Whole Number Computation, Culturally Responsive Mathematics, and Basic Fact Mastery

Vocabulary practice flashcards covering student-invented computational strategies, international algorithms, culturally responsive teaching practices, and the stages and strategies of basic math fact mastery.

Add Tens to Both Method

A European subtraction algorithm where ten is added to both the top ones digit and the bottom tens digit before subtracting (for example, in $42 - 18$, adding ten to $2$ yields $12$ and adding ten to the tens place of $18$ yields $28$, resulting in $12 - 8 = 4$ and $4 - 2 = 2$ to give $24$).

Trades

The regrouping process in multi-digit addition or subtraction where values cross a place-value threshold (such as crossing the ten or hundred boundary), resulting in carrying or borrowing.

Split Up Strategy

A student-invented addition method where multi-digit numbers are partitioned into tens and ones so that simpler components can be added separately before being recombined (for example, breaking $23 + 64$ into $20 + 60 = 80$ and $3 + 4 = 7$, then adding $80 + 7 = 87$).

Add On Strategy

A student-invented addition method where an amount is taken from one addend and added to the other to reach a friendly multiple of ten, after which the remainder is added (for example, borrowing $6$ from $23$ to make $64 + 6 = 70$, then adding the remaining $17$ to reach $87$).

Counting by Tens Strategy

A student-invented addition method where one number is decomposed into groups of tens and ones, which are then sequentially added onto the starting number (for example, splitting $23$ into $10$, $10$, and $3$ to calculate $64 + 10 = 74$, $74 + 10 = 84$, and $84 + 3 = 87$).

Compensation Strategy (Multiplication)

A mental computation method that simplifies multiplication by rounding a factor up to an easier number, calculating the product, and then subtracting the added excess (for example, solving $38 \times 4$ as $(40 \times 4) - (2 \times 4) = 160 - 8 = 152$).

Cluster Problems

A sequence of related basic and intermediate computation problems designed so that students use facts from known, simpler equations to solve a final, more complex problem (for example, using $50 \times 8$ and $4 \times 8$ to calculate $54 \times 8$).

Fair Sharing Division

A division problem structure (partitive division) in which a total set of items is distributed equally among a known number of recipients or groups to determine the size of each share.

Repeated Subtraction Division

A division problem structure (measurement division) in which a specified group size is continually subtracted from a total amount to determine how many groups can be made.

Direct Modeling

A foundational computation strategy characterized by using physical manipulatives or drawings accompanied by counting to represent and solve arithmetic problems.

Student-Invented Strategies

Flexible computational methods devised by learners that do not rely on physical materials or standard algorithms, characterized by being number- and place-value-oriented and calculated from the largest place value first ("left-handed").

Numerical Benchmarks

Personal and real-world reference points used to help students conceptualize the magnitude of large or small numbers (such as a $100\,\text{ft}$ driveway, a $5'\,6''$ wingspan, or an $8{,}000$-seat stadium).

Universality of Mathematics

The principle that while foundational mathematical concepts are universal across cultures, procedures, computational algorithms, and symbolic conventions vary across regions.

Culturally Relevant Mathematics

An instructional framework—sometimes called decolonized mathematics—that connects mathematical content to diverse cultural, ethnic, socioeconomic, and community backgrounds for all students rather than only immigrant learners.

Concrete-Pictorial-Symbolic Approach

A teaching sequence that transitions students from physical manipulatives to visual models and finally to abstract notations, maintaining academic rigor for language learners without removing linguistic context.

Word Cognates

Words across different languages that share common roots, meanings, and spelling forms (such as "division," "hexágono," "triángulo," and "circunferencia") that aid comprehension for multilingual learners.

Basic Facts (Elementary School)

Arithmetic combinations in elementary mathematics where both addends or both factors are single digits strictly less than $10$.

Basic Fact Mastery

The developmental milestone where a student can provide an accurate answer to a basic fact within approximately $3\,\text{seconds}$ without resorting to inefficient, non-automatic counting procedures.

Counting Strategies Stage

The first developmental phase of basic fact acquisition, in which students determine solutions by counting physical objects or counting verbally (such as counting all or counting on).

Reasoning Strategies Stage

The second developmental phase of basic fact acquisition, in which students use known number relationships and mathematical properties to logically deduce unknown combinations.

Mastery Stage

The third and final developmental phase in basic facts learning, characterized by the automatic, efficient production of answers through long-term memory retrieval.

Guided Intervention

An instructional approach to fact mastery where teachers design targeted story problems and facilitate class discussions that encourage students to construct and share reasoning strategies.

Think Addition

A subtraction reasoning strategy where a student determines the result of a subtraction problem by framing it as a missing-addend addition question (asking "what goes with this number to make the total?").

Down Over 10

A subtraction reasoning strategy where the subtrahend is partitioned to first subtract down to $10$, followed by subtracting the remaining portion (for example, $15 - 6 = 15 - 5 - 1 = 10 - 1 = 9$).

Take from 10

A subtraction reasoning strategy commonly used internationally where the subtrahend is subtracted directly from $10$, and that difference is added to the remaining ones of the minuend (for example, solving $15 - 8$ as $(10 - 8) + 5 = 2 + 5 = 7$).

Double and Double Again

A multiplication reasoning strategy for facts involving $4$ as a factor, where a student doubles the other factor once and then doubles that product.

Half Then Double

A multiplication reasoning strategy where an even factor is cut in half to compute a known, easier fact, after which that intermediate product is doubled (for example, calculating $6 \times 8$ by finding $3 \times 8 = 24$ and doubling $24$ to reach $48$).

Effective Drill

Targeted, brief practice activities appropriate only after reasoning strategies are firmly understood, designed to support self-monitoring, quick recall, and long-term memory retrieval without lengthy timed tests.