References

37 sources. What each found, how it is used, and which activities rest on it.

This page exists so the research behind the site can be checked rather than taken on trust. Every activity page links to the citations behind it, and every citation below links back to the activities built on it. Where a source is a practitioner document or a null result rather than a trial, it says so — nothing here borrows authority from the references around it.

The headline finding worth repeating: the strongest evidence in primary maths attaches to instructional practices, not to any brand of curriculum. That is why the design anchors on the WWC practice guide, and treats Illustrative Mathematics as a well-built plan for topic order rather than as proof of outcomes.

How to read the strength labels

Strong	Randomised trials or a WWC strong-evidence rating.	12 sources
Moderate	Real trials, smaller or narrower, or a meta-analysis with caveats.	12 sources
Limited	One study, an open trial, or correlational evidence.	2 sources
Design source	A curriculum, protocol or task specification — not an efficacy claim.	8 sources
Null result	Evidence that something does NOT work. Shapes what we avoid.	3 sources

Curriculum map

Which Illustrative Mathematics unit each activity belongs to. Unit links go straight to that unit’s lesson list.

Kindergarten

IM kindergarten scope and sequence →

Activity	IM unit	Underlying idea
◈ Counting CrewK	Unit 2: Numbers 1–10 Unit 6: Numbers 0–20	Counting, cardinality and ordinality are separable skills that develop separately.
◈ Number FriendsK	Unit 4: Understanding Addition and Subtraction Unit 5: Composing and Decomposing Numbers to 10	Part-whole structure: a quantity is composed of parts, and the bond makes that visible.
◈ Story TimeK	Unit 4: Understanding Addition and Subtraction Unit 5: Composing and Decomposing Numbers to 10	A story problem has a structure — something joins, something leaves, or two parts make a whole — and the structure is what is being learned.
◈ Shape SorterK	Unit 3: Flat Shapes All Around Us Unit 7: Solid Shapes All Around Us	A shape is defined by its properties, not its orientation.
◉ The Great RaceK	Unit 2: Numbers 1–10 Unit 6: Numbers 0–20	A linear board maps number onto space, and counting on encodes magnitude rather than moves.
◉ Ten-Frame FlashK	Unit 2: Numbers 1–10 Unit 5: Composing and Decomposing Numbers to 10	Conceptual subitizing: seeing a quantity as composed groups rather than counting units.
◉ Which Is MoreK	Unit 2: Numbers 1–10 Unit 6: Numbers 0–20	The symbolic numerical distance effect — closer numbers are harder to compare.

1st grade

IM 1st grade scope and sequence →

Activity	IM unit	Underlying idea
◈ Adding to Twenty1	Unit 1: Adding, Subtracting, and Working with Data Unit 2: Addition and Subtraction Story Problems Unit 3: Adding and Subtracting Within 20	Crossing ten is where counting on stops being efficient and a strategy must take over.
◈ All Kinds of Stories1	Unit 2: Addition and Subtraction Story Problems	Four structures cover almost every one-step story problem. A child who knows the structures can solve a problem whose wording they have never seen; a child who hunts for keywords cannot.
◈ Tens and Ones1	Unit 4: Numbers to 99 Unit 5: Adding Within 100	Place value as composed units: ten ones become one ten.
◈ Clocks and Rulers1	Unit 6: Length Measurements Within 120 Units Unit 7: Geometry and Time	Measuring is repeating a unit and counting the repeats — the same composed-unit idea as place value, in a different dress.
◈ Halves and Quarters1	Unit 7: Geometry and Time	Equal partitioning precedes fraction notation.
◉ Number Line Hop1	Unit 4: Numbers to 99 Unit 6: Length Measurements Within 120 Units	Number line estimation: mapping a numeral onto a spatial position.
◉ Make Ten Race1	Unit 3: Adding and Subtracting Within 20	Automaticity on pairs to ten frees working memory for everything built on them.

2nd grade

IM 2nd grade scope and sequence →

Activity	IM unit	Underlying idea
◈ Place Value Palace2	Unit 5: Numbers to 1,000 Unit 7: Adding and Subtracting within 1,000	Place value determines magnitude; digit-by-digit comparison is the classic failure.
◈ Carry and Borrow2	Unit 2: Adding and Subtracting within 100 Unit 7: Adding and Subtracting within 1,000	Regrouping as composing and decomposing a unit of ten.
◈ Measure and Chart2	Unit 3: Measuring Length Unit 6: Geometry, Time, and Money	A bar chart is a set of number lines standing up. Reading one is a magnitude comparison with a scale attached.
◈ Arrays and Equal Groups2	Unit 8: Equal Groups	The array turns repeated addition into a two-dimensional structure.
◉ Close to a Hundred2	Unit 1: Adding, Subtracting, and Working with Data Unit 2: Adding and Subtracting within 100 Unit 7: Adding and Subtracting within 1,000	A target-number game turns arithmetic into a decision. The child must estimate before committing, which is what makes place value matter rather than just being recited.
◉ Hundred Line Hop2	Unit 4: Addition and Subtraction on the Number Line Unit 5: Numbers to 1,000	Landmark-based estimation rather than counting.
◉ Decade Duel2	Unit 5: Numbers to 1,000	Place-value comparison, and the specific traps that defeat digit-by-digit reading.

