New Math and Common Core Math Explained

“New math” is what parents call the way US schools have taught arithmetic since the Common Core State Standards were released in 2010: the same math, taught in a different order. Children learn why a method works first, with a drawing or a model such as a ten frame or a number line, and the familiar shortcuts come later. The Common Core’s own example is first grade’s 8 + 6, taught as 8 + 2 + 4: the 2 turns the 8 into a ten, and 10 + 4 = 14 (standard 1.OA.C.6).

The old way is still required. Common Core asks for the standard algorithm, the carrying and borrowing most parents learned, by name: for adding and subtracting in grade 4 and for multiplying in grade 5, after the strategies that explain it. Four problems are solved both ways on this page, and the table of models by grade links the page that explains each one.

Why is math taught differently now?

Math is taught differently now because the Common Core standards ask schools to teach understanding, skill, and use together, rather than the procedure alone. The standards’ own summary, Key Shifts in Mathematics, tells teachers to pursue three aspects of rigor “with equal intensity”: “conceptual understanding, procedural skills and fluency, and application”. It adds that mathematics “is not a list of disconnected topics, tricks, or mnemonics”.

At a kitchen table, the difference shows up when an answer is wrong. A child taught only to “carry the 1” can follow the steps but has no way to check them. A child who knows that the carried 1 is a ten can add the tens and the ones separately, see that 47 + 38 must be more than 70, and find the step that went wrong.

Key Shifts also asks for focus. It describes the older pattern as “racing to cover many topics in a mile-wide, inch-deep curriculum” and asks for fewer topics, taught deeper. In grades K–2, the major work is addition and subtraction, so a child spends much of those years on a few ideas, such as making ten and adding tens and ones.

Research supports teaching both understanding and procedures. Researchers Bethany Rittle-Johnson, Michael Schneider, and Jon Star reviewed the studies in 2015 and concluded that each kind of knowledge supports the other: “the relations between the two types of knowledge are bidirectional”.

The drawings on your child’s homework are a stage on the way to the standard method, not a replacement for it. A child draws the reason from kindergarten to grade 3 and learns the shortcut by name from grade 4.

What is new math, and what is Common Core math?

Common Core math is a set of standards: a list of what a child should be able to do in math by the end of each grade, from kindergarten to grade 12. The National Governors Association Center for Best Practices and the Council of Chief State School Officers released the final standards on June 2, 2010.

Standards are not a curriculum. The Common Core math standards say so in their own text: “These Standards do not dictate curriculum or teaching methods.” A school district chooses the curriculum, meaning the textbook program and its lessons, such as Eureka Math, enVision, Go Math!, Illustrative Mathematics, or Bridges in Mathematics. Most of the drawings on a homework sheet come from that program. A question about what a grade should cover goes to the standards, and a question about a particular drawing goes to the teacher or the program’s family materials.

“New math” is the parents’ name for these methods, not an official one. Teachers call the methods strategies, such as making ten or counting up, and the approach as a whole is usually called Common Core math.

The 1960s New Math was a different reform. After Sputnik, the School Mathematics Study Group, founded in 1958, rebuilt school math around ideas such as sets. In the mathematician Ralph Raimi’s account, a chapter on sets opened every year’s textbook, and “the result, after twelve years, was total failure.” That reform is where the name comes from, and why Tom Lehrer’s song “New Math” turns up in a search.

Texas never adopted Common Core. House Bill 462, signed into law in June 2013, bans it, as reading educators Vasinda, Grote-Garcia, and Durham record in the Texas Journal of Literacy Education. Texas’s own standards, the Texas Essential Knowledge and Skills (TEKS), also put strategies first. First graders “apply basic fact strategies to add and subtract within 20, including making 10” (TEKS grade 1). Second graders add and subtract “using mental strategies and algorithms based on knowledge of place value” (TEKS grade 2).

New math vs. old math: the same problem solved both ways

New math and old math do the same steps; the new way writes down the step the old way hid. Each example below is a problem from one grade, solved first the way most parents learned and then the way the homework asks for it. Every one reaches the same answer.

Grade 1: 8 + 6 by making a ten

• Old way: know that 8 + 6 = 14 from memory, or count on from 8: 9, 10, 11, 12, 13, 14.
• New way: split the 6 into 2 and 4. The 2 fills the 8 up to 10, and 10 + 4 = 14.

First grade’s 1.OA.C.6 names making ten as a strategy children should use. On a ten frame, a child sees 2 counters fill the empty squares of a frame holding 8. As a number bond, the 6 splits into its parts, 2 and 4.

Grade 2: 47 + 38 by adding tens and ones

• Old way: 7 + 8 = 15, so write the 5 and carry the 1. Then 1 + 4 + 3 = 8, for 85.
• New way: add the tens, 40 + 30 = 70. Add the ones, 7 + 8 = 15. Then 70 + 15 = 85.

