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40. Planning Facilitation for Math Calculation

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40. Planning Facilitation for Math Calculation

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I am thrilled to share that I am embarking on a new project: developing an innovative line of math resources called "Escape Room Adventures"! These interactive digital puzzles are designed to engage students and facilitate the revision and practice of specific math concepts. Each adventure is centered around a unique ...

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These beautiful flash cards use spaced repetition to teach multiplication facts. Achieve fluency with just five minutes of practice per day!

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39. The Mathematical Playground Free tools, courses and manipulatives 

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39. The Mathematical Playground Free tools, courses and manipulatives 

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Discover Mathigon, the Mathematical Playground. Learning mathematics has never been so interactive and fun!

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Math: games, flashcards, roman numerals, prime numbers, multiplication

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Build your skills with math flashcards and learn more about roman numerals, prime numbers, metric conversions, multiplication, factorials, factors, fractions, and all about money.

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Practice your math facts with these flashcards. Find addition, subtraction, multiplication and division flashcards, from basic math facts to 3-digit problems.

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At youcubed, we inspire, educate and empower teachers of mathematics, transforming the latest research on mathematics learning into accessible and practical forms.

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38. Research Evidence on the Best Ways to Learn Math Facts

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38. Research Evidence on the Best Ways to Learn Math Facts

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Download PDF Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts By Jo Boaler, Professor of Mathematics Education, co-founder youcubed With the help of Cathy Williams, […]

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32 b. Multisensory Math Tracking: Numeracy to Algebra

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32 b. Multisensory Math Tracking: Numeracy to Algebra

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Multisensory Math Tracking: Numeracy to Algebra

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16. Questions and answers about addition fact research

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16. Questions and answers about addition fact research

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I started writing this two months ago and it slipped out of my grasp. I picked it up but it slipped out again. I trapped it in a corner but there was a wet spot on the floor and I hilariously fell backwards while this post flew into the air, landed behind me, and scurried away, laughing as she did it.

I had been putting together these Q&As on research with the ambition of giving teachers readable summaries of research (and maybe writers a head start also). I also wanted all my citations in place so that I could go back and write about this stuff later, perhaps even for long-suffering Maybe Edu Book #2.

I first tackled homework, grading, and I figured math facts were my home turf. I could handle that in a few weeks. No biggie.

Yes, biggie. I've been overwhelmed with the complexity of this literature, mostly (I've finally realized) because it has the most to do with learning, and so most to do with how the brain actually works. Unlike homework, which is basically a black box surrounded by a bunch of statistics, we can actually know something about how kids learn addition and multiplication.

What that means is that there are multiple literatures to juggle here. There is the cognition stuff, which comes out of cognitive scientists interesting in mathematical thinking, learning, and problem solving. Then there is the intervention literature that mostly focuses on students with learning disabilities. And then there is a much smaller literature that looks at instructional interventions for typically developing students, and to be honest I didn't find much there to get excited about.

I've decided it's time to get part of this out into the world, so you're getting what is hopefully the first half. ("Hopefully," because if this turns out to be less than a half...) It is just about addition. The next entry will be all about multiplication, and I swear to god if it takes me more than a few weeks I will lose my mind.

In the comments or via email, I would very much appreciate hearing what questions you feel were unasked or unanswered in this Q&A and what questions you have about research in general about how children learn basic arithmetic. If you see mistakes or embarrassing omissions, please let me know. (Any errors will, in the future, be attributed to you, so I hope you're taking this seriously.) I am trying to actually get this right, so please hold me to a high standard.

OK without further ado, here are some questions followed by some answers and then followed by some citations.
Q: What does it mean to "know your addition facts"?
A: It's when you can answer single-digit addition questions by heart. It's when you remember the answer to the question (quickly and easily) rather than figuring it out.
Q: Is this how most adults answer single-digit addition questions?
A: Yes, mostly. Though most of us at times use other strategies to figure them out.

Research (e.g., Ashcraft & Stazyk, 1981; Geary et al., 1987; Goldman et al., 1988; Groen & Parkman, 1972; Siegler, 1987) documents that competent NC (number combination, i.e. single-digit addition) performance involves a mix of strategies, with counting strategies and decomposition strategies serving as back-ups for primary reliance on memory-based retrieval. In fact, individuals, even adults, use varying strategies at different times to solve the same NC.

Q: How do we know this?
A: First, because adults tell us that they're remembering and not calculating. Second, from measuring how long it takes them to answer addition questions. This is called measuring their "reaction times."
Q: What is this reaction time evidence?
A: Young kids take longer to respond to 6 + 7 than they do to 2 + 3. But they don't take much longer when they respond to things such as 7 + 3 or 2 + 8. In general, as the smallest addend gets a bit larger, the reaction time gets a bit larger by a more-or-less consistent amount. This suggests that young kids are figuring out the problem using a counting strategy.

Older kids and adults also take longer to respond to 6 + 7 than they do 2 + 3, but in a different way. First, it doesn't depend only on the smallest addend but on their sum. Second, the connection between the sum and the time taken was non-linear; larger problems took a bunch longer. Finally, they were overall faster. These together suggest that adults were remembering the answers from memory (but it took longer to remember larger answers).

Groen and Parkman's analysis of their first graders' data revealed a linearly increasing reaction time (RT) function across problems...Groen and Parkman concluded that first graders performed simple addition by min counting, a process known in educational settings as ''counting on''; set the counter to the larger of the two values, then count on by ones for the value of the smaller number, the min, as in 3, 4, 5 for the problem 3 + 2.

Counting could be rejected as the basis for adults' performance to simple addition problems. Instead, we proposed that adults were searching through a memory representation for the sums of the problems. Because of the longer RTs for large problems, it seemed clear that the search through memory was somehow lengthier or more difficult for those answer

Q: Is that settled then?
A: There's still a bit of debate about whether people are just remembering or whether there is some super-fast automatic counting that happens in your brain when you try to remember a fact. It doesn't seem like this would be very important for educators though, since it manifests as something like remembering.

The conventional wisdom in cognitive psychology has long been that fact recall (a reproductive process) replaces counting and reasoning strategies (reconstructive processes), because the former is more efficient than the latter (National Mathematics Advisory Panel, 2008). That, is, "true" mental-addition experts store all basic combinations not involving zero as a specific association between the addends and their sum and directly recall such facts. Recently, proponents of a compacted (automatic and non-conscious) counting model have revived the debate by suggesting that fast reconstructive processes play a role in the retrieval of small sums.

Q: How do people memorize stuff?
A: By trying to remember it.
Q: But how do you try to remember an addition fact if you don't already know it?
A: You read it or hear it, and then try to remember it that the there's a connection between the numbers in the problem and the answer.

Miller, S. P., & Hudson, P. J. (2007). Using Evidence-Based Practices to Build Mathematics Competence Related to Conceptual, Procedural, and Declarative Knowledge. Learning Disabilities Research & Practice, 22(1), 47-57. doi:10.1111/j.1540-5826.2007.00230.x
Constant Time Delay is an evidence-based instructional procedure designed to build declarative knowledge. The procedure involves a controlling prompt that results in near-errorless student responses.

Q: When do typical students commit addition facts to memory?
A: Once they have efficient techniques for figuring them out. This process should be moving along after a few years in school, in the US by 3rd Grade.
Q: Why would students typically only memorize a fact after they have efficient strategies?
A: Less efficient strategies make it harder for us to hold in mind the numbers of the problem along with the answer.
Q: Does it actually matter if kids learn their addition facts?

In work related to algebraic reasoning in the elementary grades, Fuchs et al. (2016) determined second-grade fluency with mathematics facts predicts fourth-grade equation solving with an unknown and function tables processing. In both elementary and middle school, Lee, Ng, and Bull (2018) identified mathematics fact word problems as predictive of algebraic word-problem performance. Beyond the elementary grades, Britt and Irwin (2008) worked with middle-school students and learned a deep understanding of additive, multiplicative, and proportional relationships related to algebraic reasoning. Even at the college level, computation of whole numbers significantly predicts algebra performance ( Tolar, Lederberg, & Fletcher, 2009), and mathematics fact fluency significantly predicts success in an algebra course ( McGlaughlin, Knoop, & Holloday, 2005). This literature demonstrates a connection between arithmetic and algebra, a connection that likely needs greater emphasis in elementary and secondary school ( Banerjee, 2011) and perhaps requires explicit instructional attention in college remedial courses.

Q: So what do you do for kids that are behind or don't know them?
A: You either help kids memorize or you help them develop more efficient addition strategies. A roadblock for some students with learning disabilities seems to be learning to "count up" by the smaller addend. When compared, both an intervention focusing just on memorizing and one aimed at teaching these kids to "count up" seemed to do equally well at improving kids' addition fact retrieval.
Q: But I should probably do both if I can though, right? Both efficient counting and memorization? Along with conceptual stuff?
A: Yeah, it all seems to be valuable.
Q: What are the implications for the classroom?
A: Just to actually try to teach this stuff. Don't ignore memorization and efficiency. Don't ignore strategies either. Teach all these things, but make sure kids are actually making progress in those early years, because it absolutely makes a difference.
Q: Terrific. And I assume all of this applies in a straightforward way to multiplication as well? And also to subtraction and division?
A: Uhhhhh.

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Questions and Answers about Addition Fact Research

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I started writing this two months ago and it slipped out of my grasp. I picked it up but it slipped out again. I trapped it in a corner but there was a wet spot on the floor and I hilariously fell backwards while this post flew into the air, landed behind me, and scurried away, laughing as she did it.

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2 c. Resources for Early Numeracy & Computation Skills

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2 c. Resources for Early Numeracy & Computation Skills

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The MIND is a set of empirically-validated resources designed to supplement core math instruction and provide intensive remediation targeting early numeracy and computation skills. These materials were created to present educators with low cost, short duration interventions that effectively increase early numeracy and computation skills.

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Three scripted interventions were developed to be used in a tiered system of instruction. Facts on Fire is a school wide (Tier 1) application, Skill Remediation targets groups (Tier 2), and the Intensive Intervention program is for one on one work with individual students (Tier 3). To learn more about the MIND click the buttons located on the menu above or explore these programs by clicking the links below.

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Check back often and read our newsfeed at the bottom of this page for updates! Feel free to contact us at anytime with questions or site feedback.

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Decision Making & Intervention Resources

Brian C. Poncy, PhD. , is an Associate Professor in the College of Education at Oklahoma State University. He is a recipient of the 2005 Outstanding Dissertation Award from Division 16 (School Psychology) of the American Psychological Association and the School Psychology Review 2015 Honorable Mention Article of the Year. Dr. Poncy's research focuses on academic interventions and behavioral principles of learning, specifically in the area of mathematics. He teaches classes focusing on the design, selection, implementation, and evaluation of academic interventions and single-case research designs. He has published approximately 20 research articles and book chapters.

Please Contact via email at brian.poncy@okstate.edu with any questions or comments concerning the MIND materials or the website.

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Please contact via email at gary.duhon@okstate.edu with any questions or comments concerning the MIND materials or the website

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The MIND is a set of empirically-validated resources designed to supplement core math instruction and provide intensive remediation targeting early numeracy and computation skills. These materials were created to present educators with low cost, short duration interventions that effectively increase early numeracy and computation skills.

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13 b. Classwide Math Intervention (Grades 4-6) - Part 2 

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13 b. Classwide Math Intervention (Grades 4-6) - Part 2 

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Classwide Math Intervention (Grades 4-6) - Part 2 - Implementation Debrief and Team Sharing

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13 a. Classwide Math Intervention (Grades 4-6) - Part 1

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13 a. Classwide Math Intervention (Grades 4-6) - Part 1

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Classwide Math Intervention (Grades 4-6) - Part 1

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35. Improving Mathematical Problem Solving in Grades 4 Through 8

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35. Improving Mathematical Problem Solving in Grades 4 Through 8

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37. Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students

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37. Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students

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36. Review of Mathematics Interventions for Secondary Students With Learning Disabilities

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36. Review of Mathematics Interventions for Secondary Students With Learning Disabilities

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14. Specific learning disability in mathematics: a comprehensive review

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14. Specific learning disability in mathematics: a comprehensive review

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Abstract

Math skills are necessary for success in the childhood educational and future adult work environment. This article reviews the changing terminology for specific learning disabilities (SLD) in math and describes the emerging genetics and neuroimaging studies that relate to individuals with math disability (MD). It is important to maintain a developmental perspective on MD, as presentation changes with age, instruction, and the different models (educational and medical) of identification. Intervention requires a systematic approach to screening and remediation that has evolved with more evidence-based literature. Newer directions in behavioral, educational and novel interventions are described.

Keywords: Mathematical disability, learning disorders, numerosity, childhood, interventions

Introduction

Mathematics (math) is not only the science of numbers, but also is used in everyday life from calculating time and distance, to handling money and analyzing data to make decisions in financial planning and insurance purchasing, and is essential in the STEM (science, technology, engineering and math) fields. Infants have an innate capacity for "numerosity" or the number of things. In early childhood, counting is learned to bridge this innate capacity to more advanced math abilities like arithmetic facts and concepts ( 1). Math is a core subject taught in primary and higher education, which builds a foundation of math skills for real life situations. Numeracy is the knowledge and skills required to effectively manage and respond to the mathematical demands of diverse situations ( 2), and is a necessary skill in both blue-collar and professional work places. Analyses in both the United Kingdom (UK) and United States (US) revealed that poor numeracy skills impacted employment opportunities and wages, even in the presence of adequate literacy skills ( 3, 4). Hence, it is concerning to note that the Nation's Report Card from the National Assessment of Educational Progress (NAEP) yielded poorer results for 4 th, 8 th and 12 th graders in the US in 2015 compared to 2013 ( 5), and only 9% of US adults scored in the highest numeracy levels in a 23 country comparative study ( 6). This has led to recommendations in the US aimed at improving math education, around curricular content, learning processes, instructional practices and materials, assessment, and teacher education ( 7). New standards like the Common Core State Standards (CCSS) (currently adopted in 42 states, District of Columbia and four territories) are designed to be relevant to the real world, while preparing students for post-secondary education and careers ( 8).

Terminology utilized to describe children who have difficulties with math varies in the literature depending upon how the study populations are defined and what instruments are used. Generally speaking, the term mathematical difficulties refers to children whose poor mathematics achievement is caused by a variety of factors from poor instruction to environmental factors, and represents a broader construct than does the term math disability (MD). Children with mathematical difficulties have low average performance or poor performance in math, but not all children with mathematical difficulties will have MD, which is hypothesized to be due to an inherent weakness in mathematical cognition not attributable to sociocultural or environmental causes ( 9). The continuously changing diagnostic criteria and varying definitions between the educational and medical/mental health realms add an additional confounder between the two terms. Terms like dyscalculia and poor math achievement add to the confusion as it is unclear if the terms are meant to be synonymous or overlapping ( 10). For the purposes of this review, the terms math learning disorder, math learning disability and dyscalculia will be considered as synonymous and represented by MD.

In many international clinical settings, criteria for MD is outlined in the World Health Organization International Statistical Classification of Diseases and Related Health Problems 10 th edition (ICD-10). It is defined as a specific impairment in arithmetical skills not solely explicable on the basis of general mental retardation or of inadequate schooling, which involves mastery of basic computational skills rather than more abstract mathematical skills ( 11). For the US, the definition in the most recent revision of the Diagnostic and Statistical Manual of Mental Disorders (DSM) 5 th edition, DSM-5 ( 12) is in the single category of specific learning disorders with specifiers for the area of math (others being reading and written expression). The definition states that difficulties should have persisted for at least six months despite interventions, and skills should be substantially below those expected for age. Deficits should interfere with functioning, as confirmed by individually administered standardized achievement measures and comprehensive clinical assessment. It includes possible deficits in number sense, memorization of math facts, calculation, and math reasoning.

Across reviews of studies ( 13), and as a matter of clinical practicality, most MD is identified by the school systems (educational and school psychologists, special educators among others). As such, it is critical to understand the changing criteria for MD determinations in the education system. With passage of the original version of Individuals with Disabilities Education Act (IDEA) in 1975 when specific learning disabilities (SLD) was recognized as a disability category for special education services, the operational definition was based on ability-achievement discrepancy in 1977 ( 14). Over time, the discrepancy model was deemed flawed and lacking in validity, and in 2004 reauthorization changed the definition to be based upon inadequate response to research based interventions ( 15). Yet, reviews of research indicate that despite the abandonment of the discrepancy model, studies continue to use this to identify subjects. Furthermore, much of the research on MD appears to be focused on elementary school children and basic math skills due to an emphasis on early identification and doesn't pay adequate attention to MD in older children ( 16) where math involves complex domains like algebra, and math reasoning.

Prevalence

There are almost 2.5 million students (5%) with SLD receiving special education services in US public schools, but this number has declined in the last decade, largely due to increased use of instructional strategies, shifts in SLD identification and attention to early childhood education ( 17). There is a wide range of prevalence data in the research, depending on criteria used for identification and cut-offs for determination, as well as the country of study. Prevalence data has varied from 3-7% ( 12). With regards to gender, results have varied depending upon the criteria used for identification, and overall gender differences were not identified when using absolute thresholds or relative discrepancy criteria in defining MD ( 18- 21).

Neurobiological basis

MD is considered a neurodevelopmental disorder, involving dysfunction in specific brain regions that are implicated in math skills. Numerosity is considered the building block of math skills, and relies on visual and auditory association cortices and the parietal attention system, specifically the intraparietal sulcus (IPS) within the posterior parietal cortex (PPC) ( 22). There is a developmental sequence in childhood to math acquisition; children initially rely on procedure-based counting (which when repeated, results in associations for retrieval). This is reflected in greater activation in functional imaging studies of the dorsal basal ganglia, which is involved in working memory (WM) ( 23). Gradual development shift mostly occurs in second and third grades to retrieval from long term memory, laying the foundation for more complex math skills ( 24). Brain involvement for complex math is based in the medial temporal lobe, with connections to other brain areas, especially the hippocampus and the prefrontal cortex (PFC). Younger children exhibit greater involvement of the hippocampus, and parahippocampal gyrus ( 25, 26) compared to adults. The greater activation seen in this region in younger subjects may reflect the greater recruitment of processing resources for memorization and may also reflect novelty effects ( 27). As children mature, there is greater activation in the left PPC and lateral occipital temporal cortex along with lower activation in multiple PFC areas evincing more specialization ( 27).

Individuals with MD have reduced activation during math tasks in functional imaging studies involving the IPS ( 28, 29) and structural imaging also showed reduced gray matter in the IPS in those with MD ( 30) and reduced connectivity between parietal and occipito-temporal regions ( 31). A recent study postulated that deficient fiber projection in the superior longitudinal fasciculus (particularly adjacent to the IPS) which connects parietal, temporal and frontal regions in children with MD is related to MD via a "disconnection" or interruption of integration and control of distributed brain processes ( 32). Since research in MD is not as established as reading disorders (RD) and is ongoing, no single hypothesis accounts for all children with MD. Multiple theories exist as described below.

Core deficit hypothesis

This predicts using neuro-imaging data that numeric concepts such as quantity, magnitude, numerosity are associated with the IPS. This is further distinguished as deficit in processing number magnitude ( 33) or deficit in number sense ( 34). However, developmental literature is inconclusive and inconsistent as MD has a complex and heterogenous clinical presentation ( 35).

Deficits in general domain hypothesis

This presumes there are subtypes of MD based on impairment in underlying cognitive processes. Three subtypes are thought to be associated, deficits in; verbal WM (necessary to acquire math procedures), long-term memory (necessary for storage and retrieval of math facts), and visuospatial processing ( 36).

Deficits in domain-specific math areas

The Triple Code model ( 37) of number processing hypothesizes there are three math domains: (I) numerical quantity representation (similar to number sense); (II) visuospatial numerical representation associated with attentional shifting; and (III) auditory verbally representation (associated with math facts and retrieval). This has some neuroanatomical support with IPS corresponding to first, posterior superior parietal lobule to second, and angular gyrus (AG) and perisylvian areas to the third domain respectively.

