Does early math really matter that much before kindergarten?

Large long-term studies find that math skills at school entry are among the strongest predictors of later math and even reading achievement. These are associations, not proof that drilling preschoolers helps, but playful counting, comparing and shape talk are a low-cost bet.

What are the best things to do at home with a toddler or preschooler?

Talk about numbers in daily life (count real objects, ask "how many?", compare sizes), play simple numbered board games, and read math-rich picture books. Research on number talk and board games supports these, though the studies are mostly small.

Do board games like Chutes and Ladders actually help with math?

Linear numbered board games helped low-income preschoolers' number sense in short randomized studies; versions with colors instead of numbers did not. The effect is about understanding the number line, so games where you count spaces out loud work best.

I'm bad at math and anxious about it. Will that rub off on my child?

It can: one study found children of math-anxious parents learned less math only when those parents helped with homework often. Structured, low-stress activities such as math story apps seemed to close that gap, so how you help matters more than your own skill.

How should I help with math homework without making things worse?

Stay calm, ask your child to explain their thinking, show a worked example and compare two methods rather than taking over, and avoid saying "I'm not a math person." If the school uses unfamiliar models, the curriculum's family pages explain them.

Should my child memorize math facts, and are timed tests harmful?

Most researchers agree fact fluency matters because it frees working memory for harder problems. Whether short timed practice causes anxiety is disputed: some educators argue against it, while federal intervention guidance recommends brief timed activities to build fluency. Low-stakes, untimed-first practice is a reasonable middle path.

What is math anxiety and what helps?

Math anxiety is fear or tension about math that uses up working memory and can show up as early as first grade. Building real skill, reframing nerves, and adults modeling a calm, positive attitude are the best-supported responses; it is not a diagnosis.

Do math manipulatives and "concrete-pictorial-abstract" teaching work?

On average, manipulatives give small to moderate benefits, but only when the object clearly represents the idea and an adult helps connect it to symbols. Singapore and UK mastery programs build that concrete-to-picture-to-symbol path, and some such programs have strong evidence ratings.

Are worked examples better than just doing lots of problems?

For beginners, studying solved examples alternated with practice problems usually beats problem solving alone, and comparing two solution methods builds flexibility. As skill grows, the examples should fade.

Do math apps like Khan Academy, Zearn, DreamBox, IXL or Prodigy actually raise scores?

Some have independent randomized or quasi-experimental evidence (Zearn, DreamBox, ST Math are rated strong by Evidence for ESSA; Khan Academy has several outside trials), but effects are usually small and depend on steady weekly use. Many claims come from the companies themselves and compare heavy users to others, which cannot rule out that stronger students simply use more.

What are the math wars, and what is the fuss about Jo Boaler and California?

It is a long US argument over how much to emphasize practice and procedures versus discovery and concepts. California's 2023 framework drew fire over 8th grade algebra, data science and citations of Boaler's work; research suggests concepts and procedures build each other, so neither side alone has the full answer.

My child loves math. How can I offer more challenge?

Puzzle-rich curricula like Beast Academy and Art of Problem Solving, math circles and noncompetitive festivals, and contests such as Math Kangaroo or MATHCOUNTS all offer depth beyond school. Free problem sites like NRICH and Open Middle are a good start before paying for anything.

Will learning chess make my child better at math?

Pooled studies show at best small effects of chess lessons on math, and the better-designed studies show less. Chess is a fine hobby for focus and planning, but not a substitute for math practice.

Where can I find free, good-quality math practice and printables?

Khan Academy, NRICH, the Math Learning Center apps, K5 Learning and Math-Drills worksheets, Illustrative Mathematics and Eureka Math materials, and PBS KIDS games are all free. Short, regular sessions matter more than which site you pick.

Is screen time with Numberblocks or math shows useful for little kids?

Shows designed with math advisers, such as Numberblocks and Peg + Cat, model number ideas in a way kids enjoy, but there is little direct research on their effect on later math. Watching together and then counting real objects is the safest way to make them count.

The Big Index

Math

Number sense, facts, math anxiety, games and puzzles, and how to help with homework without tears.

