I (Kathleen) was working with a 4^{th} grader years ago. She was struggling in math class, and came in before school twice a week to do 20 minutes of math with me. (Okay, her Dad had to bribe her with a donut…) One morning I put 2 rainbow cubes on her place value chart and asked, “What happens if the ‘Times Ten Fairy’ comes and waves her wand at these two *ones*?”

Nora picked them up and considered her options. “Times ten?”

“Yup, times ten.”

She turned and put the 2 blocks in the manipulatives basket. Then she counted out 10 *new* ones blocks. Then she stopped and thought, and did it again. She put all 20 blocks in the ones column and smiled like a Cheshire cat.

I congratulated her – after all, her thinking was fundamentally correct, and she knew it. Then I scratched my head, and wondered out loud, “What about the Place Value Police? Will they let you have that many squatters in the ones column?”

She rolled her eyes and palmed the whole set of twenty, put it back in the basket and replaced it with two tens, in the 10s column.

Now my pleasure was authentic. “She’s got it now,” I remember thinking.

“Show me *times ten* again.” I cleared the chart and put down 1 ten and 1 one.

I picked up my knitting. It’s a hobby I enjoy.

Nora picked up the 1 cube, turned to the basket, deposited it, counted out 10 ones and put them in the ones column. Then she picked up the ten rod , put *it* in the basket and counted out 10 tens.

She looked at me, checking for approval, but I pretended to be fascinated by my knitting. She paused, looked at the ceiling (a common gesture in visual learners) and exchanged the ten tens for a 100 flat, and the 10 ones for a tens rod.

*Then *she beamed at me, and I stopped knitting and praised her.

“She’s got it now,” I remember thinking.

“Okay, Mademoiselle Mathématique, try this one.” I put a ten and 2 ones on the chart.

She went through the whole process again. Exchanging, counting, exchanging, placing.

I knitted furiously, so that I wouldn’t help or make suggestions, or, God forbid, *scream.*

We did 21 times ten. Same process.

22 times ten.

15 times ten …

I began to make mistakes in my knitting.

*Thirteen. *Nora bulldozed her way through THIRTEEN problems.

My knitting began to look Gordion-knottish.

On the *FOURTEENTH *problem (25 x 10), she stopped suddenly in the middle. She looked at the 10 ones in her hand, and then at the tens rod.

“Wait, Ms J! You’re just tricking me! This is easy. Times ten just makes a 1 turn into a tens rod. And the ten into a 100-waffle.”

Now she was really pleased. I felt like Annie must have felt when Helen learned how to spell water. Now* I *was the one with the Cheshire grin, and it didn’t matter that I knew I’d have to unravel a few rows of knitting.

The best part? She really DID understand. She could do it the next day and the next week and the next year.

*Could* I have simply told her “Multiply by ten, add a zero at the end”? Yes! BUT – she’d have forgotten it several times, then mixed it up with division, and *in the end, *it would have cost more time than the 20 minutes that morning before school. (Okay, 25 – she was almost late to class. Actually, because of the donut, she probably *was* late to class.)

I was fortunate to teach her again in 5^{th} grade, when we learned decimals. Multiplying and dividing by 10 was no mystery to her. Nor was Scientific Notation in 7^{th} grade, or multiplying or dividing both sides of an equation by ten in 8^{th} grade.

I vowed *never *to rush a child through the concrete phase. I’ve usually kept that vow – of course I’ve occasionally caught myself backsliding. (We humans are prone to relapsing into old ways – let’s forgive ourselves.) But that has always backfired, so I go back to knitting in silence.

Here’s the thing: Memorizing is not Understanding. Just as Dorothy couldn’t rely on Auntie Em and Uncle Henry to save her, and had to do her growing up on her own, so children need the time to connect neurons *on their own *when learning mathematics.

Corrinne and I try to plan our lessons so that students who need manipulatives longer *get them longer. *Understanding concepts is more important than “covering” them.

It takes a concerted effort to train a *whole class* to respect others’ natural working speeds without judgment. But that’s half of what education is about, isn’t it?