Procedural deficit hypothesis

This states that MD is a deficit of procedural memory involving a neural network of frontal, parietal, basal ganglia and cerebellar systems involved in storage and recall of skills. After practice, learned information is processed rapidly and automatically ( 38). However, while promising, it does not account for all cases of MD.

Genetic basis

There are thought to be a variety of intrinsic genetic contributors to MD, including genetic mutations and polymorphisms with hypothesized influences ranging from alterations in neural development to connectivity to epigenetic effects ( 39). Population-level studies take into account genetic variants on genotyping platforms and provide a lower bound estimate of heritability, while twin models capture all genetic factors and produce higher estimates due to the presence of dominance or interaction effects ( 40). Studies revealed that monozygotic and dizygotic twins of individuals with MD were twelve and eight times more likely to have MD, respectively ( 41) and >50% of siblings of individuals with MD also had MD ( 42). Math ability is thought to be influenced by many genes generating small effects across the entire spectrum of ability in genome-wide association studies ( 43) and around half of the observed correlation between math and reading ability is due to shared genetic effects ( 40). In a twin study, 60% of the genetic factors that influenced math ability also influenced reading ability and about 95% of the phenotypic correlation between the two is explained by these shared genetic influences ( 44).

Certain genetic conditions increase the risk of MD. Children with 22q11.2 deletion syndrome have deficits in calculation, math word problem solving and numerical quantities despite intact math fact retrieval ( 45). Children with Turner syndrome have intact number comprehension and processing skills, but have slower and more error-prone functioning on complex math problems ( 46). Children with fragile X syndrome have both MD and RD ( 47), while children with myelomeningocele have deficits in simple math, counting and math word problem solving despite relative strengths of basic number knowledge and language skills ( 48). Children with neurofibromatosis-1 have deficits in math calculation and math word problems and more variability in math skills than other genetic condition ( 49), and children with Williams syndrome have significant visuospatial deficits which may be responsible for their MD features ( 50).

Clinical presentation

While the theories described above are tested against cognitive neuroscience models, from a clinical perspective, it is helpful to classify presentations depending on the skills and deficits demonstrated. One such model from Karagiannakis et al. ( 51) has 4 subtypes of skills:

1. Core number: which involves numerosity, estimating numbers and quantities, number line ability, managing symbols and basic counting;

2. Memory: retrieving math facts, performing calculations, remembering rules and formulae;

3. Reasoning: grasping math concepts, complex math procedures, logical problems and problem solving;

4. Visuospatial: geometry, written calculations, graphs and tables.

Typically developing children may have an experiential knowledge of math even prior to formal school math education, where they get exposed to math facts and calculation procedures. Clinical features of a child presenting with MD () largely depends on three factors: the age/developmental age of the child, the existing math instruction and curriculum exposure, and the presence of comorbid conditions.

Table 1

Age

Examples of expected math skills

Symptoms of MD

Toddlers and preschoolers

Symbolic representation

Difficulty learning to count

Recognize and label small numbers

Difficulty sorting

Simple magnitude awareness

Difficulty corresponding numbers to objects

Simple subitizing

Difficulty with auditory memory of numbers (e.g., phone number)

Matching and naming simple shapes

Cardinal rule

One to one correspondence

Rote counting followed by meaningful counting

Kindergarten

Rapid recognition of small quantities

Difficulty counting

Mastery of counting (individually and in classes)

Difficulty subitizing

Recognition of numbers

Trouble with number recognition

Solve simple math word problems

Able to recognize and name parts of shapes

1 st-3 rd grade (early elementary)

Naming and writing large whole numbers

Difficulty with magnitude comparison

Counting forwards, backwards, skip count

Trouble learning math facts

Identifying place value of digits

Difficulty with math problem-solving skills

Simple addition and subtraction

Over reliance on ﬁnger counting for more than basic sums

Compare collections even if disparate in size and quantity

Anxiety during math tasks

Number conservation

Understand complementarity of addition and subtraction

Retrieve some arithmetical facts from memory

Classifying and sorting shapes based on attributes

4 th-8 th grade (late elementary through middle)

Numbers up to 100,000, comparing and ordering numbers

Difficulties with precision during math work

Place value of larger numbers

Difficulty remembering previously encountered patterns
Difficulty sequencing multiple steps of math problem

Understanding of fractions

Difficulty understanding real-world representation of math formulae

Multiplication and division and their inverse relationship

Anxiety during math tasks

Basic algebraic concepts to solving linear equations

Reading graphs, tables, charts

Understanding ratio, proportions, percent, unit pricing

Learning to draw, compare and classify two- and three-dimensional shapes

Learning to measure objects by length, area, weight, volume using standard units

High school

Equations and inequalities

Struggle to apply math concepts to everyday life, including money matters, estimating speed and distance

Logic and geometry

Trouble with measurements

Polynomials

Difficulty grasping information from graphs or charts

Exponential and logarithmic functions

Difficulty arriving at different approaches to same math problem

Trigonometry

Anxiety during math tasks

Sequences, series, and probability

Statistics: sampling, designing experiments, measures of center and spread

Statistics: normal distributions, confidence intervals, tests of significance, graphical displays of association

The coexistence of another condition along with the primary condition under study is considered "comorbidity" ( 52). For MD, the most prevalent comorbidity is RD, with rates as high as 70%, and correspondingly for children with RD, rates of MD can be as high as 56% ( 18, 53, 54). The rates tend to be lower when more stringent cut-offs are applied to the definitions of the various disorders ( 55), and when population samples are studied compared to identified SLD samples. Distinguishing MD from comorbid MD with RD has focused on performance on nonverbal and verbal tests with studies showing that students with RD (and MD + RD) experienced more difficulty with phonology; students with MD (and MD + RD) more difficulty with processing speed, nonverbal reasoning, and most mathematical performances ( 56). While there have been suggestions that inattention and poor planning associated with attention deficit hyperactivity disorder (ADHD) may be responsible, MD and ADHD are thought to be comorbid separate disorders that are independently transmitted in families ( 57). A Spanish study ( 58), which attempted to distinguish the cognitive profiles between ADHD and MD children, revealed that simultaneous processing was more predictive in the MD group while executive processes was predictive in ADHD group.

Math anxiety is specific to math and is a negative emotional reaction or state of discomfort involving math tasks ( 59). It is not a rare phenomenon and has been found in 4% of high school students in the UK ( 60). Children with MD can develop a negative attitude towards math in general, and avoidance of math activities associated with anxious feelings. Negative experiences with math teachers compounds the situation. Poor math achievement is strongly related to math anxiety, especially when children are expected to work rapidly towards a single correct response ( 61). There is debate whether math anxiety is a distinct entity from generalized anxiety, as some studies have shown correlation with measures of general anxiety ( 62) while other studies show measures of math anxiety correlate more with one another than with test anxiety or general anxiety ( 63). Math anxiety is thought to develop as young as first grade ( 64), and brain activity shows that math anxious children show increased amygdala activity (emotional regulation) with reduced activity in dorsolateral PFC and PPC (WM and numerical processing) ( 65).

Evaluation

Screening

This is the first step in a diagnostic process, and is usually conducted with the general student population to identify "at-risk" children (as early as kindergarten) who need to proceed to a formal diagnostic process. Gersten et al. ( 66) outlines common components included in screening batteries such as: magnitude comparison, strategic counting, retrieval of basic arithmetic facts, and (more recently) word problems and numeral recognition. Single proficiency screening measures, which are easy to administer quickly with large numbers of students, have comparable predictive validities to multiple proficiency screening measures that cover a wider range of mathematics proficiencies and skills. No single test score is found to be predictive, though performance on number line estimation ( 67) and reading numerals, number constancy, magnitude judgments of one-digit numbers, and mental addition of one-digit numbers ( 68) are most correlated with math achievement. Additionally, testing WM is recommended along with math-specific items ( 68).

In the UK, a computerized screener has been used ( 69). In the US, the Response to Intervention (RTI) model ( 70) recommends screenings on a school-wide basis at least twice yearly, using an objective tool and focused on critical math objectives for each grade level that are prescribed at the state level (or CCSS as applicable). Curriculum-based measurement (CBM) probes are empirically supported for screening ( 71) and for math generally is a single proficiency measure (digit computation). Health professionals' role in screening generally involves providing anticipatory guidance on development and behavior and surveillance for "early warning signs" of MD (especially in pre-K children) of difficulty counting loud and struggling with number recognition and rhyming ( 72). Other roles include investigating and treating potential medical problems that can affect the patient's ability to learn ( 73) while ruling out medical differential diagnoses for learning problems like seizures, anemia, thyroid disease, sensory deficits and elevated lead levels.

Diagnosis

The process of diagnosis of MD depends on the discipline of the clinician first encountered, what criteria are used, as well as local and regional regulations. In early grades (kindergarten and first grade), diagnostics probe for functioning on foundational skills like core number processing. This is reliant on exposure to, and interaction with, symbolic processing skills as well as language and spatial processing. As children get older, increasing abstract concepts are relied on for math processing, and deficits in these are used to make the diagnosis ( 74). Differences observed in young children may result from exposure to mathematics before formal schooling or from student performance on more formal mathematics in school ( 66). Diagnostics can occur either in a medical interdisciplinary or educational model depending on local and regional availability and access to such clinicians.

Specialists in interdisciplinary clinics (like Child Neurologists, Developmental-Behavioral Pediatricians, Pediatric Psychologists) can be involved, often at the request of caregivers with referrals from primary care providers, or for independent consultations in event of questions or concerns about the adequacy of school services. This may engender tension between educators and medical professionals despite the acknowledged importance of collaboration; particularly around the perception that medical clinicians foster dissension between parents and schools, don't acknowledge the competence of school personnel and make recommendations not reflecting the individual needs of the child or impact on the cost or structure of the child's education ( 75). In the medical model, history is usually elicited around the child's symptoms and current functioning in all academic areas, history of interventions or grade retention, family history of MD, presence of other developmental delays (gross, fine motor, visuomotor, language, adaptive) and behavioral symptoms (including anxiety, somatic symptoms and attentional issues). Apart from a physical and neurological examination, screening tests for learning disabilities are conducted. In the interdisciplinary team, clinical psychologists are invaluable in conducting diagnostic assessments including cognitive and academic batteries.

Neuropsychological testing (NPT) has been presumed to be broader than psychoeducational assessment and suggested as an essential part of SLD identification, as it can provide information on strengths and weaknesses, particularly if other medical conditions exist ( 76). While studies ( 77, 78) have shown that performance on neuropsychological-psychological batteries can predict academic achievement later, such testing is time intensive and many insurances (both public and private) only consider NPT medically necessary in the assessment of cognitive impairment due to medical or psychiatric conditions. They explicitly exclude coverage for educational reasons (unless a qualifying medical disorder is present like metabolic disorders, neurocutaneous disorders, traumatic brain injury) and suggest that testing be provided by school systems. NPT is of value in context of medical disorders or when function deterioration (due to neurological conditions) is not adequately explained by social-emotional or environmental factors ( 79). However, one limitation of NPT is that reports offer hypotheses about a child's level of functioning that may not account for, or be relevant to, planning interventions within the school setting and is more of a snapshot than a progressively developing picture that school personnel are privy to ( 79). There is emerging interest that technology and biopsychosocial data, such as eye-tracking data in combination with number line estimation tasks, might be a promising tool in diagnosing MD in children ( 80).

In an educational model, prior to the 2004 reauthorization of IDEA, eligibility for determination as SLD (including math) rested on the ability-achievement discrepancy model which required assessment of cognitive and academic functioning and relied on formulae that determined cut-offs. However, since 2004, local education agencies were permitted to adopt criteria that could either be the discrepancy model or an alternate based in evidence-based science. One of the latter, the RTI model ( 70) changed eligibility to students who fail to respond to increasing hierarchy of data-driven interventions based on individual student problems. However, despite the law and subsequent regulations, there is no directive regarding magnitude of achievement and progress targets for struggling learners which continues to create varying identification practices ( 81).

Another approach emphasizes the role of strengths and weaknesses in cognitive processing measured by individually administered standardized tests. This patterns of strengths and weaknesses (PSW) approach needs additional empirical evidence to determine the robustness of this model as an alternate to existing procedures ( 82). There continues to be considerable variability in the state practices using identification techniques, with 34 states continuing to use the discrepancy model, and 10 states explicitly prohibiting its use. While 45 states provide guidance on RTI implementation, only 8 states exclusively use RTI models in LD identification, but with variability regarding how to implement models. Less than half of the states allow use of PSW models, but with little information regarding identification practices. Prior to 2004, only half of the states included math reasoning as an area for LD identification, and this has increased to all presently ( 83). School eligibility evaluations are conducted by credentialed special education teachers, licensed educational psychologists, or school psychologists.

Generally, an evaluation consists of history and review of records, followed by psychometric testing for academic skills, intellectual abilities, sometimes executive function (EF), socio-emotional and behavioral assessment including qualitative information, classroom observations and questionnaires from caregivers and teachers. For MD, the assessment includes whether the student has mastery of math skills compared to the state's academic content standards despite appropriate instruction and if the impact of the disability entitles the student to specialized instruction to benefit. The school evaluators prioritize educational goals and place them in the context of the school, which is the child's natural environment ( 79).

Management

Educational Interventions

Teaching individuals with MD creates unique challenges in regular education environments, as teachers often struggle to provide individualized attention due to large class sizes, limited resources and learners with different styles. The overall goal is building knowledge and skills to develop automaticity. With students with MD, many math processes never become "automatic," and they need extra time to improve WM and extra work on cognitive functions ( 84). The co-teaching model, in which children with MD are taught in the general education classroom, involves a general education teacher delivering overall instructional content, and a second special education teacher designing and delivering more intensive interventions and learning strategies as needed. This model is built around inclusion, to foster positive interaction and behavior among students, and to foster professional development among educators, but it has been criticized for not detecting or differentiating students with MD early in their education ( 85).

While the adoption of CCSS standardizes the expected knowledge and abilities of students at a certain grade level, and allows an easier transition if students move between districts or states, some suggest that it also may cause children to fall further behind their peers, and leave gaps in important skills and general math understanding. It can create hardship for students with MD as well; since the time constraints and the expected amount of material that students are expected to master may lead to students with MD not getting enough procedural practice, and repetition of the basic concepts in order to be successful ( 86). Since CCSS is reading intensive, students with comorbid LD and MD struggle to a greater extent, and there are additional challenges transitioning from an older curriculum, and the lack of professional development educators receive during this transition ( 87).

The RTI model is a three tiered system is used for early identification (and intervention) of children who may be at risk of future educational failure. The typical representation is a pyramid () of larger numbers in a generalized education environment with progressively smaller groups receiving additional instructional supports and finally students needing intensive instruction similar to specialized education. The model has the benefit of creating accountability by encouraging and guiding practitioners to intervene earlier with the great number of children at risk of failure, and by introducing a more valid method to identify student with MD with progress monitoring and motivational strategies, particularly for students in Tier 2 and Tier 3 instruction ( 88). The impetus for RTI has been primarily for reading, with slower adoption for math, though programs have been slowly expanding over the past five to ten years. There has been debate about the value of investment in small-group intervention, whether interventions should be aligned with core curriculum, and which students are likely to benefit from intervention. Small group interventions have consistently improved students' math performance on proximal measures, that is, difficult grade level content and sophisticated topics. However, the effect on distal measures, that is building a general capacity in math, is less clear ( 89).

The National Center for Educational Evaluation and Regional Assistance recommends that instructional material should focus intensely on whole numbers in kindergarten through grade five, and on rational numbers in grades four through eight. Instruction should include at least ten minutes devoted to building fluent retrieval of facts, should be explicit and systematic, including providing models, verbalization of thought process, guided practice, corrective feedback, and frequent cumulative review, and students should have the opportunity to work with visual representations of mathematical ideas. Interestingly, in classrooms with a high percentage of students with MD, teachers are more likely to instruct with the use of manipulatives/calculators, or movement/music, or ordering and number/quantity skills, though these activities have not been associated with math achievement gains by students with MD. Students without MD benefit from both student-centered and teacher-directed activities. However, only increased classroom time doing teacher-directed activities is associated with positive gains in students with MD ( 90). One meta-analysis on teaching methods for children with MD revealed that students with lower math performance tend to respond better to instruction, and while interventions may help in many math domains, a smaller effect is seen in early numeracy and general math proficiency. Explicit teacher-led instruction, and peer-assisted learning have the largest affect, and providing instructional recommendations to teachers, and the use of technology may also have positive effects. Specific instructional components that may be beneficial for students with MD include controlling task difficulty, greater elaboration on topics, and working in a small group setting. Additionally, cognitive and metacognitive strategies computer-based interactive lessons, videos, and hands-on projects have been shown to aid math understanding and performance. Using concrete and visual representations along with teacher-facilitated instruction and virtual manipulatives have also been shown to be effective ( 91).

Three general strategies have been shown to be useful in accommodating children with MD including; improving reading skills, improving mathematical problem solving skills, and altering general instructional design () ( 92). There are several specific math intervention programs which have gained research support. For a detailed description of these programs, see: http://www.hanoverresearch.com/2015/04/06/best-practices-in-math-interventions/. A detailed list of math interventions describing the appropriate age of learners and the level of evidence is also available at the What Works Clearinghouse, an online database created by the U.S Department of Education's Institute of Education Sciences ( 93).

Table 2

Improving reading skills

Break up the text into smaller sections

Use a simple font

Do not justify the text

Use colored overlays to reduce glare

Improving mathematical problem solving skills

Photocopy math books with the relevant sections placed in order

Separate complicated problems into small steps

Use markers to highlight, and guide attention

Use color to delineate columns and rows in spreadsheets

Simplify tables

General instructional design

Supplement incomplete notes

Use posters to remind students of various basic concepts

Use flash cards

Provide flow charts to clarify procedures

Engage visual learners with manipulatives

Encourage students to move at their own pace

Teach organization, studying, and time management skills

Focus on revision prior to an examination

While emphasis is usually on implementing an intervention in the way it was researched and manualized, interventions are often slightly modified by special education teachers in response to their students' notions of mathematics, and the particular need of individual students, or small groups of students. They do this through changing the pedagogy, materials, or tasks to bridge students' prior and informal knowledge. They provide additional practice opportunities and connect procedural practices to larger concepts, provide additional time to practice in areas of difficulty, foster greater student interaction, and allow students more opportunity to think out loud and justify their thought processes ( 94). These practices should not be discouraged, because research has shown that training tailored to a child's specific needs can create positive results ( 95).

Behavioral interventions

While educational interventions comprise the bulk of effective interventions for MD, there have been behavioral interventions that are effective in improving the math skills of children with MD, in the areas of EF, cognitive tutoring, and cognitive-behavior therapy (CBT) to improve performance, decrease math anxiety, and foster a positive attitude towards math.

EF broadly refers to the processes of attention, WM, long term planning, volition, and behavior inhibition ( 96) that allows students to organize and prioritize information, monitor progress, and adapt. Students who struggle with EF may have difficulty determining key information in math word problems, performing mental math, or starting a task, and they may make careless mistakes if they fail to check their work ( 97).

Strategies that can be implemented by school personnel include teaching positive self-regulatory skills through engagement, awareness of strengths and needs, goal setting, skill mastery, and generalization. A highly structured environment and schedule, limiting distractions, and providing interventional cues can foster self-regulation. For students who have difficulty with WM, teachers should limit the number of concepts presented at any one time, and may find it helpful to group information into chunks ( 97).

Improving attention in children with low arithmetical achievement may help to improve their abilities. Strategies include using instruction in improving reaction time, sensory (visual, auditory) selectivity, and attention shift ( 98). WM is specifically related to the ability of learn new math skills and is widely accepted to be impaired in individuals with MD. WM is correlated with math performance and is indicative of future math performance. WM training shows improvement in math skills especially when involving visuospatial WM ( 99). Cognitive tutoring has been found to be successful in improving math performance, effecting neuroplasticity, and changing the brain function of children with MD. Functional imaging studies have shown that initial differences in the prefrontal, parietal and ventral temporal-occipital cortices normalized after eight weeks of one-on-one tutoring focused on strengthening conceptual and procedural knowledge to support attention, WM, visuospatial skills, and cognitive inhibition necessary for math fluency ( 100).

