Place Value Part 4: Geometry of Place Value

So far we have completed three parts of the place value series:

Part 1: The Chocolate Factory which covered the regrouping or trading aspect of place value and explored regrouping in base ten and other bases.

Part 2: Base Ten for Young Students which introduced several games and trading activities to help young children acquire a solid foundation in place value.

Part 3: The Bake Sale demonstrates the role of place value in long division.

Today, Part 4: Geometry of Place Value will explore place value within a quadratic equation. We will further show that each monomial can be modeled geometrically.


Review

The expression, 5x² + 6x + 3, appeared in The Chocolate Factory, as a summary of the chocolate packing activity. Five cases and six boxes were packed with three leftover chocolates. Where x stood for the number of chocolates per box, five cases and six boxes could represent different absolute numbers of chocolates. If x =10, or 10 chocolates per box, then ((5 times 100) + (6 times 10) + 3) chocolates, or 563 chocolates came down the conveyor belt, I Love Lucy style. In fact, this episode of I Love Lucy was the inspiration for the math activity.

If the chocolates are packed in boxes of five then the 563 means ((5 times 25) + (6 times 5) + 3 or 58 chocolates came down the conveyor belt. So a quadratic equation can be thought of as an expression of place value in any base. In fact, a polynomial of any degree can be seen as an expression of place value. Missing terms are represented by zeros. So 2x⁶ + 5x⁵+ 3x² + 7x + 2 would be 2,500,372_{base x}.

Now we can see where the analogy to place value breaks down. If x = 6, then a term like 7x would be “illegal.” Once six “boxes” had been packed, those six boxes would immediately be packed into one “case,” so in base 6 the last three terms would properly be 4x² + x + 2, either way, the last three terms represent 152 “chocolates.” Obviously I have just been speaking to adults initiated into the joys of algebra, not children.

What? No Fourth Dimension

Obviously you can use standard base ten blocks to model quadratic equations. If we assume that x represents base 10, then 4x³ + 3x² + 7x + 2, then we would use 4 large cubes, 3 flats, 2 rods, and 2 small cubes to model the expression. But if all I meant by geometry was geometric solids, I would not have meant much. The geometry is more interesting, and becomes clearer when you look at a set of blocks in a different base, say, base 5, the small cube looks the same as a base ten small cube, but the rod is five cubes long, the flat is a square of 25 cubes, the large cube is 5 flats stacked or 125 cubes.

So the rod of any set of base blocks determines the base of the set. If we look at the large cube of any base, we see that any one of the 12 edges shows x¹, the first dimension, any one of the six faces shows x², the second dimension, and the whole cube shows x³, all three dimensions. Now we run smack into a physical limitation of manipulatives; not one can show more than three dimensions. The power of math is that math is the language of imagination. We can imagine a fourth dimension, x⁴ and beyond even if we cannot model it. How fun is that?

Interestingly, we can also show x⁰ on the cube. Remember for any x, x⁰ = 1. Cubes of varying bases are all different sizes, or volume. The x1, x², and x³ is different on each cube. But since x⁰ = 1 for any base, it stands to reason that x⁰ or 1 would have an identical appearance no matter the size of the cube. In fact, it does. You can find x⁰ at any vertex, that is to say, the corner shows 1, the zero-th dimension, if you will. In fact, the vertex is a geometric point, described as having no length, width or height.

Multiplication with Base Ten Blocks

So far we have spent a great deal of time establishing that x² means x times x, and that we can show x times x geometrically, by using a flat from a base block set. The flat has a square shape which we would expect from an expression like x-squared. But let's consider a rectangle shape. Now we are not multiplying the same number by itself, x times x, the very definition of squaring, “the product obtained when a number or quantity is multiplied by itself”.

With a rectangle, we are multiplying two different numbers, x times y (or length times width, the formula for the area of a rectangle). Using the cubes from a base blocks set, we can model 5 x 3.






Math educators call this type of diagram a multiplication array. Now lets try 13 x 11.





To show the factor 13 along the top, I used a rod and three small cubes. The factor 11 is along the side with a rod and one small cube. One rod times one rod equals one flat (square, and you expected a square, right?), one rod times three small cubes equals three rods or three lengths. Then, one small cube times one rod equals one length, and one small cube times three small cubes equals three small cubes.

Combining like terms, that is, similar objects, together, we have one flat (10² or 100), four rods ((3 times 10) + (1 times 10)) or 40, and three small cubes (1 times 3, or 3) for a total of 143 which I could express as(1 x 10²) + (4 x 10) + (3 x 1) .

What if we wanted to multiply (x+3)(x+1). I am using the magenta to stand for x, a number we don't know, also called a variable.




The product is 1x² + 4x +3, and geometrically, the product is the picture of a quadratic equation showing both its factors above and to the left of the crossbars. I recommend manipulatives that elucidate the geometry of quadratic equations, available, for example, the Montessori Binomial Cube and Creative Publications Algebra Lab Gear. Remember we have shown that the magenta rod could stand for any value, that is, for any base.

We can show three factors and therefore three dimensions with the same model by standing a rod and/or stacking small cubes vertically in the corner where the crossbars intersect. If I were to stack four small cubes in that intersection, I would be modeling (4)(x+3)(x+1) or by multiplying the x-factors first, (4)(x² + 4x +3). You could think of it as stacking four layers of the x-factor product. In fact, the formula for volume is height times base, or height layers of the base.

