Which Kind of Teacher are You?

That is what the book Mathematical and Analogical Reasoning of Young Learners says there are. The three kinds of teachers are 1. theoretical, 2. experiential, and 3. intuitive. In reading over the respective descriptions and case studies of actual teachers, the categories began to remind me of those primary school reading groups, you know, the squirrels, bears and rabbits. Everybody knows the squirrels are the best and the rabbits, shall we say, are not, regardless of the attempt to camouflage the differences with neutral category names.

The researchers asked about twenty teachers of children ranging from kindergarten to third grade the following questions:

1. What does mathematical reasoning mean to you?
2. Are you familiar with the term, analogical reasoning? What does the term mean to you?
3. How competent at mathematical and analogical reasoning do you consider young children to be?
4. How did you acquire your understanding of mathematical reasoning?
5. How did you acquire your understanding of analogical reasoning?
6. How do you perceive your role in developing the children's mathematical reasoning?
7. How do you attempt to stimulate your children's mathematical reasoning?
8. What differences in reasoning ability do you see in your children?
9. How do you attempt to address these differences?
10. What kinds of mathematical reasoning abilities do you think your children will need to be successful in first and second grade?
11. Do you take these future reasoning needs of your children into account? If so, how do you do it?
12. In what ways do you think the children's out-of-school experiences contribute to the growth of their mathematical reasoning?

The beliefs and practice of the teachers of our youngest students should be examined more often. These teachers are instrumental in laying the academic foundations which may mean the difference later between academic achievement and academic frustration. According to the authors, theoretical teachers displayed substantial and detailed knowledge of mathematical and analogical reasoning, pedagogy, and cognition. Theoretical teachers believed that children are highly capable and competent. They also considered home experiences and mathematical language to be essential to the the children's development of mathematical and analogical reasoning skills.

On every question, as a group (but not necessarily as individuals) the authors found the experiential teachers and the intuitive teachers displayed less knowledge and lower expectations. Experiential teachers relied primarily on their own experience and their reflections on that experience without reference to the professional literature. Intuitive teachers were just guessing and hoping; their answers to the research questions “were brief and often included jargon” (emphasis supplied) (page 148). In fact, their responses often included the phrase, “I guess.” Although all groups emphasized the process of doing math over getting answers, the intuitive teachers “did not see a need for direct instruction. This may relate to their lack of attention to conceptual development and their emphasis on learning being “fun” (page 149). (In the near future, I will be examining the jargon and buzzwords of education, and how terminology can substitute for clear thinking.)

I mentioned earlier that I got the impression that the labels, “theoretical,” “experiential” and “intuitive,” were hardly more than euphemisms for good, middling and poor. By the end of the book, the impression was pretty much confirmed.

The knowledge, beliefs, and practices of the teachers in this study can be placed on a continuum from intuitive to experiential to theoretical. Theoretical teachers explained their knowledge and and beliefs by referencing theoretical frameworks, teaching experience, and listening to children. They provided rich examples of practices associated with effective/exemplary teachers. The experiential teachers made decisions based on knowledge that appeared to reflect their experience as opposed to the multiple sources of knowledge used by theoretical teachers. In contrast, the intuitive teachers appeared to make instructional decisions more spontaneously. Although the examples of practice the intuitive teachers provided were not necessarily ineffective, these teachers could not clearly articulate any rationale for the decisions they make regarding their instructional practices (p 167).

Since the book was written by a bunch of education researchers, the high standing of theoretical teachers may mean nothing more than theoretical teachers quote education researchers so education researchers like them. The authors recommend that future research focus on the question of whether more learning occurs in the classrooms of theoretical teachers than in the classrooms of other teachers. Maybe I am a little rankled because I know that most of the here-today-gone-tomorrow education fads that have burdened veteran teachers over the years are usually perpetrated by education researchers, researchers who may have little teaching experience of their own.

Japanese Schools are More Homogeneous than US Schools.

There are many reasons why Japanese secondary schools display a much more uniform high quality than US secondary schools, including, but not limited to:

*Top students commonly choose teaching as a career.

