Teaching Long Division So It Makes Sense

Long division is perhaps as difficult for a 4th or 5th grader as Calculus is for a high school senior. The multiple steps are so complex that they need to be drilled until they often lose all meaning for a child. Cute mnemonics and hours of drill might result in a procedural competency for many students (unfortunately not for all), but they do not build number sense, estimation skills, mental math or mastery in solving word problems.

We’ve all heard of the benefits of the curricular approach “CONCRETE > PICTORIAL> ABSTRACT” (C>P>A) — what does that look like in our classrooms, and what does it look like for long division?

Truth #1:  Children learn best using the C>P>A approach

We’ve seen this internationally, (the highest ranking countries use it) and we’ve seen it in our own classrooms. We use it because it works.

Singapore developed their C>P>A approach based on the writings of five famous math pedagogues: Piaget, Skemp, Bruner, Dienes, and Vygotsky.   Here  is an article with visual examples, and here  is a short video by Yeap Ban Har (the leading international expert in Singapore Math, in our opinion),   describing their contributions to math education.

Truth #2:  The “C>P>A”   Approach Teaches Concepts As ONE Integrated Picture.

     The Concrete = the Pictorial = the Algorithm

The video below shows the integration of the division algorithm across all 3 levels: CONCRETE > PICTORIAL> ABSTRACT.  The number bonds and the long division algorithm do nothing more than represent the picture (which uses blocks) in abstract terms. A child who forgets a step can go back and derive the steps all over again.

We all recognize the appeal of this approach, but here’s what we teachers  often get WRONG:  We spend too little time at the concrete/pictorial level. There are many reasons for this – time pressure from administrators and parents, pressure from standardized testing, poor textbooks, lack of curriculum guides showing how to teach C>P>A, lack of personal experience in our own childhoods. Mainly, though, because of Truth #3:

Truth #3:  The “C>P>A”   Approach takes T-I-M-E

Children do not learn like adults. With our more developed abstract cognitive skills, we can look at a concrete conceptual example once or twice, and integrate it into our existing algorithmic math understanding,  make connections, and make sense of it. Children, however, need a LOT more time. Some students ( traditionally labeled “good at math”), are quick, abstract thinkers, and move through the 3 levels at an almost adult speed. In fact, they might prefer the abstract.  A large number of their peers, however, understand the concrete almost exclusively, and a quick transfer to the abstract DOES NOT WORK  for them. They simply need more time. Where an adult would figure out a concept by laying out blocks 2 or 3 times in a row, a child might need to do it 14 times, or 19, or 23. If they do not get that opportunity, they discard the concrete and try to memorize the algorithm, just to keep up with their class. By 5th or 6th grade, they’ve fallen hopelessly behind, and are in danger of giving up.

Truth #4:  The Handmaiden of Number Sense Development is LANGUAGE

For children who learn visually and conceptually, the greatest ally is TALKING. In all our division problems, we write out the problem in WORDS. In the video above, 272 is written as 26 tens and 12 ones. This allows students to approach this problem with blocks – students who otherwise would not know where to begin. They can convert the words (26 tens and 12 ones) to a number by counting. Again, we adults often don’t realize the complexity of our own base-10 system, and acknowledge the amount of thinking and building that goes into its mastery.

Teaching Notes: What is a “Sandwich Unit”?

How do we give students the time needed? We use what we call a “Sandwich Unit”. This means that we introduce the CONCRETE / PICTORIAL concept using blocks and then drawings, and stay at the concrete/pictorial level for at least 2 weeks. Students only have to do one or two of these problems a day (perhaps on homework), as we go on to other topics. Finally, we return to this topic, and slowly develop the abstract representation.

“Sandwich Unit” Sequencing for Long Division:

1. Concrete/Pictorial – 2 weeks

We only actually draw circles and pass out blocks for one day (maybe two) —  the FIRST step only in the video above. We give the students 2 or 3 problems on the first day, and have them solve them using blocks. (They have to actually put away their pencils, or this won’t work well.)   At this age, they don’t have to be taught to exchange hundreds or tens in order to pass out all the blocks fairly.

Here are 2 problems that might take them some time.

  1.      13 tens and 5 ones ÷ among 3 children. (This doesn’t lend itself instantaneously to mental math, making it a good problem). The photo shows the answer, after the blocks have been passed out. Note that we use WORDS to pose the problem. This gives an advantage to concrete thinkers, and makes the abstract thinkers more willing to use blocks.

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2.     Six hundreds, 6 tens and 8 ones ÷ among 5 children.  The photo shows the answer, after the blocks have been passed out, although the remainder of 3 is off the screen.