109 entries
72 free
12 must-reads ★

Questions parents ask

Does early math really matter that much before kindergarten?
Large long-term studies find that math skills at school entry are among the strongest predictors of later math and even reading achievement. These are associations, not proof that drilling preschoolers helps, but playful counting, comparing and shape talk are a low-cost bet.
- School Readiness and Later Achievement (Duncan et al., Developmental Psychology)
- What's Past Is Prologue: Relations Between Early Mathematics Knowledge and High School Achievement (Watts, Duncan, Siegler & Davis-Kean)
- Early Math Matters: Kindergarten Number Competence and Later Mathematics Outcomes (Jordan, Kaplan, Ramineni & Locuniak)
- Which Preschool Mathematics Competencies Are Most Predictive of Fifth Grade Achievement? (Nguyen, Watts, Duncan, Clements et al.)
More in Math
What are the best things to do at home with a toddler or preschooler?
Talk about numbers in daily life (count real objects, ask "how many?", compare sizes), play simple numbered board games, and read math-rich picture books. Research on number talk and board games supports these, though the studies are mostly small.
- What counts in the development of young children's number knowledge? (Levine et al., Developmental Psychology)
- Playing linear numerical board games promotes low-income children's numerical development (Siegler & Ramani, Developmental Science)
- DREME Family Math (DREME Network, Stanford)
- Making the Most of Talk About Numbers (DREME Network, Stanford)
- Talking Math With Your Kids (Christopher Danielson)
More in Math
Do board games like Chutes and Ladders actually help with math?
Linear numbered board games helped low-income preschoolers' number sense in short randomized studies; versions with colors instead of numbers did not. The effect is about understanding the number line, so games where you count spaces out loud work best.
- Playing linear numerical board games promotes low-income children's numerical development (Siegler & Ramani, Developmental Science)
- Prime Climb (Math for Love (Dan Finkel & Katherine Cook))
- Teaching Math to Young Children (WWC Practice Guide) (What Works Clearinghouse, IES)
More in Math
I'm bad at math and anxious about it. Will that rub off on my child?
It can: one study found children of math-anxious parents learned less math only when those parents helped with homework often. Structured, low-stress activities such as math story apps seemed to close that gap, so how you help matters more than your own skill.
- Intergenerational Effects of Parents' Math Anxiety on Children's Math Achievement and Anxiety (Maloney, Ramirez, Gunderson, Levine & Beilock)
- Math at home adds up to achievement in school (Berkowitz et al., Science)
- Disassociating the relation between parents' math anxiety and children's math achievement: Long-term effects of a math app intervention (Schaeffer, Rozek, Berkowitz, Levine & Beilock)
- Ask the Cognitive Scientist: Math Anxiety: Can Teachers Help Students Reduce It? (Beilock & Willingham, American Educator)
More in Math
How should I help with math homework without making things worse?
Stay calm, ask your child to explain their thinking, show a worked example and compare two methods rather than taking over, and avoid saying "I'm not a math person." If the school uses unfamiliar models, the curriculum's family pages explain them.
- Parents' math anxiety can undermine children's math achievement (University of Chicago News)
- Does comparing solution methods facilitate conceptual and procedural knowledge? (Rittle-Johnson & Star, Journal of Educational Psychology)
- Eureka Math (Great Minds)
- Developing Math Skills (PBS KIDS for Parents) (PBS KIDS for Parents)
Should my child memorize math facts, and are timed tests harmful?
Most researchers agree fact fluency matters because it frees working memory for harder problems. Whether short timed practice causes anxiety is disputed: some educators argue against it, while federal intervention guidance recommends brief timed activities to build fluency. Low-stakes, untimed-first practice is a reasonable middle path.
- Fluency Without Fear (Jo Boaler, youcubed)
- Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades (What Works Clearinghouse, IES)
- Procedural Fluency in Mathematics (position statement) (NCTM)
- The Science of Math (The Science of Math (educator group))
What is math anxiety and what helps?
Math anxiety is fear or tension about math that uses up working memory and can show up as early as first grade. Building real skill, reframing nerves, and adults modeling a calm, positive attitude are the best-supported responses; it is not a diagnosis.
- Ask the Cognitive Scientist: Math Anxiety: Can Teachers Help Students Reduce It? (Beilock & Willingham, American Educator)
- Math anxiety causes trouble for students as early as first grade (University of Chicago News)
- Mathematics Anxiety: What Have We Learned in 60 Years? (Dowker, Sarkar & Looi, Frontiers in Psychology)
- The Maths Anxiety Trust (The Maths Anxiety Trust (UK))
Do math manipulatives and "concrete-pictorial-abstract" teaching work?
On average, manipulatives give small to moderate benefits, but only when the object clearly represents the idea and an adult helps connect it to symbols. Singapore and UK mastery programs build that concrete-to-picture-to-symbol path, and some such programs have strong evidence ratings.
- Ask the Cognitive Scientist: Do Manipulatives Help Students Learn? (Daniel T. Willingham, American Educator)
- A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives (Carbonneau, Marley & Selig, Journal of Educational Psychology)
- Five Big Ideas in Teaching for Mastery (NCETM (UK))
- Singapore math (Wikipedia)
- Polypad (Amplify (originally Mathigon))
Are worked examples better than just doing lots of problems?
For beginners, studying solved examples alternated with practice problems usually beats problem solving alone, and comparing two solution methods builds flexibility. As skill grows, the examples should fade.
- Organizing Instruction and Study to Improve Student Learning (IES Practice Guide) (Pashler et al., US Institute of Education Sciences)
- Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students (What Works Clearinghouse, IES)
- Cognitive load theory: Research that teachers really need to understand (NSW Centre for Education Statistics and Evaluation)
- Does comparing solution methods facilitate conceptual and procedural knowledge? (Rittle-Johnson & Star, Journal of Educational Psychology)
Do math apps like Khan Academy, Zearn, DreamBox, IXL or Prodigy actually raise scores?
Some have independent randomized or quasi-experimental evidence (Zearn, DreamBox, ST Math are rated strong by Evidence for ESSA; Khan Academy has several outside trials), but effects are usually small and depend on steady weekly use. Many claims come from the companies themselves and compare heavy users to others, which cannot rule out that stronger students simply use more.
- Evidence for ESSA: Math programs (Center for Research and Reform in Education, Johns Hopkins)
- Zearn: Efficacy Research Studies (Zearn)
- Multiple Studies Show Khan Academy Drives Learning Gains (Khan Academy (vendor))
- Prodigy: Education Research (Prodigy Education (vendor))
- Advocates to FTC: Prodigy Math Game Preys On Kids and Families (Fairplay (Campaign for a Commercial-Free Childhood))
What are the math wars, and what is the fuss about Jo Boaler and California?
It is a long US argument over how much to emphasize practice and procedures versus discovery and concepts. California's 2023 framework drew fire over 8th grade algebra, data science and citations of Boaler's work; research suggests concepts and procedures build each other, so neither side alone has the full answer.
- Math wars (Wikipedia)
- PROOF POINTS: Stanford's Jo Boaler talks about her new book 'MATH-ish' and takes on her critics (Jill Barshay, The Hechinger Report)
- Public Comments on the CMF (Brian Conrad, Stanford mathematician)
- California adopts controversial changes to math education (The San Francisco Standard)
- Not a One-Way Street: Bidirectional Relations Between Procedural and Conceptual Knowledge of Mathematics (Rittle-Johnson, Schneider & Star, Educational Psychology Review)
My child loves math. How can I offer more challenge?
Puzzle-rich curricula like Beast Academy and Art of Problem Solving, math circles and noncompetitive festivals, and contests such as Math Kangaroo or MATHCOUNTS all offer depth beyond school. Free problem sites like NRICH and Open Middle are a good start before paying for anything.
- Beast Academy (Art of Problem Solving)
- Art of Problem Solving (Art of Problem Solving)
- MathCircles.org (National Association of Math Circles)
- NRICH (University of Cambridge)
- Mathematical Kangaroo (Wikipedia)
Will learning chess make my child better at math?
Pooled studies show at best small effects of chess lessons on math, and the better-designed studies show less. Chess is a fine hobby for focus and planning, but not a substitute for math practice.
- Do the benefits of chess instruction transfer to academic and cognitive skills? A meta-analysis (Sala & Gobet, Educational Research Review)
- ChessKid (Chess.com)
Where can I find free, good-quality math practice and printables?
Khan Academy, NRICH, the Math Learning Center apps, K5 Learning and Math-Drills worksheets, Illustrative Mathematics and Eureka Math materials, and PBS KIDS games are all free. Short, regular sessions matter more than which site you pick.
- Khan Academy (Khan Academy (nonprofit))
- NRICH (University of Cambridge)
- Math Learning Center Apps (The Math Learning Center)
- Free Math Worksheets (K5 Learning)
- PBS KIDS Math Games (PBS KIDS)
Is screen time with Numberblocks or math shows useful for little kids?
Shows designed with math advisers, such as Numberblocks and Peg + Cat, model number ideas in a way kids enjoy, but there is little direct research on their effect on later math. Watching together and then counting real objects is the safest way to make them count.
- Numberblocks (BBC CBeebies / Blue Zoo)
- Peg + Cat (PBS KIDS)
- DREME Family Math (DREME Network, Stanford)

