# Uncovering Student Thinking in Mathematics

Uncovering Student Thinking in Mathematics, Volumes 1-3, published by Corwin Press, provide sets of grades K-12+ formative assessment probes that link key concepts and procedures in mathematics to commonly held ideas described in the research on learning. This series is similar to the Uncovering Student Ideas in Science. The primary author is Cheryl Rose Tobey, Page Keeley's mathematics colleague. These probes can be used to reveal the variety of ideas students (and practicing and prospective teachers) have about fundamental mathematics concepts and procedures. Used individually or as a set, these probes provide the diagnostic and formative tools mathematics teachers at all grades need to uncover the preconceptions students bring to their learning and inform pathways needed to build a conceptual bridge from where students are at any point in the instructional cycle to where they need to be mathematically. Each volume includes an overview of mathematics probes- what they are, how they are used, the types of ideas they elicit, and the structure and design of a probe. Each book also includes a chapter on instructional implications how to differentiate for instruction, analyzing trends in thinking, promoting math conversations, and more. Teacher notes for each probe are organized around a QUEST Cycle, which teachers can also use to engage in action research.

Uncovering Student Thinking in Mathematics (Grades K-12)- 25 Formative Assessment Probes, by Cheryl Rose, Carolyn Arline, and Leslie Minton (2007) This first volume in the series contains 25 formative assessment probes that can be used across multiple grade levels. The Foreword is written by Anne Davies. The probes are organized in 3 categories: Number and Operations Probes; Algebra, Data Analysis, and Probability Probes; and Geometry and Measurement Probes. Click here for additional information about this book.  The following is a list of probes, the concept or procedure targeted by the probe, and suggested grade levels for using the probe:

IS ONE GROUP MORE?- differentiating between magnitude and quantity (K-2)

BUILDING NUMBERS- number system and place value (K-4)

ARE THEY EQUAL?- commutative and associative properties as they relate to numbers used in addition and subtraction equations (1-5)

FRACTIONAL PARTS- part-whole relationships (3-6)

COMPARING DECIMALS- place value when comparing size of decimals (4-6)

COMPARING FRACTIONS- part-whole relationships when comparing fractions (4-8)

ORDER OF OPERATIONS- order of operations in simplifying expressions (5-11)

WHICH IS BIGGER?- effect of multiplication and division on different numbers (6-11)

ARE YOU POSITIVE?- meaning and effect of operations with negative numbers (6-10)

WHAT PERCENT IS THAT?-estimating the percent of a number (7-12)

IT'S ALL ABOUT BALANCE- meaning of an equal sign (1-5)

SEESAW- equal sign and equivalence (K-5)

STUDENTS VERSUS TEACHERS- symbolic representation (6-12)

RATE OF CHANGE (SLOPE)-graphical representation of rate of change, scale (6-12)

ARE THEY LINEAR?-representing linear relationships symbolically (7-12)

DISTANCE FROM HOME- interpretations of graphs (6-10)

GUMBALLS IN A JAR- probability, part-whole relationships (5-12)

THE SPINNERS- compound probability (9-12)

WHAT DOES A TRIANGLE LOOK LIKE?- attributes of a triangle (K-3)

WHAT DOES A RECTANGLE LOOK LIKE?- properties of a rectangle and relationship to a square (K-5)

HOW LONG IS THE PENCIL?- linear measurement (1-5)

COMPARING ANGLES- length of angle rays and angle measure (5-12)

BOX CUTTING/COMPARING- comparing the volume and surface area of a box before and after dividing it into 4 boxes (7-11)

PARALLELOGRAMS- attributes of parallelogramns (6-10)

VOLUME OF A BOX- finding volume (7-10)

Uncovering Student Thinking in Mathematics (Grades 6-12)- 30 Formative Assessment Probes for the Secondary Classroom, by Cheryl Rose and Carolyn Arline (2009) This second volume in the series contains 30 formative assessment probes that can be used across multiple grades in middle and high school. The Foreword is written by former NCTM President, Johnny Lott. The probes are organized in 3 categories: Number and Operations Probes; Geometry, Measurement, and Data Probes; and Algebra Probes. Click here for additional information about this book.  The following is a list of probes, the concept or procedure targeted by the probe, and suggested grade levels for using the probe:

WHAT IS THE VALUE OF THE DIGIT?- values of various digits of a decimal (6-9)

WHAT IS THE MEANING OF 2/3?-various meanings of a/b (6-10)

IS IT EQUIVALENT?-equivalent forms of a fraction (6-9)

IS IT AN ESTIMATE?-multiple methods of estimating the sum of a 3 digit number (6-12)

IS IT SIMPLIFIED?- using the "canceling of zeros" shortcut (6-8)

WHERE IS ONE MILLION?-locating one million on a 0-1billion number line (6-12)