The carried 1 is the ten in 15. Both methods do the same addition, and the new one writes the ten down instead of carrying it. A child may show it with base ten blocks, 7 rods and 15 small cubes, then trade 10 of the cubes for a rod. Second grade’s 2.NBT.B.5 asks children to add within 100 “using strategies based on place value”.

Grade 2: 62 − 28 by counting up on a number line

• Old way: you cannot take 8 from 2, so borrow from the 6. It becomes 5 and the 2 becomes 12. Then 12 − 8 = 4 and 5 − 2 = 3, for 34.
• New way: count up from 28 to 62. From 28 to 30 is 2, from 30 to 60 is 30, and from 60 to 62 is 2, so the difference is 2 + 30 + 2 = 34.

The child draws the jumps on an open number line, a line with no scale where only the numbers that matter are marked. Subtraction becomes “how far is it from 28 to 62?”, which uses the relationship between addition and subtraction that 2.NBT.B.5 names.

Grade 3: 6 × 14 with an area model

• Old way: 6 × 4 = 24, so write the 4 and carry the 2. Then 6 × 1 = 6, plus 2 is 8, for 84.
• New way: draw an area model, a rectangle 6 tall and 14 wide, and split the 14 into 10 and 4. The two parts are 6 × 10 = 60 and 6 × 4 = 24, and 60 + 24 = 84.

Third grade’s 3.MD.C.7c asks children to “use area models to represent the distributive property”. In grade 4 the same drawing multiplies two two-digit numbers (4.NBT.B.5).

Two more models have no shortcut to set against them. A tape diagram, also called a bar model, draws a word problem: a bar for 15 apples, split into 6 red ones and an unknown part, shows that the green apples are 15 − 6 = 9. An array draws multiplication as rows: 3 rows of 4 dots is 4 + 4 + 4 = 12, the adding of equal rows that second grade’s 2.OA.C.4 asks for.

The models on your child’s homework, by grade

Most models on K–3 math homework come from the school’s curriculum, not from the Common Core standards, which name only a few of them. We searched the full text of the Common Core math standards. It names rectangular arrays, the number line diagram, area models, and, in grade 6 only, the tape diagram. It never mentions a ten frame, a number bond, a rekenrek, a hundreds chart, or base ten blocks. The table gives each model the grades it usually appears in and where it comes from.

Model	You will see it	Where it comes from	Page
Fingers, then objects	Pre-K–K	Named in the standards: K.OA.A.1 lists “objects, fingers” among ways to show adding	Planned
Ten frame (and five frame)	Pre-K–1	Curricula, such as Bridges in Mathematics; not named in the standards	Ten frames explained
Rekenrek (bead rack)	Pre-K–1	Curricula, such as Eureka Math and Bridges in Mathematics (which calls it a number rack); not named in the standards	Rekenreks explained
Number bond (part-part-whole)	K–2	Curricula, such as Eureka Math and Singapore Math’s Dimensions Math; the standards ask children to “decompose numbers… by using objects or drawings” (K.OA.A.3)	Number bonds explained
Hundreds chart	K–2	Curricula, such as Bridges in Mathematics; not named in the standards	Planned
Number line and open number line	K–3	Named in the standards: “number line diagram” from grade 2 (2.MD.B.6)	Number lines explained
Base ten blocks	K–2	Curricula, such as Eureka Math and Bridges in Mathematics; the standards ask for “concrete models or drawings” (1.NBT.C.4, 2.NBT.B.7)	Base ten blocks explained
Tape diagram (bar model, strip diagram)	1–3	Curricula, such as Eureka Math from grade 1; Common Core names it in grade 6, and Texas names “strip diagrams” in grade 3 (TEKS 3.5(B))	Tape diagrams explained
Arrays	2–3	Named in the standards: “rectangular arrays” for adding equal rows in grade 2 (2.OA.C.4)	Arrays explained
Area model	3 (and 4–5)	Named in the standards: “area models” in grade 3 (3.MD.C.7c) and grade 4 (4.NBT.B.5)	Area models explained

Each program named in the table shows the model in its own materials. Eureka Math, from Great Minds, teaches number bonds in kindergarten, has kindergartners build a rekenrek, introduces the tape diagram in grade 1, and uses base ten blocks in grade 2. Bridges in Mathematics, from The Math Learning Center, uses ten frames and a number rack in kindergarten, base ten pieces in grade 2, and, in its publisher’s words, a hundreds chart “throughout” the program. Singapore Math’s Dimensions Math 1A gives number bonds a chapter of their own in grade 1.

For a model that is not in the table, ask the teacher for the curriculum’s family letter or parent page, which explains it in the class’s own words. Ordinary household things work as models too, such as dried beans grouped in cups of ten.

Do schools still teach the old way?

Yes. Common Core requires the standard algorithm, which is the carrying and borrowing method most parents call “the old way”. Children must “fluently add and subtract multi-digit whole numbers using the standard algorithm” in grade 4 (4.NBT.B.4). The words “standard algorithm” do not appear in the standards for grades K–3. Texas requires it earlier for multiplying and dividing.