** Footnote: Nora did the old-fashioned college prep sequence of math classes in high school (not the newly-fashionable accelerated courses, praise be). Successfully! She is now a junior at a reputable state college, with a major in Psychology and a minor in business. Go Nora!*

- The Ishango Bone
- Indigenous Number Systems
- Babylonian, Mayan, Indian and Arab and Chinese Contributions to Mathematics
- Patterns in Mathematics
- Puzzles and Word Problems

27 faces looking at us. Expectantly. Politely. Silently.

The two of us (Kathleen and Corrinne) have visited dozens of classrooms over the last few years, on a mission to help Kg-6^{th} grade teachers in their quest to improve math education in the US. The first time we failed was in a 4^{th} grade classroom, 8 years ago, where we’d posed this question:

We had expected the students to huddle together – talking, arguing, giggling – until the problem was solved. We thought they’d use guess-and-check, or push around little bits of paper with 3’s and 7’s written on them. We’ve been teaching and mentoring teachers in our own school for almost 20 years, and have grown used to the engrossing messiness of problem solving.

Instead … silence.

Not much else went well that day. So a couple weeks later, we tried it in a different 4^{th} grade classroom. This time, we provided the slips of paper with 3’s and 7’s on them, thinking that would scaffold the problem-solving attitude we wanted.

Again, not much happened. We were baffled; what was going on here?

Finally, one girl raised her hand. “Ma’am?” she ventured. “Umm, No one ever taught us that problem before.”

Ahh. Cringe. Memories came flooding back… Introducing a problem-based curriculum at our own school 17 years ago was grueling. We had to do battle with the established culture of “How Math Is Taught” – namely by explaining, practicing and drill. When we challenged that paradigm, we met with student silence, parent resistance, administrator concern, and our own self-doubt.

Thank goodness there were two of us. Alone, either one would have given up.

What makes this so hard? We teachers *mean* well. *We* *want to teach.* Something in us yearns to take the class by the hand, and say “Make an organized, 2-column list. Write ‘3¢ stamps’ as a heading on the left side, and write 1, 2, 3, 4… below that. Now label the right-hand column ‘7¢ stamps’, and in each place, put the number that would make the pair of numbers add up to 20….”

Our question to ourselves, and to teachers we work with is:

*“Who is doing the thinking here?”*

We’ve all read the research. Citizens of the 21^{st} Century will need thinking skills, creativity and brainstorming habits much more than they’ll need algorithms and computational speed.

We truly believe this –** Nowhere can we teach thinking skills better than in the math classroom**

So – Nuts and Bolts: How do we begin the change to “Building a Thinking Classroom”?

**Ask, Don’t Tell**

Answer student questions with questions! Yes, this is hard, so here’s a countdown of our favorite responses:

**#5.** “Hmmm… What do you see so far?” (note the verb!) Just get them talking. “Humph. I see two colors of blocks” is an okay answer to start with. “Oh, sure. Which color has more? How many more?” Anything you ask is okay, too. Moving from the concrete to the abstract is *hard, *and the bridge between the two levels is *talking.* Just get them talking; progress will follow.

**#4. ** “Can you see a pattern? Is it growing or shrinking? How many are in each group? Can you count them?”

**#3.** “What would happen if….? … we used simpler numbers? we acted it out? We took away the extra numbers for a minute? We lumped both numbers together for a minute?” These are not just fillers – these are responses that actually help move students forward.

*And our #2, so close to our hearts: *

**#2.** “Have you asked the blocks?” (We call base-10 blocks the “math whisperers”)

** *** * **FAQ: **But… what if students answer “I don’t know, I don’t get it” to every prompt above? **

Then we go on to our favorite response :

**#1. “I don’t know…. Let me try it with the blocks.” **

Yes, play dumb. By “not knowing” and getting the blocks, you’re modeling problem-solving behavior. It might raise their hackles the first time, but by week 2 or 3, they’re hooked. Here’s the thing – you’re giving students *respect for their inherent intelligence. *(This is very powerful – think about a time *you* didn’t feel your intelligence respected.) Here’s our take on ‘thinking’:

We want to pay tribute here to another educator whose writing has been invaluable to our own growth: Peter Liljedahl, whose website can be found here, and whose phrase “the Thinking Classroom” we’ve borrowed.