3rd grade

IM 3rd grade scope and sequence →

Activity	IM unit	Underlying idea
◈ Times Table Tower3	Unit 1: Introducing Multiplication Unit 4: Relating Multiplication to Division	Multiplicative structure, ordered by real difficulty rather than by table.
◈ Fraction Number Line3	Unit 5: Fractions as Numbers	Fraction as a number with a position, not as one whole number over another.
◈ Area and Perimeter3	Unit 2: Area and Multiplication Unit 7: Two-dimensional Shapes and Perimeter	Area is the array model measured; perimeter is a different quantity on the same figure.
◈ Round and Reckon3	Unit 3: Wrapping Up Addition and Subtraction Within 1,000	Rounding is estimation with a rule, and estimation is what lets a child notice their own wrong answer.
◈ Time and Data3	Unit 6: Measuring Length, Time, Liquid Volume, and Weight Unit 1: Introducing Multiplication	Elapsed time is a distance on a number line whose units are sixties. A scaled graph is a number line with a multiplier.
◉ Fact Family Forge3	Unit 4: Relating Multiplication to Division	Multiplication and division are one relation viewed from two directions.
◉ Array Architect3	Unit 1: Introducing Multiplication Unit 2: Area and Multiplication	Commutativity is visible in an array and invisible in a symbol string.

4th grade

IM 4th grade scope and sequence →

Activity	IM unit	Underlying idea
◈ Long Multiplication4	Unit 6: Multiplying and Dividing Multi-digit Numbers	Partial products, with the area model faded once the structure is understood.
◈ Equivalent Fractions4	Unit 2: Fraction Equivalence and Comparison	Equivalence: different names for the same position on the line.
◈ Angles and Lines4	Unit 7: Angles and Angle Measurement	An unknown deduced from a constraint rather than computed.
◈ Factor Forest4	Unit 1: Factors and Multiples	A factor pair is the two sides of a rectangle with that many squares. Primes are the numbers that can only be a single line.
◈ Times As Many4	Unit 5: Multiplicative Comparison and Measurement	Additive comparison asks how many more; multiplicative comparison asks how many times as many. Children routinely read the second as the first.
◉ Division Descent4	Unit 6: Multiplying and Dividing Multi-digit Numbers	Division as repeated subtraction of a composed unit.
◉ Decimal Drop4	Unit 4: From Hundredths to Hundred-thousands	Decimals and fractions as one idea in two notations, on one line.

5th grade

IM 5th grade scope and sequence →

Activity	IM unit	Underlying idea
◈ Fraction Foundry5	Unit 2: Fractions as Quotients and Fraction Multiplication Unit 3: Multiplying and Dividing Fractions Unit 6: More Decimal and Fraction Operations	Operating on fractions requires a common unit — the reason common denominators exist.
◈ Decimal Place5	Unit 5: Place Value Patterns and Decimal Operations Unit 6: More Decimal and Fraction Operations	Decimal place value, and the misconception that a longer decimal is larger.
◈ The Standard Algorithm5	Unit 4: Wrapping Up Multiplication and Division with Multi-Digit Numbers	The standard algorithm is the endpoint of a progression, not the starting point: it compresses partial products into a notation that only makes sense once the parts are understood.
◈ Volume and Space5	Unit 1: Finding Volume	Volume extends the area model into a third dimension.
◉ Mixed Number Line5	Unit 2: Fractions as Quotients and Fraction Multiplication Unit 3: Multiplying and Dividing Fractions	A fraction greater than one still has a single position on the line.
◉ Coordinate Quest5	Unit 7: Shapes on the Coordinate Plane	An ordered pair locates a point; the order is a convention that must be learned.

Activity to citation

Every activity and the sources it rests on.