In terms of base blocks, the product would be modeled with 4 flats, 16 rods, and 12 small cubes. If we are working in base ten, we would have 400 + 160 + 12. We can exchange 10 of the rods for a flat, and 10 of the small cubes for a rod, ending up with 5 flats, 7 rods, and 2 small cubes or 572.

If I were to stand a rod in the intersection, I am modeling 10 layers of 143 or 1430. In the upper left hand corner of the product there would be ten flats stacked which I can exchange for a large cube worth 10x10x10. Completing any other exchanges, the product would consist of 1 large cube, 4 flats, 3 rods and 0 small cubes. If I were to stand a magenta rod in that intersection. I would be modeling (x)(x+3)(x+1). The product would have 1 magenta cube, 4 magenta flats, 3 magenta rods and 0 small cubes or x³ + 4x² + 3x.


If the geometry of numbers intrigues you, check out The Fractal Geometry of Numbers.


Other Resources

Base Ten Candy Store

Base ten thinking questions (book)

Base Five Activity



Base Five worksheet

More base five worksheets

The Candle Problem: How to Damage Motivation

Herbert Kohl says we are missing the boat, motivation wise, in an open letter to Arne Duncan, Secretary of Education.

Now the mantra is high expectations and high standards. Yet, with all that zeal to produce measurable learning outcomes we have lost sight of the essential motivations to learn that moved my students. Recently I asked a number of elementary school students what they were learning about and the reactions were consistently, “We are learning how to do good on the tests.” They did not say they were learning to read.

Mr. Kohl sees a fundamental contradiction between what we say we want and we were are doing to get it.

It is hard for me to understand how educators can claim that they are creating high standards when the substance and content of learning is reduced to the mechanical task of getting a correct answer on a manufactured test.

What, for Mr. Kohl, motivates learning, at least for learning to read?

...reading is a tool, an instrument that is used for pleasure and for the acquisition of knowledge and information about the way the world works. The mastery of complex reading skills develops as students grapple with ideas, learn to understand plot and character, and develop and articulate opinions on literature.

Nowhere does Mr. Kohl mention extrinsic rewards. Teachers have observed, and Robert Slavin's research has confirmed the dissipating effect of extrinsic rewards.

Robert Slavin's position--that extrinsic rewards promote student motivation and learning--may be valid within the context of a "facts-and-skills" curriculum. However, extrinsic rewards are unnecessary when schools offer engaging learning activities; programs addressing social, ethical, and cognitive development; and a supportive environment.

Not only do extrinsic rewards fail to motivate, except in limited cases, but research has also found that extrinsic rewards actually sabotage motivation.



So what's with the ubiquitous classroom token economies and marble jars on teachers' desks? Are we deliberately sacrificing long-term benefits to students for short-term classroom management? How about pay-for-performance or merit pay? First. And foundationally, EVERYONE deserves to be paid fairly. “Getting the issue of money off the table,” as Dan Pink says.

If our society want to motivate the highest performance from teachers, then give them:

Autonomy
Mastery
Purpose

NOT merit pay.

Merit pay is inherently unfair. The bug-a-boo with merit pay is that teachers have so little control over the factors that impact student achievement. What do we say, for example, about the student I wrote about who actually scored worse after his first year with me only to leapfrog three grades the second year with me.  Should I have lost pay the first year?  I was still the same great teacher.  I had no idea his alcoholic uncle moved in with him and his mom that first year. What do you do if you are the great teacher in a school in an environment where just about everything seems to be conspiring against the kids? And what if you are lucky enough to teach in a school where kids have all kinds of advantages and their scores show it regardless of who is their teacher? Policy-makers have not figured out any equitable mechanism for awarding merit pay.

Western Education has the Wrong Mindset

Science educators know full well that school textbooks lag at least a generation behind the times. Teachers who do not take the initiative to independently keep up and supplement the textbook with current information are teaching possibly out-of-date stuff. Sad to say, the vast majority of teachers teach the book, especially at the lower grades where the lifetime foundations for critical thinking are laid.

Hans Rosling, a professor of public health, in a presentation to the US State Department, marvels that the Western world is a generation behind in its understanding of the global situation, especially regarding the developing world.

My problem is that the worldview of my students corresponds to the reality in the world the year their teachers were born.

In Dr. Roslings words, “Their mindset does not match the data set.”

We have a world that cannot be looked upon as divided.
...snip...
The world is converging.

We have completely misunderstood the HIV “epidemic.”

There is no such thing as an HIV epidemic in Africa...It's not war...It's not economy...Don't make it Africa. Don't make it a race issue. Make it a local issue and do (appropriate) preventative approaches.

See the video here Apparently, the video is not yet available on YouTube.

Dr. Rosling has made his data presentation software available for free at Gapminder.

I Love Eureka! Physics

Here is a complete episode guide.

It is pretty expensive to purchase the entire series. Here is one source.

Here are a handful of the first episodes:

Episode 1-Inertia




Episode 2-Mass



Episode 3-Speed



Episode 4-Acceleration Part 1



Episode 5-Acceleration Part 2



Here is a video player.

Enjoy.

The New School House Rock

Give a listen to Georgia teacher, Crystal Huau Mills, her students and friends performing their version of Grammar: the Musical, entitled Grammar Jammer, available on DVD.
Crustal Huau Mills wrote the lyrics and her friend, Bryan Shaw put them to music. When they were all done, they had thirteen songs.

The teacher, who is played by Crystal, falls into a dream world where her class as well as some of her co-workers are transformed. Many of the normal classroom objects come to life to help her reinforce the underlying lesson behind each song. The clock, the flag, the globe, a crayon, the computer, the ruler, her class pet, a goldfish as well as the dictionary all spring to life to help her teach the class.