Behavioral interventions are also important in addressing math anxiety, and early intervention is important as math anxiety tends to increase with age. Interventions include systematic desensitization, and cognitive behavior therapy (CBT); for example, expressive writing before a test may cause the student to reevaluate the need for worry, and therefore increase the WM available to perform specific tasks ( 101). Studies have shown that cognitive behavioral intervention with targeted tutoring may be helpful in decreasing anxiety, fostering a positive attitude towards math, and improving performance ( 102).

Novel approaches

Computer games and tutorials have been used for the past 30 years in special education to aid students with MD to improve basic math knowledge and skills ( 103). Rapid advances in broadband connectivity and ubiquity of mobile computing has led to almost every student having access to a computer connected to the internet, and increased digital content ( 104). Technology helps to emphasize important concepts, engage multiple sensory modalities, divide complex material into smaller components, and provide immediate feedback about accuracy ( 104). In one study, the use of math apps allowed both students with and without MD to make gains in math assessments. The larger gain was made in the struggling group, which helped to close the achievement gap ( 105).

Other technologies that can aid students with MD include computers and tablets with touch screens that are easier to use than traditional mouse and keyboards, and children can use them with little instruction ( 105). For students with MD, this type of feature allows for accommodations such as "talking calculators," and the easy transfer of numbers to graphs, tables, and charts to create visual aids. Now, programs containing an artificial intelligence element can interactively tailor the features and settings of these programs, to the needs and abilities of particular students. For additional information about specific programs, please see the paper by Campuzano et al. ( 106).

Recent research has shown that virtual manipulatives can be invaluable when integrated with general math concepts to ensure a meaningful learning experience to match the individual student's needs and abilities. For classrooms already equipped with computers, the cost is negligible since many virtual manipulative websites are easy to access, free of charge and don't require storage or cleaning which benefits busy teachers ( 103). Video prompting using tablet devices have the added benefit of giving educators more time to work with small groups, or one-on-one, while others work in a self-directed manner, and students benefit by gaining independence, and becoming more accountable for their own learning ( 107).

Outcomes

The importance of identification and intervention is underscored by the impact of MD on child and adult functioning. Shalev et al. ( 108) demonstrated in a longitudinal study over three and six years that MD is persistent, which is similar to other developmental disorders like ADHD and RD with educational interventions not being protective against persistence of MD. There is limited research on SLD outcomes in adults, much less with respect to MD, and it tends to focus on young adulthood with limited data points which are less helpful about trends. The National Longitudinal Transition Study- 2 only provides a snapshot of the immediate years after leaving school, lacks specific information about SLD in absence of other disabilities, and most information is about literacy and not numeracy. It also does not include persons who have dropped out of school, and describes attendance (but not completion) of two-year postsecondary schools. It does not track the types of employment and job advancement ( 109). Educational achievement is hypothesized to translate to higher SES through higher attained qualifications, improving occupational status and career development opportunities ( 110) and numeracy is associated with successful financial decision making ( 111).

Conclusions

Changing diagnostic manuals, different hypotheses regarding the core precepts of the disorder, and different criteria adopted in medical and educational systems can lead to different perspectives on MD. Advances in neuroscience and genetics offer promise in etiological determinations and understanding the neural processes underpinning MD. Clinical presentation varies depending on age and developmental status of the child, presence of other comorbid conditions, and the degree and nature of instructional methods used. The core interventions for MD continue to be educational in nature, with a varying degree of evidence-base that continues to grow. Additional avenues include behavioral interventions, which are primarily for comorbid conditions and math anxiety. Exciting new directions utilizing informational technology can help supplement educational interventions.

Acknowledgements

None.

Footnotes

Conflicts of Interest: The authors have no conflicts of interest to declare.

References

5. National Assessment of Educational Progress 2015 Mathematics Assessment: the nation's report card. Washington DC; 2015.

6. Goodman M, Finnegan R, Mohadjer L, et al. Literacy, numeracy, and problem solving in technology-rich environments among U.S. adults: results from the program for the international assessment of adult competencies 2012. Available online: https://nces.ed.gov/pubs2014/2014008.pdf

8. National Governors Association Center for Best Practices, Council of Chief State School Officers. Common core state standards for mathematics. 2010. Available online: http://www.corestandards.org/

11. World Health Organization. International statistical classification of diseases and related health problems. 10th ed. Geneva.

15. Individuals with Disabilities Education Improvement Act. 20 USC 1414(b)(6) 2004.

69. Butterworth B. Dyscalculia Screener. London: nferNelson Publishing Company Limited, 2003.

79. Sassu KA, Gelbar NW, Bray MA, et al. Exploring neuropsychology : seeking evidence of added worth to school psychology practice. School Psychology Forum 2015;9:269-76. [ Google Scholar]

88. Burton DT, Kappenberg J. Response to intervention: the teachers' role in distinguishing between mathematics difficulty and mathematics disability. Insights Learn Disabil 2010;7:53-64. [ Google Scholar]

92. Michaelson MT. An overview of dyscalculia: methods for ascertaining and accommodating dyscalculic children in the classroom. Aust Math Teach 2007;63:17-23. [ Google Scholar]

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Specific learning disability in mathematics: a comprehensive review

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Math skills are necessary for success in the childhood educational and future adult work environment. This article reviews the changing terminology for specific learning disabilities (SLD) in math and describes the emerging genetics and neuroimaging studies that relate to individuals with math disability (MD).

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7 a. 10 Key Mathematics Practices for Elementary Schools

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7 a. 10 Key Mathematics Practices for Elementary Schools

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2a. Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades

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2a. Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades

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WWC | Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades

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This practice guide provides evidence-based practices that can help teachers tailor their instructional approaches and/or their mathematics intervention programs to meet the needs of their students.

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34. Curriculum Bank - worksheets, activities, and related instructional videos 

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34. Curriculum Bank - worksheets, activities, and related instructional videos 

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Does your child struggle with numbers and arithmetic? Your child may have dyscalculia. Find out how to identify and help.

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33 b. Best Graphic Organizers to Teach Math (slides)

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33 b. Best Graphic Organizers to Teach Math (slides)

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T23Graphic Organizer Woodin.pdf | Powered by Box

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33 a. Best Graphic Organizers to Teach Math (audio only)

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33 a. Best Graphic Organizers to Teach Math (audio only)

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Dyslexia Conference Recording | Best Graphic Organizers to Teach Math

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32 a. Multisensory Math: From Arithmetic to Algebra

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32 a. Multisensory Math: From Arithmetic to Algebra

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Dyslexia Conference Recording | Multisensory Math: From Arithmetic to Algebra

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7b. 10 Key Math practices for middle and high school 

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7b. 10 Key Math practices for middle and high school 

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1. Presentation slides - Dr. Sarah Powell

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1. Presentation slides - Dr. Sarah Powell

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31. Math motivation - interventions for situations where the student is reluctant to work on math tasks

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31. Math motivation - interventions for situations where the student is reluctant to work on math tasks

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Classwide Antecedent Modifications

The context of the environment in which behaviors occur is not usually considered when analyzing a child's behavior. Instead, more attention is typically given to the consequences following that particular behavior (especially when it is a disruptive behavior being analyzed). While consequences of behaviors matter, what occurred BEFORE the problem behavior should also be considered when creating an intervention. Altering the antecedent of the target behavior has the substantial advantage of being proactive. As such, with appropriate modifications of the antecedents, a problem behavior (e.g. disruptive behavior or task demand refusal) can be avoided. This brief presents a series of classwide antecedent alterations that will change typical antecedents of problem behaviors to antecedents that prompt appropriate behaviors. See Kern and Clemens (2007) for an excellent through review of this class of intervention.

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Classwide Antecedent Modifications The context of the environment in which behaviors occur is not usually considered when analyzing a child's behavior. Instead, more attention is typically given to the consequences following that particular behavior (especially when it is a disruptive behavior being analyzed).

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30. Math generalization - interventions for situations where the student has not done the academic task that way before

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30. Math generalization - interventions for situations where the student has not done the academic task that way before

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Pirate Math is a program designed to improve the word-problem skill of elementary-aged students. Currently, there is a Pirate Math program for use in second-grade classrooms and a third-grade program for use as an individual tutoring intervention. During Pirate Math, students learn to

Focus on and identify the conceptual framework of three types of word problems (i.e., Total, Difference, and Change word problems);

Identify and ignore irrelevant information;

Translate a word problem's narrative to a mathematical equation, substituting "X" for the missing information;

Solve for the missing variable ("X") as expressed in the mathematical equation; and

Self-evaluate the reasonableness of solved equations.

Common Core Standards: Varied

Setting: Whole Class and Individual

Focus Area: Acquisition, Fluency and Generalization

Overview:
Full Brief Link

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Pirate Math is a program designed to improve the word-problem skill of elementary-aged students. Currently, there is a Pirate Math program for use in second-grade classrooms and a third-grade program for use as an individual tutoring intervention.

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29. Math proficiency - interventions for situations where the student needs to become more fluent with the math skill

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29. Math proficiency - interventions for situations where the student needs to become more fluent with the math skill

# data[].attributes.attachment_link.readable_text
Pirate Math is a program designed to improve the word-problem skill of elementary-aged students. Currently, there is a Pirate Math program for use in second-grade classrooms and a third-grade program for use as an individual tutoring intervention. During Pirate Math, students learn to

Focus on and identify the conceptual framework of three types of word problems (i.e., Total, Difference, and Change word problems);

Identify and ignore irrelevant information;

Translate a word problem's narrative to a mathematical equation, substituting "X" for the missing information;

Solve for the missing variable ("X") as expressed in the mathematical equation; and

Self-evaluate the reasonableness of solved equations.

Common Core Standards: Varied

Setting: Whole Class and Individual

Focus Area: Acquisition, Fluency and Generalization

Overview:
Full Brief Link

# data[].attributes.attachment_link.description
Pirate Math is a program designed to improve the word-problem skill of elementary-aged students. Currently, there is a Pirate Math program for use in second-grade classrooms and a third-grade program for use as an individual tutoring intervention.

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28. Math acquisition - interventions for situations where the task is too hard for the student

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28. Math acquisition - interventions for situations where the task is too hard for the student

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Concrete-Representational-Abstract (CRA)

With CRA, students work with hands-on materials that represent mathematics problems (concrete), pictorial representations of mathematics problems (representational), and mathematics problems with numbers and symbols (abstract). The teacher explicitly bridges the connection between the concrete, representational, and abstract representations of the mathematics problems. Common Core Standards: All Setting: Whole Class, Small Group or Individual Focus Area: Acquisition, Fluency and Generalization Full Brief Link

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Concrete-Representational-Abstract (CRA) With CRA, students work with hands-on materials that represent mathematics problems (concrete), pictorial representations of mathematics problems (representational), and mathematics problems with numbers and symbols (abstract).

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In TouchTimes, children use their fingers to multiply. By using two-handed gestures to create their own factors and products, children develop a strong sense of multiplicative relationships. TouchTimes offers two complementary experiences:

The Grasplify World. Here children first create, with the fingers of one hand, some form of unit, and then-through one or more taps of the other hand-scale that unit to a unit of units, a product. The Grasplify world enables experiences of multiplication that go beyond approaches based on additive thinking and repeated addition.

The Zaplify World. Here children create simultaneous bolts of lightning-horizontal from one hand, vertical from another. Orange sparks appear where these bolts cross as a product of the lightning-bolt factors. (These terms may also be displayed numerically.) This dynamic model of multiplication emphasizes the commutative property multiplication and the symmetry of multiplicative factors, and gives learners conceptual and geometric building blocks for understanding multiplicative relationships.

Since TouchTimes displays the product that results from any multiplicative expression, learners focus less on how to compute than on how multiplicative relationships work, and on how, with their hands, to coordinate and combine multiplicative factors into given multiplicative products. Learners also encounter and work with numbers, symbols and equations, and move back and forth between symbolic/numeric forms and more visual, haptic, and tactile representations of multiplicative relationships.

Children benefit from making choices and being in control. TouchTimes encourages exploration and choice-making in an environment that is structured by mathematical rules, but that is not channeled into only narrow opportunities and predetermined outcomes.

One of the best ways to support learners' experiences with TouchTimes is to encourage this open exploration, while asking them to predict or explain TouchTimes' responses in different situations. You might try to pose small challenges for them, or to ask them questions about the mathematical challenges and goals they set for themselves. Encourage them to speak as well as to act: talking through one's own understanding is one of the strongest ways to develop and solidify that understanding.

For instructions and simple, engaging activities you can use with a variety of learners, visit the Grasplify and Zaplify sections of this site.

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In TouchTimes, children use their fingers to multiply. By using two-handed gestures to create their own factors and products, children develop a strong sense of multiplicative relationships. TouchTimes offers two complementary experiences: The Grasplify World.

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In TouchCounts, children use their fingers, eyes and ears to learn to count, add and subtract. By using simple gestures to create and manipulate their own numbers, children develop a strong number sense.

TouchCounts offers two experiences:

The Numbers World. Here children create the counting numbers in sequence. The Numbers world helps young learners coordinate the various number representations to which they've been exposed-spoken numbers, written numbers, sequential finger-pointings, and depicted object collections-into a common, unified number sequence. Along the way they develop facility with enumeration and ordinality.

The Operations World. Here children create arbitrary whole numbers and explore basic number operation concepts by pushing numbers together (into new, larger numbers) or by splitting numbers apart (into new, smaller numbers). These operations give learners conceptual and hands-on building blocks for understanding cardinality, addition and subtraction.

In both worlds, learners also encounter and work with spoken and symbolic number names, one-to-one correspondence, subitization and finger knowledge.

Children benefit from making choices and being in control. TouchCounts encourages exploration and choice-making in an environment that is structured by mathematical rules, but that is not channeled into narrow opportunities or preconceived outcomes.

One of the best ways you can support learners' experience with TouchCounts is to encourage this open exploration, while asking them in different situations why they think TouchCounts responded as it did, or what they think will happen if they try something. You might try to pose small challenges for them, or to ask them questions about the mathematical challenges and goals they set for themselves. Encourage them to speak as well as to act: talking through one's own understanding is one of the strongest ways to develop and solidify that understanding.

For instructions and simple, engaging activities you can use with a variety of learners, visit the Numbers and Operations sections.

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In TouchCounts, children use their fingers, eyes and ears to learn to count, add and subtract. By using simple gestures to create and manipulate their own numbers, children develop a strong number sense. TouchCounts offers two experiences: The Numbers World. Here children create the counting numbers in sequence.

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Bedtime Math - We inspire kids to love math so they embrace and excel at it.

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When you bite into chocolate cake, it's hard to believe it was made from boring white flour and goopy eggs. Somehow when we add sugar, butter, and vanilla or chocolate, the hot oven turns them into cake. So if you had only 1 tablespoon of flour, how small a cake would you get?

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Try the for an even richer experience!

This website is not set up to display the books/stories on a smaller screen.

Please enter "Full Screen" or increase the size of your window to fully experience Math Storytime on the web.

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Try the for an even richer experience! This website is not set up to display the books/stories on a smaller screen. Please enter "Full Screen" or increase the size of your window to fully experience Math Storytime on the web.

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Testimonials

I was searching for some subtraction drills and IXL came up. Very lucky! I've rarely seen kids so intensely interested in what they are learning about.

Barry Stewart, math teacher

Hope, British Columbia

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IXL | Math and English Language Arts Practice

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IXL is the world's most popular subscription-based learning site. Used by over 13 million students, IXL provides unlimited practice in more than 6000 math and English language arts topics. Interactive questions, awards and certificates keep kids motivated as they master skills.

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27. Get the Math (Algebra in the real world)

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27. Get the Math (Algebra in the real world)

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Wolfram MathWorld: The Web’s Most Extensive Mathematics Resource

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Comprehensive encyclopedia of mathematics with 13,000 detailed entries. Continually updated, extensively illustrated, and with interactive examples.

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National Library of Virtual Manipulatives

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A digital library containing Java applets and activities for K-12 mathematics

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10. Math learning videos (elementary to high school)

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10. Math learning videos (elementary to high school)

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Mathematics is one of the fundamental 3Rs of early education. Known as "arithmetic" in those levels, it involves counting and the basic operations of addition, subtraction, multiplication, and division. As kids advance through the grade levels, they progressively deal with more complex mathematical problems. Also, more new concepts get introduced as they progress.

For the most part, math is a lot about problem-solving. Some students may view this as tedious, others would say it's such a bother. However, the point of making students solve math problems is not to annoy them or make them hate math. On the contrary, it's for them to exercise their minds. Each math problem draws on a specific concept, and repeatedly solving similar problems allows students to master the concepts. A good grasp of various principles is very important in math because each higher topic builds upon an earlier topic.

Math also builds logical thinking skills in kids. Since math problems are mostly solved through step-by-step processes, there is always a sense of order. Proving theorems in geometry is one example of such a logical process. That and other forms of mathematical problem solving equips students to think in a highly ordered manner.

Additionally, math also helps develop pattern recognition skills. For instance, in algebra, the idea of factoring makes use of patterns. If a polynomial takes a certain form, then it automatically can be simplified a certain way. Any polynomial that looks that same way can be factored similarly too.

Math is also crucial in careers like a business. Accounting, budgeting and projecting sales all require some level of competence in math.

Even everyday situations benefit a lot from math skills. Managing one's finances, for instance, obviously requires math, as with anything involving money. Cooking also involves math: proportions of ingredients, for example, as well as measurements.

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Math, Math education videos for students, teachers and parents on Teachertube.com

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Mathematics is one of the fundamental 3Rs of early education. Known as "arithmetic" in those levels, it involves counting and the basic operations of addition, subtraction, multiplication, and division. As kids advance through the grade levels, they progressively deal with more complex mathematical problems. Also, more new concepts get introduced as they progress.

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9. Fun Brain (free math games for grades 1 to 8)

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9. Fun Brain (free math games for grades 1 to 8)

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Free Fun Math Games For Kids Online | Funbrain

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Math Zone combines logic and numbers with fun and challenging math games for kids online. With a huge collection of educational maths, visit Funbrain here.

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8. EARLY MATH RESOURCES (from infancy to preschool)

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8. EARLY MATH RESOURCES (from infancy to preschool)

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From stars in the night sky to the life cycle of a butterfly, science is all around us.

We've expanded our STEM offerings for early childhood educators with earlysciencematters.org.

Visit today to learn how to set up science-rich environments for toddlers and preschoolers, download free lesson plans and more!

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From stars in the night sky to the life cycle of a butterfly, science is all around us. We've expanded our STEM offerings for early childhood educators with earlysciencematters.org. Visit today to learn how to set up science-rich environments for toddlers and preschoolers, download free lesson plans and more!

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8. 75+ free and paid Websites for Teaching and Learning Math

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8. 75+ free and paid Websites for Teaching and Learning Math

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The internet is full of incredible new ways to teach and learn math, from games and videos to lessons and even complete curricula. Teachers, students, and parents can all benefit from these online learning resources. This list of the best math websites has options for every skill level, from learning to count to advanced math like calculus. You're bound to find a new favorite!

Jump to:

These math websites provide complete standards-based math curricula. Try them if you're looking for a new way to teach math at your school. These can be good choices for homeschoolers too.

Join the 1 million teachers already using Prodigy Math in their classrooms at no cost. This fun and engaging curriculum-aligned game lets students engage in a fun wizarding world that motivates them to practice more math than ever. It contains 1,500+ standard-aligned skills. Plus you can track student progress with a teacher dashboard that provides instant feedback on areas of progress with no grading necessary.

Info: Grades 1-8; free for teachers, and monthly memberships available for parents

From McGraw-Hill, this web-based assessment and learning system uses adaptive questioning to determine students' needs. It adapts to teach kids the topics they're most ready to learn.

Info: Grades 3-12; contact them for pricing

Along with textbooks, Art of Problem Solving has a stable of robust online resources. You'll find videos, math problems from math contests, and online classes.