Activity	Grade	Sources
◈ Counting CrewK	K	Illustrative Mathematics / Kendall Hunt (2024) · What Works Clearinghouse (2023) · Nelson et al. (2019) · Geary (2011)
◈ Number FriendsK	K	Illustrative Mathematics / Kendall Hunt (2024) · What Works Clearinghouse (2023) · Clements (1999)
◈ Story TimeK	K	Institute of Education Sciences / What Works Clearinghouse (2021) · Illustrative Mathematics / Kendall Hunt (2024) · What Works Clearinghouse (2023)
◈ Shape SorterK	K	Illustrative Mathematics / Kendall Hunt (2024)
◉ The Great RaceK	K	Siegler et al. (2009) · Siegler et al. (2008) · Laski et al. (2014) · Institute of Education Sciences / What Works Clearinghouse (2021) · Schneider et al. (2018)
◉ Ten-Frame FlashK	K	Clements (1999) · What Works Clearinghouse (2023) · Qiu et al. (2021)
◉ Which Is MoreK	K	Holloway et al. (2009) · Institute of Education Sciences / What Works Clearinghouse (2021) · Qiu et al. (2021)
◈ Adding to Twenty1	1	Illustrative Mathematics / Kendall Hunt (2024) · Fuchs et al. (2012) · Codding et al. (2011) · Geary (2011)
◈ All Kinds of Stories1	1	Institute of Education Sciences / What Works Clearinghouse (2021) · Illustrative Mathematics / Kendall Hunt (2024) · van der Kleij et al. (2015)
◈ Tens and Ones1	1	Illustrative Mathematics / Kendall Hunt (2024) · Fyfe et al. (2014)
◈ Clocks and Rulers1	1	Illustrative Mathematics / Kendall Hunt (2024)
◈ Halves and Quarters1	1	Illustrative Mathematics / Kendall Hunt (2024) · Fuchs et al. (2013)
◉ Number Line Hop1	1	Institute of Education Sciences / What Works Clearinghouse (2021) · Schneider et al. (2018) · Siegler et al. (2009)
◉ Make Ten Race1	1	Fuchs et al. (2012) · Codding et al. (2011) · van der Kleij et al. (2015)
◈ Place Value Palace2	2	Illustrative Mathematics / Kendall Hunt (2024) · Holloway et al. (2009)
◈ Carry and Borrow2	2	Illustrative Mathematics / Kendall Hunt (2024) · Rohrer et al. (2020) · Barton (2018)
◈ Measure and Chart2	2	Illustrative Mathematics / Kendall Hunt (2024) · Institute of Education Sciences / What Works Clearinghouse (2021)
◈ Arrays and Equal Groups2	2	Illustrative Mathematics / Kendall Hunt (2024) · youcubed et al. (2015) · Fyfe et al. (2014)
◉ Close to a Hundred2	2	youcubed et al. (2015) · Illustrative Mathematics / Kendall Hunt (2024) · Barton (2018)
◉ Hundred Line Hop2	2	Institute of Education Sciences / What Works Clearinghouse (2021) · Schneider et al. (2018) · Geary (2011)
◉ Decade Duel2	2	Holloway et al. (2009) · Institute of Education Sciences / What Works Clearinghouse (2021)
◈ Times Table Tower3	3	Illustrative Mathematics / Kendall Hunt (2024) · Codding et al. (2011) · Fyfe et al. (2014) · Fuchs et al. (2012)
◈ Fraction Number Line3	3	Fuchs et al. (2013) · What Works Clearinghouse (2020) · Institute of Education Sciences / What Works Clearinghouse (2021) · Schneider et al. (2018)
◈ Area and Perimeter3	3	Illustrative Mathematics / Kendall Hunt (2024) · youcubed et al. (2015)
◈ Round and Reckon3	3	Illustrative Mathematics / Kendall Hunt (2024) · Institute of Education Sciences / What Works Clearinghouse (2021) · Schneider et al. (2018)
◈ Time and Data3	3	Illustrative Mathematics / Kendall Hunt (2024) · Schneider et al. (2018)
◉ Fact Family Forge3	3	Illustrative Mathematics / Kendall Hunt (2024) · Codding et al. (2011)
◉ Array Architect3	3	youcubed et al. (2015) · Fyfe et al. (2014)
◈ Long Multiplication4	4	Illustrative Mathematics / Kendall Hunt (2024) · Fyfe et al. (2014) · Barton (2018)
◈ Equivalent Fractions4	4	Fuchs et al. (2013) · What Works Clearinghouse (2020) · Institute of Education Sciences / What Works Clearinghouse (2021)
◈ Angles and Lines4	4	Illustrative Mathematics / Kendall Hunt (2024)
◈ Factor Forest4	4	Illustrative Mathematics / Kendall Hunt (2024) · youcubed et al. (2015) · Fyfe et al. (2014)
◈ Times As Many4	4	Institute of Education Sciences / What Works Clearinghouse (2021) · Illustrative Mathematics / Kendall Hunt (2024)
◉ Division Descent4	4	Illustrative Mathematics / Kendall Hunt (2024) · Codding et al. (2011) · Fuchs et al. (2012)
◉ Decimal Drop4	4	Institute of Education Sciences / What Works Clearinghouse (2021) · Schneider et al. (2018) · Riconscente (2013)
◈ Fraction Foundry5	5	Fuchs et al. (2013) · What Works Clearinghouse (2020) · Institute of Education Sciences / What Works Clearinghouse (2021)
◈ Decimal Place5	5	Illustrative Mathematics / Kendall Hunt (2024) · Holloway et al. (2009)
◈ The Standard Algorithm5	5	Illustrative Mathematics / Kendall Hunt (2024) · Fyfe et al. (2014) · Codding et al. (2011) · Fuchs et al. (2012)
◈ Volume and Space5	5	Illustrative Mathematics / Kendall Hunt (2024) · Fyfe et al. (2014)
◉ Mixed Number Line5	5	Fuchs et al. (2013) · Schneider et al. (2018) · Institute of Education Sciences / What Works Clearinghouse (2021) · Riconscente (2013)
◉ Coordinate Quest5	5	Illustrative Mathematics / Kendall Hunt (2024)

The sources

Ordered by how much weight each can carry.

