students accu-rately apply this strategy when used, 1%, 5%, and 7%, respec-tively.

Memory was not frequently used to solve these types of problems by any of the students in Grades 1, 2, or 3.

Overall, Grade 3 students used derived more frequently in subtraction than addition (33% in subtraction compared to 27% in addition) and more often than Grade 2 students (33% compared to 21%). There was also a significant improvement in the accuracy of large number mental calculation subtraction problems from Grade 2 to Grade 3 (45% to 71% for subtraction).

Usually during Grade 2 students reach a degree of proficiency with small number problems and learn to apply this knowl-edge to multi-digit problems either by ex-ecuting an algorithm or mental computa-tion (Siegler, 1998).

Mental calculation relies on students’ base ten/place value knowledge and abil-ity to decompose large numbers into small parts (Clements et al., 2004).

Typically Grade 2 students use their knowledge of the base ten system to make tens thereby simplifying the sub-traction problem (e.g., 17–7 as [17+3]–[7+3] = 20–10 = 10 (PBS, 2011).

Grade 2 students are exposed to double-digit problems using manipulatives that represent the base-ten system such as unifix cubes or stern blocks (PBS, 2011)

(Table continues)

Multiplication

Students in Grades 2 and 3 relied on both sensible (25% and 22%) and derived (11% and 16%) strategies.

When applying sensible strategies, students in Grades 2 and 3 were more often correct (15% and 9%, respectively) than when using derived (6% and 7%, re-spectively) though this difference was relatively small for both grades and through the use of both of these methods, regardless of grade, students were about equally likely to produce an incorrect answer.

Interestingly, although a low percentage of students in Grades 2 and 3 were using partitioning (i.e., 4% and 2%, respectively), they did so accurately which is im-pressive considering the difficulty of these types of problems for these grade levels.

Memory was rarely used by Grade 2 or 3 students, 3% compared to 1%, respectively.

Some students find single-digit multiplica-tion as an extension of addition and use manip-ulatives to create groups of objects (PBS, 2011). Grade 2 students are use concrete ob-jects such as chips and informal calculations such as counting by ones (e.g., "1, 2, 3, 4 in one house, 5, 6, 7, 8, in another house), skip counting (e.g., 2, 4, 6, 8), and/or repeated addi-tion (e.g., 2+2+2) to determine the product of two numbers (PBS, 2011). As students develop multiplication skills the use of concrete objects is replaced with symbolically representing multiplication as repeated addition toward the end of Grade 2 and beginning of Grade 3 (PBS, 2011).

The transition from single-digit multiplica-tion problems (e.g., 4 x 2) to double- by single-digit problems (e.g., 13 x 2) is not typically taught until the end of Grade 3 and therefore is difficult for these students.

(Table continues)

Written Number

Writing Numbers

95% of the Grade 1 students correctly represented double-digit numerals; 82% of the Grade 2 students accurately repre-sented three-digit numerals; and 77% of the Grade 3 students accurately represent four-digit numerals.

Note that Grade 3 students’ accuracy decreased a bit as it is more difficult to represent four-digit numerals than three-digit.

Beginning around Kindergarten, students are able to read multi-digit numerals up to 19 and within Grade 1, students develop the ability to read and write multi-digit nu-merals up to 99 (PBS, 2011).

In Grades 2 and 3, students further de-velop these abilities as evidenced by their accurate reading and writing of three-digit numerals and beyond (Clements et al., 2004; PBS, 2011).

Setup Problem

43% of the Grade 1 and 24% of the Grade 2 students repre-sented these types of problems horizontally (e.g., 23+14=__) whereas only 16% of Grade 1 and 31% of Grade 2 repre-sented the problems in a conventional algorithm format with one number placed on top of the other.

The decreased use of horizontal alignment from Grade 1 to Grade 2 suggests that students in the latter grade are transi-tioning toward vertical alignment of problems.

However, the results also suggest that Grade 2 students may be encountering some difficulty as there is a sharp in-crease in left alignment (i.e., from 7% in Grade 1 to 25% in Grade 2).

By Grade 3, 81% of all students are able to represent prob-lems in a vertical conventional algorithm format.

To symbolically represent numerals, stu-dents must also understand the symbols used to represent operations (e.g., +, -) and relations within an operation (e.g., =) as well as how to set up the problem (Gins-burg, 1989).

(Table continues)

Place Value

Accuracy of response for the Grades 1, 2, and 3 students in our sample was 36%, 43%, and 71%, respectively.

Student place value understanding certainly improved from Grades 1 to 3, but still appears to be challenging.

Around the middle of Grade 1, students are able to grasp place value concepts embedded within two digit numbers (Clements et al., 2004; PBS, 2011).

Grades 2 and 3 students are grappling with place value concepts embedded within three-and even four-digit numbers (Clements et al., 2004; PBS, 2011).

Written Number

Addition

Accuracy improves from Grades 2 to 3 when solving multi-digit addition problems increases (i.e., 65% to 87%, respec-tively).

The correct use of an algorithm also increases from 71% in Grade 2 when solving two- by two-digit problems to 82% in Grade 3 when solving three- by three-digit problems.

Grade 2 students are learning to solve two- by two-digit addition problems, whereas Grade 3 students are learning about three- by three-digit problems (Clements et al., 2004; PBS, 2011).

Subtraction

Of the Grade 2 students who were accurate (i.e., 14%) a good portion (i.e., 64%) correctly applied borrowing proce-dures when solving two- by two-digit subtraction problems.

The overall performance of Grade 3 students was substan-tially better: 65% were correct on three- by three-digit subtrac-tion problems and of those students who were accurate, 83% correctly applied borrowing procedures when necessary.

The results also suggest a marked decrease in the proportion of incorrect responses to problems from Grade 2 to Grade 3 (i.e., 86% to 36%, respectively).

Most students only begin understanding the complexities of two- by two-digit subtraction problems toward the middle of Grade 2 and onward (PBS, 2011).

(table continues)

Multiplication

32% of Grade 3 students were able to solve double by single-digit multiplication problems and 62% of these students used correct carrying procedures when doing so.

This finding suggests that, students are capable of successfully solving these types of difficult problems through either informal methods or correct application of an algorithm.

Some students find single-digit multiplication as an extension of addition and use manipula-tives to create groups of objects (PBS, 2011). Grade 2 students use concrete objects such as chips and informal calculations such as count-ing by ones (e.g., "1, 2, 3, 4 in one house, 5, 6, 7, 8 in another house), skip counting (e.g., 2, 4, 6, 8), and/or repeated addition (e.g., 2+2+2) to determine the product of two numbers (PBS, 2011). As students develop multiplication skills the use of concrete objects is replaced with symbolically representing multiplication as re-peated addition toward the end of Grade 2 and beginning of Grade 3 (PBS, 2011).

The transition from single-digit multiplication problems (e.g., 4×2) to double- by single-digit problems (e.g., 13×2) is not typically taught un-til the end of Grade 3 and therefore is difficult for these students.

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