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 158 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, for 4x³ + 3x² + 7x + 2, we would use 4 large cubes, 3 flats, 2 rods, and 2 small cubes to model the expression. However, 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.

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 what we 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? Why must teachers have jars ofmarbleson their desks? Are we deliberately sacrificing long-term learning benefits 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 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 a great teacher 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. Rosling's 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.

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 and 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 at the last moment I had forgotten to dryclean or mend something.

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 confusing graduation from a school of education with the process of credentialing, 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, president of the National Council for Accreditation of Teacher Education (NCATE), is delighted 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 masters in education, and more 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...