Bailey, Duncan, Cunha, Foorman & Yeager (2020)

Strong Review

Persistence and fade-out of educational intervention effects

Psychological Science in the Public Interest · link

What it found: Building Blocks’ 0.63 SD end-of-preschool impact retained only about 40% by the end of grade 1 and nearly nothing by grade 4. Decay of roughly 60% per year is typical.
How Izzi Math uses it: Why we make modest claims, and why the site is built for regular ongoing use rather than as a one-off intervention.
See it on: How to help — short and often About

Fuchs, Geary, Compton, Fuchs, Schatschneider & Hamlett (2012)

Strong Randomised trial

Effects of first-grade number knowledge tutoring with contrasting forms of practice

Journal of Educational Psychology 105(1), 58–77 · link

What it found: 648 at-risk first graders randomised to identical tutoring differing only in the final five minutes: speeded versus non-speeded practice. The speeded condition produced ES 0.51 on arithmetic, with NO difference in attitudes, motivation or effort.
How Izzi Math uses it: The cleanest experiment in the timed-practice debate, and the reason our position is not "timers are harmful". Short, self-paced, low-stakes speed practice on already-learned content is beneficial.
Practise it 5: ◈ Adding to Twenty1 ◉ Make Ten Race1 ◈ Times Table Tower3 ◉ Division Descent4 ◈ The Standard Algorithm5

Fuchs, Schumacher, Long, Namkung, Hamlett & Cirino (2013)

Strong Randomised trial

Improving at-risk learners’ understanding of fractions

Journal of Educational Psychology 105(3), 683–700 · doi:10.1037/a0032446

What it found: 259 at-risk fourth graders randomised. A measurement-interpretation (number line) fractions intervention beat a part-whole/procedural control.
How Izzi Math uses it: Why fractions are introduced on the number line rather than as shaded pizza slices — fraction-as-a-number is the interpretation that transfers.
Practise it 5: ◈ Halves and Quarters1 ◈ Fraction Number Line3 ◈ Equivalent Fractions4 ◈ Fraction Foundry5 ◉ Mixed Number Line5

Institute of Education Sciences / What Works Clearinghouse (2021)

Strong Practice guide

Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades

IES Educator’s Practice Guide WWC 2021006 · link

What it found: The only WWC maths guide in which all six recommendations reach STRONG evidence. Study counts: systematic instruction 43, representations 28, timed activities 27, word problems 18, mathematical language 16, number lines 14. Number line meta-analytic effects: rational-number computation 1.46, rational-number magnitude 1.00, whole-number computation 0.62, general achievement 0.34 — but whole-number magnitude understanding −0.05 (not significant, k=1).
How Izzi Math uses it: The backbone of the whole site. It is also the reason we treat practices as the unit of evidence rather than curricula. Note the null: the number line is overwhelmingly validated for fractions and for computation, and essentially untested for whole-number magnitude.
Practise it 15: ◈ Story TimeK ◉ The Great RaceK ◉ Which Is MoreK ◈ All Kinds of Stories1 ◉ Number Line Hop1 ◈ Measure and Chart2 ◉ Hundred Line Hop2 ◉ Decade Duel2 ◈ Fraction Number Line3 ◈ Round and Reckon3 ◈ Equivalent Fractions4 ◈ Times As Many4 ◉ Decimal Drop4 ◈ Fraction Foundry5 ◉ Mixed Number Line5

Laski & Siegler (2014)

Strong Randomised trial

Learning from number board games: you learn what you encode

Developmental Psychology · link

What it found: The only manipulation was counting on from the token’s position versus counting from one. Count-on produced roughly double the gains (number line error ~21%→~14%).
How Izzi Math uses it: Why The Great Race rejects "1, 2" and corrects it to "count on from 3: 4, 5". This is enforced in the boardmove problem type rather than left as advice.
Practise it 1: ◉ The Great RaceK

Petersen & McNeil (2013)

Strong Randomised trial

Effects of perceptually rich manipulatives on preschoolers’ counting performance: established knowledge counts

Child Development 84(3), 1020–1033 · doi:10.1111/cdev.12028

What it found: 133 preschoolers, crossing perceptual richness with established knowledge of the objects. Rich objects HELPED when knowledge of the object was low and HINDERED when it was high.
How Izzi Math uses it: The direct reason a character never changes the countable units inside a manipulative — the children most attached to Georgie are exactly the ones tennis-ball counters would hurt. Enforced by scripts/check.mjs.
Also shaped 4: ◉ Ten-Frame FlashK ◈ Number FriendsK ◈ Arrays and Equal Groups2 ◈ Counting CrewK
See it on: Which parts are evidence, and which are taste

RAND Corporation (2024)

Strong Randomised trial

Evaluation of Zearn Math

Evidence for ESSA / RAND RCT 2022–24 · link

What it found: 64 schools, around 10,000 students in grades 3–5. +0.11 SD on NWEA MAP (+0.13 for students starting below proficiency), and +0.07 non-significant on the Texas STAAR. Average across its three studies is +0.09.
How Izzi Math uses it: Our realistic ceiling. The best-evidenced digital maths programmes move attainment by about a tenth of a standard deviation, so any claim of a grade level from a light-touch home product is overselling. Stated as such on the parents page.
See it on: How to help — a realistic expectation

Rohrer, Dedrick, Hartwig & Cheung (2020)