Info: Grades 5-12; pricing varies by program

BuzzMath focuses on helping middle schoolers practice their math skills. It contains high-quality problems, gives immediate and detailed feedback, and lets students progress at their own pace. Randomly generated values let students retry problems to obtain mastery. Teachers also receive detailed results that help them guide and monitor student progress.

Info: Grades 1-9; free demo with subscription plans for students and families

Make free detailed charts of all kinds, including bar graphs, pie charts, scatter plots, and more.

Info: K-12; free

This resource from England provides math videos with associated math practice questions and worksheets. It's basic, but a good way to get no-cost practice.

Info: Grades K-12; free

This National Science Foundation-funded program helps students strengthen math skills. Students will learn to solve problems and explain their thinking using mathematician George Polya's four-step approach.

Info: Grades K-12; priced per teacher or school, with free trial available

A blend of paper and technology puts student ideas at the center of learning. Lessons pose problems that invite a variety of approaches, engaging kids more fully.

Info: Grades 6-8; contact them for pricing

An adaptive learning platform designed to complement classroom instruction and deliver results. Includes resources for teachers, student data reports, and instructive insights.

Info: Grades K-8; home users can sign up for individual or family subscriptions, schools pay per student or school

This site provides e-textbooks, answer keys, video lessons, and printables. Topics include algebra 1 and 2, geometry, and trigonometry.

Info: Grades 6-12; free lessons plans, homework sets, and videos; paid subscriptions include answer keys, assessments, and more

We love the friendly competition and game-based content offered by First in Math. Kids gain skills practice and fluency as they play games targeted toward fact proficiency and logical thinking.

Info: Grades K-8; subscriptions available for schools and parents, with 45-day free trial

This site allows kids to practice math at their own level and pace. It has more than 30,000 math questions, starting with a diagnostic that assigns material at just the right level. Front Row also features lessons, assessments, and reports for teachers.

Info: Grades K-12; basic use is free for teachers; premium subscriptions available

This site from the National Council of Teachers of Mathematics (NCTM) features complete lesson plans, mobile games for students, interactive activities, and brainteasers.

Info: Grades Pre-K-12; free

Get high-quality educational resources for teachers and students. Excellent math tasks, videos, lesson plans, and problem-based curriculum modules.

Info: Grades K-12; free

Formerly known as Edgenuity, Imagine Learning offers several online digital curricula for primary or supplementary instruction. Give students the support they need exactly when they need it.

Info: Grades Pre-K-12; priced per subject, per student

Istation makes personalized learning easy with computer-adaptive instruction, assessments, personalized data profiles, and teacher resources. Includes digital lessons and face-to-face teaching strategies.

Info: Grades Pre-K-8; priced per student

Make math about more than numbers with engaging items, real-world scenarios, and unlimited questions. Teachers choose the strand and then set up students to work independently.

Info: Grades Pre-K-12; free 30-day trial, pricing is per classroom or site

Khan Academy is on a mission to give a free, world-class education to anyone, anywhere. Their personalized learning resources are available for all ages, in a huge array of subjects.

Info: Grades K-12; free

Here's an online learning space that is engaging, supportive, and designed to get kids interested in math.

Info: Grades K-12; yearly subscription required; school and family pricing available

Through ongoing research, MIND Research Institute continues to investigate key questions about learning, mathematics, and how the brain works. ST Math is their pre-K-8 visual instructional program, helping teachers engage kids more deeply in math learning.

Info: Grades Pre-K-8; pricing based on total school enrollment

This award-winning program finds and fixes learning gaps with the power of personalized learning. The three-pronged approach features personalized learning, pinpoint assessments, and an interactive classroom.

Info: Grades K-8; pricing per student, school, and district

Stepping Stones 2.0, from Origo, is a unique and comprehensive curriculum that combines print and digital materials. It features problem-solving activities, strategies, and practice.

Info: Grades Pre-K-6; price varies by program

This organization is geared toward students, teachers, and families in low-income communities. The program features school workshops, instructional coaching, and professional learning communities.

Info: Grades K-12; try limited Family Playlists for free, request consultation for pricing

Enter an interactive, game-based learning world that motivates kids through rewards. Features include a daily challenge, step-by-step lessons, and parent connections.

Info: Grades K-5; 30-day free trial, then monthly subscription

Boost confidence, increase scores, and get ahead. Fun for enrichment or regular practice.

Info: Grades K-5; free for teachers and schools, monthly subscriptions available for parents

Math games make learning fun! Focus on specific skills, target interventions, and make assessment easy.

Info: Grades K-8; teachers can try six games for free, subscription unlocks more games and features

Tang Math strives to provide unparalleled math lessons for students as well as professional development for in-service teachers. Find games and puzzles, plus other resources like free downloads, worksheets/word problems, and math centers.

Info: Grades K-5; free

Woot Math offers adaptive practice for teaching rational numbers and related topics, such as fractions, decimals, and ratios.

Info: Grades 3-7; free tier for teachers, additional features available for extra cost

Personalized math curriculum that includes digital lessons and small-group instruction. Also has online modules, workbooks and answer keys, and professional development.

Info: Grades K-5; free for individual teachers and classrooms

These sites offer engaging videos and tools to use in your daily math instruction.

Engaging animated learning videos, games, quizzes, and activities to encourage kids on their unique learning path.

Info: Grades Pre-K-8; teacher, school, and district pricing available, as well as parent or homeschool options

Teachers prep an assignment, students work on their devices, everybody gives feedback, and the teacher sees it all!

Info: Grades K-12; free basic plan, Pro accounts unlock more features

A website that allows teachers to assign math practice content to their students. Students get immediate feedback as they complete the problems.

Info: Middle school+; free, with premium school and district plans available

An online graphing calculator that students can use for free. Includes a teacher-centric activity builder for creating digital math activities.

Info: Grades 9-12; free

An award-winning series of math apps that harness the power of digital tools to create a better, deeper, more fun learning experience. Blogs such as "Making Math Social" and "Saying No to Math Anxiety" are included as resources for teachers and parents.

Info: Grades K-6; priced per app, with free trials for teachers and bulk pricing for schools

Use hip-hop to teach math! Flocabulary offers songs, activities, and videos.

Info: Grades K-12; teacher, school, and district pricing available

Upload your own materials or build them from scratch, find something pre-made, act on live responses, and track student growth over time.

Info: Grades K-12; free, premium subscription unlocks additional features

Another graphing calculator for functions, geometry, algebra, calculus, statistics, and 3D math, along with a variety of math resources.

Info: Grades 9-12; free

You know it; your kids love it. Why not use it to teach math? Engage your students with this game-based classroom-response system played by the whole class in real time. Multiple-choice questions are projected on the screen, then students answer with their smartphone, tablet, or computer.

Info: Grades K-12; free, Kahoot+ AccessPass subscription offers premium content

Run by the University of Regina in Canada, this site offers free resources for math teachers and their students, including a database where users can search for the answers to math questions. Their "Mathematics With a Human Face" page includes information about careers in mathematics as well as profiles of mathematicians.

Info: Grades K-12; free

Teachers know that one of the best ways to make sure learning sticks is through song. Numberock features ad-free music videos of songs about math topics, like fractions, money, and integers, produced by an Emmy Award-winning studio. Numberock also has anchor charts, worksheets, comics, games, and more.

Info: Grades K-5; limited free videos, monthly subscription offers six-month free trial

Transform presentations into classroom conversations with Peardeck for Google Slides. Effortlessly build engaging instructional content, formative assessments, and interactive questions.

Info: Grades K-12; basic access is free, premium subscriptions unlock more features

Think of this site like YouTube but specifically for teachers and schools. Find videos created by other teachers, and upload your own to share.

Info: Grades K-12; free

These interactive math websites provide students with instruction and independent practice.

Two users play a game in which each player tries to connect four game pieces in a row (like Connect Four). The players answer math questions to connect the pieces. The teacher chooses how much time each player has to answer, the level of difficulty, and the type of math problem.

Info: Grades 2-8; free

Yes, math games can be cool! Check out the hundreds of games on this site and on Coolmath4kids.

Info: Grades K-12; free with ads, premium subscription removes ads and provides extra features

Figure This is a site designed to encourage families to practice math together. It includes fun and engaging math games and high-quality challenges. It even offers challenges in Spanish.

Info: Grades 6-8; free

Funbrain has been helping students learn key math concepts and develop crucial skills since 1997. Students can choose from a slew of games. The best part? It's all free!

Info: Grades Pre-K-8; free

Get the Math is about algebra in the real world. See how professionals use math in music, fashion, video games, restaurants, basketball, and special effects. Then take on interactive challenges related to those careers.

Info: Grades 6-12; free

Your students will love fun math games like Speed Math Deluxe, Mystery Math, Place Value Game, and more.

Info: Grades 3-12; free

A popular game-based site for online math-learning resources. It covers algebra, geometry, statistics, and more.

Info: Grades K-12; schools can contact them for a quote; parents pay per child

Play engaging games and find free printable worksheets at this site from Knowledge Adventure. Great for kids at home or teachers in the classroom.

Info: Grades Pre-K-8; free basic access with subscriptions for parents, teachers, and schools

There are dozens of games to explore here, organized by grade and subject.

Info: Grades Pre-K-7; free

In-depth lessons with videos, guided practice, interactive self-tests, and more.

Info: Grades 5-12; monthly or yearly subscription required

More than 425 math games, logic puzzles, and brain workouts for students to practice their math skills.

Info: Grades 1-6; free with ads, Premium removes ads and provides extra features

Most learners do best when they can see a problem walk-through, step-by-step. This site features multiple example problems, with walk-throughs by three separate instructors (including one in Spanish). They offer some basic math but are focused on advanced subjects from algebra on up.

Info: Grades 6-12; free

Thousands of original math games, workshops, and practice modules, as well as math printables!

Info: Grades 1-6; monthly or yearly subscription required

Need to master your multiplication facts? This is the site to try! Fun games and memory-building strategies will help students tackle this key skill.

Info: Grades 2-6; free, with Premium membership available for extra features

This interactive online tool helps students master basic facts. Set up as a whole-class or small-group competition, students earn ninja cards and track results on a ninja scoring poster. Two board games that teach operations are also available for purchase.

Info: Grades 2-8; pay per game or purchase teacher licenses

This is a free intervention tool designed to fill gaps in students' mental calculation skills and empower them with number fluency. Students can earn ninja belts of different colors for their skill level.

Info: Grades 2-8; free

From PBS Learning Media, middle schoolers will love this entertaining video blog. Not only does each episode cover Common Core Standards, it makes math learning culturally relevant with pop-culture references.

Info: Grades 6-9; free

Students can create study flash cards, play educational games, practice skills, collaborate with other students, and more.

Info: Grades 5-12; free, QuizletPlus provides additional features

Another resource to help students build fact fluency. Each game is tailored to students' ability levels.

Info: Grades 2-6; school and home licenses available

Tons of fun and educational online math games, from basic operations to algebra and geometry.

Info: Grades K-6; free

Simple math test activities for teachers and students, from beginning math operations to calculus. You set the skill level, number of problems, and time limit. A report, which tallies right and wrong answers, is provided after each quiz.

Info: Grades 3-12; free

How would your students like to learn multiplication while shooting hoops? They can do that and more at Toy Theater, which teaches early math concepts through game-based learning.

Info: Grades K-5; free

Upper-level math students will appreciate the no-frills information that's easy to find on this site. Get overviews and see examples of advanced math subjects.

Info: High School+; free

Xtramath is like a daily math vitamin. An interactive online tool that helps students practice and master basic arithmetic facts, it's quick and easy to use. Weekly emails provide progress reports for teachers and parents.

Info: Grades K-8; free, with premium licenses that offer additional features

These math websites provide lesson-planning resources and professional-development materials.

Formerly known as Mathalicious, this site provides supplemental math lessons. Experience an immersive, participatory approach that makes learning and teaching more rewarding.

Info: Grades 6-12; monthly or yearly subscriptions available

Find math worksheets for just about any area of study. Use them for planning lessons, review, and independent work.

Info: Grades K-6; free

Here's a reliable site for worksheets, games, and even lessons plans. They cover a variety of subjects, with lots of math options to choose from.

Info: Grades K-8; limited free access, monthly subscriptions unlock all content

This platform allows teachers to create technology-enhanced online math assessments from a huge question bank.

Info: Grades K-12; free teacher accounts

For teachers of pre-algebra through calculus. Create the math worksheets you need, exactly how you want them, in minutes. You can also create customizable homework, quizzes, and tests.

Info: Grades 8-12; free two-week trial, then single-user and site licenses available

A creative solution that aims to revive students' passion and interest in math. Mashup Math has a library of 100+ math video lessons as well as a YouTube channel that features new math video lessons every week. A free e-book of math challenges is also available.

Info: Grades K-8; free

Dynamically created math worksheets for students, teachers, and parents.

Info: Grades K-10; free with ads, paid membership removes ads

Tools for math teachers, including bell ringers and drills, math tools and manipulatives, question generators, printables, and puzzles.

Info: Grades K-12; free with ads

This National Science Foundation-supported project provides a large library of uniquely interactive, web-based virtual manipulatives and concept tutorials for math instruction.

Info: Grades K-12; free

Convert all of your paper assignments, quizzes, homework, etc., into digital activities delivered online.

Info: Grades K-12; free basic version, Pro plan adds multiple extra features

For a very affordable monthly fee, gain access to thousands of resources created by teachers like you. They cover every topic and every grade level.

Info: Grades K-12; monthly subscription

TopMarks is a U.K.-based site that provides a database of resources for teachers as well as online learning games for students.

Info: Grades Pre-K-8; free

The Varsity Learning Online Math Management System provides course templates, assignments, an online teacher's assistant, and thousands of practice problems so you can get organized, save assignments and videos on the web, share resources, and incorporate technology in your classroom.

Info: Grades: 7-12; free

More than 1,500 video lessons ranging from middle-grade math through algebra 2.

Info: Grades 6-12; free

What's on your list of the best math websites? Come share your ideas in our WeAreTeachers HELPLINE group on Facebook.

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75+ Awesome Websites for Teaching and Learning Math

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The internet is full of incredible new ways to teach and learn math, from games and videos to lessons and even complete curricula. Teachers, students, and parents can all benefit from these online learning resources. This list of the best math websites has options for every skill level, from learning to count to advanced math like calculus.

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Mathletics United States | Online Learning Math Program

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Mathletics is the award-winning online math program to help students learn math. Engaging, fun, and aligned to curricula across the United States.

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Classroom Resources - National Council of Teachers of Mathematics

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5. Evidence-based math instruction: What you need to know

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5. Evidence-based math instruction: What you need to know

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Evidence-based math instruction: What you need to know

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Learn the most effective ways to teach math to students who struggle. Understand the four types of evidence-based math instruction.

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23. Difficulty with Word Problems - Schema Instruction

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23. Difficulty with Word Problems - Schema Instruction

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Page 6: Schema Instruction

How does this practice align?

High-Leverage Practices

HLP14: Teach cognitive and metacognitive strategies to support learning and independence

CCSSM: Standards for Mathematical Practice

MP7: Look for and make use of structure.

Another effective strategy for helping students improve their mathematics performance is related to solving word problems. More specifically, it involves teaching students how to identify word problem types based on a given problem's underlying structure, or schema. Before learning about this strategy, however, it is helpful to understand why many students struggle with word problems in the first place.

Difficulty with Word Problems

Most students, especially those with mathematics difficulties and disabilities, have trouble solving word problems. This is in large part because word problems require students to:

1. Read and understand the text, including mathematics vocabulary

2. Be able to identify and separate relevant information from irrelevant information

3. Represent the problem correctly

4. Choose an appropriate strategy for solving the problem

5. Perform the computational procedures

6. Check the answer to ensure that it makes sense
(Adapted from Stevens and Powell, 2016; Jitendra, et al., 2015; Jitendra et al., 2013)

Students who experience difficulty with any of the steps listed above, such as students who struggle with mathematics, will likely arrive at an incorrect answer.

Research Shows

Students with mathematical difficulties and disabilities struggle more than their peers when solving word problems.
(Stevens & Powell, 2016; Jitendra et al., 2015; Fuchs et al., 2010)

Schema instruction-explicit instruction in identifying word problem types, representing them correctly, and using an effective method for solving them-has been found to be effective among students with mathematical difficulties and disabilities.
(Jitendra et al., 2016; Jitendra et al., 2015; Jitendra et al., 2009; Montague & Dietz, 2009; Fuchs et al., 2010)

Teaching students how to solve word problems by identifying word problem types is more effective than teaching them only to identify key words (e.g., "altogether," "difference").
(Jitendra, Griffin, Deatline-Buchman, & Sczesniak, 2007)

Word Problem Structures

To help students become more proficient at solving word problems, teachers can help students recognize the problem schema, which refers to the underlying structure of the problem or the problem type (e.g., adding or combining two or more sets, finding the difference between two sets). This, in turn, leads to an associated strategy for solving that problem type. There are two main types of schemas: additive and multiplicative. Below, we will introduce you to additive schemas before moving on to descriptions and examples of multiplicative schemas.

Additive Schemas

Additive schemas can be used for addition and subtraction problems. These schemas are effective for students in early elementary school through middle school. Below are a few examples of additive schemas used to solve word problems: total, difference, and change.

(Adapted from Stevens & Powell, 2016; Morales, Shute & Pellegrino, 1985)

For Your Information

Even when they apply the same schema to solve a word problem, students will likely approach its solution in a variety of ways. An example of this can be found below.

Problem: Emma had nine dollars. Then she earned some more money doing her chores. Now Emma has $12. How much money did she earn?

Two students, A and B, set up the problem using the change schema.

However, Student A solves the problem by subtracting 12 - 9. Student B solves the problem by counting on from 9. Although one student adds and the other subtracts, both students arrive at the correct solution. This example illustrates that the operation is secondary to the structure of the word problem.

Multiplicative Schemas

Multiplicative schemas can be used to solve multiplication and division problems. There are three main types of multiplicative schemas: equal, comparison, and ratio/proportion.

Source: Jitendra, Star, Dupuis, & Rodriguez, 2013

Combined Schemas

As students advance in school, they will encounter new kinds of mathematical problems with new underlying structures, or schema. They will also encounter multi-step mathematics problems. The example below illustrates a percent change problem which involves the combination of two schemas: a multiplicative and an additive.
Note: After the student solves for the missing value in the equation above, he enters it along with the provided information in the equation below to solve the problem.
Example: Mark is interested in buying a car. The car costs $3,200. He will receive a 10% discount if he buys the car this weekend. How much will he pay for the car?

Solution equation (to determine the amount of change):
After the student solves for the "change," which is $320, he will then create another solution equation to find the "new total."
Solution equation (to determine the "new total"):
The student determines that with the 10% discount Mark will pay $2,880.
Sarah Powell, who has conducted extensive research on schema instruction, discusses the underlying focus of this strategy (time: 2:40).

Sarah Powell, PhD
Assistant Professor, Special Education
University of Texas at Austin

Transcript: Sarah Powell, PhD

The thing with schemas is that you cannot define word problems by their operation. So you cannot describe a word problem as being a subtraction problem or being a division problem. Instead, you have to describe the word problem at a deeper level and that is describing the word problem by its schema. And sometimes I like to use the word structure. It's really important to use schemas or structures so that students have consistency with problem solving. If we teach the structure of combined problems in 1st grade and 2nd grade, students continue to see that schema in 3rd grade, 4th grade, and 5th grade. Now the numbers may get greater. So instead of adding three plus nine, they might be adding 133 plus 239. But the structure is the same. And so one of the things that we are trying to do with our math standards that guide most instruction in the United States is to provide consistency in math learning across grade levels, and the schemas really help do that, so you see that combined structure again and again.