The New School Year: My Top Ten To-do

Number 10. Go through your closet and get your own school clothes ready to go. Update or accessorize your outfits. I know I did not like wasting time trying to figure out what to wear, or discovering I had forgotten to dryclean or mend something at the last moment.

Number 9. Get to know those important unsung heroes, the backbone, of the school. The janitor, school secretary, librarian, the cafeteria ladies, the recess monitors, the school nurse.

Number 8. Figure out your rules and consequences for violation. Have a behavior management system in place. Make sure your rules and consequences are compatible with school policies and the general practices of other teachers. Talk to other teachers early to shut down efforts by students to play teachers off each other before it has a chance to begin.

Number 7. Rehearse routines with students every day until following the routines is automatic. Think about: how do I want students to enter the room and record their tardies? Do I want a student monitor to help me with roll, leading the pledge of allegiance, lunch money collection, other? What is the procedure for turning in homework? How should they set up their desks for start of class. When are pencils to be sharpened? Bathroom procedures? Papers for absent students? What kind of behavior do I expect when there is a substitute? Make sure EVERYTHING is spelled out so that they know exactly what to expect.

“Design some method to manage and keep track of daily paperwork -- especially for absent students. If you have all of your students regularly asking you for their work, you’ll lose your mind. There are so many options out there. My favorite is to have a hanging folder for each student in every class. If I pass out papers, the student at the front of the row is responsible for filing the handouts for every absent student in the appropriate folder. When the student returns they know they can look in their folder for all their work.”

Number 6. Communicate with parents before school starts.

“You can start communication with parents before the first day of school. Teachers can call home to welcome students and talk to the parents before school starts. I like to send postcards to new students introducing myself. Other teachers hold special class events such as class picnics in the park or an ice cream social before the first day. An opening letter from you on the first day of school is a wonderful way to introduce yourself to the families you will work with. Along with the letter, I also send home a family survey. The data gathered provides insight and invaluable information about my students and families right from the start. Here are some things I include in my family survey:

• What languages are spoken at home?
• Is there someone to help your child with homework?
• Emergency phone numbers, emails, updated address
• Food allergies/Health issues/Diet
• Celebrations and Cultural Awareness
• Child’s Strengths
• Special Needs
• Interests and Talents (parents love this)
• Areas of Concerns, if any
• Expectations for the year
• Questions”

I also plan for open house. I like the custom of Japanese teachers who visit the homes of every student. Take a little gift with you, maybe something the students can use in your class. Oriental Trading has tons of ideas. I like these crayon-shaped erasers.

Number 5. Write a week's worth of lesson plans for the substitute teacher BEFORE you are so sick you cannot even lift your head. I like to base my substitute teacher plans on that last “optional” chapter of the textbook, the one no one ever gets to. As a hands-on science teacher, I preferred to interrupt my regular lessons over burdening a substitute with overseeing an experiment.

Number 4. Plan your first day of class. Start out with an engaging activity that also provides students with a chance to learn and practice something to help them be successful during the year. I had my students to a simple experiment on the first day as a vehicle for teaching them lab rules and procedures in an interesting way.

Number 3. Find another teacher, whether in your grade level or field or not, to partner with, peer mentor each other, and integrate materials. You may want to integrate with more than one teacher at your grade level and with teachers in other grades.

Examples of multi-grade integration suitable for k-8 schools:
Have students create some sort of science teaching aid, like paper models of body systems, and use their teaching aid to teach younger students in another grade. Or invite a younger class to be lab partners with middle school students for a class period.

Examples of within-grade integration suitable for a middle or high school:
Coordinate spelling words with the English teacher. In my case, a word like “hypothesis” might be an extra credit word. Or combine assignments, so that a lab report written in my class get graded for data analysis and conclusions, but the same report gets graded in English class for English mechanics. Or coordinate with the math teacher to teach the metric system in math class at the same time the science teacher is teaching the metric system for gathering quantitative data.

Number 2. Get your supplemental materials together for the first unit, and make a list of the supplemental materials for subsequent units. Put a note on your calendar about a week or so before the end of a unit to remind yourself to gather the listed materials together for the next unit.

Number 1. Know your material, Read over your curriculum several times. Write out a scope and sequence for the entire year. Invariably you will make adjustments as the year progresses, but you will be able to prevent becoming bogged down if you keep an eye on the destination.

Finally, do something nice for yourself.

“Do Teachers Need Education Degrees?”

That's today's question on the New York Time's Room for Debate feature.

The debate suffers from a conflation of credentialing with the schools of education, understandable since graduation from a school of education is usually a prerequisite to a credential.

But current teacher training has a large chorus of critics, including prominent professors in education schools themselves. For example, the director of teacher education at the Harvard Graduate School of Education, Katherine Merseth, told a conference in March that of the nation’s 1,300 graduate teacher training programs, only about 100 were doing a competent job and “the others could be shut down tomorrow.”

Do you agree with Katherine Merseth? Are you a graduate of a college of education? Maybe you are old enough to have started teaching before a degree from a college of education, and possession of a credential were taken as proof of quality and competence.

See what the nine respondents and the myriad of comments (477 as I write) have to say and feel free to add your own two cents. Personally, I think it is instructive that of the nine respondents, only one is actually a school teacher. Our society does not have much use for someone who wants to be the best teacher they can be, and spend their whole life “making a difference” everyday for students. The only viable career ladder in education is outside the classroom. What is worse, many of the career ladder positions either do not require or do not value teaching experience. For example, a principal needs only three years in the classroom.