Strong Randomised trial

A randomized controlled trial of interleaved mathematics practice

Journal of Educational Psychology 112(1), 40–52 · doi:10.1037/edu0000367

What it found: Preregistered cluster RCT, 787 students, 54 classes. Four months of interleaved versus blocked assignments, then an unannounced test a month later: 61% versus 38%, d = 0.83 [0.68, 0.97]. Same problems, same total practice — only the ordering differed. Worksheets held exactly eight problems.
How Izzi Math uses it: The specification for mixed review sheets: eight problems, arranged so no two consecutive problems need the same method. Kindergarten and grade 1 sheets hold five instead, because eight problems at the type size those grades use do not fit on one page. Caveat we state openly: this was grade 7, so applying it to K–5 is an inference from mechanism either way.
Practise it 1: ◈ Carry and Borrow2

Schneider, Merz, Stricker, De Smedt, Torbeyns, Verschaffel & Luwel (2018)

Strong Meta-analysis

Associations of number line estimation with mathematical competence: a meta-analysis

Child Development 89(5) · doi:10.1111/cdev.13068

What it found: Number line estimation correlates r = .443 with mathematical competence across 263 effect sizes and 10,576 participants aged 4–14. Stronger for fractions than for whole numbers.
How Izzi Math uses it: Why the number line is the spine of the site rather than one activity among many.
Practise it 8: ◉ The Great RaceK ◉ Number Line Hop1 ◉ Hundred Line Hop2 ◈ Fraction Number Line3 ◈ Round and Reckon3 ◈ Time and Data3 ◉ Decimal Drop4 ◉ Mixed Number Line5

Siegler & Ramani (2009)

Strong Randomised trial

Playing linear number board games — but not circular ones — improves low-income preschoolers’ numerical understanding

Journal of Educational Psychology · link

What it found: N=88, four sessions totalling about an hour. Number line estimation error 29%→21% (d=1.01). The identical game on a CIRCULAR board: 29%→26% (d=0.43). Magnitude comparison 68%→77% (d=0.75). Gains held at 9 weeks.
How Izzi Math uses it: The exact specification for The Great Race: ten equal squares, numerals increasing left to right, a 1-or-2 spinner. The circular-board comparison is why no number line on this site ever loops.
Practise it 2: ◉ The Great RaceK ◉ Number Line Hop1

What Works Clearinghouse (2020)

Strong Evidence report

Fraction Face-Off! intervention report

WWC, March 2020 · link

What it found: Potentially positive effects in all three domains reviewed. Improvement index +33 (geometry and measurement), +31 (number and operations), +24 (general achievement). Author-reported effects around 1 SD on number line estimation and 1–2.5 SD on fraction calculation.
How Izzi Math uses it: The closest thing to a validated blueprint for what this site is. Its lesson shape — warm-up, explicit instruction, speed game, worksheet — is the book-plus-game-plus-printable model, already tested.
Practise it 3: ◈ Fraction Number Line3 ◈ Equivalent Fractions4 ◈ Fraction Foundry5

van der Kleij, Feskens & Eggen (2015)

Strong Meta-analysis

Effects of feedback in a computer-based learning environment on students’ learning outcomes: a meta-analysis

Review of Educational Research 85(4), 475–511 · doi:10.3102/0034654314564881

What it found: 40 studies, 70 effect sizes. Elaborated feedback g = 0.49; giving the correct answer g = 0.32; knowledge of correctness alone g = 0.05. Effects larger in mathematics than in any other subject.
How Izzi Math uses it: Every problem carries a worked explanation, and it is shown after a WRONG answer as well as a right one. Showing it only on success was the g = 0.05 case, and it was a bug.
Practise it 2: ◈ All Kinds of Stories1 ◉ Make Ten Race1

Codding, Burns & Lukito (2011)

Moderate Meta-analysis

Meta-analysis of mathematic basic-fact fluency interventions: a component analysis

Learning Disabilities Research & Practice 26(1), 36–47 · doi:10.1111/j.1540-5826.2010.00323.x

What it found: Drill and practice WITH MODELLING produced the largest effects. More than three components beat fewer than three.
How Izzi Math uses it: Why every game names and models a strategy before play rather than simply presenting items faster.
Practise it 6: ◈ Adding to Twenty1 ◉ Make Ten Race1 ◈ Times Table Tower3 ◉ Fact Family Forge3 ◉ Division Descent4 ◈ The Standard Algorithm5

Dyson, Jordan & Glutting (2013)

Moderate Randomised trial

A number sense intervention for low-income kindergartners at risk for mathematics difficulties

Journal of Learning Disabilities · link

What it found: N=121, eight weeks, 24 sessions of 30 minutes. Significant and sustained gains on every Number Sense Brief subscale. But Woodcock-Johnson Calculation was significant at posttest only, and Applied Problems was not significant at either point.
How Izzi Math uses it: Shows that early number-sense work reliably improves number sense, and that transfer distance is short. We describe activities in terms of the skill they train rather than promising general gains.
Also shaped 4: ◈ Counting CrewK ◈ Number FriendsK ◉ Ten-Frame FlashK ◈ Story TimeK

Fyfe, McNeil, Son & Goldstone (2014)

