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?

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.

** 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:

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.

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.

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.

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.

- 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.

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.

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)**

Again, **w****e 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**

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.

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

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

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

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.

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.

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.

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

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.

Feel free to use or edit as you like. (I fixed the typos!)

]]>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. *

Photos of the “Money” Problem:

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

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

Photos of classwork:

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

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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.

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.

On #1 and #2, students will need a small 100 chart to cross out values as they solve.

Mental Math: On the next warm ups, we don’t allow pencils! We find that this increases student engagement, and they can’t ‘hide behind their pencils’.

Sharky (#6 and #7) is a popular addition to the warm ups!

We humans love patterns. We reproduce them in art, music and literature, we seek them out in nature, in daily life and even in the behavior of others. They allow us to make assumptions, develop strategies and navigate uncertainty. Patterns provide a sense of order – a comforting predictability in an otherwise mysterious universe.

The study of patterns should be central to our learning of mathematics. Luckily, children find them enticing, and enjoy observing and describing predictable changes in growth.

Investigation: We ask students to lay out one tile. Is it a square? What’s its area? Could it be called 1-squared? Now lay out new tiles to show 2-squared, 3-squared and 4 squared. What do you notice?

Hmmm … do you want to change the colors of the tiles to show what you’re noticing?

3. Then students start on their classwork, which increases in difficulty as they go through it. Some students will need the whole time to complete 2 pages, while others will complete the whole 4 pages. This allows for ** differentiation by speed, **which is the only difference we find matters in math. Speed (or lack of it!) is not necessarily an indicator of math ability. Many students who need time to process concepts end up being strong math thinkers in high school and beyond!

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We decided that finding fractional *areas* of a rectangle would be a good alternative representation, since we’ve used Cuisenaire rods so often to represent fractions.

The large brown flat has 48 studs (bumps). We asked students to fill the flat completely with smaller pieces, then draw what they’d created on a white board, and label each fractional area. We asked them to repeat if time.

As they labeled the parts, we circulated, asking how they knew what that fractional part was. Almost uniformly, they all answered with a division fact. “Because 8 goes into 48 six times, so it’s a sixth.” We tried rephrasing – “You mean that SIX of those pieces fit on the flat?” Some children actually tested it to see if 6 of the yellow pieces fit on the flat. We feel that this merits reinforcement – a yellow piece IS a sixth because six of them fit on the flat. This is one more step towards tying together the two large concepts of division and fractions.

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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”.

- Group students in pairs. We then sent each pair to the white boards (we have very long white boards), but they could work at mini-white boards at their desks, too.
- One student starts as the “Builder”, the other as the “Scribe”. The builder uses Cuisenaire rods, laid out on a mini-white board or clipboard. (no pencil/pen for the builder). The scribe uses a white board pen (no blocks for the scribe) and records what the builder has built. The builder stands next to their scribe and they set to work. Of course, there’s a lot of collaboration and role-sharing at both ends of this; that’s absolutely fine
- There are hints for each word problem, but wait as long as possible before showing them. Our first goal is to MAXIMIZE THINKING TIME, not necessarily to get right answers quickly. However, we also want to avoid completely losing the students, too. Perhaps there’s a Golden Confusion Level Let them struggle and think and try, then give a little boost up.
- After each problem, ask for a little feedback. “What was the hardest wall to get over?”
- Then the partners switch roles for the next problem.

HW#6 and HW#6_answ_key

Here’s a Cuisenaire riddle/puzzle for CW, too, if there’s time: CW 10_2 Cuis_puz

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Here’s what we see:

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.

Egyptian fractions: We spent a few days answering word problems by building fractions with Cuisenaire rods. Here, for example, is a TWELVE-WIDE wall:One 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:

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

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.

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.

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However, it was hard! There were pitfalls and things we would do differently next time, but aren’t there always?

So, here goes.

- We started with this Toblerone lesson; always popular!! It takes 4 or 5 Toblerones. We use gloves and a tray during the Q&A demo, calling on students to answer each fractional question on the Powerpoint: Toblerone CW We break off the quantity each child says, compare it to the original 9-peak Toblerone (Just break off 9 peaks from a traditional 12-peak Toblerone and use it as your original, to compare the others to). Then we share the Toblerones out – about 2 peaks per child. Woohoo!
- Day 2 (this is new) starts with everyone working together to add fractions with Egyptian Fraction rules: CW Cuisenaire PowerPt . Then pass out the independent practice that continues with the same type problems: CW Practice Cuisenaire . This whole approach proved to be MUCH more difficult than we expected. Students either have memorized the fraction addition algorithm or they haven’t. Either way, their actual visual understanding of what fractions look like was shaky, even among our strongest students. Many were frustrated. We almost gave up! But now, several days later, the effort is
*paying off!* - Day 3 uses our Alpha-Beta-Gamma game. AlphaBetaGamma (here‘s an older post, too) Students will need Cuisenaire rods, and if possible, mini-white boards.

Here are the HW assignments for the week: HW#6 HW#7

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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.

On 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

On the negative side, homework

- asks parents to monitor an approach to math they are not familiar with, meaning they may undo learning done in the classroom.
- compromises family and play time because children are too busy with homework.
- might reflect the work of a parent, rather than the thought processes and progress of the child.
- is not backed by solid research, at least at the elementary level, showing its effectiveness in supporting learning.