*Teachers must rotate every three years from school to school with the express purpose of ensuring that all schools, rural or urban, rich or poor, share the teaching talent of the nation.

*The tax base for each local school is national, not local.

*Every school is required to have certain minimal basic resources, ie, every school has a fully appointed science lab, a library, an art room, physical education facilities including a gym, and more.

*Homeroom teachers visit the home of every student early in the school year.

*Every school provides a hot, highly nutritious lunch to every student.

*There is a real, national curriculum.

*Textbooks must be approved by the Ministry of Education, and schools choose only from the approved list.

*Practically every secondary student attends supplementary schools (called juku) at private expense.

*Students must pass a college entrance exam to go to college. Each year the exam is published in the local newspapers after the exam is given.

*The express goal of secondary education is to enable students to pass the entrance exam.
*And more...

I must add that the Japanese system of high schools is really three separate systems with discrete campuses. Western writers often never realize that when they think they are writing about the Japanese education system, they are not talking about a multi-track comprehensive high school such as what we have in the US. Normally Western writers research, visit and write about academic (sometimes called college-prep) high schools whether they know it or not. There are also vocational high schools attended mostly by boys with a few girls, and commercial high schools attended mostly by girls with a few boys.

When you read about the Japanese secondary education system, you are almost always reading about the academic high school system. In my experience, writers seem oblivious to the existence of the other types of Japanese high schools. I am not suggesting that the US should wholesale adopt the educational policies of Japan. However, I am suggesting that a serious discussion regarding the portability of some features of the Japanese system would be valuable. Since US society is choosing the education system it has, warts and all, it would be nice if that choice were made with eyes wide open.

Teachers Are the Most Important Variable!

Plenty of PhD types have made their careers researching and reporting on what is wrong with our education system and how to fix it. Most research leads to here-today-gone-tomorrow education fads. These fads consume precious resources and impose extra responsibilities on teachers, but in the end, education reform remains as elusive as ever. The research does not even agree on the characteristics of academic achievement except for ONE factor.

Over and over again, the most important factor contributing to the academic achievement of students is THE TEACHER. Just this week yet another testimony to the importance of the teacher appeared in “Education Week” summarizing a recent report of ten “top performing” education systems and seven other “rapidly improving” systems, and concluding that what all the systems had in common was a commitment to attracting and keeping the highest quality teachers. Encouragingly, three of the seven rapidly improving systems were in the United States: Boston, Chicago and New York.

“Top-performing systems, for instance, are typically both restrictive and selective about who is able to train as a teacher, recruiting their teachers from the top third of each group leaving secondary school.” Once top performing systems have their high quality recruits, they train them well, pay them well, and accord them professional respect and esteem.

In typical fashion, critics justified the poor performers. “Tom Loveless, a senior fellow in education at the Brookings Institution, a Washington think tank, said the report 'needed to define the variables [that affect school performance] and measure them carefully' across systems hitting the full range of performance. Identifying the practices of the better-performing school systems does not mean much if less successful systems do the same things, he said.“

That is the question, is it not? Do less successful systems, in fact, do the same things as the better-performing systems? For example, can the less successful systems show that their teachers came from the top third of each group leaving secondary school? One of the top performing systems cited in the report, Japan, because of various factors, tend to have more consistent quality of secondary schools across the country than the US. So how does the top third from one community differ from the top third of another community within the United States? There's a great research question.

The Misplaced Fascination with Technology

Educators are fascinated with technology, particularly calculators. Randall Charles, in the May/June 1999 issue of Math Education Dialogues published by the National Council of Teachers of Mathematics (page 11), is sure that not using calculators in the elementary grades “is almost certain to lead to the development of habits that are counter productive to the development of number sense, problem solving and positive dispositions.” In the same publication, Anthony Ralston suggested abolishing pencil-and-paper arithmetic, implying that American students will continue to fare poorly in international comparisons until we do.