Moderate Review

Concreteness fading in mathematics and science instruction: a systematic review

Educational Psychology Review 26(1), 9–25 · doi:10.1007/s10648-014-9249-3

What it found: Specifies the three-step enact → iconic → abstract progression over the same concept, and finds it beats staying concrete or starting abstract.
How Izzi Math uses it: Why the area model is visible on the first pages of Long Multiplication and withdrawn later, and why arrays appear before bare symbols in the times tables.
Practise it 8: ◈ Tens and Ones1 ◈ Arrays and Equal Groups2 ◈ Times Table Tower3 ◉ Array Architect3 ◈ Long Multiplication4 ◈ Factor Forest4 ◈ The Standard Algorithm5 ◈ Volume and Space5

Geary (2011)

Moderate Correlational

Cognitive predictors of achievement growth in mathematics: a five-year longitudinal study

Developmental Psychology 47(6), 1539–1552 · doi:10.1037/a0025510

What it found: 177 children from grade 1 to grade 5, controlling IQ, working memory and processing speed. The unique predictors of achievement were fluency with set size and numerals, sophisticated counting procedures, and number line placement.
How Izzi Math uses it: Why fluency and number line placement both get their own activities rather than being treated as incidental.
Practise it 3: ◈ Counting CrewK ◈ Adding to Twenty1 ◉ Hundred Line Hop2

Holloway & Ansari (2009)

Moderate Correlational

Mapping numerical magnitudes onto symbols: the numerical distance effect and individual differences in children’s mathematics achievement

Journal of Experimental Child Psychology 103, 17–29 · doi:10.1016/j.jecp.2008.04.001

What it found: The SYMBOLIC numerical distance effect, and not the non-symbolic one, relates to individual differences in six- to eight-year-olds’ maths achievement.
How Izzi Math uses it: Why Which Is More and Decade Duel compare NUMERALS, and why their difficulty is graded by numerical distance.
Practise it 4: ◉ Which Is MoreK ◈ Place Value Palace2 ◉ Decade Duel2 ◈ Decimal Place5

Nelson & McMaster (2019)

Moderate Meta-analysis

The effects of early numeracy interventions for students in preschool and early elementary grades: a motion of meta-analysis

Journal of Educational Psychology / related synthesis · link

What it found: 34 studies, 52 treatment groups, weighted g = 0.64 [0.52, 0.76]. Metaregression predicted LARGER effects for interventions including counting with one-to-one correspondence, and for durations of eight weeks or less.
How Izzi Math uses it: Why the guidance says short and focused, and why counting with one-to-one correspondence is explicit in the K activities rather than assumed.
Practise it 1: ◈ Counting CrewK

Nelson, Carter, Boedeker et al. (2024)

Moderate Meta-analysis

Mathematics interventions in informal learning environments: a meta-analysis

Review of Educational Research · link

What it found: 25 studies, 83 effect sizes, g = 0.26 [0.07, 0.45] for maths interventions delivered by caregivers. Significant moderators: intensity of caregiver training, and whether the intervention included follow-up support.
How Izzi Math uses it: The single most relevant reference for what this site actually is. It is why the "How to help" page exists and why every activity carries a note for the adult — handing over materials with no guidance picks the bottom of that interval.
See it on: How to help — the whole page exists because of this

Riconscente (2013)

Moderate Randomised trial

Results from a controlled study of the iPad fractions game Motion Math

Games and Culture · doi:10.1177/1555412013496894

What it found: 122 grade-5 students, randomised crossover, 20 minutes a day for five days: +15% on a fractions test (p<.001), plus +10% on attitude and self-efficacy.
How Izzi Math uses it: Evidence that a game helps when the mechanic IS the mathematical representation — here, a number line. It is the reason our games are number lines, arrays and comparisons rather than arithmetic wrapped in an unrelated reward loop.
Practise it 2: ◉ Decimal Drop4 ◉ Mixed Number Line5

Rohrer & Taylor (2006)

Moderate Randomised trial

The effects of overlearning and distributed practice on the retention of mathematics knowledge

Applied Cognitive Psychology · doi:10.1002/acp.1266

What it found: Three versus nine practice problems produced no difference at one or four weeks. Massed practice lost to distributed practice at the same total volume.
How Izzi Math uses it: Why sheets are short and why we tell parents that doing the same page again in a few days beats doing twice as many today.
Also shaped 2: ◈ Times Table Tower3 ◈ Carry and Borrow2
See it on: All printables — sheet length How to help — print it again later

Rohrer, Dedrick & Burgess (2014)

Moderate Randomised trial

The benefit of interleaved mathematics practice is not limited to superficially similar kinds of problems

Psychonomic Bulletin & Review 21, 1323–1330 · doi:10.3758/s13423-014-0588-3

What it found: 140 students, unannounced test two weeks later: interleaved 72% versus blocked 38%, d = 1.05 — and the benefit held even for problem types that were not superficially similar.
How Izzi Math uses it: Supports interleaving across genuinely different problem types on one sheet, not just near-neighbours.
Also shaped 3: ◈ Fraction Foundry5 ◈ Carry and Borrow2 ◈ Equivalent Fractions4
See it on: Every sheet has a mixed review version

Siegler & Ramani (2008)

Moderate Randomised trial

Playing linear numerical board games promotes low-income children’s numerical development

Developmental Science · link

What it found: The earlier experiment, whose control group played an identical game with squares varying in COLOUR rather than number. That control improved on nothing at all.
How Izzi Math uses it: Establishes that the numbers on the board are the active ingredient, not the game-ness of it. A decorative board game would do nothing.
Practise it 1: ◉ The Great RaceK