**2. Encourage Counting**

If students learn to rely on blocks, manipulatives and drawings, it means they’re functioning at the *visual* level initially. It means the answer is usually right in front of them, on the table – in the blocks, but they’ll have to **count** to distill an answer from that visual.

There’s nothing wrong with that! Real math *evolved *from counting. Any major concept you really understand can be reduced to counting. Slope in algebra? Count. Sine and Cosine? Count the degrees around a unit circle of 2 pi. Find a derivitive in calculus? Count the slope of a tangent to a curve. (isn’t YouTube great?). Our point is that there is nothing wrong with counting – math is just an abstract set of symbols developed so we *don’t have to count* – just to save time. Let them count. The human brain loves efficiency – they’ll figure out how the math just to avoid counting so much.

The one response we do make them memorize (!) has to do with drawing word problem models. “Class, why do we draw models for these problems?” Their required response: “So we can count”.

**3. Give Judicious Hints**

Even in a Thinking Classroom, students get stuck. After letting the class work for a while, we watch for signs that the task is becoming frustrating, not fun any more.

Then we call the class together and say “I wanted to share a great suggestion from one group. They said (even if it’s not true ! ) ‘What if we pretend all 20 of the stamps are 3¢ ? Could we then start switching out stamps?’ ” It doesn’t seem like much of a hint (“Try using twelve 3¢ stamps” would be a thinking-stopper). But it’s enough to get the students around 3rd base.

**4. Accept All Answers**

If students ask “Is this answer right?”, we say “I don’t know – I’m still working on it.” But we write up ALL the answers that get suggested on the board (and sometimes add a few wrong ones of our own, so the correct answer doesn’t dominate). This encourages students to look at each others’ answers and check their own work.

The post Building a Thinking Classroom first appeared on The Pi Project.]]>Last Friday, I attended a virtual conference held by the British Columbia Association of Math Teachers.

The keynote speaker, Michael Pruner, challenged us as math teachers to *slow down.* To do so, he admitted, would require us to focus less on testing and more on how children actually learn. Although admittedly disruptive to the system, he said, it is our duty as educators to bring about this change.

This was such a welcome message. Corrinne and I have been saying this for years – lonely voices in the wilderness! Many teachers have nodded at us in understanding, but added regretfully, “My district/admin/parents/curriculum/testing requirements won’t allow it.”

And yet, *WE* are the professionals. Let us claim our own expertise. We are the ones who must take this to the streets, to the board meetings, to the ballot box and to the classrooms.

Here are our suggestions for slowing down:

**1. Use Manipulatives More.**

In our own anecdotal experience (still… >60 years combined!), about one third of all students are visual learners, and therefore do poorly with the traditional ‘memorize and move on” approach to math. These children need three times – four times – eight times as much time at the concrete level as we’re currently giving them. When not given enough time, these students give up, try to memorize long enough to pass a test, and label themselves as *lifelong failures* at math. Yet – given the time they need – they *can* learn math, all the way through high school math and beyond. We’ve seen it over and over.

Because the concrete level is *where humans actually learn to make sense of mathematics.* Without it, we limp on to higher mathematics blindfolded, befuddled, reciting mysterious, magical formulas.

The pyramid at right shows the relative importance of the 3 levels of math learning.

There are many research studies that back up the effectiveness of this approach (C>P>A) to teaching math.

***HERE** is a link to a long list of studies on the value of C>P>A.

***This pdf** is from my BCAMT presentation last Friday on using manipulatives.

***This video** is of my upcoming CMC presentation on manipulatives.

**2.** **Teach in Longer Units***.*

Figuring this out was one of our greatest challenges over the last 20+ years. As teachers, don’t we all set goals that feel impossible? We want to both:

**(a)** meet the needs of the visual learners, the step-by-step learners, the ponderers, the discouraged, and

**(b)** meet the needs of the racers, the abstract thinkers, the memorizers, the gifted.

For Corrinne and me, the solution was to teach in longer units, including __one__* year-long* topic per grade level.

For 2^{nd} grade, for example, it’s 3-digit subtraction. For 5^{th} grade it’s fraction operations. We chose skills whose mastery is indispensable in the development of a child’s math understanding.