Must-reads

StudyFree
School Readiness and Later Achievement
Duncan et al., Developmental Psychology, 2007
Pooled six big long-term studies of children. Math skills at school entry predicted later math and reading better than reading or attention skills did. It shows links, not cause, but it is why early math gets so much attention.
sites.uci.edu
StudyFree
Playing linear numerical board games promotes low-income children's numerical development
Siegler & Ramani, Developmental Science, 2008
Preschoolers who played a simple numbered board game, like Chutes and Ladders with numbers 1-10, for four short sessions caught up with richer peers on number line estimates. A color-only version did not help. Cheap and easy to copy at home.
pubmed.ncbi.nlm.nih.gov
GuideFree
Teaching Math to Young Children (WWC Practice Guide)
What Works Clearinghouse, IES, 2013
US Education Department guide with five research-rated recommendations for ages 3-6: teach number and operations step by step, add geometry, patterns and measurement, use math talk, and watch progress. Written for teachers but clear enough for parents.
ies.ed.gov
ToolFree
DREME Family Math
DREME Network, Stanford
Free research-based activities, picture books guides and short videos for families with children up to about age 8, in English and Spanish. Everyday routines like cooking and walks become math moments.
familymath.stanford.edu
BlogFree
Talking Math With Your Kids
Christopher Danielson
Former teacher and dad collects real conversations with children about counting, shapes and patterns, sorted by age, with ideas for trying them at home. Also home of his books How Many? and Which One Doesn't Belong?
talkingmathwithkids.com
StudyPaid
Math at home adds up to achievement in school
Berkowitz et al., Science, 2015
Randomized trial with 587 Chicago first graders. Families using the free Bedtime Math app, even once a week, saw extra math gains over the year, most for children of math-anxious parents. University of Chicago researchers; paywalled.
doi.org
StudyFree
Intergenerational Effects of Parents' Math Anxiety on Children's Math Achievement and Anxiety
Maloney, Ramirez, Gunderson, Levine & Beilock, 2015
Among 400+ first and second graders, kids of math-anxious parents learned less math and grew more anxious, but only when those parents helped with homework often. Correlational. The takeaway: how you help matters more than how much. Free PDF.
sites.temple.edu
ArticleFree
Ask the Cognitive Scientist: Math Anxiety: Can Teachers Help Students Reduce It?
Beilock & Willingham, American Educator, 2014
Two cognitive scientists explain how anxiety eats working memory and what helps: building skill, reframing nerves, brief writing about worries before tests, and adults not saying "I'm not a math person." Free.
aft.org
ArticleFree
Ask the Cognitive Scientist: Do Manipulatives Help Students Learn?
Daniel T. Willingham, American Educator, 2017
Manipulatives like blocks often help, but only when the object clearly maps to the idea and an adult guides the link. Flashy, toy-like objects can distract. Free and practical for choosing math toys.
aft.org
ArticleFree
PROOF POINTS: Stanford's Jo Boaler talks about her new book 'MATH-ish' and takes on her critics
Jill Barshay, The Hechinger Report, 2024
Balanced explainer of the fight over Boaler's work: what she argues, what cognitive scientists counter about practice and memorization, and the 2024 citation complaint, which Stanford later declined to investigate.
hechingerreport.org
ToolFree
Evidence for ESSA: Math programs
Center for Research and Reform in Education, Johns Hopkins
Independent, free database rating math programs by the strength of their research under federal ESSA rules. Lists Zearn, DreamBox, ST Math, Math in Focus and many tutoring models. The best place to check a vendor's claims.
evidenceforessa.org
ToolFree
NRICH
University of Cambridge
Free, huge library of rich math problems, games and investigations for ages 3-18, with teacher and parent notes. Excellent for kids who want to think, not just practice.
nrich.maths.org