What Works Clearinghouse (2023)

Moderate Evidence report

Building Blocks for Math intervention report

WWC, December 2023 · link

What it found: Potentially positive effects on mathematics, Tier 2 moderate, 3,221 students, three studies meeting standards. The TRIAD scale-up produced +0.47 against another maths curriculum and +1.07 against no-treatment control.
How Izzi Math uses it: The best-evidenced early-maths curriculum there is, and the source of the subitizing and number-sense learning trajectories our K–1 content follows.
Practise it 4: ◈ Counting CrewK ◈ Number FriendsK ◈ Story TimeK ◉ Ten-Frame FlashK

Boaler & Confer (2015)

Limited Practitioner source

Fluency without fear

youcubed, Stanford · link

What it found: The primary statement of the anti-timed-testing position: time pressure blocks working memory for roughly a third of students, and fluency should be understood as flexibility rather than speed.
How Izzi Math uses it: Included because it is the most influential source on this question and shapes what parents expect. Read carefully, it cites no experiment that manipulated timing in children — its sources are adult working-memory studies and correlational anxiety work. We follow its design implications (nothing public, nothing graded, nothing compared) without adopting the claim that timing itself is harmful.
Also shaped 3: ◉ Make Ten Race1 ◉ Decade Duel2 ◉ Fact Family Forge3
See it on: How to help — on timers

Ramirez, Gunderson, Levine & Beilock (2013)

Limited Correlational

Math anxiety, working memory, and math achievement in early elementary school

Journal of Cognition and Development 14(2), 187–202 · doi:10.1080/15248372.2012.664593

What it found: 154 first and second graders. Maths anxiety related negatively to achievement only for children with HIGHER working memory — the children who rely on the most demanding strategies.
How Izzi Math uses it: Why we take anxiety seriously without assuming it only affects struggling children, and why there are no leaderboards or public scores anywhere on the site.
Also shaped 2: ◉ Make Ten Race1 ◉ Ten-Frame FlashK
See it on: How to help — do not compare them to anyone

Barton (2018)

Design source Practitioner source

Variation theory and intelligent practice

variationtheory.com / How I Wish I’d Taught Maths · link

What it found: Hold features constant and vary exactly one, so the connection between consecutive questions is visible. When everything changes from question to question, the pattern is invisible.
How Izzi Math uses it: Why generated problem sequences perturb one feature at a time inside a practice block, rather than randomising everything.
Practise it 3: ◈ Carry and Borrow2 ◉ Close to a Hundred2 ◈ Long Multiplication4

Clements (1999)

Design source Practitioner source

Subitizing: what is it? Why teach it?

Teaching Children Mathematics 5(7), 400–405 · doi:10.5951/tcm.5.7.0400

What it found: Origin of the perceptual (instant, up to about four or five) versus conceptual (seeing seven as five and two) subitizing distinction, and of the "quick images" activity in which brief exposure is essential because it prevents counting.
How Izzi Math uses it: The construct behind Ten-Frame Flash and its 900→450 ms exposure. Cited for the construct, not for efficacy — we found no trial isolating the ten-frame itself, and say so.
Practise it 2: ◈ Number FriendsK ◉ Ten-Frame FlashK

EdReports (2021)

Design source Evidence report

Kendall Hunt’s Illustrative Mathematics review

edreports.org · link

What it found: All K–5 grades meet expectations: Focus and Coherence 100%, Rigor and Mathematical Practices 100%, Usability 92%.
How Izzi Math uses it: Evidence that IM is a well-built curriculum. Explicitly NOT evidence that it raises attainment — that is a design review.
See it on: About — being honest about the evidence

Illustrative Mathematics (2024)

Design source Curriculum

IM K–5 Math (v.360)

Illustrative Mathematics / Kendall Hunt, CC BY-NC · link

What it found: A complete, free, openly licensed K–5 curriculum with a published scope and sequence per grade.
How Izzi Math uses it: The topic order for every grade on this site, and the choice of representations. A design source, not an efficacy claim.
See it on: Every grade follows this sequence

Illustrative Mathematics / Kendall Hunt (2024)

Design source Curriculum

IM K–5 Math scope and sequence, by grade

im.kendallhunt.com course guides · link

What it found: Unit titles, section titles and estimated instructional days for each of grades K–5.
How Izzi Math uses it: The specific strand names in content/activities/strands.js are derived from these.
Practise it 28: ◈ Counting CrewK ◈ Number FriendsK ◈ Story TimeK ◈ Shape SorterK ◈ Adding to Twenty1 ◈ All Kinds of Stories1 ◈ Tens and Ones1 ◈ Clocks and Rulers1 ◈ Halves and Quarters1 ◈ Place Value Palace2 ◈ Carry and Borrow2 ◈ Measure and Chart2 ◈ Arrays and Equal Groups2 ◉ Close to a Hundred2 ◈ Times Table Tower3 ◈ Area and Perimeter3 ◈ Round and Reckon3 ◈ Time and Data3 ◉ Fact Family Forge3 ◈ Long Multiplication4 ◈ Angles and Lines4 ◈ Factor Forest4 ◈ Times As Many4 ◉ Division Descent4 ◈ Decimal Place5 ◈ The Standard Algorithm5 ◈ Volume and Space5 ◉ Coordinate Quest5