We stick with that topic ALL YEAR, keeping most of it at the concrete/pictorial level. We introduce it in September with investigations, games, etc. and then we then *go on to other units as usual,* but do at least one problem a day (we call it our daily vitamin!) that reinforces the concrete/pictorial understanding of that original topic, *and we withhold the algorithm* until everyone has had a chance to build confidence at the first two levels. Because of the distance between introduction of the unit in September and its comeback in April/May, we call this a “Sandwich Unit”.

Second grade teachers – wouldn’t it be great to know *all* your incoming students mastered at least addition and subtraction to 20 (or 30?) before entering your class? Sixth grade teachers, would you rejoice if *every student *entering your grade in September had a firm grasp (at least at the pictorial level) of addition and subtraction of fractions?

Amen. Yes, this level of mastery is worth the time invested.

Here’s a summary of our approach, including examples. And our blog gives a year-long glimpse into what this looks like in a 5^{th} grade classroom.

**3. Ask, Don’t Tell. **

This one was and continues to be difficult for us. We teachers *so badly want * to help, to explain, to facilitate, to lay bricks in the Yellow Brick Road of Understanding. And yet, the truth is that *the student must build that road. *It is our job as teachers to find problems “just hard enough” to cause students to *need *manipulatives. Then it is our job to be quiet. I pretend not to know the answers, to look at my students’ blocks in focused puzzlement, to talk about what I *see, * and ask myself what I might… maybe … do next?

This takes time – if your students are not used to it, don’t expect miracles the first week. But with time, students learn to depend on their own thinking, their own ability to “ask the blocks”, their own innate intelligence, regardless of speed. Isn’t this what we want in the long run?

**4. Differentiate by Speed. **

Every one of our classwork or homework sheets has 3 levels: (1) A concrete/pictorial level, (2) A second level that transfers to word problems and algorithms, and (3) An optional challenge level.

In our experience, we’ve found that *most* (80%-ish) students often finish Level 1 and Level 2 in the time given. Half of those (40%-ish) have time to go on to the challenge level, and do at least part of it. The other 20% finishes at least Level 1, meaning that *that’s where they are, developmentally. *They still need practice at the concrete level before they can build a foundation and bridge to the abstract level.

Corrinne and I always say, “You can’t force neurons to connect.” It takes the time it takes. Here are examples. 1st grade. 5th grade.

**5. Spend Less Time on Assessment. **

One of the most striking things I ever heard Jo Boaler say was, “You can teach like a superhero, building conceptual understanding, doing everything right, meeting everyone’s needs, and then *ruin it all with one assessment. *We’ve all seen this happen. A child gets a score that is lower than that of the classmates around them, a quiz pockmarked with x’s. The child’s immediate translation is “I stink at math.”

Instead, let’s learn to tell that child, “It’s fine to just work on Level 1. That’s where we all show true understanding. Use the blocks (we call them the “Math Whisperers”) and you’ll figure it out.” Gradually, that child can conquer Level 2 as well.

Additionally, let’s keep assessments short (3-4 problems is enough), and use them for diagnostic purposes only. Feedback to parents becomes “Your child has mastered 3-digit subtraction with blocks, and is showing a growing ability to represent that understanding with drawings and mental math.” We include photos of that process. Children own their own progress, and feel proud of what they can do.

Kathleen and Corrinne are pleased to announce the launch of their new website. You can follow us there on a day-by-day basis – a whole year’s worth of daily lesson plans.

We have currently loaded lessons through September, and promise to stay at least a month ahead of the calendar. Stay tuned!

If you use any lessons, please give us feedback — how did it go? Any problems, suggestions, or improvements?

The post Download Free – A Whole Year of Lesson Plans for 4th, 5th or 6th grade math! first appeared on The Pi Project.]]>

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.

The post Asilomar Presentation first appeared on The Pi Project.]]>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

The post More Word Problems first appeared on The Pi Project.]]>

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!

The post ‘Mystery’ Warm-Ups, plus: Starting our Unit on Patterns first appeared on The Pi Project.]]>

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.

The post Fractions with Legos first appeared on The Pi Project.]]>