Now in middle school grades, you see it in a slightly different way. It might be part of a multi-step problem, but it's still there, so that every year we don't have to reteach problem solving. We just help students say, "Oh, now we're looking at a total schema but with fractions. Now here's a total schema but with decimals." And so there's a lot of consistency that's provided with the schemas. And right now problem solving is really taught grade-by-grade. So how do I solve 2nd-grade word problems, or how do I solve 5th-grade word problems? And that's not a good way of thinking about it. It's better if we focus on the schema and think about this grade-level continuum of problem solving. And they would just make problem solving so much easier for students and also easier for teachers, because then they're not going back to square one every year and starting about how do I teach problem solving in 5th grade?

I would argue that problem solving is the most important thing you have to teach, because when we look at high-stakes assessments-and that's where students show their mathematics competency-for word problems, students have to take the numbers and manipulate the numbers. It's very difficult. Problem solving should be the primary focus of the mathematics curriculum, and instead of teaching problem solving as supplementary to math instruction, problem solving should really be taught as the way that we learn mathematics. And we need to get students to be thinkers of mathematics, not just doers of mathematics.

Teaching Word Problem Structures

As when teaching any strategy, teachers should use explicit, systematic instruction when introducing schema instruction, sometimes referred to as schema-based instruction (SBI). Although the same process is used to teach any schema, for illustrative purposes, the steps for how to teach the combine schema are outlined in the box below.

Descriptions

Example

Start with one problem type or schema (e.g., combine).

Start with stories that contain all the information (i.e., no unknown quantities).

Show students how to translate the information for each problem type into a diagram (visual representation) or equation.

Teach the students how to identify combine problems.

LaTisha has 5 comic books. Riley has 3 more comic books. They have 8 comic books altogether.

Descriptions

Example

Teach the students to use the following steps:

Read the word problem.

Identify the problem type.

Translate the information into a diagram or equation that corresponds with the problem type.

Solve the problem.

One method for doing this is to use the following mnemonic:

F- Find problem type

O- Organize using a visual diagram or equation

P- Plan to solve the problem

S- Solve problem

Calla has 4 cupcakes. Jaden has 6 cupcakes. How many cupcakes do they have altogether?
Identify problem type: combine Translate into equation:

Solve problem:

Descriptions

Example

Throughout the problem-solving process, the teacher should ask students to discuss how they solved the problem.

Teacher: "Jayla, explain how you knew this problem type was combine".

(Adapted from Stevens & Powell, 2016)

Teachers should make sure that students have mastered one schema (e.g., combine) before introducing a different problem type (e.g., compare). This reduces the possibility of students confusing one schema type with another during the learning process.

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High-Leverage Practices HLP14: Teach cognitive and metacognitive strategies to support learning and independence CCSSM: Standards for Mathematical Practice MP7: Look for and make use of structure. Another effective strategy for helping students improve their mathematics performance is related to solving word problems. More specifically, it involves teaching students how to identify word problem types based on a given problem's underlying structure, or schema.

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CAEP standards for the accreditation of educators are designed to improve the quality and effectiveness not only of new instructional practitioners but also the evidence-base used to assess those qualities in the classroom.

Standard 1: Content and Pedagogical Knowledge

CEC standards encompass a wide range of ethics, standards, and practices created to help guide those who have taken on the crucial role of educating students with disabilities.

Standard 5: Instructional Planning and Strategies

The DEC Recommended Practices are designed to help improve the learning outcomes of young children (birth through age five) who have or who are at-risk for developmental delays or disabilities. Please note that, because the IRIS Center has not yet developed resources aligned with DEC Topic 8: Transition, that topic is not currently listed on this page.
Instruction

INS6. Practitioners use systematic instructional strategies with fidelity to teach skills and to promote child engagement and learning.

InTASC Model Core Teaching Standards are designed to help teachers of all grade levels and content areas to prepare their students either for college or for employment following graduation.

Standard 2: Learning Differences

Standard 8: Instructional Strategies

NCATE standards are intended to serve as professional guidelines for educators. They also overview the "organizational structures, policies, and procedures" necessary to support them.

Standard 1: Candidate Knowledge, Skills, and Professional Dispositions

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CAEP standards for the accreditation of educators are designed to improve the quality and effectiveness not only of new instructional practitioners but also the evidence-base used to assess those qualities in the classroom. Standard 1: Content and Pedagogical Knowledge CEC standards encompass a wide range of ethics, standards, and practices created to help guide those who have taken on the crucial role of educating students with disabilities.

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22. Easy ways to make math facts practice fun

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22. Easy ways to make math facts practice fun

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When it's time for math facts practice, do you automatically reach for the flash cards? That's one classic way to learn, but it's not very exciting, and some kids just don't respond to it. That's why we're big fans of these new ways to get in some math facts practice. The games, activities, and crafts here are ideal for reluctant learners and lots of fun for everyone!

1. Put together egg halves

This is a quick hands-on way to practice math facts. For more excitement, try hiding the egg halves and letting kids hunt for them before they match them up!

Learn more: Homeschool Preschool

2. Roll and multiply

This is like a simpler version of Yahtzee, and it's a cool way to practice multiplication. If you use two dice instead of one, kids can practice their facts up to 12.

Learn more: What We Do All Day

3. Compete at Multiplication Squares

If you've ever played Dots and Boxes, this will look familiar. Players roll two dice (try these polyhedral dice to expand the facts in play), and draw a line to connect two dots next to the answer. If they complete a box, they color it in with their own marker.

Learn more: Games 4 Gains/Multiplication Squares

4. Get Four in a Row

This free printable is completely editable, so you can use it for any type of math facts practice. Kids choose a problem and give the answer. If they get it right, they cover it with their marker. When they get four in a row, they win!

Learn more: Fun Learning for Kids

5. Try "Sticky Math"

Don't confuse Sticky Math with timed tests. The goal is just for kids to complete as many problems as possible in a set amount of time, then work to beat that record each time.

Learn more: Lucky Little Learners

6. Face off in Dice War

Dice games are fantastic in the classroom! With this one, kids practice their addition facts and get a little work with subitizing too. The concept is so simple: Each player rolls the dice and adds up their numbers. The highest sum wins that round. Use this game for subtraction and multiplication too.

Learn more: Miss Giraffe's Class

7. Assemble math-facts grab bags

Fill a variety of bags with collections of small objects. Kids grab a handful from two different bags, then count and add up the results. Be sure they write it all down to get practice at setting up equations. (Also, try this with subtraction and multiplication facts.)

Learn more: Susan Jones Teaching

8. Play Shut the Box

This game has been played for hundreds of years, but it's a fun and sneaky way to practice addition facts fluency! The goal is to "close" each of the numbers in the box from one to nine by rolling the dice. For instance, if a player rolls 11, they may close 1, 2, 3, and 5, as these add up to 11. If no numbers are available to add up to the dice total, play passes to the next player and continues until someone finally "shuts the box" by closing the last available number. You can play this game as people have for centuries with a specially designed box. You don't need the box, though; simply have kids write out the numbers 1 through 9 and cross them out as they play.

9. Play Math Facts War

Each student flips two cards, then adds them (or subtracts, or multiplies). The person with the highest total keeps both cards. For a tiebreaker, flip another card! See more rules at the link.

Learn more: Math Facts War/Creative Family Fun

10. Turn an egg carton into a problem generator

Using an egg carton, have students write the numbers 1 through 12 in the bottom of each depression. Place two marbles inside the egg carton and close the lid. Shake the egg carton, open the top, and then add, subtract, or multiply whichever two numbers the marbles have landed on.

Learn more: The Unlikely Homeschool

11. Assemble a domino puzzle

Dominoes are perfect for math facts practice! Keep it simple by pulling a domino from a bag, then adding, subtracting, or multiplying the two numbers.

For even more fun, print the free puzzles at the link below. Then start filling in the puzzle one piece at a time by placing a domino that adds up to the number shown in each rectangle. The trick is that regular domino rules still apply, so each number must touch another domino with the same number on that end.

Learn more: Games 4 Gains/Domino Math

12. Circle math facts in a Number Search

These number search puzzles are harder than they look! First, kids complete the addition facts. Then, they search for those equations in the puzzle. Get three free puzzles at the link, where you can purchase more if you like them.

Learn more: The Sprinkle Topped Teacher

13. Use flash cards to play Fifteen in a Row

When it comes down to it, flash cards are still one of the best ways to practice fact fluency, but a game can at least make them more fun. The goal is to lay out 15 flash cards in a row by the total of their sums (or differences, products, or dividends), from smallest to largest. Learn how it's played at the link.

Learn more: The Measured Mom

14. Make a math-facts practice wheel

All it takes is paper plates, glue, and a marker to help your students learn their math facts. Up the fun factor by having students decorate their plates any way their imagination can dream up!

Learn more: Math Facts Wheel/Creative Family Fun

15. Whack a ball to subtract

You know your elementary math students are going to love this! Build your own whack-a-mole 10-frame with a shoebox and Ping-Pong balls. Then, have kids whack the balls to practice their subtraction facts. So fun!

Learn more: Planning Playtime

16. Get a jump on your math facts practice

Lay out a grid like the one shown that has the answers to whatever set of math flash cards you're currently working with. (This teacher used masking tape; you could also do sidewalk chalk on the playground.) Two players face off, one on each side of the board. Show the flash card, and kids race to be the first to jump to the correct square with both feet inside the lines. Get all the rules at the link below.

Learn more: Teaching and Tapas

17. Run a flash card race

Tape a series of flash cards to the floor and challenge kids to see who can correctly make their way from start to finish the fastest. They can call out the answers or write them down, but they have to get it right before they move on. Kids can race side by side or work independently to beat their own best time.

Learn more: There's Just One Mommy

18. Draw Waldorf math facts flowers

This is a creative way to teach math facts. Start by drawing the center of a flower and write any number from 1 to 9 in the middle. Next, draw 12 petals around the center, labeling them 1 through 12. Last, draw another 12 petals and write the sum or product of the center number and the petal adjacent to the new petal.

Learn more: Multicultural Motherhood

19. Catch a math beach ball

Beach balls are so much fun in the classroom. Scribble numbers all over one with a Sharpie, then toss it to a student. Wherever their thumbs land, they add (or subtract, or multiply) those two numbers together before tossing the ball to the next student.

Learn more: There's Just One Mommy

20. Practice facts by stacking cups

We're not sure why, but kids simply love stacking cups. Label yours with math problems and answers, then have kids build pyramids and towers galore!

Learn more: The Kindergarten Smorgasboard

21. Design an outdoor board game

Draw a winding path and fill the spaces with math equations. Kids roll the dice and move from space to space (have them jump, skip, or twirl to mix things up). If they get the answer right, they move to the new space. If not, their turn is over. Customizable math games like this can be used at any level.

Learn more: Look! We're Learning

22. Compete at math bingo

Math facts bingo is so easy to set up and play! Give kids empty grids and ask them to write various sums, differences, products, or quotients, depending on what you're working on. Then call out math problems and have them cover the answers. First to fill in a row wins!

Learn more: Happy Go Lucky

23. Play math facts checkers

Label a checkerboard with math facts. Play checkers as usual, following the traditional rules. The twist is, you must solve the math problem you land on!

Learn more: Teach Beside Me

24. Change your students' names (temporarily)

This is so clever! Grab some name tags and write math facts on each. Give a tag to each of your students. For the remainder of the day, everyone will refer to one another by the answer to the equation on their tag (e.g., the student with the name tag that says 7×6 would be referred to as "42").

Learn more: Mr. Elementary Math

25. Match up math facts

Play Memory (also called Concentration) with math facts. Get free printable cards at the link for addition facts to get you started.

Learn more: Playdough to Plato

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25 Easy Ways to Make Math Facts Practice Fun and Effective

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When it's time for math facts practice, do you automatically reach for the flash cards? That's one classic way to learn, but it's not very exciting, and some kids just don't respond to it. That's why we're big fans of these new ways to get in some math facts practice.

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6. Over 1,500 video lessons covering Middle Grades Math through Algebra 2

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6. Over 1,500 video lessons covering Middle Grades Math through Algebra 2

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Virtual Nerd: Real math help for school and home

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Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring.

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5. Number skills - Teach Your Monster (ages 4- 6+)

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5. Number skills - Teach Your Monster (ages 4- 6+)

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About the game

Designed in collaboration with experts in early years mathematics.

Fun-filled games offer exciting new ways to practice numbers.

Aligned with Pre-K/Reception curriculum and enjoyable for ages 4-6+.

80 levels provide structured practice in addition, subtraction, counting, number bonds to 10 and more.

Watch the Teach Your Monster Number Skills trailer

At home and school

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Design your own customized monster and take it on a trip to Number Park, a fun-filled park all about numbers and mathematics.

Make number practice fun through a wide variety of games and activities, designed with early years mathematics experts to improve number skills.

Build a solid foundation in key number concepts such as number bonds to 10 and subitizing using frames and other familiar manipulatives.

Continued play leads to increased difficulty and more fun challenges.

Play the game now!

Try the game

Reviews

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Teach Your Monster Number Skills: Math games for 3 - 7 year olds

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Teach Your Monster Number Skills - The magical way to practice numbers!

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21. Math Practice: Building Math Fluency through Games

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21. Math Practice: Building Math Fluency through Games

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How do we actually build practice for our students in a way that they will enjoy? Number one my first thing is it needs to be through games. Any kind of game that you can help kids be able to learn mathematics through is so powerful. I grew up in a game-playing family and I know the power of it for me personally and I've seen it for all the kids that I work with and my own personal children as well.

Number one, games make it fun. It's way more fun to play a game than to fill out a worksheet, so games make it fun. They also provide tons of practice in a short timeframe. If you've ever done games in your classroom, kids are having so much fun playing and when it's time to stop, they'll moan and they're like, "No, we don't want to stop," and half the time they don't even realize they were doing math. If that's what you use for some of your math time they're like, "When are we going to start doing math time?" They don't even realize they're doing it. So to me, games are the number one way to help kids practice because it's fun and it gives kids tons of practice in a short timeframe.

Now here's one word of caution. Typically in a classroom we tend to do the same game with all of the kids, like we'll teach the game, now everybody goes and plays that game, and then we come back at the end of it. Now there's a time and a place for that, especially if you're going to be discussing that game. One of the hard parts though is I see often is that you play the game, let the kids play the game, but then there's no discussion afterwards.

There either needs to be discussion during while they're playing those games, walking around and having those discussions with kids, or as a whole group afterwards talking about what they noticed, what relationships they were building. What did they see happening? What were some stumbling blocks in the game? All of that kinda stuff, but oftentimes the discussion doesn't happen.

So the only time I would suggest using a whole class game where everybody's playing the same game, even if they're off in pairs, if everybody's playing the same game, the power of that is to have a discussion afterwards. Now if you're not going to be doing that, then I would also recommend that you switch it up and allow kids choice in their games. That's one of the powerful parts of games is that you can allow kids to work on a game that is building a mathematical concept they need to work on because not everybody in your classroom needs to be practicing the same concept.

Some kids might need harder numbers, so you might add an extra dice or two into their game. They might be doing a different concept even. If you're 3rd or 4th grade, you might have some kids who still need to be working on addition and subtraction, while you have some kids who are moving into multiplication and division stuff. Allowing some choice during that practice time with games is super important.

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Math Practice: Building Math Fluency through Games

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Kids still need math practice but the type of math practice that we provide for kiddos needs to change. In this video we investigate Math Practice: Building Math Fluency through Games in our quest to Build our own Math Minds so we can Build the Math Minds of our kiddos.

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Educational Technology and Mobile Learning

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Cool Math Games- Free Online Math Games for Kids and Students

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Cool Math Games is a website that offers tons of free online math games to help kids and students sharpen their math skills. Read to learn more.

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Educational Technology and Mobile Learning

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Math Playground is an educational website that offers a wide variety of math games, videos and interactive activities for students in kindergarten to grade six. The games are organized by subject and grade level and some of them are aligned to Common Core Standards. Math Playground games cover different topics including fraction games, division games, multiplication games, ratio games, pre-algebra games, geometry games, and more.

Math Playground, the premium version, enables teachers to create classes and assign games to students. Using their credentials, students can sign in to class and access the assigned games. Unfortunately, Math Playground does not offer performance tracking tools and progress reports. The games are ideal for mere math practice.

Also, the free version of the site comes loaded with ads and often times the ads are hard to decipher from games hence the importance of teachers and parents selecting and monitoring the whole game experience of their students and kids.

How to use Math Playground games in your class?

As a teacher, you can use Math Playground games to extend the learning that takes place in class and provide students with assignments to practice math concepts they learned. You can assign math games as homework for extra practice. A number of games in Math Playground are multiplayer games that students can play in class with other another, fun competitive way to engage student in math learning.

Besides math, Math Playground also features games that can help students develop skills in areas related to grammar and language learning, typing and keyboarding, spelling, geography, and more. The site also provides teachers with pre-made resources to use in class including printable practice worksheets, a modeling tool, and more.

Math games in Math Playground

Math games in Math Playground are organized by grade level (kindergarten to 6th grade) and by subject (e.g., addition games, multiplication and division games, Fractions games, Shapes and Graphs, Pre-algebra games, Money and Time games, and more). Each subject area contains several math games that help kids and students practice and learn a particular set of skills.

For instance, the Fraction games section offers several games that focus on teaching kids the concept of fractions through a number of learning games that include Bridge Builders, Brain Builders, Compare Fractions, Galaxy Pals, Add Fractions, Ordering Fractions, Multiply Fractions, and more.

Besides the general category of Math games, Math Playground games are organized into the following categories:

1. Math Arcade

Math Arcade section features a wide range of multiplayer games organized into the following sub-categories: addition and subtraction games, multiplication and division games, fraction and decimal games, Pre-algebra games, shapes and clock games, spelling games, grammar games, keyboarding games, and geography games. Kids select the category they want, browse the games it contains and click on the one they like to play.

2. Story Math

Story Math features math word problems. Using their math skills, students try to model and solve word problems. Each math problem comes with a short description containing the targeted skills and common core connections. Some problems are also accompanied with short video introduction explaining instructions to follow. Students are required to use thinking blocks to model and solve math word problems. There are Thinking Blocks Addition, Thinking Blocks Multiplication, Thinking Blocks Fractions, Thinking Blocks Ratios and more.

3. Math Videos

Math playground offers a collection of educational videos explaining math concepts in fun and engaging ways. Each video is around five minutes long and is illustrated with awesome graphics, text fonts, characters, images, and sound. Math videos are arranged into different categories including Numbers and Operations, Geometry and Measurement, Pre-algebra and Algebra. Browse through the collections and click on the video you are interested in to watch it. You can also share the video to your Google Classroom using the share button at the end of the video.

4. Logic Games

Logic games, as their name indicates, are games and puzzles that help kids and students develop logic and thinking skills including problem solving, cause and effect, sequential reasoning, analytical thinking, trial and error, and more. Examples of featured logic games include Mazes and Keys, Drop the Number, Find the Pug, Monkey Bubbles, Find the Differences, Zero Squares, Code Panda, Drop It, Logic Steps, Cross the Bridge, Robot Islands, Lightybulb, Sorting Spheres, Fox Adventurer, Brain Workouts, Reverse the Discs, and many more.

How much does Math Playground cost?

Math Playground offers both free and premium plans. The free plan includes tons of ads and offers limited access to games. The premium plan offers way more features including the ability to create classes, assign games to students, access the full library of learning games, save favourite games, provide home access for students, remove ads, and more. There are two types of premium plans: a Classroom plan which costs $9.99/month and a Family plan which costs $5.99/month.

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Math Playground - Free Math Games and Interactive Activities for K-6 Students

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Math Playground is an educational website that offers a wide variety of math games, videos and interactive activities for students in kindergarten to grade six. The games are organized by subject and grade level and some of them are aligned to Common Core Standards.

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When children have learning problems, their parents are usually the first to notice that something is just not right. Teachers often notice a child having trouble in their class. And when they notice, they want reliable information so they can help their students.

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Copyright © 2014–2022 Understood for All Inc.

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20. Fact fluency: An evidence-based math strategy

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20. Fact fluency: An evidence-based math strategy

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By Brendan R. Hodnett, MAT

Fact fluency is a foundational math skill that means mastering sums between 0-20 so students can produce the answers quickly and accurately. For example, with addition fact fluency, students can recall 1 + 4 = 5 without hesitation.