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At the 5th grade level, we don’t need to spend much more time using blocks. On the next day, we allow students to use drawings. It goes faster, and is just as conceptual.

Here 2 examples: Notice the lack of number bonds or long division. This is all based on COUNTING what you have drawn (or passed out, for anyone who prefers to use blocks) IMG_2544 3

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Again,  we only do one or two of these a day, while we go on to other lessons. For example, this week we investigated Fibonacci numbers while students did one or two division drawings on their homework.  Even children who have already learned long division at home don’t complain about 1 problem a day. 

NOTE:  Some students will understand and complete the manipulative-based problems very quickly, so any classwork needs to include review and challenge work for them. Students who don’t get to the challenge work do not have to finish it later. This is differentiation by  speed. Here is an example of classwork – 2 division problems, one fraction review problem and 2 challenge problems.    CW concrete

2. Transfer to Number Bonds (Abstract)

After 2 weeks of intermittent drawing, over 90% of our students show comfort at the pictorial level, and we move on to the recording the number bonds.

Notice that the number bonds strictly reflect the picture drawn.

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Step one: Pass out the tens and record the number bond:

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Step 2: Pass out the ones and record the number bond. Finally, record the answer by counting the blocks in each circle.

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MENTAL MATH.   Soon, we give mental math a try. Here is a Power Point with 10 problems, and prompts for the first 5.

Div Mental Math

Note:  Some children find this very hard. They lack background in number sense, and their learning style is visual, which only lends itself to mental math after LONG practice at the concrete/pictorial level.

However, we feel that number sense and mental math are extremely important. More important probably than long division. So we do one of these Power Point warm-ups maybe twice a week, and stop as soon as exhaustion or resignation become visible.

We try to make this fun – students write their answers on a mini-whiteboard and hold them up. Sometimes we have everyone change places after every 3rd problem, just to increase engagement and decrease anxiety. We pretend not to notice incorrect answers, but occasionally ask students to  explain their thinking before going on.

3. Transfer to Long Division (Abstract)

After about 2 weeks of intermittent drawing and number bonds, we introduce long division. Again, we only do 1 or 2 problems a day, while we’re working on a different unit altogether.

On the first day, we have students do all THREE steps –  blocks, number bonds and long division. They need instruction in how to record the long division. Here is a Power Point that helps engage and build procedural understanding: DIVISION story  

Here are more examples:  In the first photo, the BLUE 120 = Tens passed out.

120 is recorded 3 different ways:   (a) concrete – the number of tens rods passed out ,          (b)mental math – as one of the number bonds and       (c)  the first entry in the long division problem.

The 15 is (a) the concrete number of ones passed out, (b) one of the number bonds and (c)  the second entry in the long division problem.

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We do similar problems as homework for 2- 3 days. Then we increase the level of difficulty to include THREE number bonds.

Note: we don’t necessarily expect this level of difficulty as a mental math problem, except maybe as estimation practice. IMG_7614

 

4. Practicing at the Abstract Level

Gradually, we DROP the Concrete/Pictorial requirement. Students who still need it are encouraged to use the drawing.

Again, we only do ONE or TWO of these a day for the next 3 or 4 weeks. That provides conceptual practice, not rote memorization. Visual memory is much stronger than algorithmic memorization, and this procedure can successfully be pulled back up in moments of stress (like a quiz!)

Here is a Word document with long division practice: CW div 2 methods

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Important Notes:
1. This values the thinking, not speed and number size. You’ll notice we use SMALL numbers for a long time. It’s all about the concept, not the calculation complexity in today’s world. If a child can learn to do 1346 ÷ 11, they can later do bigger numbers if necessary. PLUS – this transfers fairly easily to decimals, since it’s so place value based.

2. This approach teaches place value and number sense.  Even  students who prefer the algorithmic approach make place value mistakes, leaving out zeroes in the answer, for example. The more concept-based the approach, the less likely this is to happen.

3. This approach develops estimation and mental math skills.  Even students who enjoy long division make mistakes around answers that have zeroes in the middle.

More Word Problems

 

1. Two More Fun Warm-ups Reviewing Word Problems

We have seen our 8th graders struggle with algebra problems that involve “the number of bills” and the “value of those bills”. They can write x + y = 27 if there are 27 five and 10 dollar bills altogether, but stumble over the value equation:  10x + 5y = 210 when told that the 27 bills add to a value of $210.

So we decided to try to start such distinctions earlier – 5th and 6th grade. Here are 2 Power-Point Warm-Ups that help students begin to make this journey.  As always, use manipualtives (we used Cuisenaire rods and Monopoly money, but any blocks will do) and give them time.