All 109

Early number sense

★School Readiness and Later AchievementDuncan et al., Developmental Psychology · 2007 · Study · Free · Ages 0–5Pooled six big long-term studies of children. Math skills at school entry predicted later math and reading better than reading or attention skills did. It shows links, not cause, but it is why early math gets so much attention.
What's Past Is Prologue: Relations Between Early Mathematics Knowledge and High School AchievementWatts, Duncan, Siegler & Davis-Kean · 2014 · Study · Free · Ages 0–5Followed children from preschool to age 15. Math knowledge before first grade still predicted high school math, even after accounting for family background and early IQ. Free full text.
Early Math Matters: Kindergarten Number Competence and Later Mathematics OutcomesJordan, Kaplan, Ramineni & Locuniak · 2009 · Study · Free · Ages 0–5Kindergarten number sense (counting, comparing quantities, simple number combinations) strongly predicted math achievement through third grade. Supports checking and building these skills early, before gaps widen. Free full text.
Which Preschool Mathematics Competencies Are Most Predictive of Fifth Grade Achievement?Nguyen, Watts, Duncan, Clements et al. · 2016 · Study · Free · Ages 0–5Looked at which preschool math skills mattered most later. More advanced counting and number skills, not just reciting numbers, best predicted fifth grade math. Useful for knowing what to practice with a preschooler.
Early Predictors of High School Mathematics AchievementSiegler, Duncan, Davis-Kean et al., Psychological Science · 2012 · Study · Paid · Ages 8–12Using US and UK data, fifth graders' knowledge of fractions and division predicted high school algebra and overall math, beyond other math skills, IQ and family income. A strong case for taking fractions seriously. Paywalled.
★Playing linear numerical board games promotes low-income children's numerical developmentSiegler & Ramani, Developmental Science · 2008 · Study · Free · Ages 0–5Preschoolers who played a simple numbered board game, like Chutes and Ladders with numbers 1-10, for four short sessions caught up with richer peers on number line estimates. A color-only version did not help. Cheap and easy to copy at home.
Experimental Evaluation of the Effects of a Research-Based Preschool Mathematics CurriculumClements & Sarama, American Educational Research Journal · 2008 · Study · Paid · Ages 0–5Randomized trial of Building Blocks, a preschool math curriculum built on learning trajectories. Children gained much more math than peers in comparison classrooms. Paywalled; the authors' free trajectory site explains the approach.
Learning TrajectoriesClements & Sarama, University of Denver · Tool · Free · Ages 0–5Free site that maps the usual steps children take in counting, shapes, adding and more, with activities for each step. Helps a parent see what comes next for their child instead of guessing.
★Teaching Math to Young Children (WWC Practice Guide)What Works Clearinghouse, IES · 2013 · Guide · Free · Ages 0–5US Education Department guide with five research-rated recommendations for ages 3-6: teach number and operations step by step, add geometry, patterns and measurement, use math talk, and watch progress. Written for teachers but clear enough for parents.

Math at home & homework help

What counts in the development of young children's number knowledge?Levine et al., Developmental Psychology · 2010 · Study · Free · Ages 0–5Recorded everyday talk in 44 homes from 14 to 30 months. Toddlers who heard more number talk from parents understood what number words mean sooner. Small sample and correlational, but the basis for most math talk advice. Free.
Making the Most of Talk About NumbersDREME Network, Stanford · Article · Free · Ages 0–5Practical tips from early math researchers: count and label sets of things, use number words a little beyond what your child knows, and watch for gestures that show more understanding than words do.
★DREME Family MathDREME Network, Stanford · Tool · Free · Ages 0–5Free research-based activities, picture books guides and short videos for families with children up to about age 8, in English and Spanish. Everyday routines like cooking and walks become math moments.
★Talking Math With Your KidsChristopher Danielson · Blog · FreeFormer teacher and dad collects real conversations with children about counting, shapes and patterns, sorted by age, with ideas for trying them at home. Also home of his books How Many? and Which One Doesn't Belong?
Bedtime MathBedtime Math Foundation · Organization · Free · Ages 5–8Nonprofit that publishes a free daily math story problem with versions for different ages, plus books, an app and family math night kits. Built on the idea that math can be as normal at bedtime as reading.
★Math at home adds up to achievement in schoolBerkowitz et al., Science · 2015 · Study · Paid · Ages 5–8Randomized trial with 587 Chicago first graders. Families using the free Bedtime Math app, even once a week, saw extra math gains over the year, most for children of math-anxious parents. University of Chicago researchers; paywalled.
Disassociating the relation between parents' math anxiety and children's math achievement: Long-term effects of a math app interventionSchaeffer, Rozek, Berkowitz, Levine & Beilock · 2018 · Study · Paid · Ages 5–8Follow-up of the Bedtime Math trial. Gains for children of math-anxious parents were still visible about two years later. One research group studying its own intervention, so wider replication would help. Paywalled.
VroomBezos Family Foundation · App · Free · Ages 0–5Free tips and app for parents of children from birth to five that turn daily routines into brain-building moments, many of them about counting, sorting and comparing. Not math-only.
Developing Math Skills (PBS KIDS for Parents)PBS KIDS for Parents · Guide · Free · Ages 0–5Short, friendly articles and activities for parents on everyday math by age, from counting games to measuring while cooking. A good first stop when you want quick ideas rather than research.
Parent Resources (youcubed)youcubed, Stanford · Guide · FreeJo Boaler's group's materials for parents: how to talk about mistakes and effort, math games, and why they discourage speed drills. Reflects one side of a live debate about fluency, which is worth knowing.