But building fact fluency isn't just about flashcards and drills. Students need to understand what they're doing, not just memorize it. And they need to practice computing to get more efficient at it. That's why building fact fluency is an important step toward thinking about math more critically.

Games like addition bingo can be a fun way for students to practice math facts, both in school and at home. With addition bingo, students roll two dice to practice adding sums up to 12. They add the numbers on the dice together and look for the sum on their bingo boards.

Students who are ready to play bingo with sums up to 20 can draw from two piles of index cards numbered 0-10 and add those numbers together.

Scroll down for tips on adapting this strategy for distance learning.

Download: Printable addition bingo boards

Read: How to use this fact fluency strategy

1 (Common Core 1.OA.C.5 Relate counting to addition and subtraction, e.g., by counting on 2 to add 2.)

1 (Common Core 1.OA.C.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.)

2 (Common Core 2.OA.B.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.)

Show students how to choose numbers from 1-12 to fill their bingo boards in random order. Make sure students know to use each number only once on the 3×3 board and that three numbers will be leftover. It helps if students write the numbers on the top of the board and cross them out as they use each one. Demonstrate with a document camera or a large version of the board.

Teaching tip: Print the bingo boards on card stock and laminate them so they can be reused. Have students use whiteboard markers to write the numbers.

1. Explain the directions. Let students know that today they'll practice adding numbers by playing bingo. Say, "Each group has a set of dice and some chips to mark off their sums. Take turns rolling the two dice and adding the two numbers. Everyone in the group should check the roller's work by counting the dots on the dice. If the sum of the two dice is on your board, you place a chip on it. The first student in the group to complete a row, column, or diagonal line says "Bingo!"

Now ask the same volunteer to roll the two dice. Let the student find the sum while you model checking the answer. This time say, "If you're having trouble thinking of the answer, use the counting on strategy to prove the sum." Walk through an example by identifying the number that is greater, then counting on the additional dots from the second die to get the sum.

3. Practice. Set a timer for 10-15 minutes. Let students play against their partner or group members as you move around the room. Observe students practicing their math facts. For those making errors, prompt them to use the counting on strategy. Most groups will play a few rounds before the time is up.

Teaching tip: Make a simple chart to keep track of student progress as you observe. For example, list students' names and create columns for if they know their facts within 5, 10, and 20. Use a column for notes about the support students might need.

For students having trouble recalling their addition facts for sums 2-12, give them oversized dice to make it easier for them to count the dots. You can also give them large print number lines to help them with counting on.

For English language learners and other students who benefit from visual supports, provide an anchor chart with the number names, the written numerals, and a visual representation of the quantities so they can connect all three together as they play.

For students ready to extend their practice, use the 5×5 bingo board and have students fill it with numbers from 0-20. Make sure students know they'll need to use three numbers twice. Create sets of index cards numbered 0-10. Each pair or small group will get two sets of 11 cards, one for each addend. As the students take turns, they select one card from each pile and calculate the sum. After the other students confirm it's correct, they each search their boards for the number. (If available, you could use two 10-sided dice instead of index cards.)

Understand: Why this strategy works

Fact fluency makes all other math problems easier - especially those with multiple steps - because students need to rely less on their working memory. When students can recall facts and do quick computation, they can focus on critical information and procedures. Otherwise, they get slowed down by basic calculations.

Many students, particularly students who learn and think differently, have trouble remembering math facts. That's why the right kind of repeated practice is so important.

Research shows that as students become more accurate with math facts and strengthen their computational fluency, they improve their ability to problem-solve in math. Playing games is an engaging way for students to build this fluency. Games give students practice with recalling math facts and thinking flexibly to solve problems.

With addition bingo, students roll the dice to create their own math problems and easily check their work. This gives teachers time to observe, measure progress, and work with students who need more support.

Playing addition bingo at home will help students practice their addition facts. Explain to families why it's so important to strengthen their child's fact fluency. Send home a sample bingo board (digital or paper) with the directions. See below for suggestions for bingo chips.

Adapt: Use for distance learning

Partner with your students' families. Find out what resources they have available and what they might need to support learning at home.

Guide small groups of students through playing bingo in a synchronous online class. Use a video conferencing tool so you can share your screen to roll virtual dice for each student's turn. Or record a video of how to play the game for asynchronous learning and suggest students play against someone at home.

Have families print the bingo board or ask students to draw their own by copying the printable. Give ideas for bingo chips that students might be able to find at home, like small game pieces, cotton balls, or coins.

If students don't have dice at home, they can use a digital option or write numbers on index cards or paper and draw two each time.

For students playing with sums up to 20, have them make two sets of paper cards for numbers 0-10. Or, if they have playing cards at home, they can use the numerals-only cards to make two piles (and leave out "0" from these practice games).

Research behind this strategy

Tell us what interests you

About the author

About the author

Brendan R. Hodnett, MAT is a special education teacher in Middletown, New Jersey, and an adjunct professor at Hunter College.

Reviewed by

Reviewed by

Sarah R. Powell, PhD is an associate professor in the department of special education at the University of Texas at Austin. She conducts research on evidence-based practices for teaching students experiencing math difficulty.

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How to Build Fact Fluency With Addition Bingo

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Fact fluency is a foundational math skill that means mastering sums between 0-20 so students can produce the answers quickly and accurately. For example, with addition fact fluency, students can recall 1 + 4 = 5 without hesitation. But building fact fluency isn't just about flashcards and drills.

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7. Seven Math Tools for Grade-Schoolers You Can Find at Home

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7. Seven Math Tools for Grade-Schoolers You Can Find at Home

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Looking for inexpensive math tools for your grade-schooler? Here are seven math learning aids you may already have around the house-and how you can use them to help younger kids work on important math skills. Working with coins can help your child practice giving the correct value to each coin.

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By Ginny Osewalt on Jan 15, 2015

In the problem 5x + 4 = 9, what exactly does the "equal" sign mean? Before you answer, let me tell you a story.

Years ago, my oldest son was struggling with a simple algebra problem, and I was trying to help him. (I was not yet a teacher at the time.) He didn't know where to even start approaching the problem. Then suddenly, I had an epiphany of how to explain it to him.

"When solving a simple algebraic equation," I told my son, "think of the equation as a balance. Envision a scale. If you do something to one side of the equation, you need to do the same thing to the other side. The equal sign (and here's the epiphany) tells you that one side must be the same value as the other side."

Slowly, a look of understanding came over his face. He subtracted 4 from both sides. Then, he divided both sides by 5. From 5x + 4 = 9, he solved for x = 1.

It was great to see him succeed. But with my epiphany, I also started to understand why I had such anxiety and underachievement as a math student in school. In fact, I realized that I had grown up with a dysfunctional relationship with the equal sign.

As a student, I incorrectly thought the equal sign meant: "here comes the answer." What I lacked was a true conceptual understanding of mathematical equivalence. In reality, the equal sign refers to equality or "the same value as." Just as four quarters are the same value as one dollar, 5 + 4 is the same value as 9. And 5x + 4 = 9 is an equation to be balanced.

A faulty interpretation of a symbol as basic and common as the equal sign can unsettle a child for many years. So can other misconceptions, like the belief that multiplying always increases a number. (This interpretation does not work well with fractions, like 1/4 x 1/4 = 1/16.) Or that when you "reduce" a fraction you are decreasing its size. (Replace "reduce" with "simplify," and 2/4 = 1/2 makes a lot more sense.)

When these kinds of early misconceptions are not addressed, they can expand, weaken confidence and create what is commonly known as math anxiety.

Procedural Knowledge vs. Conceptual Knowledge As a teacher, I've listened to parents' concerns and frustrations about how their children are learning math these days. They want to help their struggling children but are "baffled" by the material. They talk about their own math instruction and how they fluently and efficiently solved math problems using "procedural knowledge"-a set of steps, actions or procedures.

What I try to explain is that these days, we teach for conceptual knowledge. Our students don't just memorize times tables. They're expected to understand and explain their mathematical reasoning.

We want kids to demonstrate deep understanding through multiple representations, which include drawings, area models and word problems. Procedures and shortcuts are also taught, but not until the student can explain why those shortcuts work. We like our students to get the right answer, but we need them to know why it's the right answer.

The reason for teaching conceptual understanding is to help students to see connections between the math they're learning and the math they already know. This can be especially empowering for kids with learning and thinking differences. It prepares them to solve the real-world problems they will face in the future.

Yes, math is being taught differently today. It may be a little more difficult for parents at times, but it definitely can be better for kids.

Any opinions, views, information and other content contained in blogs on Understood.org are the sole responsibility of the writer of the blog, and do not necessarily reflect the views, values, opinions or beliefs of, and are not endorsed by, Understood.

Tell us what interests you

About the author

About the author

Ginny Osewalt is a dually certified elementary and special education teacher with more than 15 years of experience in general education, inclusion, resource room, and self-contained settings.

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Why Math Is Taught Differently Today | Expert Corner

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In the problem 5x + 4 = 9, what exactly does the "equal" sign mean? Before you answer, let me tell you a story. Years ago, my oldest son was struggling with a simple algebra problem, and I was trying to help him. (I was not yet a teacher at the time.)

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5. Trouble with math in kids: Your questions answered

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5. Trouble with math in kids: Your questions answered

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By The Understood Team

Math can be a hard subject for lots of kids. If your child has trouble working with numbers, you might wonder what it means and how to help. Here are our experts' answers to common questions families have about trouble with math.

Tell us what interests you

About the author

About the author

The Understood Team is made up of passionate writers and editors. Many of them have kids who learn and think differently.

Reviewed by

Reviewed by

Donna Volpitta, EdD is the founder of Pathways to Empower. Her work draws on the latest research in neuroscience, psychology, and education.

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Math Challenges for Kids: Questions Answered

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Math can be a hard subject for lots of kids. If your child has trouble working with numbers, you might wonder what it means and how to help. Here are our experts' answers to common questions families have about trouble with math. Tell us what interests you The Understood Team is made up of passionate writers and editors.

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15. Math anxiety vs. dyscalculia in kids: Comparing the signs

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15. Math anxiety vs. dyscalculia in kids: Comparing the signs

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By The Understood Team

At a glance

Dyscalculia and math anxiety are different.

Both can affect how kids do in math, and they can show up in similar ways.

Understanding the difference can help you and your child find solutions.

Dyscalculia is a learning difference that causes trouble with math. But dyscalculia isn't the same as math anxiety. Math anxiety can make kids question their abilities in math, even if they have strong skills.

It can be easy to think of dyscalculia and math anxiety as the same thing, especially because the signs can look similar. Think of it this way:

Doing math with dyscalculia is like hiking with a sprained ankle. Doing math with math anxiety is like being a physically able hiker who worries about what might happen if they try to climb the peak - self-doubt gets in the way of success.

Even though dyscalculia and math anxiety are different, the signs can overlap. And it's possible for a child to have both. This chart can help you understand what you're seeing in your child.

Signs of math anxiety

Signs of dyscalculia

Kids worry they'll do poorly on a math test, even though they understand the material and have studied.

Kids expect to do poorly on a math test because they don't understand the material, even after studying.

Kids do poorly on math tests, even after preparing for them, because anxiety gets in the way.

Kids do poorly on math tests, even after preparing for them, because they don't understand the material.

Kids can get through homework fairly easily and answer most problems correctly. But they feel anxious about doing it.

Kids spend a long time doing homework and get many of the answers wrong.

They may even make mistakes because they're so anxious: They may focus too much on some details, or have trouble focusing on others.

Kids try to avoid going to math class when there's a quiz or test.

Kids try to avoid going to math class, especially when there's a quiz or test, because they're sure they'll fail.

Kids get good grades on math homework and classwork, but not on tests.

Kids get poor grades on math homework, classwork, and tests.

Watch as an expert explains more about the difference between dyscalculia and math anxiety.

Knowing what's behind your child's difficulty with math lets you respond in the best way. Find out what to do if you're concerned your child has dyscalculia. If you think your child has anxiety, use an anxiety log to keep track of what you're seeing. If you have concerns, don't hesitate to reach out to your child's doctor.

Key takeaways

Dyscalculia is a learning difference that affects math skills such as counting, recalling math facts, and understanding math concepts.

Math anxiety is an emotional issue involving self-doubt and fear of failing.

Both can create test anxiety and lead kids to try to avoid going to math classes.

Tell us what interests you

About the author

About the author

The Understood Team is made up of passionate writers and editors. Many of them have kids who learn and think differently.

Reviewed by

Reviewed by

Jerome Schultz, PhD is a clinical neuropsychologist and lecturer in the Harvard Medical School Department of Child Psychiatry.

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Dyscalculia and math anxiety are different. Both can affect how kids do in math, and they can show up in similar ways. Understanding the difference can help you and your child find solutions. Dyscalculia is a learning difference that causes trouble with math. But dyscalculia isn't the same as math anxiety.

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By Amanda Morin

Kids start learning math the moment they start exploring the world. Each skill - from identifying shapes to counting to finding patterns - builds on what they already know.

There are certain math milestones most kids hit at roughly the same age. But keep in mind that kids develop math skills at different rates. If kids don't yet have all the skills listed for their age group, that's OK.

Here's how math skills typically develop as kids get older.

Babies (ages 0-12 months)

Begin to predict the sequence of events (like running water means bath time)

Start to understand basic cause and effect (shaking a rattle makes noise)

Begin to classify things in simple ways (some toys make noise and others don't)

Start to understand relative size (baby is small, parents are big)

Begin to understand words that describe quantities (more, bigger, enough)

Toddlers (ages 1-2 years)

Understand that numbers mean "how many" (using fingers to show how many years old they are)

Begin reciting numbers, but may skip some of them

Understand words that compare or measure things (under, behind, faster)

Match basic shapes (triangle to triangle, circle to circle)

Explore measurement by filling and emptying containers

Start seeing patterns in daily routines and in things like floor tiles

Preschoolers (ages 3-4 years)

Kindergartners (age 5 years)

Move from using hands-on methods to using paper and pencil to work out math problems

Work with money

Do addition and subtraction with regrouping (also known as borrowing)

Understand place value well enough to solve problems with decimal points

Know how to do multiplication and division, with help from fact families (collections of related math facts, like 3 × 4 = 12 and 4 × 3 = 12)

Create a number sentence or equation from a word problem

Begin basic algebra with one unknown number (like 2 + x = 10)

Use coordinates to locate points on a grid, also known as graphing ordered pairs

Work with fractions, percentages, and proportions

Work with lines, angles, types of triangles, and other basic geometric shapes

Use formulas to solve complicated problems and to find the area, perimeter, and volume of shapes

Understand that numbers can be represented in many ways (fractions, decimals, bases, and variables)

Use numbers in real-life situations (like calculating a sale price or comparing student loans)

Begin to see how math ideas build on one another

Begin to understand that some math problems don't have real-world solutions

Use mathematical language to convey thoughts and solutions

Use graphs, maps, or other representations to learn and convey information

Remember that kids develop at different paces. Some may gain some math skills later than other kids or have some that are advanced for their age.

If you're concerned about progress in math, find out why some kids have trouble with math, and next steps to take.

Tell us what interests you

About the author

About the author

Amanda Morin is the author of "The Everything Parent's Guide to Special Education" and the former director of thought leadership at Understood. As an expert and writer, she helped build Understood from its earliest days.

Reviewed by

Reviewed by

Brendan R. Hodnett, MAT is a special education teacher in Middletown, New Jersey, and an adjunct professor at Hunter College.

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Kids start learning math the moment they start exploring the world. Each skill - from identifying shapes to counting to finding patterns - builds on what they already know. There are certain math milestones most kids hit at roughly the same age. But keep in mind that kids develop math skills at different rates.

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19.. Place value with straw bundles: An evidence-based math strategy

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19.. Place value with straw bundles: An evidence-based math strategy

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By Kim Greene, MA

Place value is an important math concept for early elementary students to understand. They have to learn that the value of a digit depends on its place in a number. For example, students should understand that in the number 142, the digit 1 has a value of 1 hundred. The digit 4 has a value of 4 tens, and the digit 2 has a value of 2 ones. When students struggle to understand place value, it can hold them back in understanding more advanced math concepts.

One way to help students understand place value is to use manipulatives to show the value of each digit. You can do this using straws (or wooden craft sticks) and a place value mat. With this strategy, you use the manipulatives to explicitly teach students how to compose a number. Using the straws, they will create a multi-digit number by showing the value of each digit separately. This helps students make the connection between numbers and place values.

Scroll down for tips on adapting this strategy for distance learning.

Watch: Using bundles to teach place value

Watch this video to learn how to teach students to bundle straws or wooden craft sticks.

Download: Printable place value mat

Print this place value mat or teach students to draw their own.

Place Value Mat - Hundreds PDF

Read: How to use this place value strategy

1 (Common Core 1.NBT.B.2: Understand that the two digits of a two-digit number represent amounts of tens and ones)

2 (Common Core 2.NBT.A.1: Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones)

1-2 (Common Core Math Practice MP2: Reason abstractly and quantitatively; Common Core Math Practice MP5: Use appropriate tools strategically)

Whole class

Small groups

Individuals

1. Practice making bundles. Explain to students that they'll be using straws to help understand place value. They will count individual straws as "ones" and bundle straws together in groups of 10 as "tens." (In second grade, they'll also bundle together 10 groups of tens to make "hundreds.") Explicitly review the academic vocabulary needed for the lesson like tens, ones, place,and value.

For English language learners (ELLs): Identify words that have more than one meaning, like place. Using pictures or other supports, share the definitions of the word. For example, place could mean a physical spot, like a park. Or if a friend says "save me a place at lunch," they are asking for you to have a spot for them at the table. Make the connection that the place in place value is a spot in the number.

Model how to count 10 straws. Count them out loud and then secure them together with a rubber band to make 1 group of ten. Show that 10 ones equal 1 ten.

Then have students do the same. Prompt them to count out 10 straws and then bundle the 10 straws together to make 1 ten. Then, repeat with another 10 ones to make a second bundle of tens. If students have straws left over, have them count those straws as "ones" that can't be bundled. (For second grade, continue until students can count the bundles of tens to make 1 hundred.)

You can also offer the opportunity to all students to watch a student-friendly video about bundling.

2. Compose the number using concrete manipulatives. Tell students they will use the straws and a place value mat to compose (or "make") a number. Follow the concrete-representational-abstract (CRA) sequence of instruction, starting with the concrete. Begin by looking at the place value mat and explaining each column: hundreds, tens, and ones. Model how to use the straws on the place value mat. For instance, you can make the number 24 for first graders by putting 2 bundles of 10 in the tens column and 4 ones in the ones column. For second graders, you can compose the number 142 by stacking 10 bundles of 10 in the hundreds column and placing 4 bundles of 10 in the tens column and 2 ones in the ones column.

For ELLs: Provide additional support by using sentence frames. For example, for the number 24 you can write on the board and have students fill in the blanks: "The number is 24. There are two tens in 24. There are four ones in 24." This visual support is especially important for numbers that sound alike, such as 17 and 70.

Allow students to practice doing the same with several numbers. Give them feedback as they work. Ask students to share their thinking with their classmates after composing each number. For example, you might have students turn and talk to share what they placed in each column for the number 24.

Then, have students draw bundles in the appropriate columns on their own place value mats to make a number. For example, first graders can compose the number 53 by drawing 5 bundles of 10 in the tens column and 3 ones in the ones column. In second grade, students can compose the number 253 using similar drawings. Again, provide plenty of opportunities for practice and feedback. Invite students to explain their drawings and say the standard number.

Teaching tip: To reuse the place value mats throughout the lesson, put the mats inside dry-erase pockets and have students use dry-erase markers that can be wiped clean before moving on to the next example.

4. End with the abstract. After mastering the representational level, move on to the abstract level. On a place value mat, have students compose a number using only written numbers - like 6 tens and 7 ones make "67" or, for second grade, 4 hundreds, 3 tens, and 6 ones make "436." Provide sentence frames on the board with fill-in-the-blanks for visual support: " six tens and seven ones makes 67." To encourage flexibility, you can try moving places out of order. For instance, explain that a number has 7 ones and 3 tens and have students show 37.