Word Problem- Money

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Word Problem – Pears

Photos of the “Money” Problem:

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2. Order of Operations Lesson

This is a more traditional lesson, since it represents a convention that mathematicians follow to avoid misunderstandings.

Order of Op

Since we are also 8th grade Algebra teachers, we insist on good algebraic form – what we call the “Sacred V”.

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3. Homework

Here are all the HW assignments from the last unit, plus their Solution Keys:

HW#1  HW#2   HW#3   HW#4   HW#5    HW#6    HW#7    HW#8   HW#9

HW Keys 1-9

 

‘Mystery’ Warm-Ups, plus: Starting our Unit on Patterns

REVIEWING MADE FUN!

We found on our second quiz that many students were still struggling with word problems.  (Correct student quiz answers here.)  So we adapted some of  Steve Wyborney’s “Esti-mysteries”  to continue reviewing word problems, and students seem to enjoy these ‘mysteries’ and look forward to them.

Try these engaging warm-ups:

Important:  Take TIME when you show these. We try to slow down the process as much as possible (without totally ruining the tension!), in order to allow more students to spend the time they need thinking through the problems. Number 3 can be used as a “Number Talk’ to see how many ways students can see the problem 7 x 13. Continue reading ‘Mystery’ Warm-Ups, plus: Starting our Unit on Patterns

Fractions: Transferring From Blocks to Word Problems

Word Problems – YAY!
Our intensive use of Cuisenaire rods came to fruition now as we attempt to transfer fraction visualization to word problems. Here are the 2 lessons we spent on this so far:

CW Frac to WP1  and  CW Frac to WP2

We used a format called “Builders and Scribes”. Continue reading Fractions: Transferring From Blocks to Word Problems

Why Withhold the Fraction Algorithm?

According to this article by Jo Boaler — professor of mathematics education at Stanford and co-founder of www.youcubed.org  — math memorizers scored poorly on the international PISA test, and the U.S. has more memorizers than most other countries in the world. The highest achieving students internationally were those who thought of math as a set of connected, big ideas.

Here’s what we see:

1. A visual approach to fractions gives students better number sense, and better access to word problems.

When we require drawing, every problem becomes a word problem.   In the problem below, all students recognized that 1/2 is 6 out of 12, visually. This is a “12-peak Toblerone”, so a total of 17 twelfths (by simply counting!) . Then this student imagined moving one 12th from the top row to make the 2nd row equal to one, leaving 5/12 on top. This shows number sense! Our students can do fraction addition and subtraction mentally. More importantly, visualization helps facilitate the transfer to word problems, as below.

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Egyptian fractions: We spent a few days answering word problems by building fractions with Cuisenaire rods. Here, for example, is a TWELVE-WIDE wall:index.pngOne fourth —  the light green rod — is called one fourth because four of them fit in a whole.  The purple rod is called one third because 3 of them fit, the red is 1/6, etc.

This student had no trouble finding a way to make 11/12 with Egyptian fractions:

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After long exposure to physical representations, word problems become easier. This problem, for example, would be difficult to do with algorithms.

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How about this problem:  Erin and Kana went shopping for groceries. Each of them had an equal amount of money at first. Then Erin spent $80 and Kana spent $128. After that Kana had 4/7 of what Erin had left. How much money did Erin have left after shopping? Solve by drawing a fraction model.

This is very difficult to do without algebra. Try it yourself before looking at the answer here. Once you see the solution, it’ll make sense, and all of this will transfer to stronger algebra students in 3 years.

2. A visual approach to math is the ONLY approach that works for some students.

In the past, visual learners struggled with the algorithmic manner in which math was taught. (Challenge: randomly survey a couple dozen adults – we predict almost 1/3 of them will say they were ‘never very good at math’)

However, in the past, there were good middle class jobs available to high school graduates – jobs that are now disappearing. It is our duty to make math accessible to ALL students.
The good news is that requiring visualization of math also benefits the innately abstract math learners. Visualization skills helps students in Chemistry, Physics, Trigonometry, and other STEM subjects these students gravitate towards. Here’s  an article about visualization in physics.

 

 

 

 

 

 

Homework – How Much? How Hard?

Ah… Homework

Raise your hand if you remember hating homework as a child …   Raise your hand if you have children and hate it when they have homework … Raise your hand if, as a teacher, you have ever received homework submitted with teardrops on it 🙁     If you haven’t raised your hand yet, you were born under a lucky star, or you have a faulty memory. face-with-tears-of-joy_1f602.png

Untitled.pngOn the plus side, homework

  • offers a chance for students to independently consolidate skills they learned in a group setting
  • builds skills of responsibility and time management

Continue reading Homework – How Much? How Hard?