Math anxiety & attitudes

★Intergenerational Effects of Parents' Math Anxiety on Children's Math Achievement and AnxietyMaloney, Ramirez, Gunderson, Levine & Beilock · 2015 · Study · Free · Ages 5–8Among 400+ first and second graders, kids of math-anxious parents learned less math and grew more anxious, but only when those parents helped with homework often. Correlational. The takeaway: how you help matters more than how much. Free PDF.
Parents' math anxiety can undermine children's math achievementUniversity of Chicago News · 2015 · Article · Free · Ages 5–8Plain-language summary of the Maloney study on parents' math anxiety and homework help, with the researchers' suggestions, such as using math games and story problems instead of drilling a child you feel anxious helping.
Female teachers' math anxiety affects girls' math achievementBeilock, Gunderson, Ramirez & Levine, PNAS · 2010 · Study · Free · Ages 5–8In 17 first and second grade classrooms, the more math-anxious the (female) teacher, the more girls came to believe boys are better at math, and the lower those girls scored by year's end. Small and early, but widely cited.
Math anxiety causes trouble for students as early as first gradeUniversity of Chicago News · 2012 · Article · Free · Ages 5–8Summary of Ramirez et al. (2013). Math anxiety already shows up in first and second grade, and it hurt the highest working-memory kids most, setting them back almost half a year.
Mathematics Anxiety: What Have We Learned in 60 Years?Dowker, Sarkar & Looi, Frontiers in Psychology · 2016 · Review · FreeReadable open-access review of math anxiety research: what it is, how it relates to performance in both directions, gender patterns, and what interventions have been tried. Good background for curious parents.
★Ask the Cognitive Scientist: Math Anxiety: Can Teachers Help Students Reduce It?Beilock & Willingham, American Educator · 2014 · Article · FreeTwo cognitive scientists explain how anxiety eats working memory and what helps: building skill, reframing nerves, brief writing about worries before tests, and adults not saying "I'm not a math person." Free.
The Maths Anxiety TrustThe Maths Anxiety Trust (UK) · Organization · FreeUK charity with guides for parents and teachers on spotting and easing math anxiety. Practical, low-key advice on language, homework and confidence.
Choke: What the Secrets of the Brain Reveal About Getting It Right When You Have ToSian Beilock · 2010 · Book · PaidA cognitive scientist's popular book on why people freeze under pressure, with chapters on math anxiety and test performance and research-based ways to cope. For adults.

How math is taught: the evidence

Assisting Students Struggling with Mathematics: Intervention in the Elementary GradesWhat Works Clearinghouse, IES · 2021 · Guide · Free · Ages 5–8Research-rated advice for helping K-6 kids who struggle: systematic instruction, clear math language, visual representations, the number line, word-problem routines and timed activities to build fact fluency. Free.
Improving Mathematical Problem Solving in Grades 4 Through 8What Works Clearinghouse, IES · 2012 · Guide · Free · Ages 8–12Five recommendations for teaching problem solving: use routine and non-routine problems, help students monitor their thinking, teach visual representations, compare several strategies, and connect ideas to notation.
Developing Effective Fractions Instruction for Kindergarten Through 8th GradeWhat Works Clearinghouse, IES · 2010 · Guide · Free · Ages 8–12Guide on teaching fractions, including starting from fair sharing, using number lines to show fractions as numbers, and explaining why procedures work. Fractions are a known stumbling block that predicts later algebra.
Teaching Strategies for Improving Algebra Knowledge in Middle and High School StudentsWhat Works Clearinghouse, IES · 2015 · Guide · Free · TeensRecommends having students study solved problems (worked examples), learn the structure of algebra, and compare different solution strategies. Useful when a teen asks for help with algebra homework.
Does comparing solution methods facilitate conceptual and procedural knowledge?Rittle-Johnson & Star, Journal of Educational Psychology · 2007 · Study · Paid · Ages 8–12Seventh graders who compared two worked solutions side by side learned more and became more flexible than those who studied the same solutions one at a time. A simple trick for homework help: show two ways and compare. Paywalled.
Not a One-Way Street: Bidirectional Relations Between Procedural and Conceptual Knowledge of MathematicsRittle-Johnson, Schneider & Star, Educational Psychology Review · 2015 · Review · PaidReview finding that knowing procedures and understanding concepts build each other over time. A useful corrective to the math wars idea that you must choose understanding or practice.
★Ask the Cognitive Scientist: Do Manipulatives Help Students Learn?Daniel T. Willingham, American Educator · 2017 · Article · Free · Ages 5–8Manipulatives like blocks often help, but only when the object clearly maps to the idea and an adult guides the link. Flashy, toy-like objects can distract. Free and practical for choosing math toys.
A meta-analysis of the efficacy of teaching mathematics with concrete manipulativesCarbonneau, Marley & Selig, Journal of Educational Psychology · 2013 · Meta-analysis · Paid · Ages 5–8Pooled 55 studies. Teaching with manipulatives had small to moderate benefits on average, depending on how much guidance teachers gave and what was being learned. Paywalled.
Five Big Ideas in Teaching for MasteryNCETM (UK) · Guide · Free · Ages 5–8England's national math teaching center explains its mastery approach, influenced by Shanghai and Singapore: coherence, representation and structure (concrete-pictorial-abstract), mathematical thinking, fluency and variation.
NCETMNational Centre for Excellence in the Teaching of Mathematics · Organization · FreeUK government-funded body supporting math teaching in England through Maths Hubs and free materials. Its pages on mastery and on number fluency are useful outside the UK too.
NCTMNational Council of Teachers of Mathematics · Organization · FreeThe main US association of math teachers. Publishes standards, position statements and classroom resources, including free problems and interactive tools. Its views shaped the reform side of the math wars.
Procedural Fluency in Mathematics (position statement)NCTM · 2023 · Guide · FreeNCTM's view that fluency means using procedures accurately, efficiently and flexibly, built on understanding. Useful to see how teacher groups define the fluency that the timed-tests debate is about.
Singapore mathWikipedia · Article · Free · Ages 5–8Overview of the Singapore approach: fewer topics in more depth, a concrete-pictorial-abstract sequence, and bar models for word problems. Explains why US programs like Math in Focus borrow it.
Math in FocusHoughton Mifflin Harcourt · App · Paid · Ages 5–8A US K-8 curriculum adapted from Singapore's materials, built on concrete-pictorial-abstract teaching. Evidence for ESSA lists it as having strong evidence. Sold to schools by its publisher.
Singapore Math (Primary Mathematics, Dimensions Math)Singapore Math Inc. · App · Paid · Ages 5–8Publisher of Singapore-style workbooks popular with homeschoolers and some schools, including Dimensions Math and Primary Mathematics. Books are inexpensive; parent guides help adults teach bar models.
Common Core State Standards for MathematicsCommon Core State Standards Initiative · 2010 · Guide · FreeThe grade-by-grade math standards many US states use or adapted. Reading your child's grade page shows what school expects, including why kids draw models before using the standard method.
The Science of MathThe Science of Math (educator group) · Organization · FreeEducator-led group pushing for explicit instruction, practice and fact fluency based on cognitive science, in the spirit of the science of reading. A clear voice on the traditional side of the debates.