If students struggle at the abstract level, allow them to go back to using bundles and then pictures. Emphasize the connection between the bundled objects, the representational pictures, and the numerical symbols. Move on to the abstract level once students demonstrate understanding.

Teaching tip: You can continue to practice place value by finding real-world numbers all around your students. During lessons in other subjects, point to numbers you see, like numbered pages in a picture book or a street address on an envelope. Challenge students to make the numbers by place value. You can also challenge students to find two- or three-digit numbers in their own lives and bring those examples back to class for practice.

Understand: Why this strategy works

Some students struggle to make a connection between our base 10 number system and the language we use for numbers. That's because the language we use for numbers doesn't directly translate. For example, we write "216," not "200 10 6."

Composing numbers using this strategy will help students make the connection between the number system and language. It shows them that a number is made up of the sum of its parts.

Also, students benefit from the use of multiple models, including straw bundles, place value disks, and drawings. Students also learn from support and feedback as they move from concrete to abstract representations of a number.

Improving students' understanding of place value lays the foundation for future math work, including addition with regrouping in second grade, and multiplication, fractions, and decimals in upper elementary grades.

Share with families that students are learning about place value in school. It may be different from how family members learned math when they were younger. That's because math is taught differently today.

You might also share the following video with families to help them understand how their children are learning to compose and decompose numbers.

Adapt: Use for distance learning

Partner with your students' families. Find out what resources they have available and what they might need to support learning at home.

Guide students through the steps in a synchronous online class or record a video for asynchronous learning. Either way, use UDL as you're planning the lesson.

Tell students to print the place value mat or draw their own.

Students can save paper by using a pencil and erasing their work to use the mat over and over. Or they can put the mat inside a Ziploc bag and use a marker that can be wiped off (if they have these supplies).

Give ideas for manipulatives that students might be able to find at home, like crayons, toothpicks, or small game pieces bundled in snack bags.

Research behind this strategy

About the author

Reviewed by

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How to Teach Place Value With Straw Bundles

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Place value is an important math concept for early elementary students to understand. They have to learn that the value of a digit depends on its place in a number. For example, students should understand that in the number 142, the digit 1 has a value of 1 hundred.

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Daily Numeracy Tasks from @DonnaMorgan8 on Twitter

Click the blue links to get information about or copies of the games.

Tiny Polka Dot

Math for Love is an excellent resource website, and I highly recommend their two games, Tiny Polka Dot and Prime Climb. They have also done an amazing thing, which is to release an open, free version of Tiny Polka Dots that parents can print at home, along with many ideas for games to play with the cards. I highly recommend this resource for math learners from 4-8 (and some of the games are good for fluency practice for even older students). You can find the free resource here. or go to their website if you want to order the boxed set.

Earth Day Graphing

Jill Pelto is an amazing scientist/artist who fashions beautiful if disconcerting images from climate data. You can find a great lesson on this from NASA, or just download the climate data and graph paper here. Challenge students to graph the data and then make an image that relates to climate change from the graph. More advanced
students can go to the NOAA and choose the date range and which global data they wish to map. Younger students might be interested in making simple picture graphs of the types of garbage in their homes or that they see on a walk.

Shapes Walk

G o for a shapes walk. How many shapes can you see? Keep a notebook. Classify them, for older students, measure angles, for even older students, calculate the height of trees using trig! Check out The Walking Curriculum for this and more ideas.

Got Some Bored Students?

Try these math challenges from the CEMC.

Have fun and develop math skills at home. Here are some weekly plans with questions and fluency activities:
Helping Your Child Learn Skill Building Math Games 10 Playful Math Activities Ultimate Tic Tac Toe Fluency Without Fear
Here are some ideas for supporting your students in their math learning, and developing good connections and communication skills:
Brochure for Parents and Students
Information for Parents about the Graduation Numeracy Assessment

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Click the blue links to get information about or copies of the games. Math for Love is an excellent resource website, and I highly recommend their two games, Tiny Polka Dot and Prime Climb.

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Check out our last Numeracy Newsletter for the 2021-2022 school year!

This is a great rich task to introduce the idea of patterning and algebra. This comes from Jo Boaler's YouCubed site. She suggests to lead this activity as follows:

1. Ask "How do you see the shapes growing?" (Looking at shape and number)

2. Record variety of responses on chart paper. Then ask students to use their method to determine the number in the 100th case. What would it look like? How do the numbers relate to the visual solution?

3. Ask students to make a table of the results. Ask students to
use both the number pattern they see in their table and their visuals to
predict how many squares there would be in the 4th and 5th case. Even
though they have already thought about the hundredth case adding
more rows to their table can help them see the functional growth in both
the numbers and the shape. Keep encouraging students to connect the
numbers and visuals. Ask them where they see the extra squares in their
pictures and in the numbers in the table they have made.

4. Could you tell me how many squares there would be in any case, if I just told you the case number? Eg if I told you it was case 500 how many squares would there be? Write a sentence to describe how many squares there would be with any case number. This can lead to asking students to use a variable instead of the words.

Access this task and the handouts to go with it on the YouCubed site here.

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Check out our last Numeracy Newsletter for the 2021-2022 school year! This is a great rich task to introduce the idea of patterning and algebra. This comes from Jo Boaler's YouCubed site. She suggests to lead this activity as follows: Ask "How do you see the shapes growing?"

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1. Intensive Intervention in Mathematics Course Content (free and excellent)

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1. Intensive Intervention in Mathematics Course Content (free and excellent)

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Intensive instruction was recently identified as a high-leverage practice in special education, and DBI is a research based approach to delivering intensive instruction across content areas (NCII, 2013). This course provides learners with an opportunity to extend their understanding of intensive instruction through in-depth exposure to DBI in mathematics, complete with exemplars from actual classroom teachers.

NCII, through a collaboration with the University of Connecticut and the National Center on Leadership in Intensive Intervention and with support from the CEEDAR Center, developed course content focused on enhancing educators' skills in intensive mathematics intervention. The course includes eight modules that can support faculty and professional development providers with instructing pre-service and in-service educators who are learning to implement intensive mathematics intervention through data-based individualization (DBI). The content in this course complements concepts covered in the Features of Explicit Instruction Course and so we suggest that users complete both courses.

Modules 1-8: Intensive Intervention in Mathematics

There are 8 modules within this course. Each course has an introduction, three parts, and a conclusion.

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Intensive Mathematics Intervention: Training Modules | NCII

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Explore eight in-depth training modules to help enhance educator skills on implementing intensive mathematics intervention through data-based individualization.

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Pirate Math Equation Quest is an intensive word-problem intervention for elementary children.

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Misconceptions Conceptual Procedural — The Science of Math

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Citation: Advocates for the Science of Math (2021). Common misconceptions: Conceptual and procedural understanding. Authors.

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The CUBES Math Strategy is a great tool for students to break down and solve those tricky word problems. Includes a free CUBES math strategy template!

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Universal Screening in Mathematics - The Science of Math

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Academic screening is necessary to meet the needs of all learners and is commonly used in Multi-Tiered Systems of Support (MTSS). It generally occurs 2-3 times per year, using brief paper and pencil or computer-based assessments delivered in the classroom by the teacher.

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We provide resources related to effective math instruction. Our goal is to ensure that all students, regardless of background or status, have equitable access to high-quality math instruction. To guide the Science of Math, we rely on well-researched instructional strategies and research about how st

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8. a. Intro to Concept Circles, a thinking routine by Dr. Cathy Marks Krpan

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8. a. Intro to Concept Circles, a thinking routine by Dr. Cathy Marks Krpan

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Intro to Concept Circles, a thinking routine by Dr. Cathy Marks Krpan

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15. Using the Placemat Strategy to Support Student Learning in Mathematics

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15. Using the Placemat Strategy to Support Student Learning in Mathematics

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Using the  Placemat Strategy to Support Student  Learning  in Mathematics

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Introduction

Key Elements of Thinking Aloud

Students focus on one step at a time and take time to fully comprehend the problem before trying to reach the final answer.

Students talk about:

All the things they notice

Every decision they make

Why they belive their choices are correct

—>

When students verbalize what they know, it helps them to reflect upon and clarify the problem they are trying to solve, and to focus on solving it one step at a time. “Thinking aloud” requires students to talk through the details of the problem, the decisions they have made as they try to solve the problem, and the reasoning behind those decisions. Struggling students, in particular, can benefit from slowing down and articulating the problem-solving process, because it gives them time to focus on the key parts of the problem. This helps them to more fully comprehend the problem before they try to solve it. The graphic lists what students should be thinking and talking about as they delve into problems.

Thinking aloud helps students, especially those who struggle with mathematics, to clarify their ideas, identify what they do and do not understand, and learn from others when they hear how their peers think about and approach the problem. It also helps the teacher to monitor students’ progress as part of the formative assessment process.

Using Evidence-Based Practice

Teachers can help their students to organize information and find and use patterns by adopting three categories of evidence-based practices: provide clear explanations , give students strategies and models , and provide ongoing formative assessment . The chart below provides a concrete teaching strategy for each of these categories.

Evidence-Based Practice | Strategy

Provide clear explanations

When you demonstrate problem solving, model the thinking aloud process by explicitly stating what it means to explain your reasoning. Include all the decisions you make, even the very small ones (e.g., which numbers to work with and what operations to use). Employ technology to support note taking and create visualizations of the big ideas as you work to give students a clearer understanding of your thinking process.

Give students strategies and models

Give your students a series of prompts—e.g., questions or sentence starters—to guide them through the process of thinking aloud. Make sure you include questions that require them to justify their decisions, such as:

I know that…

I’m trying to figure out…

One thing I can try is…

I want to try … because …

Provide ongoing formative assessment

When a student engages in thinking aloud, invite a peer to listen and comment on the content while you concentrate on the student’s use of the strategy (i.e., check to make sure the student shows an understanding of why those steps were needed). With struggling students, a small group activity might work better because it will give students a chance to hear others before sharing their thinking.

Technology Resources

Teachers and students can use online and offline tools that allow students to practice this “thinking aloud” strategy. They can create problems with natural stopping points, experiment, consider, and then decide their next steps. This chart shows a variety of tools that you could use in your classroom.

Audio recording a student’s thinking aloud (using a recording device on a computer, tablet, or smart phone) allows the student, the teacher, and the student’s peers to review the thinking process so that revisions and edits can be made. Teachers can also consider posting podcasts on their class website to show how different students thought about the problem and tried to solve it. Making comparisons allows students to understand how their peers think.

Virtual manipulatives also have a role to play in helping students. This short video, “Virtual Manipulatives ” provides background information manipulatives and explains how they can be integrated into instruction. Imagine students manipulating problems and trying out solutions that might work. These virtual manipulatives offer students varied practice in thinking aloud for each new iteration.

In the Classroom

When school started last week, Mrs. Jefferson devoted some time to figuring out where her fifth-grade students were in terms of their mathematical understanding. She found out that they know their addition and subtraction facts, and that most of them remember their multiplication facts. As she expected, however, the students are reluctant to describe aloud the steps they follow to find an answer. When she asks them to explain the steps they follow in their work, her students say they don’t know or they just look at her, clearly unsure what to say.

To ensure that her students are college and career ready, she is focusing on the following Common Core State Standards:

CCSS.Math.5.OA.3
Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

CCSS.Math.MP2
Reason abstractly and quantitatively.

She is planning a lesson with a specific objective: Students will begin practicing thinking aloud so that they can effectively employ this strategy as they move forward and learn the mathematical language necessary to communicate their thinking.

Mrs. Jefferson will take advantage of her interactive whiteboard (as well as students’ individual whiteboards) to communicate visually with the class and to allow the class to communicate visually with her. She will use a classroom response system to collect anonymous feedback from her students. A colleague recently told her about a particularly effective applet, Whole Number Cruncher , which can provide numerical patterns.

She has found that formative assessment truly helps her strengthen her teaching. She will observe her students as they use the thinking aloud strategy, particularly when they use Whole Number Cruncher . She will elicit comments from students about the learning task and will read students’ critiques of the strategy.

Her lesson plan is divided into three sections—launch, learning task, and closure—and is outlined below.

Lesson Plan

Launch

Introduce the mathematical content and strategy goals for the lesson.

Model the thinking aloud process for students before the groups start work.

Learning Task

Have students work in pairs.

Remind students to be supportive listeners.

Circulate around the room as pairs work, providing prompts and feedback as needed.

After all the pairs have worked on two rules, bring the class back together for a discussion.

Give students a handout with tables based on pairs of sequences.

Divide students into groups of three or four and give each group a different table to work on.

Circulate around the room as the pairs work, providing prompts and suggestions, as needed.

Closure

Bring the class back together for a final discussion.

Ask students to share how thinking aloud can help them in the future.

Have all students write a paragraph about what they liked and disliked about thinking aloud.

Online Teacher Resources

This article draws from the PowerUp WHAT WORKS website, particularly the Thinking Aloud Instructional Strategy Guide . PowerUp is a free, teacher-friendly website that requires no log in or registration. The Instructional Strategy Guide on thinking aloud includes a brief overview with an accompanying slide show; a list of the relevant mathematics Common Core State Standards; evidence-based teaching strategies to differentiate instruction using technology; short videos; two Lessons in Action, and links to resources that will help you use technology to support mathematics instruction. If you want to dig deeper into the research foundation behind best practices in the use of virtual manipulatives, take a look at our Tech Research Brief on the topic. If you are responsible for professional development, see PowerUp Your Professional Development for helpful ideas and materials for using the thinking aloud resources. Want more information? See PowerUp WHAT WORKS .

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Students focus on one step at a time and take time to fully comprehend the problem before trying to reach the final answer. Students talk about: All the things they notice Every decision they make Why they belive their choices are correct -> When students verbalize what they know, it helps them to reflect upon and clarify the problem they are trying to solve, and to focus on solving it one step at a time.

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14 a. Why Use Mathematical Think Alouds?

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14 a. Why Use Mathematical Think Alouds?

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Students often hold misconceptions about effective thinkers. They may believe that these people never struggle; that they never second-guess their ideas; that "they get it right" immediately. Think alouds can help dispel these misconceptions by opening up discussions about self-talk and enabling all students to access mathematical thinking.

Self-talk is simply what we say to ourselves when we think. It can be covert or overt but regardless of the form it takes there's a direct correlation in educational research between effective self-talk and strong metacognition skills. If we want our students to become strong thinkers, we need to ensure that they are aware of how their own self-talk directs their learning. It has been documented that learners who struggle often lack effective self-talk. If we need to improve our thinking skills, we need to have some awareness of our own self-talk-you can't improve something you're not aware of. What we say (or don't say) to ourselves has an impact on our decision making, mindset and learning process.

As teachers we often use think alouds in language arts to help our students become aware of the different kinds of self-talk that take place when someone reads or writes a text. You may think out loud and share your self-talk as you write or read a text. The goal is to ensure that all of your leaners have access to what effective readers and writers are thinking as they engage in a specific activity. Think-alouds in language arts help our students become stronger readers and writers by simply enabling them to understand what effective self-talk is and how they can use it to direct their own learning. So why don't we use think alouds in mathematics?

Mathematical think alouds, like think alouds in language arts, reveal the self-talk that leads to specific decisions and actions. Through the use of self-talk, students navigate thinking and make decisions. For many students who struggle, mathematical think alouds pull back the curtain on learning processes and allow them to access parts of thinking that are often inferred but invisible to them. Think alouds render this powerful thinking more concrete and accessible.

In mathematics, we often invite students to share their thinking and describe their mathematical ideas to others. We may also invite students to share their solutions as part of a whole-class discussion. These are critical learning experiences. When students communicate their thinking, they have opportunities to rehearse/edit their ideas and those of their peers. This process deepens mathematical understanding and also helps students to appreciate that mathematics is indeed a social activity.

What I believe is missing in our work is the exploration of the self-talk, which is what takes place before final solutions/strategies are shared. I'm not referring to someone describing, after the fact, what they thought as they completed an activity. I am referring to the raw, messy self-talk, in quotations, that takes place in our brain as we navigate the complex journey of thinking. In my book, Teaching Math with Meaning, I explore examples of self-talk. In some cases, there may be several self-talk statements that might fall under the same category, based on how they're used.

Modelling a short, one-minute think aloud as you do a math activity enables students to have access to the part of thinking that they often don't hear (or see). It's helpful to include the reasons for the thinking you share in your think aloud. For example, if I state, "I think I should model this using pattern blocks" it's helpful to add the reason for my decision (which is also part of my self-talk): "because this will help me see the area of the shapes more clearly." This helps students make connections between our thinking and the actions/decisions we make.

Thinking is messy! Mathematical think alouds can enable our students to have a better understanding of this messiness. The goal of implementing think alouds is not to provide them with a script to follow, rather it is about helping our students become aware of their own self-talk and understand how it can help them to become strong mathematical thinkers.

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Why Use Mathematical Think Alouds? - Cathy Marks Krpan

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Students often hold misconceptions about effective thinkers. They may believe that these people never struggle; that they never second-guess their ideas; that "they get it right" immediately. Think alouds can help dispel these misconceptions by opening up discussions about self-talk and enabling all students to access mathematical thinking.

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Sample Concept Circle Activities - Cathy Marks Kspan

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For Introduction to Concept Circles, Click Here The concept circle I use most often has 8 sections with the smaller circle in the middle. Students do not have to use all of the sections, but having 8 provides flexibility. What is placed in the outside sections relates in some way to what is placed in [...]

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Teaching your child about numbers and math can be easy and fun! A rekenrek is a great beginning math tool to help your child become fluent with numbers. Lear...

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12. Mental Math Strategies you NEED to teach in 1st and 2nd Grade // mental math for kids

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12. Mental Math Strategies you NEED to teach in 1st and 2nd Grade // mental math for kids

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Mental Math Strategies you NEED to teach in 1st and 2nd Grade // mental math for kids

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11. Beginning of the Year Math Activities for Kindergarten, First, and Second Grade!

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11. Beginning of the Year Math Activities for Kindergarten, First, and Second Grade!

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Beginning of the Year Math Activities for Kindergarten, First, and Second Grade!

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10. Outdoor Math Games for K-2 // fun outdoor math activities for kids

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10. Outdoor Math Games for K-2 // fun outdoor math activities for kids

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Outdoor Math Games for K-2 // fun outdoor math activities for kids

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11. Dr. Daniel Ansari on the science in teaching math

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11. Dr. Daniel Ansari on the science in teaching math

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Dr. Daniel Ansari on the science in teaching math

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10. Numerical Cognition Laboratory - News

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10. Numerical Cognition Laboratory - News

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*unless stated otherwise, all posts are by Bea Goffin
February 7, 2022 Tsz Tan (Nathan) Lau Talks about his recent PNAS publication
Nathan Lau, a postdoc in the Numerical Cognition Lab, has garnered a lot of attention with his most recent PNAS publication "Disentangling the individual and contextual effects of math anxiety: A global perspective. He was interviewed by The Guardian and spoke to Jeff Renaud of Western News. Congratulations Nathan!

Daniel Ansari receives prestigious Tier 1 Canada Research Chair

Another momentous achievement for Daniel Ansari! He is the recipient of a Tier 1 Canada Research Chair of Developmental Cognitive Neuroscience and Learning in the Faculties of Education and Social Science. To find out more about this honour, please check out the following in the Western News here. Congratulations Daniel!

Lien Peters wins CHRI Trainee Award

Lien Peters' most recent accomplishment includes her becoming the recipient of the Post-Doctoral Trainee Award from the Children's Health Research Institute ((CHRI), a part of Lawson Research Institute. Well earned Lien!

Daniel Ansari new CIFAR Fellow

Daniel Ansari has recently been appointed as a Fellow of the CIFAR program in Child and Brain Development. His appointment will be in effect from July 1st until June 30, 2024. CIFAR is a Canadian-based global charitable organization that seeks to address science's and humanity's most pressing questions through the bringing together of great minds. To find out more about CIFAR, please click here. Congratulations Daniel!