As in many schools, we are expected to give homework, so we had to go back and weigh our thoughts in terms of our BASIC BELIEF:

**Math should be taught through understanding, not memorization. **In the 21st Century, students will not need speedy calculations skills nearly as much as they will need big-picture thinking, problem-solving skills, and creativity.

So, here are our conclusions, compromises and solutions:

1. Less is More. We set a maximum limit of 20 minutes for each math homework assignment, and students have TWO days to complete it (It’s due every other day). This is enough to build independent organizational skills (one homework benefit). And it allows for other activities – school is important, but the physical, personal and social development of children is far more important than completing a homework assignment. If a student spends 20 minutes without finishing, they STOP, no penalty.

2. It’s All About the Thinking. We also do *not* *penalize* *mistakes* on homework. We say, “OK, did you spend a solid 20 minutes, thinking? wondering? Did you get stuck? **That’s ok – that’s part of learning.” **We then spend the time needed to correct homework **in class. **By this time, students have skin in the game – they’re interested in finding out *how* to solve that problem that eluded them alone.

3. Offer Challenge. Our homework has three levels.

- Level 1 is concrete/pictorial, and solidifies students’ visual understanding of key concepts. (Throughout our fifth grade year, much of Level 1 covers operations with fractions.) For some students, this is easy and done quickly, giving them more time for Levels 2 and 3. (In the long run, these abstract thinkers still benefit from these small doses of visual underpinning of their conceptual understanding). For other students, Level 1 is time-consuming, but
*absolutely essential*to their progress. It’s okay to have to draw one or two fraction problems every day for months, if that’s what it takes to internalize understanding. In the mean time, they often excel in concrete investigations done in class, and their confidence grows. These students are usually visual learners, humans who a generation or two ago would have been excluded from grade-level math achievement and soon relegated to dead-end, remedial courses. A huge fraction of our adult population today readily confesses to “not being very good at math”.

- Level 2 is at grade level, and transfers students’ visual understanding to the traditional, abstract mastery of mathematics. About 90% of our students complete both Levels 1 and 2 on their homework. Students only go on to Level 3 if they have time and interest.
- Level 3 is the challenge level, and is there to give our fast-working students a feeling of struggle and challenge. On a survey yesterday (an exit ticket), several of our fastest students said that Level 3 has not been hard enough, so we’re listening, and will notch it up a little

Why not just move fast students *ahead* in the curriculum?

- We realize the importance of higher-level math skills for students who end up going into STEM fields, but their push toward accelerated math should come in
*high school.*Success in today’s high schools is improved greatly by a solid conceptual basis from K-8, and from the independent ability to doggedly THINK things through. Challenge problems with depth lend a greater mental acuity than speeded-up algorithmic memorization.

With all that in mind, here are our HW assignments 2 – 5 and here is a key to all four of them. HW#2 HW#3 HW#4 HW#5 HW keys 2_5

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- Party Seating. The PowerPoint is here , the worksheet is here, and here are answers and teacher feedback. A wonderful low-floor, high-ceiling activity. Everyone has access, since one method of solution is
*counting.*Everyone began to see patterns quickly and found shortcuts, which is what math IS, we think! Source: https://nrich.maths.org/7228 - Knights of King Arthur’s Court. The slides , worksheet and answers are here. Again, the activity is accessible to all because students can draw circles of increasing sizes and count. We started with 8 volunteer students standing in a circle, and counted them off – “Stay, Go, Stay, Go, …” until only one student is left. Then we tried it with 5 volunteers. After that, students can draw the circles and count off themselves. Again, most students noticed a pattern quickly, and had great questions about WHY that pattern occurs. Source: Ask Dr. Math
- Four Fours. Here is the PowerPoint. Source; Youcubed. “Soon after setting the challenge the board area becomes full of students putting up their solutions, then returning to their seat to look for more. For students, it is a very safe and non threatening activity. It builds number sense and is a fun challenge. This task is also a really nice way of helping students become comfortable sharing their work in front of the class.” (from YouCubed) Here are 3 photos
- Folding Fraction Strips. Look at our post from last year about this activity: WEEK ONE – Activity Three – Fraction Strips. This activity always takes longer and is more difficult than we expect. The folding (especially of thirds) is inexact and challenging. We will use the strips next week for the game “Capture the Circle”.

Over and over, through our combined decades of teaching, we have witnessed the COLOSSAL importance of * AFFECT* in mathematics instruction.

We feel that a child who comes into our class thinking “I stink at math. I’m dumb. I can’t wait to get out of here. Can I get a bathroom pass? Can I copy? Can I hide? Can I become invisible?” *might as well be outside at recess. *At least they’d get exercise.

Any learning they do pick up half-willingly will usually be shallow, disconnected and fragile. Forgotten soon, misfiled, irretrievable.

Therefore, a*ny time spent in September improving confidence is repaid tenfold later*. So we choose investigations that build that confidence. Investigations that are accessible to all students. Investigations complex enough to make fifth graders feel smart.

We believe every human brain comes with 3 basic characteristics:

- A love of patterns
- A love of stories
- A fascination with the mystery of the world around us

The MATH classroom can hold a WEALTH of all that which humans find most satisfying! Let’s aim for that truth.

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