Math wars & debates

Fluency Without FearJo Boaler, youcubed · 2015 · Article · Free · Ages 5–8Boaler's widely shared argument that timed fact tests cause anxiety and that facts are better learned through number sense and games. Influential but contested; critics say it overstates the evidence against timed practice.
Timed Tests and the Development of Math Anxiety (opinion)Education Week · 2012 · Article · Free + paid · Ages 5–8An opinion piece arguing timed tests can trigger math anxiety in young children. One side of the timed-tests debate; research on short, low-stakes timed practice is more mixed.
★PROOF POINTS: Stanford's Jo Boaler talks about her new book 'MATH-ish' and takes on her criticsJill Barshay, The Hechinger Report · 2024 · Article · FreeBalanced explainer of the fight over Boaler's work: what she argues, what cognitive scientists counter about practice and memorization, and the 2024 citation complaint, which Stanford later declined to investigate.
California math matters: Stanford looks into complaints on professor Jo BoalerThe Stanford Daily · 2024 · Article · FreeStudent newspaper report on the anonymous 2024 complaint alleging that Boaler misrepresented research citations, including in the California framework, and her response. Allegations, not findings.
Public Comments on the CMFBrian Conrad, Stanford mathematician · 2023 · Blog · Free · TeensA Stanford math professor's detailed public comments criticizing the California Math Framework's citations, data science pathway and guidance on algebra. Strongly argued; the most-cited critic's own words.
California adopts controversial changes to math educationThe San Francisco Standard · 2023 · Article · Free · TeensNews report on the State Board's 2023 adoption of the revised framework after years of fights over 8th grade algebra, data science and detracking. Good quick context.
Math warsWikipedia · Article · FreeHistory of the long US argument between "traditional" math (procedures, practice) and "reform" math (discovery, concepts) since the 1989 NCTM standards. Helps parents decode school debates.
The Nation's Report Card: Mathematics 2024NAEP, National Center for Education Statistics · 2025 · Data · Free · Ages 8–12Official US results for 4th and 8th grade math. Scores remained below pre-pandemic levels and the gap between high and low scorers grew. The backdrop for current math policy fights.

Program efficacy

★Evidence for ESSA: Math programsCenter for Research and Reform in Education, Johns Hopkins · Tool · FreeIndependent, free database rating math programs by the strength of their research under federal ESSA rules. Lists Zearn, DreamBox, ST Math, Math in Focus and many tutoring models. The best place to check a vendor's claims.
Zearn: Efficacy Research StudiesZearn · Report · Free · Ages 5–8Zearn's research page, including a RAND randomized trial in Texas (small gains on MAP, not significant on the state test) and a Johns Hopkins study in Louisiana. Usage studies on the page appear to be Zearn's own. Zearn is a nonprofit.
Zearn MathZearn (nonprofit) · App · Free + paid · Ages 5–8Online K-8 math lessons with videos, digital manipulatives and practice that track grade-level curricula. Free for teachers and families at the basic level; schools pay for more. Rated strong by Evidence for ESSA.
Harvard Study Suggests DreamBox Math Impact on Student Learning OutcomesDiscovery Education (vendor summary of Harvard CEPR study) · 2016 · Report · Free · Ages 5–8Vendor summary of a Harvard Center for Education Policy Research study of about 3,000 students: more weekly DreamBox time was linked to higher MAP scores. Not a randomized trial; DreamBox is now owned by Discovery Education.
DreamBox MathDreamBox by Discovery Education · App · Paid · Ages 5–8Adaptive K-8 math program that adjusts lessons as a child works, with lots of virtual manipulatives. Mostly sold to schools. Rated strong by Evidence for ESSA for grades K-5.
ST MathMIND Education (nonprofit) · App · Paid · Ages 5–8Visual, puzzle-based PreK-8 math program in which a penguin named JiJi moves through problems with little text. Rated strong for grades 3-6 by Evidence for ESSA. Mainly for schools; a homeschool version exists.
Khan AcademyKhan Academy (nonprofit) · App · FreeFree videos and mastery practice from early math through calculus, plus Khanmigo, its AI tutor. Widely used in US schools. A good free default for practice at home.
Multiple Studies Show Khan Academy Drives Learning GainsKhan Academy (vendor) · 2026 · Report · FreeKhan Academy's own roundup of studies, sorted by ESSA tier, including outside randomized trials in the US, India, Brazil and El Salvador. Effects are mostly small and depend on regular use. Written by the organization.
IXL ResearchIXL Learning (vendor) · Report · FreeIXL's research page with studies comparing schools that use IXL heavily to similar schools. These are mostly commissioned or company studies and correlational, not randomized trials. IXL itself is a large paid practice site.
IXLIXL Learning · App · PaidLarge paid practice site with thousands of skills from preschool to high school, with instant feedback and a score that rises as you get questions right. Some families find the score system stressful; it suits short, regular sessions.
Prodigy MathProdigy Education · Game · Free + paid · Ages 5–8Fantasy role-playing game where kids answer math questions to battle and level up. Free to play, with a paid membership for extras. Kids love it; how much math happens per minute varies.
Prodigy: Education ResearchProdigy Education (vendor) · Report · Free · Ages 5–8Prodigy's own research page and state efficacy white papers, such as a California report that heavy users outperformed non-users. These are company reports comparing users, which cannot rule out that stronger students play more.
Advocates to FTC: Prodigy Math Game Preys On Kids and FamiliesFairplay (Campaign for a Commercial-Free Childhood) · 2021 · Article · Free · Ages 5–8A coalition of 22 groups asked the FTC to investigate Prodigy's upselling to kids and its learning claims, saying kids could see more ads than math questions. An advocacy complaint, not a ruling; worth reading before choosing.

