By Hanna A. Kubas and James B. Hale

Mathematics. Some love it, some loathe it, but there are many myths about math achievement and math learning disabilities (LDs). The old belief – boys are naturally better at math than girls – may be more a consequence of teacher differences or societal expectations than individual differences in math skill (Lindberg, Hyde, Petersen, & Linn, 2010).

Similarly, the old belief that reading is a left brain task, and math is a right brain task, is not a useful dichotomy as clearly multiple shared and distinct brain regions explain these academic domains (e.g., Ashkenazi, Black, Abrams, Hoeft, & Menon, 2013).

Math is a language with symbols that represent quantity facts instead of language facts (i.e., vocabulary), so rules (syntax) are important for both (Maruyama, Pallier, Jobert, Sigman, & Dehaene, 2012). You might be surprised to learn that approximately 7% of school-aged children have a LD in mathematics (Geary, Hoard, Nugent, & Bailey 2012).

Let’s first explore the fundamental skills needed for math achievement.

## Number Sense / Numerical Knowledge

Children develop knowledge of quantity even before math instruction in schools, and kindergarten number sense is predictive of math computation and problem solving skills in elementary school (Jordan et al., 2010). These basic math skills include understanding of number magnitudes, relations, and operations (e.g., adding). Children link basic number sense to symbolic representations of quantity (numbers); the math “language”. Poor early number sense predicts math LDs in later grades (Mazzocco & Thompson, 2005).

## Math Computation vs. Math Fluency

Children often rely on various strategies when solving simple calculation problems, but math computation requires caring out a sequence of steps on paper or in your mind (working memory) to arrive at an answer. Math fluency refers to how quickly and accurately students can answer simple math problems without having to compute an answer (i.e., from memory 6 x 6 = 36), with no “steps”, calculation, or number sense needed.

Children with fluency deficits often use immature counting strategies and often do not shift from computation to storing and retrieving math facts from memory, taking more time to provide an answer. Difficulty with retrieval of math facts is a weakness/deficit associated with math LDs (Geary et al., 2007; Gersten, Jordan, & Flojo, 2005). Without math fact automaticity, working memory may be taxed when doing computation, and the child “loses his place” in the problem while computing each part to arrive at a final answer.

## Developmental Sequence of Math Skills

1. Finger Counting Strategies: Students first display both addends/numbers with their fingers; this is the most immature strategy.

2. Verbal Counting Strategies: Next, students begin to develop basic adding skills and typically go through three phases.

• Sum: counting both addends/numbers starting from 1, this is a beginning math counting skill;
• Max:counting from the smaller number; and finally
• Min:counting from the larger number (most efficient strategy).

3. Decomposition (Splitting) Strategies: Students learn that a whole can be decomposed into parts in different ways, a good problem solving strategy for unknown math facts

4. Automatic Retrieval from Long-Term Memory: Students become faster and more efficient at pairing problems they see with correct answers stored in long-term memory (as is the case with sight word reading), no computation is required

## The Role of Visual-Spatial Skills

Basic arithmetic skills are factual, detailed “left hemisphere” functions (similar to basic reading), but Byron Rourke (2001) discovered many students with nonverbal or “right hemisphere” LDs had math calculation problems, suggesting left was verbal and right nonverbal.

Students need “right hemisphere” visual/spatial skills to align numbers when setting up multistep math problems, they need to need to be able to understand and spatially represent relationships and magnitude between numbers, and they need to be able to interpret spatially represented information (Geary, 2013).

Neuropsychology has also taught us that children with visual/spatial problems may neglect the left side of stimuli (the left visual field is contralateral to the right hemisphere) (Hale & Fiorello, 2004; Rourke, 2000).

## Math Reasoning and Problem-Solving

Word problems require both receptive and expressive language skills, unlike simple calculation, so students with language-based LDs may struggle even if math skills are good. Students must translate math problem sentences/words into numbers and equations, so they must identify what the sentences are asking them to do in terms of calculation, and then perform the calculation

Students with LDs are typically poor strategic learners and problem solvers, and often manifest strategy deficits that hinder performance, particularly on tasks that require higher level processing (Montague, 2008). So there is a strong relationship between fluid reasoning, executive functioning, and quantitative reasoning (Hale et al., 2008). Students with LDs often benefit from explicit instruction in selecting, applying, monitoring, evaluating use of appropriate strategies to solve word problems.

