Regrouping is the first place primary maths asks a child to hold two ideas in one column at the same time, and it is exactly where many children quietly stop understanding maths and start surviving it. The failures almost never come from the algorithm; they come from teaching the algorithm before the child owns place value. This guide walks the full sequence: the language to install first, the concrete-to-abstract path, the "no-regrouping bridge set" that quietly prevents the infamous wall, step-by-step carrying and borrowing, an error-analysis table for the five classic mistakes, and a practice ladder with real problem counts.
Place-Value Language Comes First
Before any vertical addition appears, children need to say numbers the way they are built. 47 is not "forty-seven"; it is "four tens and seven ones". Drill this as a game for a week: teacher says 32, child says "three tens, two ones", then reverse. Two non-negotiable language rules for the whole unit:
- Say "trade", not "carry" or "borrow" — at first. Ten ones become one ten. One ten becomes ten ones. Nothing is carried anywhere, and nothing is ever given back, which is why "borrow" quietly confuses children who take words literally.
- Name the columns every single time. "Ones column: 7 and 6" — not "7 plus 6". The words are the concept; the algorithm is just the words written down fast.
Concrete → Representational → Abstract
The concrete-representational-abstract (CRA) sequence is old because it works. Do not compress it.
- Concrete (about 3–5 lessons): real bundles. Ten straws banded into tens, base-ten blocks, or bundles of ten pencils. Solve 26 + 17 with objects: build 2 tens 6 ones and 1 ten 7 ones, push the ones together, and physically band ten loose ones into a new ten. The band is the "regroup".
- Representational (3–5 lessons): drawings. Squares for hundreds, sticks for tens, dots for ones. Same problem, now on paper: draw it, circle ten dots, draw an arrow carrying the circled group to the tens column.
- Abstract (ongoing): digits only. The written 1 above the tens column is introduced last, and the child should be able to explain what it stands for ("that is the ten I made from ten ones").
The test for moving down the ladder: cover the blocks and ask the child to predict what will happen to the ones. If they can say "there will be thirteen, so a new ten", they are ready for paper.
The No-Regrouping Bridge Set
Here is the mistake that builds the wall: the first worksheet a child meets contains both new columns-procedure and new regrouping decisions at once. When the answer is wrong, nobody can tell which part failed — the procedure or the regroup. So the first set must be no-regrouping only: 34 + 25, 58 − 23, 41 + 36 — every column comfortably under ten. This bridge set exists to make the column procedure automatic while regrouping is still zero-percent of the load. Stay on it until a page of ten problems is error-free two days in a row. Then — and this is the second half that teachers skip — introduce a mixed decision set: some problems regroup, some do not, and the child's first job is to sort them into "trade / no trade" piles with a pencil mark before solving anything. Deciding whether to regroup is itself a skill, and it deserves its own practice instead of ambush.
Carrying in Addition, Step by Step
Worked example, 57 + 36, with the exact language to use:
- "Ones column: 7 and 6. Seven and six make thirteen."
- "Thirteen is one ten and three ones." (Say it before writing anything.)
- "The three stays in the ones house. The one ten — we made it ourselves — moves to the tens column." Write a small 1 above the tens.
- "Tens column: 5 tens, 3 tens, and our new ten — that is 9 tens."
- Read the answer the place-value way: "9 tens and 3 ones — ninety-three."
The common failure here is writing the whole 13 small above the column, which wrecks the tens addition. If you see a two-digit number squeezed above the tens column, stop drilling and go back to the representational stage for two days; that error means the ten is not yet real to the child.
Borrowing in Subtraction, Step by Step
Worked example, 52 − 37:
- "Ones column: 2 take away 7. Two is not enough — we need more ones."
- "Trade one of the 5 tens for ten ones." Cross out the 5, write 4; give the 2 a new left-hand digit: 12.
- "Now the ones house holds 12 ones. 12 take away 7 is 5."
- "Tens house: 4 tens take away 3 tens is 1 ten."
- "1 ten and 5 ones — fifteen."
Teach the zero trap (503 − 128) explicitly, a full week after regular borrowing: the tens column holds nothing to trade, so the trade has to come from the hundreds — hundreds become tens, tens become ones. Model 500 − 128 with real notes or blocks once, slowly; children who have only ever seen single-step trades will otherwise write nonsense in the middle column forever.