Lien Peters recipient of a CHRI grant

Lien Peters has been awarded a Chidren's Health Research Grant for her research proposal entitled "The role of cognitive, neural and environmental factors in the development of arithmetic and reading: An intergenerational approach". Congratulations Lien on a successful application!

Daniel Ansari addresses early maths teaching in TES podcast

Daniel Ansari was interviewed by , an education business supporting teachers, schools and school staff about how children learn to count and how teachers can help. To read the interview and listen to the podcast, please click here.

Celia Goffin and Moriah Sokolowski interviewed at MCLS Conference

Celia Goffin and Moriah Sokolowski spoke to Stephen Hurley from voicEd Radio at the Mathematical Cognition and Learning Society Conference in Ottawa. To listen to their interview, please click here.

Daniel Ansari interviewed for "the Pulse" Podcast

Daniel Ansari took part in an interview about Dyscalculia on , a podcast that examines questions regarding health, science and innovation. To read or listen to the interview, please click .

Eric Wilkey newest recipient of the Banting Postdoctoral Fellowship

Eric Wilkey is the second recipient in the Numerical Cognition Lab to be awarded the prestigious Banting Postdoctoral Fellowship! Only 70 fellowships are awarded each year in Canada to postdoctoral students who show the greatest promise to contribute to the country's economic, social and research‑based growth. To read more about this, please click here . So well-deserved Eric!
Michael Slipenkyj receives the R.B. Liddy Gold Meda
In addition to his scholarship, Michael was also awarded the R.B. Liddy Gold Medal, given to the student with the highest average in Psychology. The recipient of this award must have achieved: 1. the highest average, 80% or greater in the module, 2. an average of 80% or greater in fourth year based on a course load of 5.0 courses, 3. a cumulative average within the top 10% of the student graduating from the department. Well done Michael!

Michael Slipenkyj awarded the Clark and Mary J. Wright Scholarship

Michael Slipenkyj is the recipient of the Clark and Mary J. Wright Scholarship, awarded to a fourth-year student with an "A" average who, upon graduation, will be entering a graduate program. The Department of Psychology selects the recipient. Michael will be heading off to Georgetown University to pursue his graduate training in the fall of 2019. Congratulations Michael on this achievement!

Daniel Ansari is recipient of the 2018 Marion Welchman International Award

At the recent British Dyslexia Association International Conference in Telford, UK, Daniel Ansari received the 2018 Marion Welchman International Award To find out more, please click here. Congratulations Daniel!
Daniel Ansari awarded the prestigious Jacobs Foundation Advanced Research Fellowship
​Daniel Ansari has been awarded a Jacobs Foundation Advanced Research Fellowship for 2018-2020. This Fellowship is a globally competitive one for mid-career researchers whose research programs are related to improving the living conditions of children and youth and, in particular, whose work is of an interdisciplinary nature. To read the Western News article about the award and Daniel Ansari's work, please click . Congratulations Daniel on this huge achievement!
Daniel Ansari speaks to Anna Maria Tremonti of CBC's The Current about math anxiety
Daniel Ansari, along with Ian VanderBurgh, from the University of Waterloo, talk about the effect of math anxiety on the development of children's mathematical skills. For more details and to listen to the episode, please click here.

Moriah Sokolowski is the Numerical Cognition Lab's newest award recipient

Moriah is the recipient of the 2016 Marilyn (Pack) McClelland Award. The Marilyn (Pack) McClelland is awarded to a full-time graduate student who is conducting research in Developmental or Clinical Psychology relating to children. Selection is based on academic achievement (minimum 78% average), research productivity and quality of research publications. This award was established by a gift from Dr. W.J. McClelland.

Congratulations Moriah!
Daniel Ansari's Interview on BOLD (Blog on Learning and Development)
Daniel Ansari discusses the development of numerical and mathematical abilities with Meeri Kim from the Blog on Learning and Development. To read the blog, please click .
Daniel Ansari discusses math anxiety in The Learning Exchange
Erin Maloney, Jonathan Fugelsang and Daniel Ansari discuss the role of math anxiety in mathematical success. For more details about this blog, please click here.

Cognitive Neuroscience at Western receives major federal grant support

Western University has received $66 million dollar grant from the government of Canada through the Canada First Excellence Research Fund. The Canada First Research Excellence Fund helps competitively selected Canadian postsecondary institutions turn their key strengths into world-leading capabilities. This is the largest grant in the University's history! It will support cognitive neuroscience research at Western with a particular focus on better understanding disorders of brain function. For more information see: http://www.uwo.ca/brainscan/

Celia Goffin and Anna Matejko Recipients of Department of Psychology Awards

Celia Goffin and Anna Matejko recently received awards from the Department of Psychology in recognition of their achievements!Award for Best Master's Thesis. Each year, Psychology gives this award to the most outstanding Master's thesis in any of the following subject areas in the Graduate Psychology Program: clinical, developmental, education, industrial/organizational, measurement, personality or social psychology. Students who are graduating in the fall or who graduated in the spring of the same calendar year will be eligible for consideration. After graduation the recipient must immediately continue as a full-time student in a doctoral program in Psychology at any university. The Graduate Studies Committee in the Department of Psychology selects the recipient.

Anna is the recipient of the 2015 Marilyn (Pack) McClelland Award. The Marilyn (Pack) McClelland is awarded to a full-time graduate student who is conducting research in Developmental or Clinical Psychology relating to children. Selection is based on academic achievement
Celia has received the 2015 Leola E. Neal (minimum 78% average), research productivity and quality of research publications. This award was established by a gift from Dr. W.J. McClelland.

Congratulations to Anna and Celia!

Post-doctoral fellow Ian Lyons named a "Rising Star" by the APS

Dr. Ian Lyons was named as a 2015 'Rising Star' by the Association for Psychological Science (APS). For more details, please click .

As stated on the APS website : "The Rising Star designation recognizesoutstanding psychological scientists in the earliest stages of their research career post-PhD whose innovative work has already advanced the field and signals great potential for their continued contributions." Congratulations Ian!

On a related note, Daniel Ansari also received notification that he has been awarded Fellow Status by APS for his "outstanding contributions to the science of psychology...". For more details, please click . Congratulations Daniel!

Two Western students were among 25 Overall Winners of The Undergraduate Awards. They received a Gold Medal from President of Ireland, Michael D. Higgins, at an international medal presentation ceremony in Dublin.which brought the winners and other top performers of The Undergraduate Awards 2015 programme from around the world to Ireland to recognise their achievement and encourage cross-discipline and cross-border collaboration. Nov. 12, 2015
​Western Students Receive Gold Medal from President of Ireland​
Post By: Lara@undergraduateawards.com

The UA Global Summit 2015 featured talks and workshops run by philosopher AC Grayling, astrophysicist Dame Jocelyn Bell Burnell, genome sequencer Craig Ventor, the Guinness World Record-holder for World's Youngest University Professor, Alia Sabur, and many more. The event gathered 150 of the top performing entrants
to The Undergraduate Awards 2015: the 25 winners and 125 Highly Commended Entrants. "With two out of 25 winners, and a further 26 Highly Commended Entrants, Western was one of the top performing universities in our awards programme this year," commented Louise Hodgson, Executive Director of The Undergraduate Awards. "Jane and Michelle have done extraordinarily well to receive this medal, which places them as global leaders in their research fields."
Cited as the ultimate champion of high-potential undergraduates, and often referred to as a "junior Nobel Prize", The Undergraduate Awards is the world's largest academic awards programme, recognising excellent research and original work across the sciences, humanities, business and creative arts.

This year, The Undergraduate Awards (UA) received 5,117 submissions from undergraduates in 255 universities across 39 countries. Winners are the top performers in each of the 25 category.

The winners from Western University are: Jane Hutchison in the Education category; and Michelle Harder in the Literature Pre-1710 category.

The Gold Medal Presentation Ceremony was part of a four-day UA Global Summit,

Daniel Ansari Talks to the "180 with Jim Brown"

Daniel Ansari talked to the "180 with Jim Brown" on CBC Radio 1 about mathematics education and research. To hear the interview, please click here.

Daniel Ansari Talks to CBC Radio in Vancourver about Developmental Dyscalculia

Daniel Ansari gave an interview about Developmental Dyscalculia on "The Early Edition" on CBC Radio One. To hear the interview, please click here.

Opening of the Mary J. Wright Research and Education Centre at Merrymount

Members of the Numerical cognition Laboratory are excited about the opening of the Mary J. Wright Research and Education Centre at Merrymount. This new centre will allow for cross-disciplinary research and the application of evidence-based tools for screening and intervention. See here for more information.

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Jane Hutchison named Overall Winner of 2015 Undergraduate Awards

After earlier receiving notice that her submission to the Undergraduate Awards had been selected as one of the highly-commended entries (see news item below), Jane received the outstanding news that her paper entitled: "Bringing Neuroscience to the Classroom: A Case for the Value of Mind, Brain and Education" was selected as the overall Winner in the Education category of the Undergraduate Awards

Jane will now travel to Dublin to receive her Award at the Undergraduate Awards Global Summit.
Western News
See Western's media coverage as well as Julie McMullin's (Western's VP International) article in the . Congratulations Jane!

Jane Hutchison Highly Commended Entrant in The Undergraduate Awards

Jane Hutchison is a Highly Commended Entrant in The Undergraduate Awards, having been identified as an outstanding academic at the international level. This year, they received 5,117 submissions from students in 255 universities across 39 countries. Her paper, entitled "Bringing Neuroscience to the Classroom: A Case for the Value of Mind, Brain and Education", made it to the top 10% of all submissions in the Education category, representing Western University, as assessed by a panel of academics from universities around the world. As a Highly Commended Entrant, Jane has the opportunity to go to Dublin in November to collect her certificate and to meet fellow Highly Commended Entrants. Congratulations Jane on this fantastic achievement!
​

Jane Hutchison recognized for Honors Thesis
Post By: Ahmad Mousa
Jane Hutchison has been awarded the W. J. McClelland Award from Western University. The W.J. McClelland Award is awarded for the best fourth year Honors Thesis in Psychology. Jane has also received a certificate of Academic Excellence from the Canadian Psychological Association (CPA) for her outstanding thesis. Jane's Honors Thesis can be viewed . Congratulations Jane!

Graduate students secure external funding All Graduate Students in the Numerical Cognition Laboratory supported by external funding for the coming academic years! It's a record year of external graduate student funding for our lab: both Moriah Sokolowski and Celia Goffin have been awarded NSERC (Natural Sciences and Engineering Council of Canada) Doctoral Scholarships. Moreover, Anna Matejko has been awarded an OGS (Ontario Graduate Scholarship) grant for her final year of doctoral studies (after having been supported by the Vanier Canada Graduate Scholarship for the past 3 years). Last, but not least, incoming PhD student Zachary Hawes has been awarded an SSHRC (Social Sciences and Humanities Research Council of Canada) Doctoral Scholarship. Congratulations to all!

May 11, 2015
Daniel Ansari named 'postdoctoral supervisor of the year'
Post By: Ahmad Mousa
Daniel Ansari has been awarded the award for 'postdoctoral supervisor of the year' by the School for Graduate and Postdoctoral Studies (SGPS) at Western University. This award is given to Postdoctoral Supervisors who have demonstrated exemplary support for postdoctoral scholars at Western by going above and beyond supervisory expectations. These mentors have given their time and put forth extra effort to ensure a successful experience for postdoctoral scholars.

​​​​​ ​February 19, 2015
Daniel Ansari named among 'world-class talent'
Post By: Jason Winders of Western News

Psychology professor Daniel Ansari and Medical Biophysics professor Michael Kovacs were named two of 34 recipients of Natural Sciences and Engineering Research Council of Canada (NSERC) awards recognizing the work of outstanding Canadian scientists and engineers.

"As Canadians, we can look with pride and honour today at this group of scientists and engineers who embody world-class talent and innovation," said B. Mario Pinto, NSERC president, at a ceremony recognizing the scholars this week. "Their search for knowledge through inspired scholarly inquiry will help Canada take its place as a world leader."

In a recent article published in the London Free Press , Ansari emphasized the importance of work in the field: "School-entry numerical skills are a more important predictor of future academic achievement than are early reading and socio-emotional skills - around 5% of children have (developmental dyscalculia) in acquiring even the most basic numerical skills, putting them at a significant disadvantage".

A Canada Research Chair in Developmental Cognitive Neuroscience, Tier 2, Ansari was presented the E.W.R. Steacie Memorial Fellowship, one of six awarded to enhance the career development of outstanding and highly promising scientists and engineers who are faculty members of Canadian universities.

​Please click here to read the rest of this story.

​September 16, 2014

Daniel Ansari named as inaugural member of 'The College'
Post By: Western University Communications Staff
Three Western professors, and one King's University College professor, have been named among 91 inaugural members of the Royal Society of Canada's College of New Scholars, Artists and Scientists. Those named to the College represent the emerging generation of scholarly, scientific and artistic leadership in Canada.

Together, the members will address issues of particular concern to new scholars, artists and scientists, for the advancement of understanding and the benefit of society, taking advantage of the interdisciplinary approaches fostered by the establishment of the College.

Daniel Ansari - Department of Psychology, Numerical Cognition Lab

​Robert Ventresca - Department of History, King's University College
Please click here to read the rest of this story.
September 1, 2014

Does your child understand numbers?
Post By: Nathalie Laflamme of Montreal Families.ca
Math can be a tricky thing, especially for a child. Struggling with a basic concept can lead to a domino effect of confusion, since the next lesson will be more difficult when directly related. When it comes to learning, all children are different. So how can parents make sure that their child is on the right track?

The Numeracy Screener, a free test that is available for download and only takes two to four minutes to complete, can help parents determine whether their child needs some extra help in math. This test is aimed at students from kindergarten to Grade 3.
Please click here to read the rest of this story.
August 14, 2014

Numerical Cognition Laboratory post-doc Ian Lyons awarded Banting scholarship

The Numerical Cognition Lab is very proud to announce that Ian Lyons, a post-doctoral fellow at the lab, has been awarded with the prestigious NSERC Banting Scholarship. Ian is the only recipient of the award at Western University and one of only three recipients in Psychology. Congratulations Ian!

A full list of the award recipients can be found here.

June 19, 2014

Test puts math prep on par with language
Post By: Paul Mayne of Western News
For some kids, numbers simply don't add up. But now, thanks to a test developed by Western researchers, teachers and parents will have an early warning that extra help may be needed.

"Numeracy is one of the most important early development skills that children learn," said Western Psychology professor Daniel Ansari. "Individual differences in children's early numeracy skills are critical predictors of their later educational achievement - as strong as literacy skills. Yet, we do not have very many good tools for assessing how well children are doing, early in their development."

A two-minute test may show if a child needs early help in math - and it's free online
Post By: Erin Anderssen of The Globe and Mail
It's basic math: seven is larger than five. Three dots represent less than eight dots. But what if some kindergarten students miss this step? Eventually, they land in Grade 3, facing multiplication, and math is a muddle.

Research has shown that a strong understanding of numerical quantity - both in symbols and actual numbers - is an important early predictor for future math success. That's because math is a subject that builds upon itself; students who miss a key step quickly fall behind.

Welcome to the New Home of the Numerical Cognition Laboratory

Welcome to the new web site of the Numerical Cognition Laboratory at the Western University. Please check this page for the most recent updates from the lab!

Graduate students connect education and neuroscience

Post By: Western Communications Staff
Western graduate students Stephanie Budgen and Anna Matejko's work in the area of cognitive neuroscience is now part of a new web-based, open-access science journal launched by Nature Publishing Group recently at the USA Science and Engineering Festival in Washington, D.C.

Frontiers for Young Minds aims to involve children and young people in the research process.

Under the supervision of Daniel Ansari, Canada Research Chair in Developmental Cognitive Neuroscience, Budgen submitted When your brain can't do 2+2: A case of Developmental Dyscalculia, which explores the developmental struggle an alarming number of children have learning simple mathematics; while Matejko published White matter counts: Brain connections help us do 2+2, which reviews the emerging technologies used to explore how brain structure relates to math learning. Budgen and Matejko have kick-started promising academic careers, working with Ansari in Western's Department of Psychology and the Brain & Mind Institute, by participating in the highly competitive Latin American School for Education (LA School) the past two academic years.

Budgen just returned from her two-week study in Punta del Este, Uruguay, while Matejko attended in 2013 when it was held in Bahia, Brazil. In each respective class, the pair were the only students from Canada to be selected to participate in this international school. Matejko and her classmates from last year's LA School have already generated a paper, which was recently published by Trends in Neuroscience and Education. Along with co-authors from the U.K., Germany and Latin America, the researchers propose the emergence of a new and distinct field of study that combines neuroscience and education.

This field would use a multidisciplinary approach that would integrate both researchers and teachers to answer questions such as, How can we identify children who will struggle in school? This is in contrast to the current approach where education and neuroscience are largely two distinct fields. Poor communication between researchers and educators has been ineffective in the past, leading to issues such as neuromyths. This multidisciplinary approach would attempt to revolutionize this issue and answer educationally relevant questions.​

October 26, 2013

The root of the problem: This is your brain on math

Post By: Ivan Semeniuk of The Globe and Mail
​"The first game we're going to play is where you see the squares and you have to tell me which side has more of them, okay?" says Anna Matejko.
She is talking to six-year-old Olivia Hall, whose head is so deep in the maw of an fMRI machine that only her feet can be seen sticking out from the giant magnetic imaging device.

It's an odd way to spend a Sunday morning, but the Grade 1 student has come to University Hospital in London, Ont., so Ms. Matejko can study the fleeting neural fireworks going off in Olivia's brain as she responds to a series of simple counting and number problems.

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unless stated otherwise, all posts are by Bea Goffin February 7, 2022 Tsz Tan (Nathan) Lau Talks about his recent PNAS publication Nathan Lau, a postdoc in the Numerical Cognition Lab, has garnered a lot of attention with his most recent PNAS publication "Disentangling the individual and contextual effects of math anxiety: A global perspective.

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COPYRIGHT © NUMERICAL COGNITION LABORATORY. ALL RIGHTS RESERVED. PRIVACY POLICY
Submit the scores and compare the child's performance to a norm of Canadian children's scores from senior kindergarten through to grade 3

Administer the paper and pencil test and mark the scores

- Directed by Dr. Daniel Ansari, consisting of graduate students and post-doctoral fellows from the fields of Psychology, Neuroscience, and Education

- We aim to better understand what predicts success and failures in early math learning

- Focus of our research is to gain a better understanding of how children develop numerical and mathematical competencies

Tests are provided in PDF format

with detailed assembly instructions

A 2-4 minute test composed of two parts:

1. Symbolic (i.e. numerical digits)

2. Non-Symbolic (i.e. arrays of dots)

- Developed by Drs. Nadia Nosworthy & Daniel Ansari in the Numerical Cognition Laboratory at Western University

- Measures a child's ability to understand numerical magnitude (quantity)
- Correlations between performance on the Numeracy Screener and arithmetic skills have been demonstrated and published in scientific literature
- These findings show that performance on the Numeracy Screener can explain individual differences in children's arithmetic skills

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Numeracy Screener | Home | Numerical Cognition Laboratory

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The Numerical Cognition Laboratory Numeracy Screener. A test that measures a child's ability to understand numerical magnitude. Supported by scientific research.

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4. Understanding Dyscalculia: Symptoms Explained

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4. Understanding Dyscalculia: Symptoms Explained

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National Center for Learning Disabilities (NCLD)

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5. What is Dyscalculia? - Maths Dyslexia - Simple Explanation and Solution

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5. What is Dyscalculia? - Maths Dyslexia - Simple Explanation and Solution

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What is Dyscalculia? - Maths Dyslexia - Simple Explanation and Solution

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17. Dyscalculia: Teaching Strategies & Modifications

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17. Dyscalculia: Teaching Strategies & Modifications

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Dyscalculia: Teaching Strategies & Modifications

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2. Dr. Daniel Ansari - Building Blocks of Mathematical Competence

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2. Dr. Daniel Ansari - Building Blocks of Mathematical Competence

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Dr. Daniel Ansari - Building Blocks of Mathematical Competence