A teacher who waits too long to get on the career ladder may find it an unwelcoming place. Such teachers applying for positions outside the classroom may be rejected with a dismissive, “All you have ever done is teach” comment.

Anyway, here is a potpourri of excerpts from the debate:

Michael Goldstein wonders if someday proven experience might trump an embossed piece of paper.

Many education schools have already been wrestling with their mission. Is it to do education research and pose larger questions? Or is it to train 22-year-old schoolteachers to be ready for Day 1 in September?

If merit pay indeed becomes more common, then teachers are likely in turn to become more demanding customers — they will want more practical guidance.
One result may be a new labor market in education schools, where top veteran schoolteachers, those who know how to map backward from an algebra final or how to enlist challenging kids, are prized as lecturers, in lieu of ivory tower theorists.

On the other hand, Margaret Crocco thinks practical training is exactly what the colleges of education offer.

What T.F.A. represents for some parents are young people with knowledge, skills, intelligence and ambition. These parents may assume that such attributes aren’t found in those who enter teaching through traditional teacher preparation programs, which typically invest more time in education courses — addressing the “how” of teaching — than does Teach for America. As far as these parents are concerned, teaching boils down to talking

Patrick Welsh, the only practicing teacher on the panel, gets right to the point.

The credentialing game in public education may have once been a well-meaning effort to create some measurable criteria to maintain standards, but it has turned into an absurd process that forces both teachers and administrators to waste time jumping through hoops that have little or no relation to their job performance...

bureaucrats, obsessed with rules and numbers, would rather hire a mediocre but “fully certified” prospect than the brightest, most promising applicant who lacked the “education” courses...

one of the brightest... teachers in the school ... was told he would not be certified unless he took a basic composition course, a low-level course he had been exempted from at the University of Virginia on the basis of his Advanced Placement score in high school.

I understand that young man's frustration. I was denied a math credential in one state because I did not have College Algebra in my transcript. Never mind that I had been exempted by the college placement exam.

Mr. Welsh's recommendation? “hire enthusiastic candidates who exhibit knowledge and love of their subject and a passion for communicating that knowledge and love to students” credential or no credential.

Jeffrey Mirel allows that maybe colleges of education deserve criticism, but they are improving.

Attacked for being purveyors of progressive educational snake oil, for providing inadequate instruction for pre-service teachers, and for pervasive anti-intellectualism, schools and colleges of education are among the favorite targets of educational reformers...

For a long time ed schools did not focus specifically on how to teach challenging content to all students. But that is changing.

Colleges of education need to start by being more selective about the applicants they accept.

Some of those applicants may actually be practicing teachers going for their masters. Arthur Levine laments the motivation of some of those applicants.

This system lacks quality control and too often encourages universities to offer quick, low quality graduate programs in order to attract those teachers who may be more interested in salary bumps than professional development.

James G. Cibulka is president of the National Council for Accreditation of Teacher Education in Washington is just happy that the NCATE is having such a big influence.

About half of our accredited institutions have aligned their master’s programs with NCATE’s propositions, and some have designed master’s programs to help prepare candidates for board assessments.

If you think teacher credentialing is more about state indoctrination than best practices, Martin Kozloff, a professor of education himself, is inclined to agree.

a master’s degree in most education subfields further stamps in the “progressive,” “child-centered,” “constructivist,” “developmentally appropriate,” postmodernist, pseudo-liberationist baloney that infects the undergraduate curriculum, and which leaves graduating ed students unprepared to provide their own students with coherent, logically sequenced instruction...

And if you ask graduating master’s students who have managed to escape indoctrination (because they are fortunately endowed with a wide streak of skepticism), they will tell you that they learned nothing new. Yes, many teachers with master’s degrees in education are more skilled teachers. But this is not because they got a master’s degree. They went for a master’s degree because they are intelligent, were already skilled teachers (self-taught), and had the gumption to go back to school.

I know when I went back to school for my masters, I was young and idealistic, and just wanted to be the best teacher I could be. I wish someone had told me what a waste of time and money the masters degree would be, especially a master in education, and especially a masters in curriculum development (as opposed to school administration). The masters degree has rendered many an out-of district teacher virtually unemployable as the receiving district does not want to pay the higher salary. I'm not the only one who feels this way.

Finally, some common sense from Linda Mikels, the principal of Sixth Street Prep School, a charter elementary school in Victorville, Calif.

The art and skill of effective pedagogy is arguably equally critical to effective classroom instruction. While most aspiring teachers hope to develop these skills through university coursework, in reality the most effective training is acquired through an apprenticeship at a high-performing school with a highly effective classroom teacher. As with most trades, the craft of effective pedagogy is one that is best developed in the context of the “workplace.”

In other news, Bill Gates notices the obvious.

“We don’t know the answers because we’re not even asking the right questions and making the right measurements,..Better teachers are more likely to result in higher achievement than other approaches such as lowering class size...

Place Value Part 3: The Bake Sale

Place value is such a fundamental concept that we ensure the students recognize place value and its significance wherever it occurs. An activity I call “The Bake Sale” highlights place value in the operation of division. I will present just one example. Of course, teachers can have as many examples as groups within the classroom. The groups should not be too large, not more than three of four students per group.

The scenario: They are getting ready for a bake sale. They have a platter of cookies and they want to make sure they will have enough cellophane bags to package the cookies. In today's example, the platter has 173 cookies and they will be packing 6 cookies to a bag. I use beans for cookies and little squares of paper for the bags. So the students would start with 173 precounted beans.

