Table S1. CBM Mathematics Measure Description and Exemplary Items

Kindergarten
Description / Range / # of Items / Example
Counting / Students count as high as he/she can, starting from 1. / 1-120 / 120 / 1, 2, 3…
Missing
Number / Two numbers and a blank are presented in a forward counting sequence by 1’s, with the location of the blank varying. Students identify which number is missing from the sequence. / Numbers range from 0-20. / 54 / [ _, 3, 4]
Number
Identification / Students identify printed numbers from a set of randomly-assigned digits. / 0-100; 50% of the numbers are between 0-20; 30% are between 21-50; and 20% are between 51-100. / 84 / 2, 6, 18…
Quantity
Discrimination / Students point to or say which number in a pair is larger. The order of the numbers is alternated, in terms of placing the larger number first or second between sets. / 67% are between 0-10 and 33% are between 0-20. / 63 / [7 2]
Grade 1
Description / Range / # of Items / Example
Counting / Students count as high as he/she can, starting from 1. / 1-120 / 120 / 1, 2, 3…
Missing
Number / Two numbers and a blank are presented in a forward counting sequence by 1’s, with the location of the blank varying. Students identify which number is missing from the sequence. / Numbers range from 0-20. / 54 / [9, _, 11]
Next
Number / Students say the number that follows the number verbally presented by the teacher, ranging from 0-100. / 50% of the numbers between 0-20; 30% between 21-50; and 20% between 51-100. / 84 / “What comes after 34?”
Number
Facts / Verbally presented addition problemswith digits ≤10 and sums less than 20. / 60 / “How much is four plus two?”
Number
Identification / Students identify printed numbers from a set of randomly-assigned digits. / 0-100; 50% of the numbers are between 0-20; 30% are between 21-50; and 20% are between 51-100. / 84 / 5, 84, 17...
Quantity
Discrimination / Students point to or say which number in a pair is larger. The order of the numbers is alternated, in terms of placing the larger number first or second between sets. / 67% are between 0-10 and 33% are between 0-20. / 63 / [ 7 2 ]
Grade 2
Description / Range / # of Items / Example
Computation / Addition and subtraction problems; Scoring is the correct number of digits in the responses. / Addition:
  • 2 by 1, with regrouping (4 items)
  • 2 by 1, without regrouping (2 items)
  • 2 by 2, without regrouping (2 items)
  • 2 by 2, with regrouping (2 items)
  • 3 single digits, without regrouping (2 items)
  • 3 double digits without regrouping (2 items)
Subtraction:
  • 2 by 1, without regrouping (3 items)
  • 2 by 1, with regrouping (3 items)
  • 2 by 2, without regrouping (4 items)
  • 2 by 2, with regrouping (2 items)
/ 24 / 28+5
Concepts / A variety of items assessing the student’s understanding of fundamental number concepts, including principles, place value, fractions, and symbolic notations. / Principles (8 items)
  • Commutative principle of addition - 2 single-digit and 2 double- by single-digit items (4 items)
  • Inverse principle with double-digit numbers (4 items)
Place value (4 items)
  • Identify the place value of a number in the thousands, hundreds, tens, or ones position (4 items)
Fractions (16 items)
  • Match the fraction to its model (3 items)
  • Match the fraction to its word form (3 items)
  • Select the fraction that represents the shaded model (4 items)
  • Writing fractions. Show a model with a portion shaded and ask the student to write the fraction (6 items)
Notations (2 items)
  • Write <, >, or = in each blank e.g., 316 ___ 968; 98 ____225 (2 item in one box)
/ 24 / “What place value does the 6 hold in the number 6891 and what value does the 8 hold in the same number?”
Missing
Number / Each item is three numbers and a blank, presented in a forward counting sequence, with the location of the blank varying. Students write the missing number in the sequence. The position of the blank is rotated throughout the items. /
  • Non-sequential decade items with odd and even numbers between 0 and 99 (10 items)
  • Sequential by decade, items containing 0 (4 items)
  • Skip counting by 2, items containing odd and even numbers between 30 and 99 (8 items)
  • Skip counting by 3, items containing odd and even double- and single-digit items between 0 and 99 (8 items)
  • Skip counting by 4, items containing odd and even double- and single-digits between 20 and 99 (8 items)
  • Skip counting by 5, items containing numbers between 50 and 99 (5 items)
  • Skip counting by 6, items containing odd and even double- and single-digits between 0 and 30 (7 items)
/ 50 / [ _,75, 77]
Number
Facts / Addition and subtraction items using randomly selected single-digit problems and double-digit (no larger than 18); the position of the larger number in addition alternates. / 30 Addition items; 30 Subtraction items; equal number of single and double digit problems / 60 / 13+5; 9-5
Quantity
Discrimination / Whole numbers and operations are presented in pairs, and the student circles which number or quantity is larger. The order of placing the larger quantity first or second between sets is alternated. /
  • 12 whole number comparisons
  • 14 items comparing whole numbers to addition operations using whole numbers
  • 10 items comparing addition operations
  • 14 items comparing whole numbers to subtraction operations using whole numbers
  • 10 items comparing subtraction operations
/ 60 / [(72) 66+5]
Grade 3
Description / Range / # of Items / Example
Computation / Addition, subtraction, multiplication, and division problems; Scoring is the correct number of digits in the responses. / Addition:
  • 3 by 2, with two regrouping (2 items)
  • 3 by 3, with two regrouping (2 items)
  • 2 by 2, with regrouping (2 items)
Subtraction:
  • 3 by 2, with one regrouping (2 items)
  • 3 by 3, with one regrouping (2 items)
  • 2 by 2, with regrouping (2 items)
Multiplication:
  • 2 by 1, without grouping (7 items)
Division:
  • 2 by 1, without remainders (5 items)
/ 24 / 468+59; 73-57; 54×2; 35/7
Concepts / A variety of items assessing the student’s understanding of fundamental number concepts, including principles, place value, fractions, and symbolic notations. / Principles (11 items)
  • Commutative principle of addition - 1 double- by double-digit item (2 items)
  • Commutative principle of multiplication - single-digit and 2 double- by double digit items (3 items)
  • Inverse principle of addition and subtraction with double-digit numbers (2 items)
  • Inverse principle of multiplication and division with single digit multiplicative results and double-digit division results (2 items)
  • Associative property of multiplication, 1 single digit and 1 double digit (2 items)
Place value (7 items)
  • Identify the place value of a number in the ten thousands, thousands, hundreds, tens, or ones position (5 items)
  • Round numbers, 1 item contains numbers between 40 and 99 and the other contains numbers between 101 and 199 (2 items)
Derived Facts (2 items)
  • Using knowledge of derived facts to solve adjacent problem (e.g., if 37+56=93 then 37+57=__?) (2 items)
Fractions (10 items)
  • Writing fractions. Show a model with a portion shaded and ask the student to write the fraction (6 items)
  • Write <, >, or = in each blank, e.g., 1/3 ___ 2/3; 3/4 ____2/4 (2 fractions in one box; 4 items)
/ 30 / “If 8×9=72 then __/8=9?
Missing
Number / Each item is three numbers and a blank, presented in a forward counting sequence, with the location of the blank varying. Students write the missing number in the sequence. The position of the blank is rotated throughout the items. /
  • Non-sequential decade items with odd and even numbers between 0 and 99 (10 items)
  • Sequential by decade items with numbers between 0 and 99 (4 items)
  • Skip counting by 5, items containing numbers between 30 and 99 (8 items)
  • Skip counting by 2, items containing odd and even numbers between 50 and 99 (6 items)
  • Skip counting by 3, items containing odd and even numbers between 0 and 99 (8 items)
  • Skip counting by 4, items containing odd and even numbers between 30 and 99 (7 items)
  • Skip counting by 6, items containing odd and even numbers between 50 and 99 (7 items)
/ 50 / [5, 11, __]
Number
Facts / Addition, subtraction, and multiplication items using randomly selected single-digit problems and double-digit (no larger than 18); the position of the larger number in addition alternates. / 15 Addition items; 15 Subtraction items; 30 Multiplication items / 60 / 9+5; 8-7; 9×3
Quantity
Discrimination / Whole numbers, operations, fractions, and decimals are presented in pairs, and the student circles which number or quantity is larger. The order of placing the larger quantity first or second between sets is alternated. /
  • 6 whole number comparisons
  • 8 items comparing whole numbers to addition operations using whole numbers
  • 8 items comparing whole numbers to subtraction operations using whole numbers
  • 14 items comparing whole numbers to multiplication operations using whole numbers
  • 14 items comparing fractions with different denominators
  • 10 items comparing decimals
/ 60 / [(55) 9×6]