Enrichment, games & competitions

Beast AcademyArt of Problem Solving · App · Paid · Ages 5–8Comic-book style math curriculum for ages 6-13, online or in books, aimed at kids who like puzzles and challenge. Harder than typical school math, with an emphasis on reasoning over speed.
Art of Problem SolvingArt of Problem Solving · Organization · Some free · Ages 8–12Books, online classes and community for math-loving middle and high schoolers, with deep problem solving and contest preparation. Many US competition winners came through AoPS.
MathCircles.orgNational Association of Math Circles · Community · Free · Ages 8–12Directory and resources for math circles, informal groups where mathematicians and kids explore puzzles together. Find a local circle or materials to start one with other families.
Julia Robinson Mathematics FestivalJRMF · Organization · Some free · Ages 8–12Nonprofit that runs noncompetitive math festivals with hands-on puzzle tables for K-12 kids, in person and online. A friendly way to show kids math is more than worksheets.
MATHCOUNTSMATHCOUNTS Foundation · Organization · Some free · Ages 8–12US middle school math programs: a competition series, a noncompetitive club program, and a video challenge. Free problem sets are on the site.
Mathematical KangarooWikipedia · Article · FreeOverview of Math Kangaroo, a large international multiple-choice contest for grades 1-12 held each spring, known for puzzle-like questions young kids can try. The US organizer's own site did not load during this check.
★NRICHUniversity of Cambridge · Tool · FreeFree, huge library of rich math problems, games and investigations for ages 3-18, with teacher and parent notes. Excellent for kids who want to think, not just practice.
NumberblocksBBC CBeebies / Blue Zoo · 2017 · Video · Free · Ages 0–5Animated preschool series where characters made of blocks are numbers, showing how numbers combine and split. Made with math education advisers, and free on BBC iPlayer and YouTube in many places.
Peg + CatPBS KIDS · Video · Free · Ages 0–5PBS preschool show and games in which Peg and her cat solve problems with counting, shapes and simple math. Free and ad-free on PBS KIDS.
MathigonMathigon (now part of Amplify) · Tool · Free · Ages 8–12Free interactive math textbook and "mathematical playground" with lovely visuals for middle and high school topics. Its virtual manipulatives tool, Polypad, is a separate free site.
PolypadAmplify (originally Mathigon) · Tool · FreeFree virtual manipulatives canvas: base-ten blocks, fraction bars, pattern blocks, algebra tiles and more. Good for showing a child the picture step between objects and symbols.
DesmosDesmos Studio · Tool · Free · TeensFree online graphing and scientific calculators used in many state tests, plus a geometry tool. Kids can make art from equations. The classroom lessons moved to Amplify in 2022.
BrilliantBrilliant · App · Free + paid · TeensPaid app of short, interactive lessons in math, logic and coding, suited to curious teens and adults. Emphasizes problem solving over drills.
Math AcademyMath Academy · App · Paid · Ages 8–12Paid adaptive online course for motivated students from about grade 4 through university math, built on spaced review and mastery. Grew out of a Pasadena school program for advanced students.
DragonBoxDragonBox (Kahoot!) · App · Paid · Ages 5–8Paid puzzle apps that sneak in algebra, geometry and number ideas through games for young kids and up. The algebra app lets children balance equations before they know the symbols.
Prime ClimbMath for Love (Dan Finkel & Katherine Cook) · 2014 · Game · Paid · Ages 8–12Colorful board game for about ages 8+ where you add, subtract, multiply and divide to move pawns, with prime factors shown in colors. From Math for Love, which also makes Tiny Polka Dot for younger kids.
Do the benefits of chess instruction transfer to academic and cognitive skills? A meta-analysisSala & Gobet, Educational Research Review · 2016 · Meta-analysis · Paid · Ages 5–8Pooled studies of chess lessons in school. Average effects on math and thinking looked positive but small, and were weaker in better-designed studies. Play chess for fun, not as a math shortcut. Paywalled.
ChessKidChess.com · App · Free + paid · Ages 5–8Kid-safe chess site and app with lessons, puzzles and moderated games. Chess builds patience and planning; do not expect it to raise math scores by itself.
SolveMe PuzzlesEducation Development Center · Tool · Free · Ages 8–12Free puzzles, including balance mobiles, that build algebraic thinking without symbols first. Children find the weight of each shape, which is algebra in disguise.
Which One Doesn't Belong?Mary Bourassa and contributors · Tool · FreeFree sets of four pictures, shapes or numbers where every choice can be the odd one out. Great dinner-table prompts because there is no single right answer, only good reasons.
Estimation 180Andrew Stadel · Tool · Free · Ages 8–12A free photo-based estimation challenge for every school day: how many, how tall, how heavy. Builds number sense and is quick enough for breakfast.
Open MiddleRobert Kaplinsky & Nanette Johnson · Tool · Free · Ages 8–12Free problems with one answer but many ways to get there, often using the digits 1-9 once. Pushes kids past rote procedure into reasoning, sorted by grade.