The standard algorithm for	Common Core	Texas (TEKS)
Adding and subtracting	Grade 4 (4.NBT.B.4)	Grade 4 (4.4(A))
Multiplying	Grade 5 (5.NBT.B.5)	Grade 3 for two digits by one (3.4(G)); grade 4 for two digits by two (4.4(D))
Dividing	Grade 6 (6.NS.B.2)	Grade 4 for up to four digits by one (4.4(F))

The other grades come from the Common Core standards for grade 5 and grade 6, and from TEKS grade 3 and TEKS grade 4. Texas also asks for “algorithms”, without the word standard, from second grade (2.4(B)).

Before grade 4, children add and subtract “using concrete models or drawings” and then “relate the strategy to a written method”, in first grade’s 1.NBT.C.4 and again for numbers to 1,000 in second grade’s 2.NBT.B.7. By grade 4, the shortcut arrives already explained. Mathematics educators Karen Fuson and Sybilla Beckmann describe the progression in the NCSM Journal: students “explain the reasoning used in a written method with visual models. Then, in a later grade, students move to using the standard algorithm fluently with no visual models.”

For tonight’s homework, this means a child in grades 1–3 who is told to “use the model” is at the stage before the standard algorithm, not on a path that replaces it. Showing them how to carry is not wrong, but it skips the stage their teacher is marking, so the sheet will still ask for the drawing.

Is new math better than old math?

The research supports teaching both understanding and procedures, which Common Core requires, and has not settled which should come first. Rittle-Johnson, Schneider, and Star’s 2015 review found that “alternative orderings of instruction on concepts and procedures have rarely been compared, with limited empirical support for one ordering of instruction over another.”

The evidence for the models themselves is stronger. The What Works Clearinghouse, the US Department of Education’s research review, rates “a well-chosen set of concrete and semi-concrete representations” as having strong evidence, from 28 studies, in its 2021 practice guide Assisting Students Struggling with Mathematics. The drawings it names include strip diagrams, arrays, and number lines, and the objects include ten frames and base ten blocks. The guide is written for children who struggle with math, so it supports the models for those children rather than settling the question for every child.

The criticism is real too. Parents object to homework they cannot help with, and some states left the standards. Reporter Jo Napolitano of The 74 wrote in August 2025 that Oklahoma and Florida “are among those that dropped the standards”. In the same article, math experts say “dozens of states still use the standards, some by their original name and others under new monikers.”

New math is not harder arithmetic. A problem looks longer because more of its steps are written down, but the numbers and the answer are the same. The method is the school’s choice, and your child’s job is the problem in front of them, done the way the class is learning it.

How can parents help with new math homework?

Help with new math homework by letting your child show their method first, then checking the answer your own way. A parent does not need to know the method before the session starts. The child has just been taught it, and explaining it back is part of what the class is practicing: second grade’s 2.NBT.B.9 asks children to “explain why addition and subtraction strategies work”.

1. 1

   Ask your child to show you their method first

   Before you say anything about how you would do it, ask your child to solve the problem and explain each step. Explaining is part of the assignment: second graders are asked to explain why addition and subtraction strategies work (Common Core 2.NBT.B.9).

2. 2

   Find the model on the page

   Look for the drawing the worksheet asks for, such as a ten frame, a number bond, a number line, or an area model, and look it up in the models table on this page, which links the page that explains each one.

3. 3

   Check the answer your way, privately

   Work the problem your own way on scrap paper, then ask your child to check theirs. If the answers match, the homework is done. If they differ, look for the step where the two methods part, which is most often a ten that was made or traded.

4. 4

   Show your method as a second way, and connect it

   If you show the way you learned, present it as another way to the same answer, and link the two: the 1 you carry is the ten your child wrote down. Do not ask your child to hand in your method when the sheet asks for theirs.

5. 5

   Write a note to the teacher when neither of you can follow it

   Write a short note saying which problem stopped you and at which step. Ask for the curriculum's family letter or parent page, which explains the methods the class is using in the words the teacher uses.

6. 6

   Talk about numbers away from the worksheet

   Spend a few minutes a day on math talk outside homework: make ten with fingers in the car, split a price into tens and ones at the shop, or ask how many more steps to the door. The same ideas come back on the worksheet.

The last step needs no worksheet. Everyday talk about how many, how many more, and how many tens is practice in number sense, which every model on the homework draws on.

What if new math homework frustrates my child (or me)?

When new math homework ends in frustration, a step is usually missing, and the child is not failing at math. Three moves get a session back on track:

• Go back one model. Move from the number line to counters, or from the drawing to real objects, and work up again from the one your child can do.
• Stop while it is still going well. End on a problem your child solved, and pick up the hard one tomorrow.
• Say “I learned it a different way” rather than “this is silly”. A child who hears an adult dismiss the method has a reason to stop trying it.

If math itself is hard for your child, and not only the method, talk to the teacher about what your child can and cannot do yet.