However, from the naive student's point of view, technology often looks more like magic than anything else. For example, a popular software program called Geometer's Sketchpad claims that you can know a square is really a square if you drag a corner with the mouse, and while expanding and contracting it maintains the shape of a square. There are two problems here: circular reasoning and dependence on appearance to draw mathematical conclusions. Then students are presented with two apparent squares. When you drag on a corner of the first square it indeed only expands or contracts. When you drag on a corner of the second square, it transforms into any number of other quadrilaterals of varying shapes with varying lengths of sides. The second square is obviously not a “real” square. However, the essential difference between the two original apparent squares does not reside in the properties of squares but in software code. The computer code (written by some programmer) makes the first square expand and contract, while maintaining its square-looking shape. Different computer code makes the second square wildly transform itself. From the student's point of view it was all magical. Magical and mystical mathematical processes do not promote solid math reasoning ability.

A few years ago I challenged the National Council of Teachers of Mathematics (NCTM) on two occasions six months apart to provide me with a list of the research studies they claimed supported their recommendation on page 78 of the Principles and Standards published in 2000 that even the youngest children should learn to use calculators. Their publications and press releases often asserted that calculators promote mathematics reasoning among the youngest students. NCTM was unable to provide the list on either occasion. I conducted my own diligent search. I found that the most that could be concluded from the research was that calculators could not be positively shown to promote the acquisition of mathematical reasoning skills in our youngest students.

Funny thing—generations of students the world over managed to acquire mathematical reasoning skills in the hundreds of years before calculators and other fascinating technologies were ever invented. Technology is not necessary. It might be nice but any specific technological tool needs to be strictly evaluated and well chosen by elementary teachers skilled in the teaching of mathematics.

Place Value and Algebraic Thinking

Surprisingly, an important foundation to algebraic thinking is the ability to simply read a number correctly. Teachers often blow off reading numbers, but when mathematical expressions are correctly read, the math speaks. It almost screams, “This is how you solve me!” Even before students encounter mathematical expressions,simply reading a number can clarify the difference between mere digits, and the number itself. For example, 392.67 as digits is “three nine two point six seven”, and as a number, “three hundred ninety-two and sixty-seven hundredths.” The worst habit is mixing digits and numbers, or putting "and" in the wrong place. The worst way to read the example is "three hundred and ninety-two point six seven," yet math teachers model the poor reading of numbers everyday. Because the placement of the word “and” marks the boundary between the wholes and the part of a whole, students need to develop facility with the use of “and”.

Another important by-product of the chocolate factory activity is a keener sense of rounding. They learn that rounding a number is a matter of distinguishing place value. After work with cases, boxes, and leftovers (loose ones), children can readily accept the renaming of the place value columns as “10 x 10,” “10,” and “1,” or even, for example, “7 x 7,” “7,” and “1” depending on the base. Then it is an easy precursor to exponents to rename the columns as 10^2, 10^1, 10^0 or 7^2, 7^1, 7^0 (or whatever, depending on the base).

Students often practice writing in expanded notation without ever grasping the 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 this could 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 10^1) + (3 x 10^0). If x = 7, then the expanded notation is (5 x 7^2) + (6 x 7^1) + (3 x 7^0). Rewriting numbers as polynomial expressions often makes calculations in different bases much easier, and the Chocolate Factory activity enhances such algebraic understanding.

The Chocolate Factory: Place Value in Algebraic Thinking

Even good math students may begin studying algebra with deficiencies in their understanding of place value. The following activity offers middle school students concrete, hands-on experience with the concept of place value and an opportunity to express the concept algebraically. In the process, students clarify their understanding of the difference between numeral and number while enhancing their number sense. Hopefully students will be able to replace the blind procedure of borrowing and carrying with genuine understanding.

Although students may easily name the place-value column for any particular digit in a number, they often do not understand the significance of the names and they cannot explain mathematically why regrouping works. Students need help in formulating a mathematical explanation to replace the non-math explanation they usually hold. In an activity called “The Chocolate Factory”, students pack chocolates and then tally the number of boxes and cases.