## Strategies for Promoting Math Computation and Fluency

Note : Your understanding of foundational mathematical concepts and skills is critical for targeted interventions that are developed, implemented, monitored, evaluated, and modified until treatment efficacy is obtained!

Remember: early identification and intervention are key!

### Strategic Number Counting

Fuchs et al. 2009

Goal: Improve counting strategies (e.g., MIN; decomposition) to efficiently pair problem stems and answers

Skills Targeted: Explicitly teach math counting strategies when number sense or algorithm adherence is limited

Target Age Group: Elementary students struggling with basic computation and quantity-number association

Description:

• Direct instruction of efficient counting (g., MIN for addition), followed by guided practice.
• For two-number subtraction, students start at ‘minus number’ and count up to ‘starting number,’ tallying numbers
• Flashcards used to math fact encoding, storage, and/or retrieval deficits. Optional number line can enhance method.

Empirical Support:

• Fuchs et al. (2009) found strategic counting led to better math fact fluency compared to control groups, even better if combined with intensive drill and practice
• Strategic counting with and without deliberate practice better math fluency, with deliberate practice better than controls (Fuchs et al., 2010)

### Drill and Practice

Fuchs et al. 2008

Goal: Drill and practice interventions help children quickly and accurately recall simple math facts

Skills Targeted: Practice and repetition of math fact calculations

Target Age Group: Students struggling with basic math facts, especially with limited automaticity

Description:

• May be paper-and-pencil and/or computerized drill and practice in either a game or drill format, typically includes modeling, practice, frequent administration, and brief, timed practice, self-management, and reinforcement
• Drill and practice with math problem solving strategies may be more effective
• Software to ensure correct student response; math facts appear for 1-3 seconds, and students reproduce the whole equation and answer from short-term memory
• Students visually encode both the number question and answer for long-term memory storage

Empirical Support:

• Connection between math fact rehearsal and increased fact retention and generalization (Burns, 2005; Codding et al., 2010; Duhon et al., 2012)
• Promotes efficient paring of problems and the correct answers (Fuchs et al., 2008)
• Computer versions improve math fact retrieval fluency (Burns et al., 2010; Slavin & Lake, 2008)

### Cover-Copy-Compare

Skinner et al. 1997

Goal: Improve accuracy and speed in basic math facts

Skills Targeted: Students taught self-management through modeling, guided practice, and corrective feedback

Target Age Group: Students learning basic math facts, those with executive, sequential, or integration problems

Description:

• Students learn 5-step strategy to solve simple math equations and self-evaluating correct responses
• Students look at math problem, cover it, copy it, and evaluate response to compare to original
• For errors, brief error correction procedure undertaken before next item introduced
• Strategy requires little teaching time or student training

Empirical Support:

• CCC procedures enhance math accuracy and fluency across general education (Codding et al., 2009; Grafman & Cates, 2010) and special education (Poncy et al., 2007; Skinner et al., 1997)
• Meta-analysis of many studies shows CCC improves math performance, especially when coupled with other evidence-based methods (e.g., token economies, goal setting, correct digits, increased response opportunity; Joseph et al., 2012)

### Detect-Practice-Repair

Poncy, Skinner & O’Mara, 2006

Goal: Promote efficient basic math fact practice targeting problems not completed accurately and/or fluently

Skills Targeted: Encoding and retrieval of math facts from long-term memory

Target Age Group: Students developing basic math facts, may be useful for executive memory difficulties

Description:

• DPR is a 3-phase test-teach-test procedure for individualizing math fact instruction for basic fact groups (e.g., addition)
• (1) Detect phase - metronome determined rate to determine automatic (< 2 seconds) vs. slow (>2 second) math fact responding
• (2) Practice phase using Cover-Copy-Compare (CCC; see description above)
• (3) Repair phase using 1-minute math sprint with items requiring practice embedded in automatic ones

Empirical Support:

• DPR validated across grades, skills, and research designs (Poncy et al., 2013)
• Improves subtraction, multiplication, and division fluency (Axtell et al., 2009; Poncy et al., 2006; 2010; Parkhurst et al., 2010)
• Differentiation possible because DPR targets specific difficulties (Poncy et al., 2013)