Table S2. Description of Subtests within Each Clinical Interview Domain Measure

Domain / Subtest / Description* / Example
Counting / 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?
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 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?
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 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?
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 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. / How much is 3 times 1?
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?
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 13 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 S3. Profile Definitions

Small Numbers

Question 1 / Question 2 / Strategy / Profile
Correct / Correct / Memory (Expressed Strategy) + Sensible (Alternate Strategy) / Expert
Sensible (Expressed Strategy)
Memory (Expressed Strategy) / Mechanical
Other (Expressed Strategy) / Ordinary
Correct
(Incorrect) / Incorrect
(Correct) / Memory (Expressed Strategy) + Sensible (Alternate Strategy)
Sensible (Expressed Strategy)
Memory (Expressed Strategy) / Mechanical
Other (Expressed Strategy) / Struggling
Incorrect / Incorrect / Memory (Expressed Strategy) + Sensible (Alternate Strategy)
Sensible (Expressed Strategy)
Memory (Expressed Strategy) / Lost
Other (Expressed Strategy)

Mental Calculation

Question 1 / Question 2 / Strategy / Profile
Correct / Correct / Sensible / Expert
Other / Ordinary
Correct
(Incorrect) / Incorrect
(Correct) / Sensible
Other / Struggling
Incorrect / Incorrect / Sensible
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