The first concept I want them to see is division as repeated subtraction. They are to remove 6 beans at a time, just as if they were really packing cookies, and place them on a square of paper. As they do so they place a tally mark. Very young children would have a specially designed “worksheet” for recoding each “bag.” For example, a page of squares that the students color as they “pack” each “bag.” When they are through, the number of squares with beans and the number of tally marks or colored squares on the worksheet should be the same.

Older students will want to cut to the chase and simply perform the long division. But one purpose of this activity is to help students see the math behind the procedure, and besides in real life, they really would be subtracting 6 cookies at a time, repeatedly, until there were no longer enough cookies to pack a bag.

They should have 28 bags with 5 cookies left over. Some older students already know that the “real” answer is 28 and 5/6, or even 28.83... or 28.83 depending on what decisions they make. Some will be sure that the answer is 29 because they learned to round somewhere along the way. Some of them may believe an answer with a remainder (as in 28 R5) is juvenile, and not as good an answer as some of the other possibilities. Students must always be reminded that math is the servant, not the master.

Later in the activity students will see that the “juvenile” answer is the most useful answer.

Once they have determined the answer, it is time to revisit the standard algorithm with a variation. Rewrite the division problem like this:




The green lines show the place value columns. In a class discussion, we establish that a 2 goes above the 7, not because 6 goes into 17 twice, but because the 7 is in the tens’ place, 6 is going into 170 (17 tens) 20 times. The 2 is really a twenty. Students need to be reminded continually what the numerals really signify as they complete calculations. Otherwise, students are merely manipulating abstract, meaningless symbols.

Because we are writing the division problem with Arabic numerals, naturally each digit and its columns represent a place value. Since 6 roundly goes into 170 twenty times, meaning we can show 20 repeated subtractions in one step, we write a 20, not a 2, over the 173. Since we have filled 20 bags at once with 6 cookies per bag, we have removed or subtracted 20 x 6, or 120 cookies from the platter. We show this very concrete action by subtracting 120 from 173, leaving 53 cookies on the platter. We remove enough cookies to fill eight more bags, that is 48 cookies, leaving 5 cookies on the platter, not enough to fill a bag. We needed 28 bags.

Although not “wrong,” 28 and 5/6, 28.833, 28.83 or 29 have no practical utility in this scenario. Students will have an easier time evaluating the reasonableness of an answer if they are encouraged to keep the context and the numbers together. When the round to 29, they are saying 29 what? 29 bags. By the end of the activity, it should be clear that 5/6 of a bag is not helpful and that rounding serves no useful purpose. I require students to write their answers in complete English sentences. The answer to this problem is not “28,” or even “28 bags,” but something like “we needed 28 bags to pack the cookies.”

The finished problem would look like this:






The format looks a little different than the standard algorithm, but the significance of place value is preserved. This type of format did not have a name when I first started using it, or perhaps I mistakenly thought at the time that it was an innovation of mine. I was little surprised when the format began appearing in textbooks as “scaffolding.”

Incidentally, at every opportunity we should insist that students read numerals correctly. Simply reading numerals correctly can prevent confusion. “And” marks the spot between “wholes” and “parts.” Although the answers with fractional parts served no real purpose in this activity, of course there are other contexts where the fractional part is important. In any case, some of the other possible answers would be read “twenty eight and five-sixths,” “twenty eight and eighty three hundredths.” I would use “twenty eight point eight three” only for dictation purposes, not for mathematical purposes.

Again I encourage you to send me an email request to join the email list. The email list is different from the blog subscription. My report on calculator use research is over twenty pages long, so naturally I will not be posting it to the blog. The substitute teaching course is more suited to an email format because of its interactive qualities. Those of you who have subscribed to the blog, thank you! If you subscribed to the blog hoping to receive reports or courses, you will need to take the extra step of sending an email because the blog subscription only notifies of a new postings to the blog. The blog subscription is unrelated to reports or courses. Thanks again.

Place Value Part 2: Base Ten for Young Students

One of the most fundamental mathematical concepts, yet one of the most poorly understood, is place value. The typical primary school lesson presents only a superficial, nominal understanding of place value. Students learn only to correctly name the place-value columns, or identify the digit in a given column, but they often do not understand the significance of the column names.


In Part 1, The Chocolate Factory, I introduced a middle school activity for rebuilding often weak base ten foundational concepts. The activity extends understanding to place value in other bases. In Part 2, I will introduce activities suitable for much younger children. Young children can construct the meaning of base ten place value through many activities and games.

There is some evidence from Jean Piaget's work as illustrated in the video, that base ten is conceptually out of reach for very young children. If there is demand, I will present some activities that help young children explore “Two Land” and “Three Land.” Years ago I field tested a unit called “The Land of Hand” which of course would be “Five Land” in the terminology of the video.




Today I am going to concentrate on base ten, or “Ten Land.”
1. Morning Circle
Many kindergarten and first grade teachers have a regular morning circle time when they gather the children and go through a structured routine of talking about the calendar, the season, birthdays and other topics using a set of visual materials that are permanently on display.



The teacher prepares a display of three horizontal pockets with transparent envelopes on the front of each pocket. On the side is a cup full of Popsicle sticks and a stack of cards numbered with the digits from 0 to 9. Pocket charts can also be purchased from various vendors. Every morning the teacher takes one Popsicle stick and places it in the far right pocket (as you face the display). Each day the teacher replaces the card in the envelope to reflect the number of sticks in the pocket.