### Reciprocal Peer Tutoring

Fuchs et al., 2008

Goal: Peer tutoring procedure includes explicit timing, immediate response feedback, and overcorrection

Skills Targeted: Basic math fact retrieval and automaticity through constant engagement in dyads

Target Age Group: All students, but especially useful for students with poor attention or persistence

Description:

• Students are paired up and take turns serving as the “tutor”
• Flashcards with problem on one side (e.g., 2 x 3 = ___) and answer on other side (e.g., 6)
• Student tutors shows flashcard, tutee responds verbally
• Tutor states either “correct,” (and puts in correct stack) or “incorrect” (and puts in incorrect stack)
• If incorrect, tutee writes problem and correct answer 3 times on paper
• Roles change after 2 minutes; then students complete 1 minute math probes and grade each other

Empirical Support:

• Multicomponent approach + other evidence-based efforts improve math fact rates (Rhymer et al., 2000)
• Improves math achievement, engagement, and prosocial interactions (Rohrbeck et al., 2003)
• Improves achievement, self-concept, and attitudes (Bowman-Perrot et al., 2013; Tsuei, 2012)

## Strategies for Promoting Math Problem-Solving

### Schema Theory Instruction

Jitendra et al. 2002

Goal: Teaches mathematical problem structures, strategies to solve, and transfer to solve novel problems

Skills Targeted: Expanding student math problem solving schemas

Target Age Group: Students in any grade learning math problem solving skills, helps conceptual “gestalt”

Description:

• Encourages math problem solving schemas for word problems, identifying new, unfamiliar, or unnecessary information, and grouping novel problem features into broad schema for strategy use
• Explicit instruction in recognizing, understanding, and solving problems based on mathematical structures; can be used with schema-broadening instruction for generalization (e.g., Fuchs et al., 2008)

Empirical Support:

• Randomized controlled trials show improved math word problem solving (Fuchs et al., 2008, 2009)
• Schema-based approach generalizes into better math word problem solving (Jitendra et al., 2002; 2007; Xin, Jitendra, & Deatline-Buchman, 2005)

### Fast Draw

Mercer & Miller, 1992

Goal: Self-regulated strategy instruction method for increasing math problem solving skills

Skills Targeted: Targets self-teaching, self-monitoring, and self-support strategies for identifying salient math words in sentences, determining and completing operation, and checking accuracy

Target Age Group: Students struggling with executive monitoring and evaluation skills

Description:

• Teaches 8-step math word problems strategy and self-regulation
• The mnemonic FAST DRAW cues students, can use as checklist

Empirical Support:

• Increases math achievement and improves math attitude (Tok & Keskin, 2012)
• Increases math achievement in math LD (Miller & Mercer, 1997; Cassel & Reid, 1996)

### Cognitive Strategy Instruction

Montague & Dietz, 2009

Goal: Teach multiple cognitive strategies to enhance math problem solving skills

Skills Targeted: Focuses on cognitive processes, including executive functions (self-regulation/metacognition)

Target Age Group: Useful for differentiating instruction based on processing weaknesses

Description:

• Teaches 7-step cognitive strategy for solving math word problems, with 3-step metacognitive self-coaching routine for each step
• Direct instruction includes structured lesson plans, cognitive modeling, guided practice cues and prompts, distributed practice, frequent teacher-student interaction, immediate corrective feedback, positive reinforcement, overlearning, and mastery
• 7-step process includes:
1. Read the problem for understanding
2. Paraphrase the problem in your own words
3. Visualize a picture or a diagram to accompany the written problem
4. Hypothesize a plan to solve the problem
7. Check your answer to make sure everything is right Say, Ask, Check metacognitive routine in each of the 7-step cognitive processes
• Say requires self-talk to identify and direct self when solving problem
• Ask requires self-questioning, promoting self-talk internal dialogue
• Check requires self-monitoring strategy for checking understanding and accuracy

Empirical Support:

• Self-regulation strategies foster math problem solving in meta-analyses (Kroesbergen & van Luit, 2003)
• Cognitive strategy instruction increases math problem solving skills in general education (Mercer & Miller, 1992, Montague et al. 2011) and ADHD and LD (Iseman & Naglieri, 2011)

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