Free practice & printables

Free Math WorksheetsK5 Learning · Tool · Free · Ages 5–8Printable worksheets from kindergarten to grade 6, sorted by grade and topic with answer keys. Handy for quick, short practice; the company also sells a paid online program.
Math-DrillsMath-Drills.com · Tool · Free · Ages 5–8Thousands of free printable math worksheets, from fact drills to fractions, with answer keys. Useful for short untimed fact practice at home.
Math is FunRod Pierce · Tool · FreeLong-running free site that explains math topics simply, with examples, puzzles and practice. A good place for a parent to brush up before helping with homework.
Math Learning Center AppsThe Math Learning Center · Tool · Free · Ages 5–8Free virtual manipulative apps: number racks, number lines, fractions, geoboards, money and more. Works in a browser and is great for showing ideas visually.
Illustrative MathematicsIllustrative Mathematics · Tool · Some freeProblem-based K-12 math curriculum whose core materials are openly licensed and free to view. If your child's school uses it, the family pages explain each unit.
Eureka MathGreat Minds · Tool · Some free · Ages 5–8Widely used K-12 curriculum; original Eureka Math materials and parent tip sheets are free to download. Helps parents understand homework that uses tape diagrams and number bonds.
PBS KIDS Math GamesPBS KIDS · Game · Free · Ages 0–5Free, ad-free math games from PBS shows like Peg + Cat, Odd Squad and Cyberchase, covering counting, shapes, measuring and patterns for preschool and early elementary kids.
Duolingo MathDuolingo · 2022 · App · Free + paid · Ages 5–8Short, game-style math practice inside the Duolingo app, from elementary arithmetic up. Free with ads or paid for extras. Good for brief daily practice; no independent efficacy studies yet.
SplashLearnSplashLearn · App · Free + paid · Ages 5–8Game-based math and reading practice for preschool to grade 5. Free for teachers; families get a limited free tier and pay for full access.
Math AnticsMath Antics · Video · Free + paid · Ages 8–12Clear, friendly videos explaining basic math, from place value to fractions and pre-algebra, with worksheets. Free on YouTube; extras are paid.

Blogs, books, podcasts & video

Teaching Math With ExamplesMichael Pershan · 2021 · Book · Paid · Ages 8–12A teacher's practical book (John Catt) on using worked examples well in math class, drawing on cognitive science. His old blog "Teaching With Problems" is closed; this homepage lists the book.
Denise Gaskins' Let's Play MathDenise Gaskins · Blog · Free · Ages 5–8Homeschooling mom and author shares math games, living-math books and playful activities for families, plus her books of math games. Warm and practical.
Math with Bad DrawingsBen Orlin · Blog · Free · TeensFunny, insightful blog by a math teacher and author of popular books that explain math through stick figures. Good for teens and parents who think math is dull.
Mr Barton MathsCraig Barton · Podcast · Free · Ages 8–12UK teacher's podcast and free resources centered on worked examples, explicit teaching and cognitive science. Author of "How I Wish I'd Taught Maths."
Math Ed PodcastSamuel Otten · Podcast · FreeLong-running interviews with math education researchers about their studies. Good for parents who want to hear the research from the people who did it.
3Blue1BrownGrant Sanderson · Video · Free · TeensBeautifully animated videos that explain why math works, from fractions-level puzzles to calculus and neural networks. Free; best for curious teens.
NumberphileBrady Haran · Video · Free · TeensFree YouTube series of mathematicians chatting about numbers, puzzles and famous problems. Shows kids that math has stories and personalities.
Mathematical MindsetsJo Boaler · 2022 · Book · PaidPopular book (2nd edition) arguing for open tasks, mistakes as learning and less speed pressure. Inspiring to many teachers and parents; critics say some research claims are overstated, so read it alongside the debates.
How Not to Be WrongJordan Ellenberg · 2014 · Book · Paid · TeensBestselling book by a mathematician showing how math thinking helps with everyday questions, from lotteries to polls. For parents and older teens who want to see why math matters.
Math CurseJon Scieszka & Lane Smith · 1995 · Novel · Paid · Ages 5–8Funny picture book about a girl who starts seeing every part of her day as a math problem. A favorite read-aloud for ages 6-10 and a good opener for math talk.
The Number DevilHans Magnus Enzensberger · 1997 · Novel · Paid · Ages 8–12A novel in which a boy who hates math meets a devil in his dreams who shows him primes, triangles and infinity. Well-loved for ages 10 and up.
Sir Cumference Math AdventuresCindy Neuschwander, Charlesbridge · Novel · Paid · Ages 5–8Picture-book series set in a punny King Arthur world that explains geometry and measurement ideas through stories. Good for ages 7-10.