Parrish (2010)

Design source Practitioner source

Number Talks

Math Solutions (protocol document) · link

What it found: A five- to fifteen-minute conversation around purposefully crafted problems solved mentally, in which all answers are recorded before any is evaluated.
How Izzi Math uses it: The source of the "hold the answer, then explain how you saw it" framing used in the subitizing and fact activities.
Also shaped 2: ◉ Ten-Frame FlashK ◈ Number FriendsK

Yeatman Lab, Stanford Brain Development and Education Lab (2026)

Design source Assessment instrument

ROAM — Rapid Online Assessment of Math

roam-apps.web.app; item corpora on storage.googleapis.com/roam-apps · link

What it found: Four adaptive tasks: ALPACA (core maths, 4PL IRT over 339 items, preK to calculus), MagPI (symbolic comparison binned by ratio and by place-value trap, plus number line estimation over 0-20, 0-100, 0-1 and 0-2), ARF (single-digit fact retrieval, separately parameterised for sum, minus, mult and div) and CALF (multi-digit procedure, with an explicit skill field naming carry and borrow).
How Izzi Math uses it: The source of the DIFFICULTY PROGRESSIONS, not of the topic order. Read off the published item corpora rather than inferred: CALF’s four named carry/borrow skills became the four stages of Carry and Borrow; ARF’s band walk set the order of the times tables; MagPI’s own number line targets and comparison bins are used directly. Two structural facts mattered most — fraction placement on a number line is literally a MagPI construct, which is why it is the flagship; and ARF and CALF measure different things, which is why the recommender will not drill a procedure on top of missing facts. Izzi Math is practice, not assessment: it produces no score and predicts none.
Also shaped 10: ◈ Carry and Borrow2 ◈ Times Table Tower3 ◉ Number Line Hop1 ◉ Hundred Line Hop2 ◈ Fraction Number Line3 ◉ Mixed Number Line5 ◉ Which Is MoreK ◉ Decade Duel2 ◈ Fraction Foundry5 ◉ The Great RaceK
See it on: The score-to-practice page

youcubed, Stanford (2015)

Design source Practitioner source

How Close to 100?

youcubed.org tasks · link

What it found: Two dice and a blank 10×10 grid: build the array the dice describe anywhere on the grid, record the number sentence, and try to fill the grid.
How Izzi Math uses it: The task behind Array Architect and the array work in grade 2.
Practise it 5: ◈ Arrays and Equal Groups2 ◉ Close to a Hundred2 ◈ Area and Perimeter3 ◉ Array Architect3 ◈ Factor Forest4

Kim et al. (2018)

Null result Randomised trial

Comprehensive early numeracy home training and transfer to symbolic mathematics

Frontiers in Psychology 9:1775 · doi:10.3389/fpsyg.2018.01775

What it found: 56 first graders, six weeks of home training at 30 minutes a day on numerosity comparison, number line, approximate arithmetic and symbol-numerosity mapping. ANS acuity improved; there was no transfer to symbolic or exact calculation.
How Izzi Math uses it: A home-delivered null on the same design we would otherwise have been tempted by. Corroborates the meta-analysis above.
Also shaped 2: ◉ Which Is MoreK ◉ Ten-Frame FlashK

Massachusetts Department of Elementary and Secondary Education (CURATE) (2024)

Null result Evidence report

CURATE panel review: Illustrative Mathematics K–5 (2021)

doe.mass.edu, October 2024 · link

What it found: "High-quality studies of student learning impacts are not yet available for Illustrative Mathematics K–5 (2021)." No rating given on Impact on Learning.
How Izzi Math uses it: Why we describe IM as a well-vetted plan rather than an efficacy-proven programme. The +0.18 to +0.50 SD figures often quoted for IM are from grades 6–8.
See it on: About — being honest about the evidence

Qiu, Chen, Wan & Bailey (2021)

Null result Meta-analysis

Doubting the reliability of ANS training effects on symbolic mathematics: a multilevel meta-analysis

Journal of Experimental Psychology: Learning, Memory and Cognition · link

What it found: Approximate number system training transfers to symbolic mathematics at g = .11 (not significant), and −.04 after publication-bias correction. No moderation by training type or duration.
How Izzi Math uses it: Why there is no dot-cloud comparison game on this site. Where dots appear they are exact, small, structured, and mapped to a numeral.
Practise it 2: ◉ Ten-Frame FlashK ◉ Which Is MoreK

What we deliberately do not build

Null results are as useful as positive ones. These sources shape what is absent from the site.

Qiu et al. (2021) — Why there is no dot-cloud comparison game on this site. Where dots appear they are exact, small, structured, and mapped to a numeral.
Kim et al. (2018) — A home-delivered null on the same design we would otherwise have been tempted by. Corroborates the meta-analysis above.
Massachusetts Department of Elementary and Secondary Education (CURATE) (2024) — Why we describe IM as a well-vetted plan rather than an efficacy-proven programme. The +0.18 to +0.50 SD figures often quoted for IM are from grades 6–8.