On the tenth day, the teacher places the tenth stick in the pocket and then makes a show of pointing out there are ten sticks. The teacher then bundles up the ten sticks with a rubber band and places the bundle in the middle pocket. The pocket envelopes should now show (empty, 1, 0) representing 1 bundle of ten sticks and 0 single sticks. The teacher goes through the Popsicle stick routine every day.

On the hundredth day, a celebration day in many schools, the teacher gathers the 10 bundles, ties them together with a piece of yarn and places the whole bundle in the far left pocket and changes the display to show (1,0,0) representing 1 packet of 10 bundles, 0 bundles of 10 sticks, and 0 single sticks. The teacher continues the routine until the last day of school at which point the display should show something like (1, 8, 5).

For step-by-step instructions on how to prepare a complete circle time (sometimes called calendar time) display, see this excerpt from the Center for Innovation in Education. Michael Naylor uses the calendar to build number sense.

2. Trading Activities and Games

Playing games is a natural way for children to acquire all sorts of different aspects of number sense. Years ago I checked a book out of the library that was chock full of wonderful tutoring games. The book has long since gone out of print but no matter. I found the author, Peggy Kaye's website. Here is my version of a game she calls "Fifty Wins."

The teacher creates two boards on heavy card stock, one for each player. Each player also has a die. I recommend using extra large die if you can find them. Each player also has a collection of 50+ beans, pennies, or other counters. My own modification involves using the board at first, then doing away with the board and playing with pennies and dimes.




Each child casts their die in turn, and draws the number of counters that matches the number of dots on their die, placing one counter in each of the small squares of which there are nine. Upon accumulating the tenth counter, they transfer ten counters to one of the five squares. The first person to get fifty counters wins. Children learn there can never be more than nine in the one's place, and that the ten's place is precisely groups of ten. If three big squares are filled and none of the little squares, they can see very clearly 3 (groups of ten) 0 or 30.

A modification I have made is to use poker chips for counters. I change the design of the board so that the long rectangle is outlined in one color (say blue) and the squares are outlined in another color (say red). Then as the child accumulates 10 blue chips, the child exchanges the 10 blue chips for one red chip and places it in one of the red squares. The poker chip modification leads quite naturally in the penny-dime modification I mentioned earlier. I have also used the same poker chips with the same color signification for "The Chocolate Factory" activity, blue for leftovers, red for boxes, white for cases.

Another modification of mine which may be considered a weakening of the game is the use of a die to generate numbers. The original game uses a spinner where some of the fields say “Win 10.” At the beginning the child will dutifully count out ten beans and place them one by one in the small squares, only to have to transfer the entire group of ten to a big square. Very soon the child counts out the ten beans and straightway places them in a big square. The opportunity to realize a group of ten in one turn is lost when die are used, but I suppose you could use a set of two dice. I like the die because the child does not have to read words or numerals. With die, the child has only to match, by one-to-one correspondence, the beans to the die spots. There is no need to reference numerals at all, so the game stays squarely focused on number and avoid number/numeral conflation.


“Make Fifty” is just one example of what is known as a “trading activity.” Cuisenaire rods also work well for trading activities. Every ten cubes makes one rod. Any base-ten block set goes one step further where every ten rods makes one flat, and every ten flats makes one cube. Many base ten block worksheets can be adapted to active lessons.

All manipulatives have limitations and some researchers are concerned about the limitations of base ten blocks. Nevertheless, with a good mix of activities, the teacher can address the differing learning styles of each student.

Stuff to Avoid
Worksheets
Generally speaking, worksheets should be avoided. Nevertheless, I like to design special worksheets as data recording instruments for math labs utilizing base-ten blocks and Cuisenaire rods. Students can learn a lot of math without writing numerals. In fact, a foundation of math reasoning skills without reliance on numerals helps children acquire the concept of the difference between numbers and culturally-determined symbols for numbers such as Arabic numerals. Schools “accidentally on purpose” teach children to confuse number and symbol. Cuisenaire has a few such worksheets along this idea, but I have some problems with the worksheet design. Maybe I'll collect my math lab worksheets into some kind of cohesive with comprehensive directions for using them with children and make them available.

Computer-Based Materials

Too many of the computer-based materials, animated mathematics and virtual manipulatives, though so appealing to adults, often has a magical quality to young children. Regrouping happens before their very eyes but they do not understand the mathematical concept and mechanism. They do not get from the computer what I call the psychology of numbers, or how numbers behave. It's just a lot of cool special effects without specific mathematical concept acquisition benefit.

Calculators

Despite the National Council of Teachers of Mathematics (NCTM) claims to the contrary, calculator studies with the youngest students show no advantage in the development of children's number sense. In 2002, I conducted a major survey of research, research critiques, case studies, and editorials. I periodically asked NCTM to provide me a list of what they characterized as supporting research, but they never did. I found no basis for NCTM's assertion that research backed their recommendation for calculator use in the earliest grades. I found that calculator usage need not hinder the development of math reasoning skills, but it may in fact do so. Teachers report that children become overly dependent on the calculator and have difficulty learning to evaluate the reasonableness of their answers. They trust the calculator more than themselves.

Links and Invitation

The following is a list of links of base ten lessons. They are presented as is. Many exemplify what I believe are the main weaknesses of most base ten teaching.
Lesson plans reviewed by teachers:
Crayola Tally Sticks:
Applet: but better off using concrete manipulatives.
Applet:
An Unreviewed Collection of various resources:
A favorite resource for getting teaching ideas:
Vendor:



Instead of posting this series to my blog, I would like to send it out as emails. The html formatting requirements for the blog are daunting, and frankly, overwhelming. I mean, it can take me hours to produce a table with html, not to mention creating illustrations, and all of the other audio-visual aids that go beyond mere web links. If you would like to join the email list, please send me an email.