Error Analysis: The Classic Five
When a page comes back wrong, diagnose before drilling. Five errors cover nearly everything:
| Error | What it looks like | What it means | Fix |
|---|---|---|---|
| Smaller-from-larger | 23 − 15 = 12 (wrote 5 − 3 in the ones) | Child subtracts whichever digit is on top; place value is decoration | Back to concrete: physically try to take 7 ones from 2 ones |
| Whole-number carry | Writes 13 above the tens column | "Thirteen" written, never unbundled into 1 ten | Representational stage: circle ten dots, move the circle |
| Trade without reduce | 52 − 37: crosses the 5, then still does 5 − 3 | Procedure memorised as gestures; meaning lost | "Point and say" — finger on every changed digit before solving |
| Zero trap | 503 − 128 goes wrong in the tens column | Never saw a two-step trade | One slow modeled example with blocks; then 3 targeted problems |
| Unneeded regroup | 58 − 23 gets a borrowed 1 | Child regroups reflexively — decision skill never practised | Mixed decision set: sort first, solve second |
The Practice Ladder
Small sets, often, beat giant pages rarely. Ten to twenty problems per day at the current rung, moving up only on two consecutive error-free days:
| Rung | Problem type | Per set | Mastery check |
|---|---|---|---|
| 1 | 2-digit, no regrouping (bridge set) | 10 | Zero errors two days running |
| 2 | 2-digit, carry only | 10 | Says the trade aloud while solving |
| 3 | 2-digit, borrow only | 10 | Reduce-and-rewrite happens unprompted |
| 4 | 2-digit mixed (decision set) | 15 | Sorts trade/no-trade correctly before solving |
| 5 | 3-digit, mixed, with carrying | 15 | Zero errors two days running |
| 6 | 3-digit with zero trap | 10 | Explains the double trade in words |
| 7 | Mixed 2–3 digit, both operations | 20 | Under time-free conditions, 90%+ correct |
Two supporting tools make this cheap to run: the math drill generator prints any rung with a "no regrouping / with regrouping" toggle and an answer key, and the number-bonds generator rebuilds the "ten ones become one ten" reflex at the bonds-to-10 level that everything above stands on — both free, printable PDFs, no sign-up. Regrouping is not a trick to survive. Taught with the right language, the right objects and a patient ladder, it is simply the day place value stops being a poster on the wall and becomes something a child can do with their hands.
进位与退位是小学数学第一次要求孩子在同一列里同时装下两个想法的地方，也恰恰是许多孩子悄悄地「不再理解数学、开始死记数学」的地方。失败几乎从不来自计算程序本身，而来自孩子在还没有真正拥有位值概念之前，就被教了计算程序。本指南走完整个序列：先要安装的语言、从具体到抽象的路径、悄悄预防「那堵墙」的「无进退位桥接题组」、手把手的进位与退位步骤、五大经典错误的诊断表，以及带真实题量的练习阶梯。