HOW LOW CAN YOU GO?- how various integers and exponents affect the value of numerical expressions (7-12)

WHAT'S YOUR SUBTRACTION STRATEGY?- applying and understanding a variety of subtraction strategies (6-12)

WHAT'S YOUR MULTIPLICATION STRATEGY?- applying and understanding a variety of multiplication strategies (6-12)

WHAT'S YOUR DIVISION STRATEGY?- applying and understanding a variety of division strategies (6-12)

WHAT'S THE MEASURE?- measure of a line with changes in the intervals or a non-zero starting point (6-12)

ARE AREA AND PERIMETER RELATED?- figures with the same perimeter but different areas (6-11)

WHAT'S THE AREA?- how a change in dimension affects the area of a figure (6-10)

WHAT'S THE CAPACITY?-how a change in dimension affects the volume of a figure (6-12)

IS IT TRANSFORMED?- recognizing dilations as types of transformations (6-12)

ARE THEY SIMILAR?- determining whether two figures are similar (8-12)

WHAT DO YOU MEAN?- understanding mean and how is is affected by changes to a data set (7-12)

NAME OF THE GRAPH?- graph interpretation (6-8)

GRAPH CONSTRUCTION- construction of a graph and use of interval scale (7-10)

EQUAL OR NOT EQUAL?- equivalent expressions in the form of familiar formulas (6-12)

IS IT THE SAME AS A + B?- use of "key words" when writing expressions (6-12)

M & N's- equality and relationships among quantities (6-12)

WHAT'S THE SUBSTITUTE?- evaluating an expression in the form "ax" (6-9)

IS IT TRUE?-operation implied by concatenation of literal symbols and numbers (7-9)

SOLVING EQUATIONS- methods and notations in solving for an unknown (6-9)

CORRECT REPRESENTATION OF THE INEQUALITY?- representations of an inequality (7-11)

IS IT A VARIABLE?- literal symbol used as a variable (6-12)

BINOMIAL EXPANSION- using the distributive law when multipling algebraic binomials (8-11)

Uncovering Student Thinking in Mathematics (Grades K-5)- 25 Formative Assessment Probes for the Elementary Classroom, by Cheryl Rose Tobey and Leslie Minton (2010) This third volume in the series contains 25 formative assessment probes that can be used across multiple grades in elementary school. The Foreword is written by former NSTA President and Author of the best-selling series Uncovering Student Ideas in Science, Page Keeley. The probes are organized in 4 categories: Place Value, Number Charts, and Number Lines; Parts and Wholes and Equality; Computation and Estimation; and Measurement, Geometry, and Data Probes. Click here for additional information about this book.  The following is a list of probes, the concept or procedure targeted by the probe, and suggested grade levels for using the probe:

HOW MANY STARS?- value of a digit as it moves from a single to a double-digit number (K-3)

WHAT'S THE NUMBER?- translating numbers from verbal to symbolic representation (K-3)

WHAT IS THE VALUE OF THE PLACE?- values of various digits of a number (K-5)

WHAT NUMBER IS THAT?- estimating a value on a number line given the value of the endpoints (K-5)

WHAT NUMBER IS THAT?- missing values in a 100's chart (2-5)

WHAT IS THE VALUE OF THE DIGIT?- values of digits in a decimal (3-5)

EQUAL TO 4?- meaning of an equal sign (2-5)

CRAYON COUNT- fractional parts of a whole (2-5)

IS 1/4 OF THE WHOLE SHADED?- identifying 1/4 of a whole (3-5)

GRANOLA BAR- fractional parts of a whole (3-5)

IS IT EQUIVALENT?- equivalent forms of a fraction (4-5)

IS IT SIMPLIFIED?- use of "canceling of zeros" shortcut (3-5)

HOW MANY DOTS?- using structure of ten when combining collections (K-2)

PLAY BALL- identfying part-party-whole combinations (1-3)

WHAT'S YOUR SUBTRACTION STRATEGY?- applying and understanding a variety of subtraction strategies (3-5)

WHAT'S YOUR MULTIPLICATION STRATEGY?- applying and understanding a variety of multiplication strategies (3-5)

WHAT'S YOUR DIVISION STRATEGY?- applying and understanding a variety of division strategies (3-5)

IS IT AN ESTIMATE?- estimating the sum of three 2-digit numbers (3-5)

WHAT'S THE AREA?- determining area without the typical length by width labeling (3-5)

WHAT'S THE MEASURE?- measure of a line given a change in the interval or non-zero starting point (3-5)

GRAPH CHOICES-graphical representation of a numerical situation (2-5)

THE MEDIAN- median and how it is affected by changes in a data set (3-5)

All of the books in this series can be ordered through Corwin Press, Amazon or other on-line book sellers.

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