If you use the subscribe feature on the blog you will receive notice of postings to the blog, but you will not receive the email-based information. I do not know how to set up an email capture form with double opt-in. Therefore I am requesting that if you would like to join the email list, please send an email to the email address at the top of the blog.

Lesson Plan: The Chocolate Factory or Place Value in Algebraic Thinking

Because students typically have fuzzy notions of place value, they may be able to correctly name the place-value columns, but they often do not understand the significance of the names. For example, they cannot give a mathematical explanation of why regrouping works. One reason may be that they rarely receive mathematical explanations.

The explanations may certainly be chock full of numbers yet without having the least connection to the way the numbers work. A good example is the standard method for finding 10% of a number: just move the decimal place one digit to the left. The method is nothing but a trick, and our children learn to mistake performance of tricks for understanding of math.

Students need help in constructing mathematical explanations. In an activity I call “The Chocolate Factory,” students pack chocolates in boxes, then in cases, while keeping a tally. At the end of the activity, students will be able to trade and regroup in order to add or subtract.

I usually use beans instead of chocolate because it is less messy and less tempting. I explain that the students are working for Hershey Chocolate Company packing chocolates as chocolate pieces roll down the conveyor belt a la a famous “I Love Lucy” episode. The number of chocolates in each group is simulated by drawing a card from a shuffled deck with no picture cards. A specially made set of number cards with spots but no numerals would be better. Students pack the pieces into boxes of ten pieces each, then pack the boxes into cases of ten boxes each, keeping a running tally in a table on the blackboard.

Draw	Cases	Boxes	Leftovers
1			/////
2			////////
Result		/	///
3			//////
Result		/	/////////
4			//
Result		//	/
	etc. until, say,
Result	/////	//////	///

Each pair of students shares a set-up: 100 beans, a container capable of holding ten beans to represent boxes, and a larger container to hold ten “boxes.” The teacher explains that the rule of the game is that a “box” can only hold ten beans. Once a box is filled, they begin filling another box, and so on until they have ten boxes. Ten boxes are then packed into a case.

The teacher shuffles the cards and holds the deck face down. The teacher uses any suitable method to select a student to pick a card. The student takes a card from the deck (a five-spot in the example) and shows it to the class. Each pair counts out five beans and puts them in a “box.” The teacher records the five as tally marks in the “leftovers” column. Another student picks a card (an eight spot). The students count out eight beans and the teacher records the tally in the “leftover” column The students use the beans to fill a box, pointing out that they have one full box and three leftovers. The teacher records the result with one tally mark in the “box” column, and three tally marks in the “leftover” column.

It is important to give students experience with “Cases, Boxes and Leftovers” before renaming these columns “100’s, 10’s and 1’s.” Another advantage to using the column names, “cases, boxes, leftovers” is that the activity can be recycled later for teaching any base. I have found it is more helpful to rename the “ones” place “leftovers”. Then it is easy to explain that there are leftovers when the amount is insufficient to fill a box. Thus, there will never be 10 leftovers, because 10 will fill a box, thereby adding 1 to the tally in the “Boxes” column. Converting the final tally in the table to numerals yields 563. Students readily understand that as they accumulate 10 boxes, they transfer those boxes as 1 case and put a tally mark in the “Case” column. It is often at this very point the light bulbs go on, and students see the why carrying works for the first time.

Then we repeat the activity, but the cards now simulate consumed chocolate (yum). A student draws perhaps an eight-spot to stand for eating eight pieces. Students will naturally want to open a box to accomplish this. As they take a box, they erase a tally mark and dump the 10 chocolates (beans) with the leftovers, and record ten more tally marks for a total of 13 tally marks in the leftover column. They continue subtracting in this way. This activity is very similar to most trading activities, but seems to be more effective at building the concept of place value because we avoid giving the columns numerical names at the outset.

With older students we simultaneously keep a record of this computation in the standard algorithm. Again, students often understand regrouping for the first time. We expand and repeat the activity with other groupings which I have carefully planned in advance. I tell students that they have done such a good job that now they work for a more expensive chocolate company, perhaps Ghirardelli, where chocolates are packed in boxes of 5 pieces, and cases of 5 boxes. I give the students 158 chocolates, knowing full well they will again end up with 5 cases, 6 boxes, and 3 leftovers, the same tally as for the Hershey exercise.

In the ensuing class discussion, we talk about why the first 563 (10 to a box) has more chocolate pieces than the second 563 (5 to a box). Students discover that neatly lining up their addition and subtraction columns is not merely for neatness sake, but because the columns have real meaning. Students find they can work just as readily in other bases as long as they remember the basis (pun intended) of the groupings.

If working in base ten, I prefer to name the columns from right to left “leftovers, 10, 100, 1000,” etc. As the Chocolate Factory activity illustrates, the “ones” are “ones” only because there are not enough of them to make a “ten”. They are the ungrouped leftovers, whether in base ten or any other base. In fact, students get very comfortable with working in a variety of bases and discover that for any base (b), the column names will be (from right to left) “(leftovers), (b), (b x b), (b x b x b), and so on. For example, they would name the base 7 columns “(leftovers), (7), (7 x 7), (7 x 7 x 7), and so on.