先立位值语言
在任何竖式加法出现之前，孩子需要按数字的构成方式把它说出来。47 不是「四十七」，而是「四个十和七个一」。用一周时间把它当游戏练：老师说 32，孩子说「三个十、两个一」，然后反过来。整个单元有两条不可妥协的语言规则：
- 先说「换」，不说「进位」或「借位」。十个一变成一个十；一个十变回十个一。没有任何东西被「carry」到别处，也没有任何东西要「还」——这正是「借」字悄悄坑害按字面理解的孩子的地方。
- 每一次都点名数位。「个位：7 和 6」——而不是「7 加 6」。语言就是概念本身；竖式只是把语言快速写下来的方式。
具体 → 表象 → 抽象
「具体—表象—抽象」（CRA）序列古老，因为它有效。不要压缩它。
- 具体（约 3–5 节课）：真实捆扎。十根吸管用皮筋扎成一捆的「十」，位值块，或十支一扎的铅笔。用实物解 26 + 17：摆出 2 个十 6 个一和 1 个十 7 个一，把零散的「一」推到一起，然后真的用皮筋把十个散的扎成一个新「十」。扎皮筋这个动作，就是「进位」。
- 表象（3–5 节课）：画出来。百位画方块、十位画竖条、个位画圆点。同一道题，现在在纸上：画出来，圈住十个圆点，画一个箭头把这圈「十」搬到十位列。
- 抽象（长期）：只剩数字。十位上方那个小小的 1 最后才登场，而且孩子要能解释它代表什么（「那是我用十个一换来的一个十」）。
能否降一级台阶的测试：把积木盖住，让孩子预测「一」会发生什么。如果孩子能说出「会是十三，所以要出一个新的十」，就可以上纸笔了。
无进退位桥接题组
筑起「那堵墙」的错误是：孩子遇到的第一张练习纸，同时装着全新的竖式程序和全新的进退位判断。答案错了，没人说得清是程序错了还是进退位错了。所以第一个题组必须只有无进退位的题：34 + 25、58 − 23、41 + 36——每一列都稳稳地小于十。这个桥接题组存在的意义，是在进退位负担为零的前提下，把竖式程序练到自动化。在上面待到「十道题连续两天全对」再走。然后——这是很多老师跳过的后半段——引入混合判断题组：有的题要进退位，有的不要，孩子的第一个任务是先用铅笔在旁边做记号，把它们分成「要换 / 不用换」两堆，再动笔算。要不要进退位本身就是一个技能，它值得拥有自己的练习，而不是被伏击。
加法进位，一步一步
例题 57 + 36，配可照说的语言：
- 「个位：7 和 6。七加六，是十三。」
- 「十三是一个十和三个一。」（先说出来，再动笔。）
- 「三个一留在个位的小房子里。这个新造出来的十，搬到十位列。」在十位上方写一个小小的 1。
- 「十位：5 个十、3 个十，加上我们新造的 1 个十——是 9 个十。」
- 按位值读答案：「9 个十和 3 个一——九十三。」
这里的经典失败是把整个 13 小小地写在十位上方，这会把十位的加法搅得一团糟。一旦看到两位数被挤在十位上方，停止刷题，退回表象阶段两天；这个错误说明「十」对孩子还不是真的。
减法退位，一步一步
例题 52 − 37：
- 「个位：2 减 7。2 不够——我们需要更多的『一』。」
- 「拿 5 个十里的 1 个十，换成十个一。」划掉 5 写 4；给个位的 2 添上新邻居，变成 12。
- 「现在个位小房子里有 12 个一。12 减 7 是 5。」
- 「十位小房子：4 个十减 3 个十，是 1 个十。」
- 「1 个十和 5 个一——十五。」
「零陷阱」（503 − 128）要在常规退位之后再整整一周、专门教：十位上没有东西可换，所以必须到百位去换——百换成十，十再换成一。用真钞或积木把 500 − 128 完整慢速演示一次；只见过单步交换的孩子，否则会在中间那列永远写出荒唐的东西。
错误分析：经典五连
当一页纸错着回来，先诊断，再刷题。五个错误几乎覆盖一切：
| 错误 | 长什么样 | 说明什么 | 修法 |
|---|---|---|---|
| 大减小 | 23 − 15 = 12（个位写了 5 − 3） | 孩子永远用上面那位减下面那位；位值只是装饰 | 回到实物：真的试着从 2 个一里拿走 7 个一 |
| 整个数上房 | 把 13 整个写在十位上方 | 「十三」被写下来了，但从未拆成 1 个十 | 表象阶段：圈十个点，把这圈搬走 |
| 换而不减 | 52 − 37：划掉了 5，却仍然算 5 − 3 | 程序被记成了手势，含义丢了 | 「指读」——动笔前手指点过每一个被改动的数字并说出 |
| 零陷阱 | 503 − 128 在十位出错 | 从未见过两步交换 | 用积木慢速示范一次；随后 3 道针对性练习 |
| 不该换乱换 | 58 − 23 被退了位 | 孩子条件反射地退位——判断技能从没练过 | 混合判断题组：先分堆，后解题 |
练习阶梯
小题量、高频率，胜过大页、偶一为之。在当前台阶每天做十到二十题，连续两天全对才升一级：
| 台阶 | 题型 | 每组题量 | 过关标准 |
|---|---|---|---|
| 1 | 两位数无进退位（桥接组） | 10 | 连续两天零错误 |
| 2 | 两位数只进位 | 10 | 解题时能说出交换过程 |
| 3 | 两位数只退位 | 10 | 无需提醒即完成「划掉重写」 |
| 4 | 两位数混合（判断组） | 15 | 动笔前正确分好「换 / 不换」 |
| 5 | 三位数混合，含进位 | 15 | 连续两天零错误 |
| 6 | 三位数含零陷阱 | 10 | 能用语言解释两步交换 |
| 7 | 两三位数混合，两种运算 | 20 | 不限时条件下正确率 90% 以上 |
两个配套工具让这一切低成本运转：本站的口算生成器可以打印任意台阶，带「无进退位 / 含进退位」开关和答案页；数字键生成器则在「凑十」层面重建「十个一变成一个十」的反射——上面的一切都立在它之上。两者都免费、可打印 PDF、无需注册。进位退位不是要熬过去的难关。用对的语言、对的事物和一座耐心的阶梯去教，它只是「位值从墙上的海报变成孩子手上的动作」的那一天。