Practically speaking, I usually use beans instead of chocolate. I tell students they are packaging chocolates as the chocolates come down an assembly line. Each piece of chocolate is a “one” or a unit. Students pack the pieces into boxes of ten pieces each, then pack the boxes into cases of ten boxes each, keeping a tally in a table:

I give students opportunities to gain experience adding and subtracting “Cases,” “Boxes” and Loose Chocolates.” Another advantage to using the column names as shown in the table is that the activity can be recycled to teach any base. I have found it is more helpful to rename the “ones” place “loose chocolates”. Then it is easy to explain that there are loose chocolates when there are not enough chocolates to fill a box. There will never be 10 “loose chocolates”, because 10 will fill a box, thereby adding 1 to the tally in the “Boxes” column.

If the above table represents base 10, converting the above tally to numerals yields 562. 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. Often at this point the light bulbs go on, and students see the concept of carrying for the first time. We continue packing chocolates and "carrying" for a while. Then we "eating" the chocolates, metaphorically speaking. Perhaps starting from the above tally, I ask them to take out 8 pieces of chocolate. They 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 loose chocolates, resulting in a total of 12 tally marks in the “loose chocolates” column. This activity is very similar to other trading activities used to teach place value, but seems to be more effective at building the concept of place value because we avoid giving the columns numerical names at the outset.

Simultaneously, we keep a record of each subtraction in the standard algorithm. Students will often understand regrouping for the first time as they compare the physical packing of the chocolates with the ongoing mathematical representation. Three things are going on at the same time, the packing activity, the data record in terms of a tally and the data record in terms of the standard algorithm. We expand and repeat the activity with other groupings which I have carefully planned in advance. For example, if we repeat with groupings of 7, then 7 chocolates make a box, and 7 boxes make a case. I give the students 289 chocolates, knowing full well they will again end up with 5 cases, 6 boxes, and 2 "loose ones". Sometimes I like to use the word "leftovers" for"loose ones."

Students instantly want to know how they got the same apparent number, 562. In the ensuing class discussion, we talk about why the first 562 (10 to a box) has more chocolate pieces than the second 562 (7 to a box). We discover that the reason we line up columns for addition and subtraction is not merely for neatness sake, but because the grouping determines the mathematical meaning of the columns. Students find they can work just as readily in other bases as long as they remember the basis of the grouping (pun intended). I prefer to name the columns from right to left in base ten as: loose, 10, 100, 1000 etc. If, we are working in, for example, base 7, the column names are "loose", 7, 49, 343, etc. Eventually, we can work in base y with column names y^0, y^1, y^2, y^3, etc. As the Chocolate Factory activity illustrates, the “ones” are “ones” only because there are not enough of them to make the next grouping level. They are the ungrouped loose ones, whether in base ten or any other base.

The Vital Place of Place Value

Perhaps one of the most important foundational concepts in mathematics is place value. As the Massachusetts Department of Education rightly observes, “The subtly powerful invention known as place value enables all (my emphasis) of modern mathematics, science, and engineering. A thorough understanding removes the mystery from computational algorithms, decimals, estimation, scientific notation, and—later—polynomials” (Massachusetts Department of Education (2007). In fact, it is when students first meet polynomials in algebra, that the lack of a proper grounding in place value becomes painfully apparent. Most likely a significant number of the difficulties that students experience with math may be traced to place value.

I reviewed the state standards of various states with regard to place value. I looked for an explicit reference to “regrouping,” the current term for what we used to call “borrowing” and “carrying.” My survey of state standards resulted in a mixed bag. Some states require students to do little more than name the place value of a particular digit. Other states expect students to use various means to model place value. Alaska asks students to not only perform the operations of addition and subtraction, but to explain those operations.

State standards have their utility, but apparently whatever the specific state standard, students are able to follow the regrouping recipe without having any real understanding of why the recipe works. In fact, adults of all ages add and subtract by mindlessly following the recipe. Most adults, and of course, all children could do with a solid grounding in place value.

I have a number of activities I use to make place value explicit. Tomorrow I will tell you about an activity I like to call “The Chocolate Factory.”