I like using the parenthesis early on so students become familiar with the parenthesis holding a number just as cupped hands hold an apple, and that the number have different appearances but still be equivalent. In practice I often go beyond leftovers, boxes, and cases, and extend the activity to crates, trucks and warehouses. Just like the the song from School House Rock says, “Don't you worry 'bout the big numbers, they're just bigger, that's all.”

Later the columns can be renamed with exponents, 10^2, 10^1, 10^0 or 7^2, 7^1, 7^0. Then it is a small step to b2, b1, b0, then x2, x1, x0. Students often practice writing in expanded notation without ever grasping real significance of what they are doing. In algebra, many polynomial expressions are really bases in disguise. For example, the base 10 tally and the base 7 tally were both 563. Algebraically both would be expressed as 5x^2 + 6x + 3, where x is the base. The algebraic expression is nothing more than expanded notation. If x is 10, then the expanded notation is (5 x 10^2) + (6 x 10) + (3 x 1). If x = 7, then the expanded notation is (5 x 7^2) + (6 x 7) + (3 x 1). Rewriting numbers as polynomial expressions often makes calculations in different bases much easier, and The Chocolate Factory activity enhances such algebraic understanding.

There is a vendor, Digi-Block, who sells a manipulative that would be ideal for the base 10 chocolate factory. The set has pieces, boxes and cases. Each box holds exactly 10 pieces, and each case holds exactly 10 boxes. Its advantage is that students are prevented from overpacking or underpacking. I have usually had to rely on materials I have scrounged: beans, little cough syrup cups for boxes, and little containers to hold 10 cough syrup cups. If you are looking for basic base ten blocks, Nasco probably has the most complete assortment anywhere.

Whole-System Reform

Yeah, that's what we need for education in America—whole system reform. But it sounds daunting and overwhelming. Is whole system reform even possible?Opposing ideologies argue themselves into stalemate, and the upshot is nothing changes. Teachers ride the roller coaster of one educational fad after another. So a few schools here and there garner media attention for their success in raising the academic of achievement of their students, but results seem immune to wholesale transfer. A great strategy with proven results in one school fails dismally in another.

Actually, America has experienced a form of whole-system change. “Reform” is the wrong word. The change has been gradual and insidious, taking decades to get where we are today. Decades ago, a strong liberal arts education was the objective of any student dreaming of a bright future and social mobility.

Is whole-system reform even a reasonable goal? Wasn't whole-system reform the goal of No Child Left Behind?

Ontario, Canada thinks it is. In fact, they say they have accomplished whole-system reform.

We have done (whole-system reform) in Ontario, Canada, where we have had the opportunity since 2003 to implement new policies and practices across the system-all 4,000 elementary schools, 900 secondary schools, and the 72 districts that serve 2 million students. Following five years of stagnation and low morale, from 1998 to 2003, the impact of the new strategies has been dramatic: Higher-order literacy and numeracy have increased by 10 percentage points across the system; the high school graduation rate has risen 9 percentage points, from 68 percent to 77 percent; the morale of teachers and principals has improved; and the public's confidence in the system is up.

For the Canadians in Ontario, whole-system reform does not mean taking on every single issue. It means diligently accomplishing a set of “core policies and strategies.”

Whole-system reform is possible, but it must be tackled directly. There are no single-factor solutions. By implementing a core of fundamental components, system leaders can get results in fairly short order, and build on those results for sustainable futures.

Ontario worked on six “fundamental components.”

1.The entire teaching profession.
2.A small number of ambitious priorities-literacy, numeracy, and high school graduation.
3.The two-way street between instruction and assessment.
4.Distributive coordinated leadership at all levels of the system.
5.Focused, mostly nonpunitive, comprehensive, relentless intervention strategy.
6.Use money to drive reform only in the service of the previous five fundamentals.

Those six fundamentals seem pretty comprehensive and the report lacks specific details. What exactly did everyone do to accomplish the fundamentals?

The only way to get whole-system reform is by motivating and mobilizing the vast majority of people in the system.

There we are, the crux of the problem—motivating and mobilizing the vast majority of people in the system. Do the leaders simply order mobilization by fiat or did they motivate individual buy-in?

One major piece is the student success program. Instead of restricting curriculum as we have so often done here in the US, Ontario believes expanding the curriculum is the way to go. Ontario students can choose a specialist major in a number of fields.

Specialist High Skills Majors are now available in:
Agriculture
Arts and Culture
Aviation/Aerospace
Business
Community Safety and Emergency Services
Construction
Energy
The Environment
Forestry
Health and Wellness
Hospitality and Tourism
Horticulture and Landscaping
Information and Communications Technology
Manufacturing
Mining
Transportation

Students can choose a work coop situation. Back in the day, my own high school in California offered work coops. Maybe it's time to bring them back.

Our students are plugged in anyway. What about offering high school students online courses? Ontario offers fifty of them.

How about this idea? Dual credits.

Students participate in apprenticeship training and postsecondary courses, earning dual credits that count towards both their high school diploma and their postsecondary diploma, degree or apprenticeship certification.

Clearly, Ontario's main strategy for motivating success is to give students a rich variety of choices. Meanwhile, many American schools have been eliminating choices and electives. American universities have been following suit, so that students must strictly follow a curricular flow chart if they expect to graduate, and the number of available electives has been reduced as more and more classes become required in order to ensure, as one example, exposure to multicultural information. Breadth is no longer built into a liberal arts education. With the emphasis on meeting the market demand for job training in the university, liberal arts may be a dying concept.